From 8018e923c75eb5504310864f821972f794b7d554 Mon Sep 17 00:00:00 2001 From: Benjamin Barenblat Date: Wed, 13 Feb 2019 20:40:51 -0500 Subject: New upstream version 8.8.0+1.gbp069dc3b --- .gitignore | 19 + AAC.v | 1141 ------------------------------------------- CHANGELOG | 28 -- COPYING | 674 -------------------------- COPYING.LESSER | 165 ------- Caveats.v | 372 -------------- Instances.v | 260 ---------- LICENSE | 172 ++++++- Make | 23 - Makefile | 14 +- README | 55 --- README.md | 117 +++++ Tutorial.v | 399 --------------- _CoqProject | 28 ++ aac.mlpack | 7 - aac_rewrite.ml4 | 529 -------------------- aac_rewrite.mli | 9 - coq.ml | 611 ----------------------- coq.mli | 232 --------- default.nix | 28 ++ description | 15 - helper.ml | 41 -- helper.mli | 33 -- matcher.ml | 1293 ------------------------------------------------- matcher.mli | 189 -------- meta.yml | 109 +++++ opam | 30 ++ print.ml | 101 ---- print.mli | 23 - search_monad.ml | 70 --- search_monad.mli | 42 -- src/aac.ml4 | 79 +++ src/aac_plugin.mlpack | 8 + src/aac_rewrite.ml | 425 ++++++++++++++++ src/aac_rewrite.mli | 24 + src/coq.ml | 594 +++++++++++++++++++++++ src/coq.mli | 232 +++++++++ src/helper.ml | 41 ++ src/helper.mli | 33 ++ src/matcher.ml | 1293 +++++++++++++++++++++++++++++++++++++++++++++++++ src/matcher.mli | 189 ++++++++ src/print.ml | 104 ++++ src/print.mli | 23 + src/search_monad.ml | 70 +++ src/search_monad.mli | 42 ++ src/theory.ml | 1153 +++++++++++++++++++++++++++++++++++++++++++ src/theory.mli | 197 ++++++++ theories/AAC.v | 901 ++++++++++++++++++++++++++++++++++ theories/Caveats.v | 376 ++++++++++++++ theories/Instances.v | 261 ++++++++++ theories/Tutorial.v | 401 +++++++++++++++ theories/Utils.v | 257 ++++++++++ theory.ml | 1149 ------------------------------------------- theory.mli | 197 -------- 54 files changed, 7209 insertions(+), 7669 deletions(-) create mode 100644 .gitignore delete mode 100644 AAC.v delete mode 100644 CHANGELOG delete mode 100644 COPYING delete mode 100644 COPYING.LESSER delete mode 100644 Caveats.v delete mode 100644 Instances.v delete mode 100644 Make delete mode 100644 README create mode 100644 README.md delete mode 100644 Tutorial.v create mode 100644 _CoqProject delete mode 100644 aac.mlpack delete mode 100644 aac_rewrite.ml4 delete mode 100644 aac_rewrite.mli delete mode 100755 coq.ml delete mode 100644 coq.mli create mode 100644 default.nix delete mode 100644 description delete mode 100644 helper.ml delete mode 100644 helper.mli delete mode 100644 matcher.ml delete mode 100644 matcher.mli create mode 100644 meta.yml create mode 100644 opam delete mode 100644 print.ml delete mode 100644 print.mli delete mode 100644 search_monad.ml delete mode 100644 search_monad.mli create mode 100644 src/aac.ml4 create mode 100644 src/aac_plugin.mlpack create mode 100644 src/aac_rewrite.ml create mode 100644 src/aac_rewrite.mli create mode 100644 src/coq.ml create mode 100644 src/coq.mli create mode 100644 src/helper.ml create mode 100644 src/helper.mli create mode 100644 src/matcher.ml create mode 100644 src/matcher.mli create mode 100644 src/print.ml create mode 100644 src/print.mli create mode 100644 src/search_monad.ml create mode 100644 src/search_monad.mli create mode 100644 src/theory.ml create mode 100644 src/theory.mli create mode 100644 theories/AAC.v create mode 100644 theories/Caveats.v create mode 100644 theories/Instances.v create mode 100644 theories/Tutorial.v create mode 100644 theories/Utils.v delete mode 100644 theory.ml delete mode 100644 theory.mli diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..73bd09d --- /dev/null +++ b/.gitignore @@ -0,0 +1,19 @@ +*.aux +*.a +*.cma +*.cmi +*.cmo +*.cmx +*.cmxa +*.cmxs +*.glob +*.native +*.o +*.d +*.vio +*.vo +.coq-native + +.merlin +Makefile.coq +Makefile.coq.conf diff --git a/AAC.v b/AAC.v deleted file mode 100644 index 5feb0b6..0000000 --- a/AAC.v +++ /dev/null @@ -1,1141 +0,0 @@ -(***************************************************************************) -(* This is part of aac_tactics, it is distributed under the terms of the *) -(* GNU Lesser General Public License version 3 *) -(* (see file LICENSE for more details) *) -(* *) -(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) -(***************************************************************************) - -(** * Theory file for the aac_rewrite tactic - - We define several base classes to package associative and possibly - commutative operators, and define a data-type for reified (or - quoted) expressions (with morphisms). - - We then define a reflexive decision procedure to decide the - equality of reified terms: first normalise reified terms, then - compare them. This allows us to close transitivity steps - automatically, in the [aac_rewrite] tactic. - - We restrict ourselves to the case where all symbols operate on a - single fixed type. In particular, this means that we cannot handle - situations like - - [H: forall x y, nat_of_pos (pos_of_nat (x) + y) + x = ....] - - where one occurrence of [+] operates on nat while the other one - operates on positive. *) - -Require Import Arith NArith. -Require Import List. -Require Import FMapPositive FMapFacts. -Require Import RelationClasses Equality. -Require Export Morphisms. - -Set Implicit Arguments. -Set Asymmetric Patterns. - -Local Open Scope signature_scope. - -(** * Environments for the reification process: we use positive maps to index elements *) - -Section sigma. - Definition sigma := PositiveMap.t. - Definition sigma_get A (null : A) (map : sigma A) (n : positive) : A := - match PositiveMap.find n map with - | None => null - | Some x => x - end. - Definition sigma_add := @PositiveMap.add. - Definition sigma_empty := @PositiveMap.empty. -End sigma. - - -(** * Classes for properties of operators *) - -Class Associative (X:Type) (R:relation X) (dot: X -> X -> X) := - law_assoc : forall x y z, R (dot x (dot y z)) (dot (dot x y) z). -Class Commutative (X:Type) (R: relation X) (plus: X -> X -> X) := - law_comm: forall x y, R (plus x y) (plus y x). -Class Unit (X:Type) (R:relation X) (op : X -> X -> X) (unit:X) := { - law_neutral_left: forall x, R (op unit x) x; - law_neutral_right: forall x, R (op x unit) x -}. - - -(** Class used to find the equivalence relation on which operations - are A or AC, starting from the relation appearing in the goal *) - -Class AAC_lift X (R: relation X) (E : relation X) := { - aac_lift_equivalence : Equivalence E; - aac_list_proper : Proper (E ==> E ==> iff) R -}. - -(** simple instances, when we have a subrelation, or an equivalence *) - -Instance aac_lift_subrelation {X} {R} {E} {HE: Equivalence E} - {HR: @Transitive X R} {HER: subrelation E R}: AAC_lift R E | 3. -Proof. - constructor; trivial. - intros ? ? H ? ? H'. split; intro G. - rewrite <- H, G. apply HER, H'. - rewrite H, G. apply HER. symmetry. apply H'. -Qed. - -Instance aac_lift_proper {X} {R : relation X} {E} {HE: Equivalence E} - {HR: Proper (E==>E==>iff) R}: AAC_lift R E | 4 := {}. - - - -Module Internal. - -(** * Utilities for the evaluation function *) - -Section copy. - - Context {X} {R} {HR: @Equivalence X R} {plus} - (op: Associative R plus) (op': Commutative R plus) (po: Proper (R ==> R ==> R) plus). - - (* copy n x = x+...+x (n times) *) - Fixpoint copy' n x := match n with - | xH => x - | xI n => let xn := copy' n x in plus (plus xn xn) x - | xO n => let xn := copy' n x in (plus xn xn) - end. - Definition copy n x := Prect (fun _ => X) x (fun _ xn => plus x xn) n. - - Lemma copy_plus : forall n m x, R (copy (n+m) x) (plus (copy n x) (copy m x)). - Proof. - unfold copy. - induction n using Pind; intros m x. - rewrite Prect_base. rewrite <- Pplus_one_succ_l. rewrite Prect_succ. reflexivity. - rewrite Pplus_succ_permute_l. rewrite 2Prect_succ. rewrite IHn. apply op. - Qed. - Lemma copy_xH : forall x, R (copy 1 x) x. - Proof. intros; unfold copy; rewrite Prect_base. reflexivity. Qed. - Lemma copy_Psucc : forall n x, R (copy (Psucc n) x) (plus x (copy n x)). - Proof. intros; unfold copy; rewrite Prect_succ. reflexivity. Qed. - - Global Instance copy_compat n: Proper (R ==> R) (copy n). - Proof. - unfold copy. - induction n using Pind; intros x y H. - rewrite 2Prect_base. assumption. - rewrite 2Prect_succ. apply po; auto. - Qed. - -End copy. - -(** * Utilities for positive numbers - which we use as: - - indices for morphisms and symbols - - multiplicity of terms in sums *) - -Local Notation idx := positive. - -Fixpoint eq_idx_bool i j := - match i,j with - | xH, xH => true - | xO i, xO j => eq_idx_bool i j - | xI i, xI j => eq_idx_bool i j - | _, _ => false - end. - -Fixpoint idx_compare i j := - match i,j with - | xH, xH => Eq - | xH, _ => Lt - | _, xH => Gt - | xO i, xO j => idx_compare i j - | xI i, xI j => idx_compare i j - | xI _, xO _ => Gt - | xO _, xI _ => Lt - end. - -Local Notation pos_compare := idx_compare (only parsing). - -(** Specification predicate for boolean binary functions *) -Inductive decide_spec {A} {B} (R : A -> B -> Prop) (x : A) (y : B) : bool -> Prop := -| decide_true : R x y -> decide_spec R x y true -| decide_false : ~(R x y) -> decide_spec R x y false. - -Lemma eq_idx_spec : forall i j, decide_spec (@eq _) i j (eq_idx_bool i j). -Proof. - induction i; destruct j; simpl; try (constructor; congruence). - case (IHi j); constructor; congruence. - case (IHi j); constructor; congruence. -Qed. - -(** weak specification predicate for comparison functions: only the 'Eq' case is specified *) -Inductive compare_weak_spec A: A -> A -> comparison -> Prop := -| pcws_eq: forall i, compare_weak_spec i i Eq -| pcws_lt: forall i j, compare_weak_spec i j Lt -| pcws_gt: forall i j, compare_weak_spec i j Gt. - -Lemma pos_compare_weak_spec: forall i j, compare_weak_spec i j (pos_compare i j). -Proof. induction i; destruct j; simpl; try constructor; case (IHi j); intros; constructor. Qed. - -Lemma idx_compare_reflect_eq: forall i j, idx_compare i j = Eq -> i=j. -Proof. intros i j. case (pos_compare_weak_spec i j); intros; congruence. Qed. - -(** * Dependent types utilities *) - -Local Notation cast T H u := (eq_rect _ T u _ H). - -Section dep. - Variable U: Type. - Variable T: U -> Type. - - Lemma cast_eq: (forall u v: U, {u=v}+{u<>v}) -> - forall A (H: A=A) (u: T A), cast T H u = u. - Proof. intros. rewrite <- Eqdep_dec.eq_rect_eq_dec; trivial. Qed. - - Variable f: forall A B, T A -> T B -> comparison. - Definition reflect_eqdep := forall A u B v (H: A=B), @f A B u v = Eq -> cast T H u = v. - - (* these lemmas have to remain transparent to get structural recursion - in the lemma [tcompare_weak_spec] below *) - Lemma reflect_eqdep_eq: reflect_eqdep -> - forall A u v, @f A A u v = Eq -> u = v. - Proof. intros H A u v He. apply (H _ _ _ _ eq_refl He). Defined. - - Lemma reflect_eqdep_weak_spec: reflect_eqdep -> - forall A u v, compare_weak_spec u v (@f A A u v). - Proof. - intros. case_eq (f u v); try constructor. - intro H'. apply reflect_eqdep_eq in H'. subst. constructor. assumption. - Defined. -End dep. - - - -(** * Utilities about (non-empty) lists and multisets *) - -Inductive nelist (A : Type) : Type := -| nil : A -> nelist A -| cons : A -> nelist A -> nelist A. - -Local Notation "x :: y" := (cons x y). - -Fixpoint nelist_map (A B: Type) (f: A -> B) l := - match l with - | nil x => nil ( f x) - | cons x l => cons ( f x) (nelist_map f l) - end. - -Fixpoint appne A l l' : nelist A := - match l with - nil x => cons x l' - | cons t q => cons t (appne A q l') - end. - -Local Notation "x ++ y" := (appne x y). - -(** finite multisets are represented with ordered lists with multiplicities *) -Definition mset A := nelist (A*positive). - -(** lexicographic composition of comparisons (this is a notation to keep it lazy) *) -Local Notation lex e f := (match e with Eq => f | _ => e end). - - -Section lists. - - (** comparison functions *) - - Section c. - Variables A B: Type. - Variable compare: A -> B -> comparison. - Fixpoint list_compare h k := - match h,k with - | nil x, nil y => compare x y - | nil x, _ => Lt - | _, nil x => Gt - | u::h, v::k => lex (compare u v) (list_compare h k) - end. - End c. - Definition mset_compare A B compare: mset A -> mset B -> comparison := - list_compare (fun un vm => - let '(u,n) := un in - let '(v,m) := vm in - lex (compare u v) (pos_compare n m)). - - Section list_compare_weak_spec. - Variable A: Type. - Variable compare: A -> A -> comparison. - Hypothesis Hcompare: forall u v, compare_weak_spec u v (compare u v). - (* this lemma has to remain transparent to get structural recursion - in the lemma [tcompare_weak_spec] below *) - Lemma list_compare_weak_spec: forall h k, - compare_weak_spec h k (list_compare compare h k). - Proof. - induction h as [|u h IHh]; destruct k as [|v k]; simpl; try constructor. - - case (Hcompare a a0 ); try constructor. - case (Hcompare u v ); try constructor. - case (IHh k); intros; constructor. - Defined. - End list_compare_weak_spec. - - Section mset_compare_weak_spec. - Variable A: Type. - Variable compare: A -> A -> comparison. - Hypothesis Hcompare: forall u v, compare_weak_spec u v (compare u v). - (* this lemma has to remain transparent to get structural recursion - in the lemma [tcompare_weak_spec] below *) - Lemma mset_compare_weak_spec: forall h k, - compare_weak_spec h k (mset_compare compare h k). - Proof. - apply list_compare_weak_spec. - intros [u n] [v m]. - case (Hcompare u v); try constructor. - case (pos_compare_weak_spec n m); try constructor. - Defined. - End mset_compare_weak_spec. - - (** (sorted) merging functions *) - - Section m. - Variable A: Type. - Variable compare: A -> A -> comparison. - Definition insert n1 h1 := - let fix insert_aux l2 := - match l2 with - | nil (h2,n2) => - match compare h1 h2 with - | Eq => nil (h1,Pplus n1 n2) - | Lt => (h1,n1):: nil (h2,n2) - | Gt => (h2,n2):: nil (h1,n1) - end - | (h2,n2)::t2 => - match compare h1 h2 with - | Eq => (h1,Pplus n1 n2):: t2 - | Lt => (h1,n1)::l2 - | Gt => (h2,n2)::insert_aux t2 - end - end - in insert_aux. - - Fixpoint merge_msets l1 := - match l1 with - | nil (h1,n1) => fun l2 => insert n1 h1 l2 - | (h1,n1)::t1 => - let fix merge_aux l2 := - match l2 with - | nil (h2,n2) => - match compare h1 h2 with - | Eq => (h1,Pplus n1 n2) :: t1 - | Lt => (h1,n1):: merge_msets t1 l2 - | Gt => (h2,n2):: l1 - end - | (h2,n2)::t2 => - match compare h1 h2 with - | Eq => (h1,Pplus n1 n2)::merge_msets t1 t2 - | Lt => (h1,n1)::merge_msets t1 l2 - | Gt => (h2,n2)::merge_aux t2 - end - end - in merge_aux - end. - - (** interpretation of a list with a constant and a binary operation *) - - Variable B: Type. - Variable map: A -> B. - Variable b2: B -> B -> B. - Fixpoint fold_map l := - match l with - | nil x => map x - | u::l => b2 (map u) (fold_map l) - end. - - (** mapping and merging *) - - Variable merge: A -> nelist B -> nelist B. - Fixpoint merge_map (l: nelist A): nelist B := - match l with - | nil x => nil (map x) - | u::l => merge u (merge_map l) - end. - - Variable ret : A -> B. - Variable bind : A -> B -> B. - Fixpoint fold_map' (l : nelist A) : B := - match l with - | nil x => ret x - | u::l => bind u (fold_map' l) - end. - - End m. -End lists. - -(** * Packaging structures *) - -(** ** free symbols *) - -Module Sym. - Section t. - Context {X} {R : relation X} . - - (** type of an arity *) - Fixpoint type_of (n: nat) := - match n with - | O => X - | S n => X -> type_of n - end. - - (** relation to be preserved at an arity *) - Fixpoint rel_of n : relation (type_of n) := - match n with - | O => R - | S n => respectful R (rel_of n) - end. - - (** a symbol package contains an arity, - a value of the corresponding type, - and a proof that the value is a proper morphism *) - Record pack : Type := mkPack { - ar : nat; - value :> type_of ar; - morph : Proper (rel_of ar) value - }. - - (** helper to build default values, when filling reification environments *) - Definition null: pack := mkPack 1 (fun x => x) (fun _ _ H => H). - - End t. - -End Sym. - -(** ** binary operations *) - -Module Bin. - Section t. - Context {X} {R: relation X}. - - Record pack := mk_pack { - value:> X -> X -> X; - compat: Proper (R ==> R ==> R) value; - assoc: Associative R value; - comm: option (Commutative R value) - }. - End t. - (* See #Instances.v# for concrete instances of these classes. *) - -End Bin. - - -(** * Reification, normalisation, and decision *) - -Section s. - Context {X} {R: relation X} {E: @Equivalence X R}. - Infix "==" := R (at level 80). - - (* We use environments to store the various operators and the - morphisms.*) - - Variable e_sym: idx -> @Sym.pack X R. - Variable e_bin: idx -> @Bin.pack X R. - - - (** packaging units (depends on e_bin) *) - - Record unit_of u := mk_unit_for { - uf_idx: idx; - uf_desc: Unit R (Bin.value (e_bin uf_idx)) u - }. - - Record unit_pack := mk_unit_pack { - u_value:> X; - u_desc: list (unit_of u_value) - }. - Variable e_unit: positive -> unit_pack. - - Hint Resolve e_bin e_unit: typeclass_instances. - - (** ** Almost normalised syntax - a term in [T] is in normal form if: - - sums do not contain sums - - products do not contain products - - there are no unary sums or products - - lists and msets are lexicographically sorted according to the order we define below - - [vT n] denotes the set of term vectors of size [n] (the mutual dependency could be removed), - - Note that [T] and [vT] depend on the [e_sym] environment (which - contains, among other things, the arity of symbols) - *) - - Inductive T: Type := - | sum: idx -> mset T -> T - | prd: idx -> nelist T -> T - | sym: forall i, vT (Sym.ar (e_sym i)) -> T - | unit : idx -> T - with vT: nat -> Type := - | vnil: vT O - | vcons: forall n, T -> vT n -> vT (S n). - - - (** lexicographic rpo over the normalised syntax *) - Fixpoint compare (u v: T) := - match u,v with - | sum i l, sum j vs => lex (idx_compare i j) (mset_compare compare l vs) - | prd i l, prd j vs => lex (idx_compare i j) (list_compare compare l vs) - | sym i l, sym j vs => lex (idx_compare i j) (vcompare l vs) - | unit i , unit j => idx_compare i j - | unit _ , _ => Lt - | _ , unit _ => Gt - | sum _ _, _ => Lt - | _ , sum _ _ => Gt - | prd _ _, _ => Lt - | _ , prd _ _ => Gt - - end - with vcompare i j (us: vT i) (vs: vT j) := - match us,vs with - | vnil, vnil => Eq - | vnil, _ => Lt - | _, vnil => Gt - | vcons _ u us, vcons _ v vs => lex (compare u v) (vcompare us vs) - end. - - - - (** ** Evaluation from syntax to the abstract domain *) - - Fixpoint eval u: X := - match u with - | sum i l => let o := Bin.value (e_bin i) in - fold_map (fun un => let '(u,n):=un in @copy _ o n (eval u)) o l - | prd i l => fold_map eval (Bin.value (e_bin i)) l - | sym i v => eval_aux v (Sym.value (e_sym i)) - | unit i => e_unit i - end - with eval_aux i (v: vT i): Sym.type_of i -> X := - match v with - | vnil => fun f => f - | vcons _ u v => fun f => eval_aux v (f (eval u)) - end. - - (** we need to show that compare reflects equality (this is because - we work with msets rather than lists with arities) *) - Lemma tcompare_weak_spec: forall (u v : T), compare_weak_spec u v (compare u v) - with vcompare_reflect_eqdep: forall i us j vs (H: i=j), vcompare us vs = Eq -> cast vT H us = vs. - Proof. - induction u. - destruct v; simpl; try constructor. - case (pos_compare_weak_spec p p0); intros; try constructor. - case (mset_compare_weak_spec compare tcompare_weak_spec m m0); intros; try constructor. - destruct v; simpl; try constructor. - case (pos_compare_weak_spec p p0); intros; try constructor. - case (list_compare_weak_spec compare tcompare_weak_spec n n0); intros; try constructor. - destruct v0; simpl; try constructor. - case_eq (idx_compare i i0); intro Hi; try constructor. - apply idx_compare_reflect_eq in Hi. symmetry in Hi. subst. (* the [symmetry] is required ! *) - case_eq (vcompare v v0); intro Hv; try constructor. - rewrite <- (vcompare_reflect_eqdep _ _ _ _ eq_refl Hv). constructor. - destruct v; simpl; try constructor. - case_eq (idx_compare p p0); intro Hi; try constructor. - apply idx_compare_reflect_eq in Hi. symmetry in Hi. subst. constructor. - - induction us; destruct vs; simpl; intros H Huv; try discriminate. - apply cast_eq, eq_nat_dec. - injection H; intro Hn. - revert Huv; case (tcompare_weak_spec t t0); intros; try discriminate. - symmetry in Hn. subst. (* symmetry required *) - rewrite <- (IHus _ _ eq_refl Huv). - apply cast_eq, eq_nat_dec. - Qed. - - Instance eval_aux_compat i (l: vT i): Proper (@Sym.rel_of X R i ==> R) (eval_aux l). - Proof. - induction l; simpl; repeat intro. - assumption. - apply IHl, H. reflexivity. - Qed. - - - (* is [i] a unit for [j] ? *) - Definition is_unit_of j i := - List.existsb (fun p => eq_idx_bool j (uf_idx p)) (u_desc (e_unit i)). - - (* is [i] commutative ? *) - Definition is_commutative i := - match Bin.comm (e_bin i) with Some _ => true | None => false end. - - - (** ** Normalisation *) - - Inductive discr {A} : Type := - | Is_op : A -> discr - | Is_unit : idx -> discr - | Is_nothing : discr . - - (* This is called sum in the std lib *) - Inductive m {A} {B} := - | left : A -> m - | right : B -> m. - - Definition comp A B (merge : B -> B -> B) (l : B) (l' : @m A B) : @m A B := - match l' with - | left _ => right l - | right l' => right (merge l l') - end. - - (** auxiliary functions, to clean up sums *) - - Section sums. - Variable i : idx. - Variable is_unit : idx -> bool. - - Definition sum' (u: mset T): T := - match u with - | nil (u,xH) => u - | _ => sum i u - end. - - Definition is_sum (u: T) : @discr (mset T) := - match u with - | sum j l => if eq_idx_bool j i then Is_op l else Is_nothing - | unit j => if is_unit j then Is_unit j else Is_nothing - | u => Is_nothing - end. - - Definition copy_mset n (l: mset T): mset T := - match n with - | xH => l - | _ => nelist_map (fun vm => let '(v,m):=vm in (v,Pmult n m)) l - end. - - Definition return_sum u n := - match is_sum u with - | Is_nothing => right (nil (u,n)) - | Is_op l' => right (copy_mset n l') - | Is_unit j => left j - end. - - Definition add_to_sum u n (l : @m idx (mset T)) := - match is_sum u with - | Is_nothing => comp (merge_msets compare) (nil (u,n)) l - | Is_op l' => comp (merge_msets compare) (copy_mset n l') l - | Is_unit _ => l - end. - - - Definition norm_msets_ norm (l: mset T) := - fold_map' - (fun un => let '(u,n) := un in return_sum (norm u) n) - (fun un l => let '(u,n) := un in add_to_sum (norm u) n l) l. - - - End sums. - - (** similar functions for products *) - - Section prds. - - Variable i : idx. - Variable is_unit : idx -> bool. - Definition prd' (u: nelist T): T := - match u with - | nil u => u - | _ => prd i u - end. - - Definition is_prd (u: T) : @discr (nelist T) := - match u with - | prd j l => if eq_idx_bool j i then Is_op l else Is_nothing - | unit j => if is_unit j then Is_unit j else Is_nothing - | u => Is_nothing - end. - - - Definition return_prd u := - match is_prd u with - | Is_nothing => right (nil (u)) - | Is_op l' => right (l') - | Is_unit j => left j - end. - - Definition add_to_prd u (l : @m idx (nelist T)) := - match is_prd u with - | Is_nothing => comp (@appne T) (nil (u)) l - | Is_op l' => comp (@appne T) (l') l - | Is_unit _ => l - end. - - Definition norm_lists_ norm (l : nelist T) := - fold_map' - (fun u => return_prd (norm u)) - (fun u l => add_to_prd (norm u) l) l. - - - End prds. - - - Definition run_list x := match x with - | left n => nil (unit n) - | right l => l - end. - - Definition norm_lists norm i l := - let is_unit := is_unit_of i in - run_list (norm_lists_ i is_unit norm l). - - Definition run_msets x := match x with - | left n => nil (unit n, xH) - | right l => l - end. - - Definition norm_msets norm i l := - let is_unit := is_unit_of i in - run_msets (norm_msets_ i is_unit norm l). - - Fixpoint norm u {struct u}:= - match u with - | sum i l => if is_commutative i then sum' i (norm_msets norm i l) else u - | prd i l => prd' i (norm_lists norm i l) - | sym i l => sym i (vnorm l) - | unit i => unit i - end - with vnorm i (l: vT i): vT i := - match l with - | vnil => vnil - | vcons _ u l => vcons (norm u) (vnorm l) - end. - - (** ** Correctness *) - - Lemma is_unit_of_Unit : forall i j : idx, - is_unit_of i j = true -> Unit R (Bin.value (e_bin i)) (eval (unit j)). - Proof. - intros. unfold is_unit_of in H. - rewrite existsb_exists in H. - destruct H as [x [H H']]. - revert H' ; case (eq_idx_spec); [intros H' _ ; subst| intros _ H'; discriminate]. - simpl. destruct x. simpl. auto. - Qed. - - Instance Binvalue_Commutative i (H : is_commutative i = true) : Commutative R (@Bin.value _ _ (e_bin i) ). - Proof. - unfold is_commutative in H. - destruct (Bin.comm (e_bin i)); auto. - discriminate. - Qed. - - Instance Binvalue_Associative i :Associative R (@Bin.value _ _ (e_bin i) ). - Proof. - destruct ((e_bin i)); auto. - Qed. - - Instance Binvalue_Proper i : Proper (R ==> R ==> R) (@Bin.value _ _ (e_bin i) ). - Proof. - destruct ((e_bin i)); auto. - Qed. - Hint Resolve Binvalue_Proper Binvalue_Associative Binvalue_Commutative. - - (** auxiliary lemmas about sums *) - - Hint Resolve is_unit_of_Unit. - Section sum_correctness. - Variable i : idx. - Variable is_unit : idx -> bool. - Hypothesis is_unit_sum_Unit : forall j, is_unit j = true-> @Unit X R (Bin.value (e_bin i)) (eval (unit j)). - - Inductive is_sum_spec_ind : T -> @discr (mset T) -> Prop := - | is_sum_spec_op : forall j l, j = i -> is_sum_spec_ind (sum j l) (Is_op l) - | is_sum_spec_unit : forall j, is_unit j = true -> is_sum_spec_ind (unit j) (Is_unit j) - | is_sum_spec_nothing : forall u, is_sum_spec_ind u (Is_nothing). - - Lemma is_sum_spec u : is_sum_spec_ind u (is_sum i is_unit u). - Proof. - unfold is_sum; case u; intros; try constructor. - case_eq (eq_idx_bool p i); intros; subst; try constructor; auto. - revert H. case eq_idx_spec; try discriminate. auto. - case_eq (is_unit p); intros; try constructor. auto. - Qed. - - Instance assoc : @Associative X R (Bin.value (e_bin i)). - Proof. - destruct (e_bin i). simpl. assumption. - Qed. - Instance proper : Proper (R ==> R ==> R)(Bin.value (e_bin i)). - Proof. - destruct (e_bin i). simpl. assumption. - Qed. - Hypothesis comm : @Commutative X R (Bin.value (e_bin i)). - - Lemma sum'_sum : forall (l: mset T), eval (sum' i l) ==eval (sum i l) . - Proof. - intros [[a n] | [a n] l]; destruct n; simpl; reflexivity. - Qed. - - Lemma eval_sum_nil x: - eval (sum i (nil (x,xH))) == (eval x). - Proof. rewrite <- sum'_sum. reflexivity. Qed. - - Lemma eval_sum_cons : forall n a (l: mset T), - (eval (sum i ((a,n)::l))) == (@Bin.value _ _ (e_bin i) (@copy _ (@Bin.value _ _ (e_bin i)) n (eval a)) (eval (sum i l))). - Proof. - intros n a [[? ? ]|[b m] l]; simpl; reflexivity. - Qed. - - Inductive compat_sum_unit : @m idx (mset T) -> Prop := - | csu_left : forall x, is_unit x = true-> compat_sum_unit (left x) - | csu_right : forall m, compat_sum_unit (right m) - . - - Lemma compat_sum_unit_return x n : compat_sum_unit (return_sum i is_unit x n). - Proof. - unfold return_sum. - case is_sum_spec; intros; try constructor; auto. - Qed. - - Lemma compat_sum_unit_add : forall x n h, - compat_sum_unit h - -> - compat_sum_unit - (add_to_sum i (is_unit_of i) x n - h). - Proof. - unfold add_to_sum;intros; inversion H; - case_eq (is_sum i (is_unit_of i) x); - intros; simpl; try constructor || eauto. apply H0. - Qed. - - (* Hint Resolve copy_plus. : this lags because of the inference of the implicit arguments *) - Hint Extern 5 (copy (?n + ?m) (eval ?a) == Bin.value (copy ?n (eval ?a)) (copy ?m (eval ?a))) => apply copy_plus. - Hint Extern 5 (?x == ?x) => reflexivity. - Hint Extern 5 ( Bin.value ?x ?y == Bin.value ?y ?x) => apply Bin.comm. - - Lemma eval_merge_bin : forall (h k: mset T), - eval (sum i (merge_msets compare h k)) == @Bin.value _ _ (e_bin i) (eval (sum i h)) (eval (sum i k)). - Proof. - induction h as [[a n]|[a n] h IHh]; intro k. - simpl. - induction k as [[b m]|[b m] k IHk]; simpl. - destruct (tcompare_weak_spec a b) as [a|a b|a b]; simpl; auto. apply copy_plus; auto. - - destruct (tcompare_weak_spec a b) as [a|a b|a b]; simpl; auto. - rewrite copy_plus,law_assoc; auto. - rewrite IHk; clear IHk. rewrite 2 law_assoc. apply proper. apply law_comm. reflexivity. - - induction k as [[b m]|[b m] k IHk]; simpl; simpl in IHh. - destruct (tcompare_weak_spec a b) as [a|a b|a b]; simpl. - rewrite (law_comm _ (copy m (eval a))), law_assoc, <- copy_plus, Pplus_comm; auto. - rewrite <- law_assoc, IHh. reflexivity. - rewrite law_comm. reflexivity. - - simpl in IHk. - destruct (tcompare_weak_spec a b) as [a|a b|a b]; simpl. - rewrite IHh; clear IHh. rewrite 2 law_assoc. rewrite (law_comm _ (copy m (eval a))). rewrite law_assoc, <- copy_plus, Pplus_comm; auto. - rewrite IHh; clear IHh. simpl. rewrite law_assoc. reflexivity. - rewrite 2 (law_comm (copy m (eval b))). rewrite law_assoc. apply proper; [ | reflexivity]. - rewrite <- IHk. reflexivity. - Qed. - - - Lemma copy_mset' n (l: mset T): copy_mset n l = nelist_map (fun vm => let '(v,m):=vm in (v,Pmult n m)) l. - Proof. - unfold copy_mset. destruct n; try reflexivity. - - simpl. induction l as [|[a l] IHl]; simpl; try congruence. destruct a. reflexivity. - Qed. - - - Lemma copy_mset_succ n (l: mset T): eval (sum i (copy_mset (Psucc n) l)) == @Bin.value _ _ (e_bin i) (eval (sum i l)) (eval (sum i (copy_mset n l))). - Proof. - rewrite 2 copy_mset'. - - induction l as [[a m]|[a m] l IHl]. - simpl eval. rewrite <- copy_plus; auto. rewrite Pmult_Sn_m. reflexivity. - - simpl nelist_map. rewrite ! eval_sum_cons. rewrite IHl. clear IHl. - rewrite Pmult_Sn_m. rewrite copy_plus; auto. rewrite <- !law_assoc. - apply Binvalue_Proper; try reflexivity. - rewrite law_comm . rewrite <- !law_assoc. apply proper; try reflexivity. - apply law_comm. - Qed. - - Lemma copy_mset_copy : forall n (m : mset T), eval (sum i (copy_mset n m)) == @copy _ (@Bin.value _ _ (e_bin i)) n (eval (sum i m)). - Proof. - induction n using Pind; intros. - - unfold copy_mset. rewrite copy_xH. reflexivity. - rewrite copy_mset_succ. rewrite copy_Psucc. rewrite IHn. reflexivity. - Qed. - - Instance compat_sum_unit_Unit : forall p, compat_sum_unit (left p) -> - @Unit X R (Bin.value (e_bin i)) (eval (unit p)). - Proof. - intros. - inversion H. subst. auto. - Qed. - - Lemma copy_n_unit : forall j n, is_unit j = true -> - eval (unit j) == @copy _ (Bin.value (e_bin i)) n (eval (unit j)). - Proof. - intros. - induction n using Prect. - rewrite copy_xH. reflexivity. - rewrite copy_Psucc. rewrite <- IHn. apply is_unit_sum_Unit in H. rewrite law_neutral_left. reflexivity. - Qed. - - - Lemma z0 l r (H : compat_sum_unit r): - eval (sum i (run_msets (comp (merge_msets compare) l r))) == eval (sum i ((merge_msets compare) (l) (run_msets r))). - Proof. - unfold comp. unfold run_msets. - case_eq r; intros; subst. - rewrite eval_merge_bin; auto. - rewrite eval_sum_nil. - apply compat_sum_unit_Unit in H. rewrite law_neutral_right. reflexivity. - reflexivity. - Qed. - - Lemma z1 : forall n x, - eval (sum i (run_msets (return_sum i (is_unit) x n ))) - == @copy _ (@Bin.value _ _ (e_bin i)) n (eval x). - Proof. - intros. unfold return_sum. unfold run_msets. - case (is_sum_spec); intros; subst. - rewrite copy_mset_copy. - reflexivity. - - rewrite eval_sum_nil. apply copy_n_unit. auto. - reflexivity. - Qed. - Lemma z2 : forall u n x, compat_sum_unit x -> - eval (sum i ( run_msets - (add_to_sum i (is_unit) u n x))) - == - @Bin.value _ _ (e_bin i) (@copy _ (@Bin.value _ _ (e_bin i)) n (eval u)) (eval (sum i (run_msets x))). - Proof. - intros u n x Hix . - unfold add_to_sum. - case is_sum_spec; intros; subst. - - rewrite z0 by auto. rewrite eval_merge_bin. rewrite copy_mset_copy. reflexivity. - rewrite <- copy_n_unit by assumption. - apply is_unit_sum_Unit in H. - rewrite law_neutral_left. reflexivity. - - - rewrite z0 by auto. rewrite eval_merge_bin. reflexivity. - Qed. - End sum_correctness. - Lemma eval_norm_msets i norm - (Comm : Commutative R (Bin.value (e_bin i))) - (Hnorm: forall u, eval (norm u) == eval u) : forall h, eval (sum i (norm_msets norm i h)) == eval (sum i h). - Proof. - unfold norm_msets. - assert (H : - forall h : mset T, - eval (sum i (run_msets (norm_msets_ i (is_unit_of i) norm h))) == eval (sum i h) /\ compat_sum_unit (is_unit_of i) (norm_msets_ i (is_unit_of i) norm h)). - - induction h as [[a n] | [a n] h [IHh IHh']]; simpl norm_msets_; split. - rewrite z1 by auto. rewrite Hnorm . reflexivity. auto. - apply compat_sum_unit_return. - - rewrite z2 by auto. rewrite IHh. - rewrite eval_sum_cons. rewrite Hnorm. reflexivity. apply compat_sum_unit_add, IHh'. - - apply H. - Defined. - - (** auxiliary lemmas about products *) - - Section prd_correctness. - Variable i : idx. - Variable is_unit : idx -> bool. - Hypothesis is_unit_prd_Unit : forall j, is_unit j = true-> @Unit X R (Bin.value (e_bin i)) (eval (unit j)). - - Inductive is_prd_spec_ind : T -> @discr (nelist T) -> Prop := - | is_prd_spec_op : - forall j l, j = i -> is_prd_spec_ind (prd j l) (Is_op l) - | is_prd_spec_unit : - forall j, is_unit j = true -> is_prd_spec_ind (unit j) (Is_unit j) - | is_prd_spec_nothing : - forall u, is_prd_spec_ind u (Is_nothing). - - Lemma is_prd_spec u : is_prd_spec_ind u (is_prd i is_unit u). - Proof. - unfold is_prd; case u; intros; try constructor. - case (eq_idx_spec); intros; subst; try constructor; auto. - case_eq (is_unit p); intros; try constructor; auto. - Qed. - - Lemma prd'_prd : forall (l: nelist T), eval (prd' i l) == eval (prd i l). - Proof. - intros [?|? [|? ?]]; simpl; reflexivity. - Qed. - - - Lemma eval_prd_nil x: eval (prd i (nil x)) == eval x. - Proof. - rewrite <- prd'_prd. simpl. reflexivity. - Qed. - Lemma eval_prd_cons a : forall (l: nelist T), eval (prd i (a::l)) == @Bin.value _ _ (e_bin i) (eval a) (eval (prd i l)). - Proof. - intros [|b l]; simpl; reflexivity. - Qed. - Lemma eval_prd_app : forall (h k: nelist T), eval (prd i (h++k)) == @Bin.value _ _ (e_bin i) (eval (prd i h)) (eval (prd i k)). - Proof. - induction h; intro k. simpl; try reflexivity. - simpl appne. rewrite 2 eval_prd_cons, IHh, law_assoc. reflexivity. - Qed. - - Inductive compat_prd_unit : @m idx (nelist T) -> Prop := - | cpu_left : forall x, is_unit x = true -> compat_prd_unit (left x) - | cpu_right : forall m, compat_prd_unit (right m) - . - - Lemma compat_prd_unit_return x: compat_prd_unit (return_prd i is_unit x). - Proof. - unfold return_prd. - case (is_prd_spec); intros; try constructor; auto. - Qed. - - Lemma compat_prd_unit_add : forall x h, - compat_prd_unit h - -> - compat_prd_unit - (add_to_prd i is_unit x - h). - Proof. - intros. - unfold add_to_prd. - unfold comp. - case (is_prd_spec); intros; try constructor; auto. - unfold comp; case h; try constructor. - unfold comp; case h; try constructor. - Qed. - - - Instance compat_prd_Unit : forall p, compat_prd_unit (left p) -> - @Unit X R (Bin.value (e_bin i)) (eval (unit p)). - Proof. - intros. - inversion H; subst. apply is_unit_prd_Unit. assumption. - Qed. - - Lemma z0' : forall l (r: @m idx (nelist T)), compat_prd_unit r -> - eval (prd i (run_list (comp (@appne T) l r))) == eval (prd i ((appne (l) (run_list r)))). - Proof. - intros. - unfold comp. unfold run_list. case_eq r; intros; auto; subst. - rewrite eval_prd_app. - rewrite eval_prd_nil. - apply compat_prd_Unit in H. rewrite law_neutral_right. reflexivity. - reflexivity. - Qed. - - Lemma z1' a : eval (prd i (run_list (return_prd i is_unit a))) == eval (prd i (nil a)). - Proof. - intros. unfold return_prd. unfold run_list. - case (is_prd_spec); intros; subst; reflexivity. - Qed. - Lemma z2' : forall u x, compat_prd_unit x -> - eval (prd i ( run_list - (add_to_prd i is_unit u x))) == @Bin.value _ _ (e_bin i) (eval u) (eval (prd i (run_list x))). - Proof. - intros u x Hix. - unfold add_to_prd. - case (is_prd_spec); intros; subst. - rewrite z0' by auto. rewrite eval_prd_app. reflexivity. - apply is_unit_prd_Unit in H. rewrite law_neutral_left. reflexivity. - rewrite z0' by auto. rewrite eval_prd_app. reflexivity. - Qed. - - End prd_correctness. - - - - - Lemma eval_norm_lists i (Hnorm: forall u, eval (norm u) == eval u) : forall h, eval (prd i (norm_lists norm i h)) == eval (prd i h). - Proof. - unfold norm_lists. - assert (H : forall h : nelist T, - eval (prd i (run_list (norm_lists_ i (is_unit_of i) norm h))) == - eval (prd i h) - /\ compat_prd_unit (is_unit_of i) (norm_lists_ i (is_unit_of i) norm h)). - - - induction h as [a | a h [IHh IHh']]; simpl norm_lists_; split. - rewrite z1'. simpl. apply Hnorm. - - apply compat_prd_unit_return. - - rewrite z2'. rewrite IHh. rewrite eval_prd_cons. rewrite Hnorm. reflexivity. apply is_unit_of_Unit. - auto. - - apply compat_prd_unit_add. auto. - apply H. - Defined. - - (** correctness of the normalisation function *) - - Theorem eval_norm: forall u, eval (norm u) == eval u - with eval_norm_aux: forall i (l: vT i) (f: Sym.type_of i), - Proper (@Sym.rel_of X R i) f -> eval_aux (vnorm l) f == eval_aux l f. - Proof. - induction u as [ p m | p l | ? | ?]; simpl norm. - case_eq (is_commutative p); intros. - rewrite sum'_sum. - apply eval_norm_msets; auto. - reflexivity. - - rewrite prd'_prd. - apply eval_norm_lists; auto. - - apply eval_norm_aux, Sym.morph. - - reflexivity. - - induction l; simpl; intros f Hf. reflexivity. - rewrite eval_norm. apply IHl, Hf; reflexivity. - Qed. - - - (** corollaries, for goal normalisation or decision *) - - Lemma normalise : forall (u v: T), eval (norm u) == eval (norm v) -> eval u == eval v. - Proof. intros u v. rewrite 2 eval_norm. trivial. Qed. - - Lemma compare_reflect_eq: forall u v, compare u v = Eq -> eval u == eval v. - Proof. intros u v. case (tcompare_weak_spec u v); intros; try congruence. reflexivity. Qed. - - Lemma decide: forall (u v: T), compare (norm u) (norm v) = Eq -> eval u == eval v. - Proof. intros u v H. apply normalise. apply compare_reflect_eq. apply H. Qed. - - Lemma lift_normalise {S} {H : AAC_lift S R}: - forall (u v: T), (let x := norm u in let y := norm v in - S (eval x) (eval y)) -> S (eval u) (eval v). - Proof. destruct H. intros u v; simpl; rewrite 2 eval_norm. trivial. Qed. - -End s. -End Internal. - -Local Ltac internal_normalize := - let x := fresh in let y := fresh in - intro x; intro y; vm_compute in x; vm_compute in y; unfold x; unfold y; - compute [Internal.eval Internal.fold_map Internal.copy Prect]; simpl. - - -(** * Lemmas for performing transitivity steps - given an instance of AAC_lift *) - -Section t. - Context `{AAC_lift}. - - Lemma lift_transitivity_left (y x z : X): E x y -> R y z -> R x z. - Proof. destruct H as [Hequiv Hproper]; intros G;rewrite G. trivial. Qed. - - Lemma lift_transitivity_right (y x z : X): E y z -> R x y -> R x z. - Proof. destruct H as [Hequiv Hproper]; intros G. rewrite G. trivial. Qed. - - Lemma lift_reflexivity {HR :Reflexive R}: forall x y, E x y -> R x y. - Proof. destruct H. intros ? ? G. rewrite G. reflexivity. Qed. - -End t. - -Declare ML Module "aac". diff --git a/CHANGELOG b/CHANGELOG deleted file mode 100644 index 84b2aae..0000000 --- a/CHANGELOG +++ /dev/null @@ -1,28 +0,0 @@ -AAC_tactics 0.2-pl2 : ------------------ - -- Improved the handling of nullifiable patterns. - -AAC_tactics 0.2.1 : ------------------ - -- backport of some debian patches (thanks to S. Glondu) - -AAC_tactics 0.2 : ------------------ - -- Several operators can share a given unit (like max and plus sharing zero) -- Added some support to rewrite in inequations (using inequations) -- Better priting functions for aac_instances -- Overhauled inference of morphisms and operators : - * Lift the previous requirement to have at leat one AC and one A operator - * Binary operations are infered before morphisms (hence List.assoc can be recognized as being Associative) -- Should now be able to handle goal with evars (but this is not unification modulo AC) -- Added several new instances of Associative and Commutative operators -- The old syntax to declare AC and A operators is no longer supported -- The tactics do not abstract the proof they built (was troublesome if evars appeared) - -AAC_tactics 0.1 : ------------------ -Initial release - diff --git a/COPYING b/COPYING deleted file mode 100644 index 94a9ed0..0000000 --- a/COPYING +++ /dev/null @@ -1,674 +0,0 @@ - GNU GENERAL PUBLIC LICENSE - Version 3, 29 June 2007 - - Copyright (C) 2007 Free Software Foundation, Inc. - Everyone is permitted to copy and distribute verbatim copies - of this license document, but changing it is not allowed. - - Preamble - - The GNU General Public License is a free, copyleft license for -software and other kinds of works. - - The licenses for most software and other practical works are designed -to take away your freedom to share and change the works. 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If the Library as you -received it does not specify a version number of the GNU Lesser -General Public License, you may choose any version of the GNU Lesser -General Public License ever published by the Free Software Foundation. - - If the Library as you received it specifies that a proxy can decide -whether future versions of the GNU Lesser General Public License shall -apply, that proxy's public statement of acceptance of any version is -permanent authorization for you to choose that version for the -Library. diff --git a/Caveats.v b/Caveats.v deleted file mode 100644 index a7967cc..0000000 --- a/Caveats.v +++ /dev/null @@ -1,372 +0,0 @@ -(***************************************************************************) -(* This is part of aac_tactics, it is distributed under the terms of the *) -(* GNU Lesser General Public License version 3 *) -(* (see file LICENSE for more details) *) -(* *) -(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) -(***************************************************************************) - -(** * Currently known caveats and limitations of the [aac_tactics] library. - - Depending on your installation, either uncomment the following two - lines, or add them to your .coqrc files, replacing "." - with the path to the [aac_tactics] library -*) - -Require Import AAC. -Require Instances. - -(** ** Limitations *) - -(** *** 1. Dependent parameters - The type of the rewriting hypothesis must be of the form - - [forall (x_1: T_1) ... (x_n: T_n), R l r], - - where [R] is a relation over some type [T] and such that for all - variable [x_i] appearing in the left-hand side ([l]), we actually - have [T_i]=[T]. The goal should be of the form [S g d], where [S] - is a relation on [T]. - - In other words, we cannot instantiate arguments of an exogeneous - type. *) - -Section parameters. - - Context {X} {R} {E: @Equivalence X R} - {plus} {plus_A: Associative R plus} {plus_C: Commutative R plus} - {plus_Proper: Proper (R ==> R ==> R) plus} - {zero} {Zero: Unit R plus zero}. - - Notation "x == y" := (R x y) (at level 70). - Notation "x + y" := (plus x y) (at level 50, left associativity). - Notation "0" := (zero). - - Variable f: nat -> X -> X. - - (** in [Hf], the parameter [n] has type [nat], it cannot be instantiated automatically *) - Hypothesis Hf: forall n x, f n x + x == x. - Hypothesis Hf': forall n, Proper (R ==> R) (f n). - - Goal forall a b k, a + f k (b+a) + b == a+b. - intros. - Fail aac_rewrite Hf. (** [aac_rewrite] does not instantiate [n] automatically *) - aac_rewrite (Hf k). (** of course, this argument can be given explicitly *) - aac_reflexivity. - Qed. - - (** for the same reason, we cannot handle higher-order parameters (here, [g])*) - Hypothesis H : forall g x y, g x + g y == g (x + y). - Variable g : X -> X. - Hypothesis Hg : Proper (R ==> R) g. - Goal forall a b c, g a + g b + g c == g (a + b + c). - intros. - Fail aac_rewrite H. - do 2 aac_rewrite (H g). aac_reflexivity. - Qed. - -End parameters. - - -(** *** 2. Exogeneous morphisms - - We do not handle `exogeneous' morphisms: morphisms that move from - type [T] to some other type [T']. *) - -Section morphism. - Require Import NArith Minus. - Open Scope nat_scope. - - (** Typically, although [N_of_nat] is a proper morphism from - [@eq nat] to [@eq N], we cannot rewrite under [N_of_nat] *) - Goal forall a b: nat, N_of_nat (a+b-(b+a)) = 0%N. - intros. - Fail aac_rewrite minus_diag. - Abort. - - - (* More generally, this prevents us from rewriting under - propositional contexts *) - Context {P} {HP : Proper (@eq nat ==> iff) P}. - Hypothesis H : P 0. - - Goal forall a b, P (a + b - (b + a)). - intros a b. - Fail aac_rewrite minus_diag. - (** a solution is to introduce an evar to replace the part to be - rewritten. This tiresome process should be improved in the - future. Here, it can be done using eapply and the morphism. *) - eapply HP. - aac_rewrite minus_diag. - reflexivity. - exact H. - Qed. - - Goal forall a b, a+b-(b+a) = 0 /\ b-b = 0. - intros. - (** similarly, we need to bring equations to the toplevel before - being able to rewrite *) - Fail aac_rewrite minus_diag. - split; aac_rewrite minus_diag; reflexivity. - Qed. - -End morphism. - - -(** *** 3. Treatment of variance with inequations. - - We do not take variance into account when we compute the set of - solutions to a matching problem modulo AC. As a consequence, - [aac_instances] may propose solutions for which [aac_rewrite] will - fail, due to the lack of adequate morphisms *) - -Section ineq. - - Require Import ZArith. - Import Instances.Z. - Open Scope Z_scope. - - Instance Zplus_incr: Proper (Zle ==> Zle ==> Zle) Zplus. - Proof. intros ? ? H ? ? H'. apply Zplus_le_compat; assumption. Qed. - - Hypothesis H: forall x, x+x <= x. - Goal forall a b c, c + - (a + a) + b + b <= c. - intros. - (** this fails because the first solution is not valid ([Zopp] is not increasing), *) - Fail aac_rewrite H. - aac_instances H. - (** on the contrary, the second solution is valid: *) - aac_rewrite H at 1. - (** Currently, we cannot filter out such invalid solutions in an easy way; - this should be fixed in the future *) - Abort. - -End ineq. - - - -(** ** Caveats *) - -(** *** 1. Special treatment for units. - [S O] is considered as a unit for multiplication whenever a [Peano.mult] - appears in the goal. The downside is that [S x] does not match [1], - and [1] does not match [S(0+0)] whenever [Peano.mult] appears in - the goal. *) - -Section Peano. - Import Instances.Peano. - - Hypothesis H : forall x, x + S x = S (x+x). - - Goal 1 = 1. - (** ok (no multiplication around), [x] is instantiated with [O] *) - aacu_rewrite H. - Abort. - - Goal 1*1 = 1. - (** fails since 1 is seen as a unit, not the application of the - morphism [S] to the constant [O] *) - Fail aacu_rewrite H. - Abort. - - Hypothesis H': forall x, x+1 = 1+x. - - Goal forall a, a+S(0+0) = 1+a. - (** ok (no multiplication around), [x] is instantiated with [a]*) - intro. aac_rewrite H'. - Abort. - - Goal forall a, a*a+S(0+0) = 1+a*a. - (** fails: although [S(0+0)] is understood as the application of - the morphism [S] to the constant [O], it is not recognised - as the unit [S O] of multiplication *) - intro. Fail aac_rewrite H'. - Abort. - - (** More generally, similar counter-intuitive behaviours can appear - when declaring an applied morphism as an unit. *) - -End Peano. - - - -(** *** 2. Existential variables. -We implemented an algorithm for _matching_ modulo AC, not for -_unifying_ modulo AC. As a consequence, existential variables -appearing in a goal are considered as constants, they will not be -instantiated. *) - -Section evars. - Require Import ZArith. - Import Instances.Z. - - Variable P: Prop. - Hypothesis H: forall x y, x+y+x = x -> P. - Hypothesis idem: forall x, x+x = x. - Goal P. - eapply H. - aac_rewrite idem. (** this works: [x] is instantiated with an evar *) - instantiate (2 := 0). - symmetry. aac_reflexivity. (** this does work but there are remaining evars in the end *) - Abort. - - Hypothesis H': forall x, 3+x = x -> P. - Goal P. - eapply H'. - Fail aac_rewrite idem. (** this fails since we do not instantiate evars *) - Abort. -End evars. - - -(** *** 3. Distinction between [aac_rewrite] and [aacu_rewrite] *) - -Section U. - Context {X} {R} {E: @Equivalence X R} - {dot} {dot_A: Associative R dot} {dot_Proper: Proper (R ==> R ==> R) dot} - {one} {One: Unit R dot one}. - - Infix "==" := R (at level 70). - Infix "*" := dot. - Notation "1" := one. - - (** In some situations, the [aac_rewrite] tactic allows - instantiations of a variable with a unit, when the variable occurs - directly under a function symbol: *) - - Variable f : X -> X. - Hypothesis Hf : Proper (R ==> R) f. - Hypothesis dot_inv_left : forall x, f x*x == x. - Goal f 1 == 1. - aac_rewrite dot_inv_left. reflexivity. - Qed. - - (** This behaviour seems desirable in most situations: these - solutions with units are less peculiar than the other ones, since - the unit comes from the goal. However, this policy is not properly - enforced for now (hard to do with the current algorithm): *) - - Hypothesis dot_inv_right : forall x, x*f x == x. - Goal f 1 == 1. - Fail aac_rewrite dot_inv_right. - aacu_rewrite dot_inv_right. reflexivity. - Qed. - -End U. - -(** *** 4. Rewriting units *) -Section V. - Context {X} {R} {E: @Equivalence X R} - {dot} {dot_A: Associative R dot} {dot_Proper: Proper (R ==> R ==> R) dot} - {one} {One: Unit R dot one}. - - Infix "==" := R (at level 70). - Infix "*" := dot. - Notation "1" := one. - - (** [aac_rewrite] uses the symbols appearing in the goal and the - hypothesis to infer the AC and A operations. In the following - example, [dot] appears neither in the left-hand-side of the goal, - nor in the right-hand side of the hypothesis. Hence, 1 is not - recognised as a unit. To circumvent this problem, we can force - [aac_rewrite] to take into account a given operation, by giving - it an extra argument. This extra argument seems useful only in - this peculiar case. *) - - Lemma inv_unique: forall x y y', x*y == 1 -> y'*x == 1 -> y==y'. - Proof. - intros x y y' Hxy Hy'x. - aac_instances <- Hy'x [dot]. - aac_rewrite <- Hy'x at 1 [dot]. - aac_rewrite Hxy. aac_reflexivity. - Qed. -End V. - -(** *** 5. Rewriting too much things. *) -Section W. - Variables a b c: nat. - Hypothesis H: 0 = c. - - Goal b*(a+a) <= b*(c+a+a+1). - - (** [aac_rewrite] finds a pattern modulo AC that matches a given - hypothesis, and then makes a call to [setoid_rewrite]. This - [setoid_rewrite] can unfortunately make several rewrites (in a - non-intuitive way: below, the [1] in the right-hand side is - rewritten into [S c]) *) - aac_rewrite H. - - (** To this end, we provide a companion tactic to [aac_rewrite] - and [aacu_rewrite], that makes the transitivity step, but not the - setoid_rewrite: - - This allows the user to select the relevant occurrences in which - to rewrite. *) - aac_pattern H at 2. setoid_rewrite H at 1. - Abort. - -End W. - -(** *** 6. Rewriting nullifiable patterns. *) -Section Z. - -(** If the pattern of the rewritten hypothesis does not contain "hard" -symbols (like constants, function symbols, AC or A symbols without -units), there can be infinitely many subterms such that the pattern -matches: it is possible to build "subterms" modulo ACU that make the -size of the term increase (by making neutral elements appear in a -layered fashion). Hence, we settled with heuristics to propose only -"some" of these solutions. In such cases, the tactic displays a -(conservative) warning. *) - -Variables a b c: nat. -Variable f: nat -> nat. -Hypothesis H0: forall x, 0 = x - x. -Hypothesis H1: forall x, 1 = x * x. - -Goal a+b*c = c. - aac_instances H0. - (** In this case, only three solutions are proposed, while there are - infinitely many solutions. E.g. - - a+b*c*(1+[]) - - a+b*c*(1+0*(1+ [])) - - ... - *) -Abort. - -(** **** If the pattern is a unit or can be instanciated to be equal - to a unit: - - The heuristic is to make the unit appear at each possible position - in the term, e.g. transforming [a] into [1*a] and [a*1], but this - process is not recursive (we will not transform [1*a]) into - [(1+0*1)*a] *) - -Goal a+b+c = c. - - aac_instances H0 [mult]. - (** 1 solution, we miss solutions like [(a+b+c*(1+0*(1+[])))] and so on *) - - aac_instances H1 [mult]. - (** 7 solutions, we miss solutions like [(a+b+c+0*(1+0*[]))]*) -Abort. - -(** *** Another example of the former case is the following, where the hypothesis can be instanciated to be equal to [1] *) -Hypothesis H : forall x y, (x+y)*x = x*x + y *x. -Goal a*a+b*a + c = c. - (** Here, only one solution if we use the aac_instance tactic *) - aac_instances <- H. - - (** There are 8 solutions using aacu_instances (but, here, - there are infinitely many different solutions). We miss e.g. [a*a +b*a - + (x*x + y*x)*c], which seems to be more peculiar. *) - aacu_instances <- H. - - (** The 7 last solutions are the same as if we were matching [1] *) - aacu_instances H1. Abort. - -(** The behavior of the tactic is not satisfying in this case. It is -still unclear how to handle properly this kind of situation : we plan -to investigate on this in the future *) - -End Z. - diff --git a/Instances.v b/Instances.v deleted file mode 100644 index 55dffbb..0000000 --- a/Instances.v +++ /dev/null @@ -1,260 +0,0 @@ -(***************************************************************************) -(* This is part of aac_tactics, it is distributed under the terms of the *) -(* GNU Lesser General Public License version 3 *) -(* (see file LICENSE for more details) *) -(* *) -(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) -(***************************************************************************) - -Require List. -Require Arith NArith Max Min. -Require ZArith Zminmax. -Require QArith Qminmax. -Require Relations. - -Require Export AAC. - -(** Instances for aac_rewrite.*) - - -(* This one is not declared as an instance: this interferes badly with setoid_rewrite *) -Lemma eq_subr {X} {R} `{@Reflexive X R}: subrelation eq R. -Proof. intros x y ->. reflexivity. Qed. - -(* At the moment, all the instances are exported even if they are packaged into modules. Even using LocalInstances in fact*) - -Module Peano. - Import Arith NArith Max Min. - Instance aac_plus_Assoc : Associative eq plus := plus_assoc. - Instance aac_plus_Comm : Commutative eq plus := plus_comm. - - Instance aac_mult_Comm : Commutative eq mult := mult_comm. - Instance aac_mult_Assoc : Associative eq mult := mult_assoc. - - Instance aac_min_Comm : Commutative eq min := min_comm. - Instance aac_min_Assoc : Associative eq min := min_assoc. - - Instance aac_max_Comm : Commutative eq max := max_comm. - Instance aac_max_Assoc : Associative eq max := max_assoc. - - Instance aac_one : Unit eq mult 1 := Build_Unit eq mult 1 mult_1_l mult_1_r. - Instance aac_zero_plus : Unit eq plus O := Build_Unit eq plus (O) plus_0_l plus_0_r. - Instance aac_zero_max : Unit eq max O := Build_Unit eq max 0 max_0_l max_0_r. - - - (* We also provide liftings from le to eq *) - Instance preorder_le : PreOrder le := Build_PreOrder _ le_refl le_trans. - Instance lift_le_eq : AAC_lift le eq := Build_AAC_lift eq_equivalence _. - -End Peano. - - -Module Z. - Import ZArith Zminmax. - Open Scope Z_scope. - Instance aac_Zplus_Assoc : Associative eq Zplus := Zplus_assoc. - Instance aac_Zplus_Comm : Commutative eq Zplus := Zplus_comm. - - Instance aac_Zmult_Comm : Commutative eq Zmult := Zmult_comm. - Instance aac_Zmult_Assoc : Associative eq Zmult := Zmult_assoc. - - Instance aac_Zmin_Comm : Commutative eq Zmin := Zmin_comm. - Instance aac_Zmin_Assoc : Associative eq Zmin := Zmin_assoc. - - Instance aac_Zmax_Comm : Commutative eq Zmax := Zmax_comm. - Instance aac_Zmax_Assoc : Associative eq Zmax := Zmax_assoc. - - Instance aac_one : Unit eq Zmult 1 := Build_Unit eq Zmult 1 Zmult_1_l Zmult_1_r. - Instance aac_zero_Zplus : Unit eq Zplus 0 := Build_Unit eq Zplus 0 Zplus_0_l Zplus_0_r. - - (* We also provide liftings from le to eq *) - Instance preorder_Zle : PreOrder Zle := Build_PreOrder _ Zle_refl Zle_trans. - Instance lift_le_eq : AAC_lift Zle eq := Build_AAC_lift eq_equivalence _. - -End Z. - -Module Lists. - Import List. - Instance aac_append_Assoc {A} : Associative eq (@app A) := @app_assoc A. - Instance aac_nil_append {A} : @Unit (list A) eq (@app A) (@nil A) := Build_Unit _ (@app A) (@nil A) (@app_nil_l A) (@app_nil_r A). - Instance aac_append_Proper {A} : Proper (eq ==> eq ==> eq) (@app A). - Proof. - repeat intro. - subst. - reflexivity. - Qed. -End Lists. - - -Module N. - Import NArith. - Open Scope N_scope. - Instance aac_Nplus_Assoc : Associative eq Nplus := Nplus_assoc. - Instance aac_Nplus_Comm : Commutative eq Nplus := Nplus_comm. - - Instance aac_Nmult_Comm : Commutative eq Nmult := Nmult_comm. - Instance aac_Nmult_Assoc : Associative eq Nmult := Nmult_assoc. - - Instance aac_Nmin_Comm : Commutative eq Nmin := N.min_comm. - Instance aac_Nmin_Assoc : Associative eq Nmin := N.min_assoc. - - Instance aac_Nmax_Comm : Commutative eq Nmax := N.max_comm. - Instance aac_Nmax_Assoc : Associative eq Nmax := N.max_assoc. - - Instance aac_one : Unit eq Nmult (1)%N := Build_Unit eq Nmult (1)%N Nmult_1_l Nmult_1_r. - Instance aac_zero : Unit eq Nplus (0)%N := Build_Unit eq Nplus (0)%N Nplus_0_l Nplus_0_r. - Instance aac_zero_max : Unit eq Nmax 0 := Build_Unit eq Nmax 0 N.max_0_l N.max_0_r. - - (* We also provide liftings from le to eq *) - Instance preorder_le : PreOrder Nle := Build_PreOrder Nle N.le_refl N.le_trans. - Instance lift_le_eq : AAC_lift Nle eq := Build_AAC_lift eq_equivalence _. - -End N. - -Module P. - Import NArith. - Open Scope positive_scope. - Instance aac_Pplus_Assoc : Associative eq Pplus := Pplus_assoc. - Instance aac_Pplus_Comm : Commutative eq Pplus := Pplus_comm. - - Instance aac_Pmult_Comm : Commutative eq Pmult := Pmult_comm. - Instance aac_Pmult_Assoc : Associative eq Pmult := Pmult_assoc. - - Instance aac_Pmin_Comm : Commutative eq Pmin := Pos.min_comm. - Instance aac_Pmin_Assoc : Associative eq Pmin := Pos.min_assoc. - - Instance aac_Pmax_Comm : Commutative eq Pmax := Pos.max_comm. - Instance aac_Pmax_Assoc : Associative eq Pmax := Pos.max_assoc. - - Instance aac_one : Unit eq Pmult 1 := Build_Unit eq Pmult 1 _ Pmult_1_r. - intros; reflexivity. Qed. (* TODO : add this lemma in the stdlib *) - Instance aac_one_max : Unit eq Pmax 1 := Build_Unit eq Pmax 1 Pos.max_1_l Pos.max_1_r. - - (* We also provide liftings from le to eq *) - Instance preorder_le : PreOrder Ple := Build_PreOrder Ple Pos.le_refl Pos.le_trans. - Instance lift_le_eq : AAC_lift Ple eq := Build_AAC_lift eq_equivalence _. -End P. - -Module Q. - Import QArith Qminmax. - Instance aac_Qplus_Assoc : Associative Qeq Qplus := Qplus_assoc. - Instance aac_Qplus_Comm : Commutative Qeq Qplus := Qplus_comm. - - Instance aac_Qmult_Comm : Commutative Qeq Qmult := Qmult_comm. - Instance aac_Qmult_Assoc : Associative Qeq Qmult := Qmult_assoc. - - Instance aac_Qmin_Comm : Commutative Qeq Qmin := Q.min_comm. - Instance aac_Qmin_Assoc : Associative Qeq Qmin := Q.min_assoc. - - Instance aac_Qmax_Comm : Commutative Qeq Qmax := Q.max_comm. - Instance aac_Qmax_Assoc : Associative Qeq Qmax := Q.max_assoc. - - Instance aac_one : Unit Qeq Qmult 1 := Build_Unit Qeq Qmult 1 Qmult_1_l Qmult_1_r. - Instance aac_zero_Qplus : Unit Qeq Qplus 0 := Build_Unit Qeq Qplus 0 Qplus_0_l Qplus_0_r. - - (* We also provide liftings from le to eq *) - Instance preorder_le : PreOrder Qle := Build_PreOrder Qle Qle_refl Qle_trans. - Instance lift_le_eq : AAC_lift Qle Qeq := Build_AAC_lift QOrderedType.QOrder.TO.eq_equiv _. - -End Q. - -Module Prop_ops. - Instance aac_or_Assoc : Associative iff or. Proof. unfold Associative; tauto. Qed. - Instance aac_or_Comm : Commutative iff or. Proof. unfold Commutative; tauto. Qed. - Instance aac_and_Assoc : Associative iff and. Proof. unfold Associative; tauto. Qed. - Instance aac_and_Comm : Commutative iff and. Proof. unfold Commutative; tauto. Qed. - Instance aac_True : Unit iff or False. Proof. constructor; firstorder. Qed. - Instance aac_False : Unit iff and True. Proof. constructor; firstorder. Qed. - - Program Instance aac_not_compat : Proper (iff ==> iff) not. - Solve All Obligations with firstorder. - - Instance lift_impl_iff : AAC_lift Basics.impl iff := Build_AAC_lift _ _. -End Prop_ops. - -Module Bool. - Instance aac_orb_Assoc : Associative eq orb. Proof. unfold Associative; firstorder. Qed. - Instance aac_orb_Comm : Commutative eq orb. Proof. unfold Commutative; firstorder. Qed. - Instance aac_andb_Assoc : Associative eq andb. Proof. unfold Associative; firstorder. Qed. - Instance aac_andb_Comm : Commutative eq andb. Proof. unfold Commutative; firstorder. Qed. - Instance aac_true : Unit eq orb false. Proof. constructor; firstorder. Qed. - Instance aac_false : Unit eq andb true. Proof. constructor; intros [|];firstorder. Qed. - - Instance negb_compat : Proper (eq ==> eq) negb. Proof. intros [|] [|]; auto. Qed. -End Bool. - -Module Relations. - Import Relations.Relations. - Section defs. - Variable T : Type. - Variables R S: relation T. - Definition inter : relation T := fun x y => R x y /\ S x y. - Definition compo : relation T := fun x y => exists z : T, R x z /\ S z y. - Definition negr : relation T := fun x y => ~ R x y. - (* union and converse are already defined in the standard library *) - - Definition bot : relation T := fun _ _ => False. - Definition top : relation T := fun _ _ => True. - End defs. - - Instance eq_same_relation T : Equivalence (same_relation T). Proof. firstorder. Qed. - - Instance aac_union_Comm T : Commutative (same_relation T) (union T). Proof. unfold Commutative; compute; intuition. Qed. - Instance aac_union_Assoc T : Associative (same_relation T) (union T). Proof. unfold Associative; compute; intuition. Qed. - Instance aac_bot T : Unit (same_relation T) (union T) (bot T). Proof. constructor; compute; intuition. Qed. - - Instance aac_inter_Comm T : Commutative (same_relation T) (inter T). Proof. unfold Commutative; compute; intuition. Qed. - Instance aac_inter_Assoc T : Associative (same_relation T) (inter T). Proof. unfold Associative; compute; intuition. Qed. - Instance aac_top T : Unit (same_relation T) (inter T) (top T). Proof. constructor; compute; intuition. Qed. - - (* note that we use [eq] directly as a neutral element for composition *) - Instance aac_compo T : Associative (same_relation T) (compo T). Proof. unfold Associative; compute; firstorder. Qed. - Instance aac_eq T : Unit (same_relation T) (compo T) (eq). Proof. compute; firstorder subst; trivial. Qed. - - Instance negr_compat T : Proper (same_relation T ==> same_relation T) (negr T). - Proof. compute. firstorder. Qed. - - Instance transp_compat T : Proper (same_relation T ==> same_relation T) (transp T). - Proof. compute. firstorder. Qed. - - Instance clos_trans_incr T : Proper (inclusion T ==> inclusion T) (clos_trans T). - Proof. - intros R S H x y Hxy. induction Hxy. - constructor 1. apply H. assumption. - econstructor 2; eauto 3. - Qed. - Instance clos_trans_compat T: Proper (same_relation T ==> same_relation T) (clos_trans T). - Proof. intros R S H; split; apply clos_trans_incr, H. Qed. - - Instance clos_refl_trans_incr T : Proper (inclusion T ==> inclusion T) (clos_refl_trans T). - Proof. - intros R S H x y Hxy. induction Hxy. - constructor 1. apply H. assumption. - constructor 2. - econstructor 3; eauto 3. - Qed. - Instance clos_refl_trans_compat T : Proper (same_relation T ==> same_relation T) (clos_refl_trans T). - Proof. intros R S H; split; apply clos_refl_trans_incr, H. Qed. - - Instance preorder_inclusion T : PreOrder (inclusion T). - Proof. constructor; unfold Reflexive, Transitive, inclusion; intuition. Qed. - - Instance lift_inclusion_same_relation T: AAC_lift (inclusion T) (same_relation T) := - Build_AAC_lift (eq_same_relation T) _. - Proof. firstorder. Qed. - -End Relations. - -Module All. - Export Peano. - Export Z. - Export P. - Export N. - Export Prop_ops. - Export Bool. - Export Relations. -End All. - -(* Here, we should not see any instance of our classes. - Print HintDb typeclass_instances. -*) diff --git a/LICENSE b/LICENSE index 1be0ca5..fb8f8b7 100644 --- a/LICENSE +++ b/LICENSE @@ -1,4 +1,6 @@ -The aac_tactics plugin library is free software: you can redistribute +Copyright (C) 2009-2018 Thomas Braibant, Damien Pous, Fabian Kunze + +The AAC tactics plugin library is free software: you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. @@ -8,6 +10,168 @@ WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. 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If the Library as you +received it does not specify a version number of the GNU Lesser +General Public License, you may choose any version of the GNU Lesser +General Public License ever published by the Free Software Foundation. + + If the Library as you received it specifies that a proxy can decide +whether future versions of the GNU Lesser General Public License shall +apply, that proxy's public statement of acceptance of any version is +permanent authorization for you to choose that version for the +Library. diff --git a/Make b/Make deleted file mode 100644 index 3804332..0000000 --- a/Make +++ /dev/null @@ -1,23 +0,0 @@ --I . --R . AAC_tactics - -coq.mli -helper.mli -search_monad.mli -matcher.mli -theory.mli -print.mli -aac_rewrite.mli -coq.ml -helper.ml -search_monad.ml -matcher.ml -theory.ml -print.ml -aac_rewrite.ml4 -aac.mlpack - -AAC.v -Instances.v -Tutorial.v -Caveats.v diff --git a/Makefile b/Makefile index abf2ec6..3e77020 100644 --- a/Makefile +++ b/Makefile @@ -1,16 +1,16 @@ all: Makefile.coq - +make -f Makefile.coq all + +$(MAKE) -f Makefile.coq all clean: Makefile.coq - +make -f Makefile.coq clean - rm -f Makefile.coq + +$(MAKE) -f Makefile.coq cleanall + rm -f Makefile.coq Makefile.coq.conf -Makefile.coq: Make - $(COQBIN)coq_makefile -f Make -o Makefile.coq +Makefile.coq: _CoqProject + $(COQBIN)coq_makefile -f _CoqProject -o Makefile.coq -Make: ; +_CoqProject Makefile: ; %: Makefile.coq - +make -f Makefile.coq $@ + +$(MAKE) -f Makefile.coq $@ .PHONY: all clean diff --git a/README b/README deleted file mode 100644 index 38fc514..0000000 --- a/README +++ /dev/null @@ -1,55 +0,0 @@ - - aac_tactics - =========== - - Thomas Braibant & Damien Pous - -Laboratoire d'Informatique de Grenoble (UMR 5217), INRIA, CNRS, France - - -FOREWORD -======== - -This plugin provides tactics for rewriting universally quantified -equations, modulo associativity and commutativity of some operators. - -INSTALLATION -============ - -opam repo add coq-released https://coq.inria.fr/opam/released -opam install coq-aac-tactics - -DOCUMENTATION -============= - -Please refer to Tutorial.v for a succinct introduction on how to use -this plugin. - -To understand the inner-working of the tactic, please refer to the -.mli files as the main source of information on each .ml -file. Alternatively, [make world] generates ocamldoc/coqdoc -documentation in directories doc/ and html/, respectively. - -File Instances.v defines several instances for frequent use-cases of -this plugin, that should allow you to use it out-of-the-shelf. Namely, -we have instances for: - -- Peano naturals (Import Instances.Peano) -- Z binary numbers (Import Instances.Z) -- N binary numbers (Import Instances.N) -- P binary numbers (Import Instances.P) -- Rationnal numbers (Import Instances.Q) -- Prop (Import Instances.Prop_ops) -- Booleans (Import Instances.Bool) -- Relations (Import Instances.Relations) -- All of the above (Import Instances.All) - - -ACKNOWLEDGEMENTS -================ - -We are grateful to Evelyne Contejean, Hugo Herbelin, Assia Mahboubi -and Matthieu Sozeau for highly instructive discussions. - -This plugin took inspiration from the plugin tutorial "constructors", -distributed under the LGPL 2.1, copyrighted by Matthieu Sozeau diff --git a/README.md b/README.md new file mode 100644 index 0000000..c70f1bf --- /dev/null +++ b/README.md @@ -0,0 +1,117 @@ +# AAC tactics + +[![Travis][travis-shield]][travis-link] +[![Contributing][contributing-shield]][contributing-link] +[![Code of Conduct][conduct-shield]][conduct-link] +[![Gitter][gitter-shield]][gitter-link] +[![DOI][doi-shield]][doi-link] + +[doi-shield]: https://zenodo.org/badge/DOI/10.1007/978-3-642-25379-9_14.svg +[doi-link]: https://doi.org/10.1007/978-3-642-25379-9_14 + +[travis-shield]: https://travis-ci.com/coq-community/aac-tactics.svg?branch=master +[travis-link]: https://travis-ci.com/coq-community/aac-tactics/builds + +[contributing-shield]: https://img.shields.io/badge/contributions-welcome-%23f7931e.svg +[contributing-link]: https://github.com/coq-community/manifesto/blob/master/CONTRIBUTING.md + +[conduct-shield]: https://img.shields.io/badge/%E2%9D%A4-code%20of%20conduct-%23f15a24.svg +[conduct-link]: https://github.com/coq-community/manifesto/blob/master/CODE_OF_CONDUCT.md + +[gitter-shield]: https://img.shields.io/badge/chat-on%20gitter-%23c1272d.svg +[gitter-link]: https://gitter.im/coq-community/Lobby + +This Coq plugin provides tactics for rewriting universally quantified +equations, modulo associativity and commutativity of some operator. +The tactics can be applied for custom operators by registering the +operators and their properties as type class instances. Many common +operator instances, such as for Z binary arithmetic and booleans, are +provided with the plugin. + + +More details about the project can be found in the paper +[Tactics for Reasoning modulo AC in Coq](https://arxiv.org/abs/1106.4448). + +## Meta + +- Author(s): + - Thomas Braibant (initial) + - Damien Pous (initial) + - Fabian Kunze +- Coq-community maintainer(s): + - Fabian Kunze ([**@fakusb**](https://github.com/fakusb)) + - Karl Palmskog ([**@palmskog**](https://github.com/palmskog)) +- License: [GNU Lesser General Public License v3.0 or later](LICENSE) +- Compatible Coq versions: Coq 8.9 (use the corresponding branch or release for other Coq versions) +- Compatible OCaml versions: all versions supported by Coq +- Additional dependencies: none + +## Building and installation instructions + +The easiest way to install the latest released version is via +[OPAM](https://opam.ocaml.org/doc/Install.html): + +```shell +opam repo add coq-released https://coq.inria.fr/opam/released +opam install coq-aac-tactics +``` + +To instead build and install manually, do: + +``` shell +git clone https://github.com/coq-community/aac-tactics +cd aac-tactics +make # or make -j +make install +``` + +After installation, the included modules are available under +the `AAC_tactics` namespace. + +## Documentation + +The following example shows an application of the tactics for reasoning over Z binary numbers: +```coq +Require Import AAC_tactics.AAC. +Require AAC_tactics.Instances. +Require Import ZArith. + +Section ZOpp. + Import Instances.Z. + Variables a b c : Z. + Hypothesis H: forall x, x + Z.opp x = 0. + + Goal a + b + c + Z.opp (c + a) = b. + aac_rewrite H. + aac_reflexivity. + Qed. +End ZOpp. +``` + +The file [Tutorial.v](theories/Tutorial.v) provides a succinct introduction +and more examples of how to use this plugin. + +The file [Instances.v](theories/Instances.v) defines several type class instances +for frequent use-cases of this plugin, that should allow you to use it off-the-shelf. +Namely, it contains instances for: + +- Peano naturals (`Import Instances.Peano.`) +- Z binary numbers (`Import Instances.Z.`) +- N binary numbers (`Import Instances.N.`) +- P binary numbers (`Import Instances.P.`) +- Rational numbers (`Import Instances.Q.`) +- Prop (`Import Instances.Prop_ops.`) +- Booleans (`Import Instances.Bool.`) +- Relations (`Import Instances.Relations.`) +- all of the above (`Import Instances.All.`) + +To understand the inner workings of the tactics, please refer to +the `.mli` files as the main source of information on each `.ml` file. + +## Acknowledgements + +The initial authors are grateful to Evelyne Contejean, Hugo Herbelin, +Assia Mahboubi, and Matthieu Sozeau for highly instructive discussions. +The plugin took inspiration from the plugin tutorial "constructors" by +Matthieu Sozeau, distributed under the LGPL 2.1. + diff --git a/Tutorial.v b/Tutorial.v deleted file mode 100644 index 76006ca..0000000 --- a/Tutorial.v +++ /dev/null @@ -1,399 +0,0 @@ -(***************************************************************************) -(* This is part of aac_tactics, it is distributed under the terms of the *) -(* GNU Lesser General Public License version 3 *) -(* (see file LICENSE for more details) *) -(* *) -(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) -(***************************************************************************) - -(** * Tutorial for using the [aac_tactics] library. - - Depending on your installation, either modify the following two - lines, or add them to your .coqrc files, replacing "." with the - path to the [aac_tactics] library. *) - -Add Rec LoadPath "." as AAC_tactics. -Add ML Path ".". -Require Import AAC. -Require Instances. - -(** ** Introductory example - - Here is a first example with relative numbers ([Z]): one can - rewrite an universally quantified hypothesis modulo the - associativity and commutativity of [Zplus]. *) - -Section introduction. - - Import ZArith. - Import Instances.Z. - - Variables a b c : Z. - Hypothesis H: forall x, x + Zopp x = 0. - Goal a + b + c + Zopp (c + a) = b. - aac_rewrite H. - aac_reflexivity. - Qed. - Goal a + c + Zopp (b + a + Zopp b) = c. - do 2 aac_rewrite H. - reflexivity. - Qed. - - (** Notes: - - the tactic handles arbitrary function symbols like [Zopp] (as - long as they are proper morphisms w.r.t. the considered - equivalence relation); - - here, ring would have done the job. - *) - -End introduction. - - -(** ** Usage - - One can also work in an abstract context, with arbitrary - associative and commutative operators. (Note that one can declare - several operations of each kind.) *) - -Section base. - Context {X} {R} {E: Equivalence R} - {plus} - {dot} - {zero} - {one} - {dot_A: @Associative X R dot } - {plus_A: @Associative X R plus } - {plus_C: @Commutative X R plus } - {dot_Proper :Proper (R ==> R ==> R) dot} - {plus_Proper :Proper (R ==> R ==> R) plus} - {Zero : Unit R plus zero} - {One : Unit R dot one} - . - - Notation "x == y" := (R x y) (at level 70). - Notation "x * y" := (dot x y) (at level 40, left associativity). - Notation "1" := (one). - Notation "x + y" := (plus x y) (at level 50, left associativity). - Notation "0" := (zero). - - (** In the very first example, [ring] would have solved the - goal. Here, since [dot] does not necessarily distribute over [plus], - it is not possible to rely on it. *) - - Section reminder. - Hypothesis H : forall x, x * x == x. - Variables a b c : X. - - Goal (a+b+c)*(c+a+b) == a+b+c. - aac_rewrite H. - aac_reflexivity. - Qed. - - (** The tactic starts by normalising terms, so that trailing units - are always eliminated. *) - - Goal ((a+b)+0+c)*((c+a)+b*1) == a+b+c. - aac_rewrite H. - aac_reflexivity. - Qed. - End reminder. - - (** The tactic can deal with "proper" morphisms of arbitrary arity - (here [f] and [g], or [Zopp] earlier): it rewrites under such - morphisms ([g]), and, more importantly, it is able to reorder - terms modulo AC under these morphisms ([f]). *) - - Section morphisms. - Variable f : X -> X -> X. - Hypothesis Hf : Proper (R ==> R ==> R) f. - Variable g : X -> X. - Hypothesis Hg : Proper (R ==> R) g. - - Variable a b: X. - Hypothesis H : forall x y, x+f (b+y) x == y+x. - Goal g ((f (a+b) a) + a) == g (a+a). - aac_rewrite H. - reflexivity. - Qed. - End morphisms. - - (** *** Selecting what and where to rewrite - - There are sometimes several solutions to the matching problem. We - now show how to interact with the tactic to select the desired - one. *) - - Section occurrence. - Variable f : X -> X. - Variable a : X. - Hypothesis Hf : Proper (R ==> R) f. - Hypothesis H : forall x, x + x == x. - - Goal f(a+a)+f(a+a) == f a. - (** In case there are several possible solutions, one can print - the different solutions using the [aac_instances] tactic (in - proof-general, look at buffer *coq* ): *) - aac_instances H. - (** the default choice is the occurrence with the smallest - possible context (number 0), but one can choose the desired - one; *) - aac_rewrite H at 1. - (** now the goal is [f a + f a == f a], there is only one solution. *) - aac_rewrite H. - reflexivity. - Qed. - - End occurrence. - - Section subst. - Variables a b c d : X. - Hypothesis H: forall x y, a*x*y*b == a*(x+y)*b. - Hypothesis H': forall x, x + x == x. - - Goal a*c*d*c*d*b == a*c*d*b. - (** Here, there is only one possible occurrence, but several substitutions; *) - aac_instances H. - (** one can select them with the proper keyword. *) - aac_rewrite H subst 1. - aac_rewrite H'. - aac_reflexivity. - Qed. - End subst. - - (** As expected, one can use both keywords together to select the - occurrence and the substitution. We also provide a keyword to - specify that the rewrite should be done in the right-hand side of - the equation. *) - - Section both. - Variables a b c d : X. - Hypothesis H: forall x y, a*x*y*b == a*(x+y)*b. - Hypothesis H': forall x, x + x == x. - - Goal a*c*d*c*d*b*b == a*(c*d*b)*b. - aac_instances H. - aac_rewrite H at 1 subst 1. - aac_instances H. - aac_rewrite H. - aac_rewrite H'. - aac_rewrite H at 0 subst 1 in_right. - aac_reflexivity. - Qed. - - End both. - - (** *** Distinction between [aac_rewrite] and [aacu_rewrite]: - - [aac_rewrite] rejects solutions in which variables are instantiated - by units, while the companion tactic, [aacu_rewrite] allows such - solutions. *) - - Section dealing_with_units. - Variables a b c: X. - Hypothesis H: forall x, a*x*a == x. - Goal a*a == 1. - (** Here, [x] must be instantiated with [1], so that the [aac_*] - tactics give no solutions; *) - try aac_instances H. - (** while we get solutions with the [aacu_*] tactics. *) - aacu_instances H. - aacu_rewrite H. - reflexivity. - Qed. - - (** We introduced this distinction because it allows us to rule - out dummy cases in common situations: *) - - Hypothesis H': forall x y z, x*y + x*z == x*(y+z). - Goal a*b*c + a*c + a*b == a*(c+b*(1+c)). - (** 6 solutions without units, *) - aac_instances H'. - aac_rewrite H' at 0. - (** more than 52 with units. *) - aacu_instances H'. - Abort. - - End dealing_with_units. -End base. - -(** *** Declaring instances - - To use one's own operations: it suffices to declare them as - instances of our classes. (Note that the following instances are - already declared in file [Instances.v].) *) - -Section Peano. - Require Import Arith. - - Instance aac_plus_Assoc : Associative eq plus := plus_assoc. - Instance aac_plus_Comm : Commutative eq plus := plus_comm. - - Instance aac_one : Unit eq mult 1 := - Build_Unit eq mult 1 mult_1_l mult_1_r. - Instance aac_zero_plus : Unit eq plus O := - Build_Unit eq plus (O) plus_0_l plus_0_r. - - - (** Two (or more) operations may share the same units: in the - following example, [0] is understood as the unit of [max] as well as - the unit of [plus]. *) - - Instance aac_max_Comm : Commutative eq Max.max := Max.max_comm. - Instance aac_max_Assoc : Associative eq Max.max := Max.max_assoc. - - Instance aac_zero_max : Unit eq Max.max O := - Build_Unit eq Max.max 0 Max.max_0_l Max.max_0_r. - - Variable a b c : nat. - Goal Max.max (a + 0) 0 = a. - aac_reflexivity. - Qed. - - (** Furthermore, several operators can be mixed: *) - - Hypothesis H : forall x y z, Max.max (x + y) (x + z) = x+ Max.max y z. - - Goal Max.max (a + b) (c + (a * 1)) = Max.max c b + a. - aac_instances H. aac_rewrite H. aac_reflexivity. - Qed. - Goal Max.max (a + b) (c + Max.max (a*1+0) 0) = a + Max.max b c. - aac_instances H. aac_rewrite H. aac_reflexivity. - Qed. - - - (** *** Working with inequations - - To be able to use the tactics, the goal must be a relation [R] - applied to two arguments, and the rewritten hypothesis must end - with a relation [Q] applied to two arguments. These relations are - not necessarily equivalences, but they should be related - according to the occurrence where the rewrite takes place; we - leave this check to the underlying call to [setoid_rewrite]. *) - - (** One can rewrite equations in the left member of inequations, *) - Goal (forall x, x + x = x) -> a + b + b + a <= a + b. - intro Hx. - aac_rewrite Hx. - reflexivity. - Qed. - - (** or in the right member of inequations, using the [in_right] keyword *) - Goal (forall x, x + x = x) -> a + b <= a + b + b + a. - intro Hx. - aac_rewrite Hx in_right. - reflexivity. - Qed. - - (** Similarly, one can rewrite inequations in inequations, *) - Goal (forall x, x + x <= x) -> a + b + b + a <= a + b. - intro Hx. - aac_rewrite Hx. - reflexivity. - Qed. - - (** possibly in the right-hand side. *) - Goal (forall x, x <= x + x) -> a + b <= a + b + b + a. - intro Hx. - aac_rewrite <- Hx in_right. - reflexivity. - Qed. - - (** [aac_reflexivity] deals with "trivial" inequations too *) - Goal Max.max (a + b) (c + a) <= Max.max (b + a) (c + 1*a). - aac_reflexivity. - Qed. - - (** In the last three examples, there were no equivalence relation - involved in the goal. However, we actually had to guess the - equivalence relation with respect to which the operators - ([plus,max,0]) were AC. In this case, it was Leibniz equality - [eq] so that it was automatically inferred; more generally, one - can specify which equivalence relation to use by declaring - instances of the [AAC_lift] type class: *) - - Instance lift_le_eq : AAC_lift le eq := {}. - (** (This instance is automatically inferred because [eq] is always a - valid candidate, here for [le].) *) - - -End Peano. - - -(** *** Normalising goals - - We also provide a tactic to normalise terms modulo AC. This - normalisation is the one we use internally. *) - -Section AAC_normalise. - - Import Instances.Z. - Require Import ZArith. - Open Scope Z_scope. - - Variable a b c d : Z. - Goal a + (b + c*c*d) + a + 0 + d*1 = a. - aac_normalise. - Abort. - -End AAC_normalise. - - - - -(** ** Examples from the web page *) -Section Examples. - - Import Instances.Z. - Require Import ZArith. - Open Scope Z_scope. - - (** *** Reverse triangle inequality *) - - Lemma Zabs_triangle : forall x y, Zabs (x + y) <= Zabs x + Zabs y . - Proof Zabs_triangle. - - Lemma Zplus_opp_r : forall x, x + -x = 0. - Proof Zplus_opp_r. - - (** The following morphisms are required to perform the required rewrites *) - Instance Zminus_compat : Proper (Zge ==> Zle) Zopp. - Proof. intros x y. omega. Qed. - - Instance Proper_Zplus : Proper (Zle ==> Zle ==> Zle) Zplus. - Proof. firstorder. Qed. - - Goal forall a b, Zabs a - Zabs b <= Zabs (a - b). - intros. unfold Zminus. - aac_instances <- (Zminus_diag b). - aac_rewrite <- (Zminus_diag b) at 3. - unfold Zminus. - aac_rewrite Zabs_triangle. - aac_rewrite Zplus_opp_r. - aac_reflexivity. - Qed. - - - (** *** Pythagorean triples *) - - Notation "x ^2" := (x*x) (at level 40). - Notation "2 ⋅ x" := (x+x) (at level 41). - - Lemma Hbin1: forall x y, (x+y)^2 = x^2 + y^2 + 2⋅x*y. Proof. intros; ring. Qed. - Lemma Hbin2: forall x y, x^2 + y^2 = (x+y)^2 + -(2⋅x*y). Proof. intros; ring. Qed. - Lemma Hopp : forall x, x + -x = 0. Proof Zplus_opp_r. - - Variables a b c : Z. - Hypothesis H : c^2 + 2⋅(a+1)*b = (a+1+b)^2. - Goal a^2 + b^2 + 2⋅a + 1 = c^2. - aacu_rewrite <- Hbin1. - rewrite Hbin2. - aac_rewrite <- H. - aac_rewrite Hopp. - aac_reflexivity. - Qed. - - (** Note: after the [aac_rewrite <- H], one could use [ring] to close the proof.*) - -End Examples. - - diff --git a/_CoqProject b/_CoqProject new file mode 100644 index 0000000..f803b8d --- /dev/null +++ b/_CoqProject @@ -0,0 +1,28 @@ +-Q theories AAC_tactics +-Q src AAC_tactics +-I src + +-arg -w -arg +default + +src/coq.mli +src/helper.mli +src/search_monad.mli +src/matcher.mli +src/theory.mli +src/print.mli +src/aac_rewrite.mli +src/coq.ml +src/helper.ml +src/search_monad.ml +src/matcher.ml +src/theory.ml +src/print.ml +src/aac_rewrite.ml +src/aac.ml4 +src/aac_plugin.mlpack + +theories/Utils.v +theories/AAC.v +theories/Instances.v +theories/Tutorial.v +theories/Caveats.v diff --git a/aac.mlpack b/aac.mlpack deleted file mode 100644 index 80cb22c..0000000 --- a/aac.mlpack +++ /dev/null @@ -1,7 +0,0 @@ -Coq -Helper -Search_monad -Matcher -Theory -Print -Aac_rewrite diff --git a/aac_rewrite.ml4 b/aac_rewrite.ml4 deleted file mode 100644 index 1f57c0b..0000000 --- a/aac_rewrite.ml4 +++ /dev/null @@ -1,529 +0,0 @@ -(***************************************************************************) -(* This is part of aac_tactics, it is distributed under the terms of the *) -(* GNU Lesser General Public License version 3 *) -(* (see file LICENSE for more details) *) -(* *) -(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) -(***************************************************************************) - -(** aac_rewrite -- rewriting modulo *) - -open Pcoq.Prim -open Pcoq.Constr -open Stdarg -open Constrarg - -DECLARE PLUGIN "aac" - -module Control = struct - let debug = false - let printing = false - let time = false -end - -module Debug = Helper.Debug (Control) -open Debug - -let time_tac msg tac = - if Control.time then Coq.tclTIME msg tac else tac - -let tac_or_exn tac exn msg = fun gl -> - try tac gl with e -> - pr_constr "last goal" (Tacmach.pf_concl gl); - exn msg e - - -let retype = Coq.retype - -(* helper to be used with the previous function: raise a new anomaly - except if a another one was previously raised *) -let push_anomaly msg = function - | e when CErrors.is_anomaly e -> raise e - | _ -> Coq.anomaly msg - -module M = Matcher - -open Term -open Names -open Coqlib -open Proof_type - -(** The various kind of relation we can encounter, as a hierarchy *) -type rew_relation = - | Bare of Coq.Relation.t - | Transitive of Coq.Transitive.t - | Equivalence of Coq.Equivalence.t - -(** {!promote try to go higher in the aforementionned hierarchy} *) -let promote (rlt : Coq.Relation.t) (k : rew_relation -> Proof_type.tactic) = - try Coq.Equivalence.cps_from_relation rlt - (fun e -> k (Equivalence e)) - with - | Not_found -> - begin - try Coq.Transitive.cps_from_relation rlt - (fun trans -> k (Transitive trans)) - with - |Not_found -> k (Bare rlt) - end - - -(* - Various situations: - p == q |- left == right : rewrite <- -> - p <= q |- left <= right : rewrite -> - p <= q |- left == right : failure - p == q |- left <= right : rewrite <- -> - - Not handled - p <= q |- left >= right : failure -*) - -(** aac_lift : the ideal type beyond AAC.v/Lift - - A base relation r, together with an equivalence relation, and the - proof that the former lifts to the later. Howver, we have to - ensure manually the invariant : r.carrier == e.carrier, and that - lift connects the two things *) -type aac_lift = - { - r : Coq.Relation.t; - e : Coq.Equivalence.t; - lift : Term.constr - } - -type rewinfo = - { - hypinfo : Coq.Rewrite.hypinfo; - - in_left : bool; (** are we rewriting in the left hand-sie of the goal *) - pattern : constr; - subject : constr; - morph_rlt : Coq.Relation.t; (** the relation we look for in morphism *) - eqt : Coq.Equivalence.t; (** the equivalence we use as workbase *) - rlt : Coq.Relation.t; (** the relation in the goal *) - lifting: aac_lift - } - -let infer_lifting (rlt: Coq.Relation.t) (k : lift:aac_lift -> Proof_type.tactic) : Proof_type.tactic = - Coq.cps_evar_relation rlt.Coq.Relation.carrier (fun e -> - let lift_ty = - mkApp (Lazy.force Theory.Stubs.lift, - [| - rlt.Coq.Relation.carrier; - rlt.Coq.Relation.r; - e - |] - ) in - Coq.cps_resolve_one_typeclass ~error:"Cannot infer a lifting" - lift_ty (fun lift goal -> - let x = rlt.Coq.Relation.carrier in - let r = rlt.Coq.Relation.r in - let eq = (Coq.nf_evar goal e) in - let equiv = Coq.lapp Theory.Stubs.lift_proj_equivalence [| x;r;eq; lift |] in - let lift = - { - r = rlt; - e = Coq.Equivalence.make x eq equiv; - lift = lift; - } - in - k ~lift:lift goal - )) - -(** Builds a rewinfo, once and for all *) -let dispatch in_left (left,right,rlt) hypinfo (k: rewinfo -> Proof_type.tactic ) : Proof_type.tactic= - let l2r = hypinfo.Coq.Rewrite.l2r in - infer_lifting rlt - (fun ~lift -> - let eq = lift.e in - k { - hypinfo = hypinfo; - in_left = in_left; - pattern = if l2r then hypinfo.Coq.Rewrite.left else hypinfo.Coq.Rewrite.right; - subject = if in_left then left else right; - morph_rlt = Coq.Equivalence.to_relation eq; - eqt = eq; - lifting = lift; - rlt = rlt - } - ) - - - -(** {1 Tactics} *) - - -(** Build the reifiers, the reified terms, and the evaluation fonction *) -let handle eqt zero envs (t : Matcher.Terms.t) (t' : Matcher.Terms.t) k = - - let (x,r,_) = Coq.Equivalence.split eqt in - Theory.Trans.mk_reifier (Coq.Equivalence.to_relation eqt) zero envs - (fun (maps, reifier) -> - (* fold through a term and reify *) - let t = Theory.Trans.reif_constr_of_t reifier t in - let t' = Theory.Trans.reif_constr_of_t reifier t' in - (* Some letins *) - let eval = (mkApp (Lazy.force Theory.Stubs.eval, [|x;r; maps.Theory.Trans.env_sym; maps.Theory.Trans.env_bin; maps.Theory.Trans.env_units|])) in - - Coq.cps_mk_letin "eval" eval (fun eval -> - Coq.cps_mk_letin "left" t (fun t -> - Coq.cps_mk_letin "right" t' (fun t' -> - k maps eval t t')))) - -(** [by_aac_reflexivity] is a sub-tactic that closes a sub-goal that - is merely a proof of equality of two terms modulo AAC *) -let by_aac_reflexivity zero - eqt envs (t : Matcher.Terms.t) (t' : Matcher.Terms.t) : Proof_type.tactic = - handle eqt zero envs t t' - (fun maps eval t t' -> - let (x,r,e) = Coq.Equivalence.split eqt in - let decision_thm = Coq.lapp Theory.Stubs.decide_thm - [|x;r;e; - maps.Theory.Trans.env_sym; - maps.Theory.Trans.env_bin; - maps.Theory.Trans.env_units; - t;t'; - |] - in - (* This convert is required to deal with evars in a proper - way *) - let convert_to = mkApp (r, [| mkApp (eval,[| t |]); mkApp (eval, [|t'|])|]) in - let convert = Proofview.V82.of_tactic (Tactics.convert_concl convert_to Term.VMcast) in - let apply_tac = Proofview.V82.of_tactic (Tactics.apply decision_thm) in - (Tacticals.tclTHENLIST - [ retype decision_thm; retype convert_to; - convert ; - tac_or_exn apply_tac Coq.user_error "unification failure"; - tac_or_exn (time_tac "vm_norm" (Proofview.V82.of_tactic (Tactics.normalise_in_concl))) Coq.anomaly "vm_compute failure"; - Tacticals.tclORELSE (Proofview.V82.of_tactic Tactics.reflexivity) - (Tacticals.tclFAIL 0 (Pp.str "Not an equality modulo A/AC")) - ]) - ) - -let by_aac_normalise zero lift ir t t' = - let eqt = lift.e in - let rlt = lift.r in - handle eqt zero ir t t' - (fun maps eval t t' -> - let (x,r,e) = Coq.Equivalence.split eqt in - let normalise_thm = Coq.lapp Theory.Stubs.lift_normalise_thm - [|x;r;e; - maps.Theory.Trans.env_sym; - maps.Theory.Trans.env_bin; - maps.Theory.Trans.env_units; - rlt.Coq.Relation.r; - lift.lift; - t;t'; - |] - in - (* This convert is required to deal with evars in a proper - way *) - let convert_to = mkApp (rlt.Coq.Relation.r, [| mkApp (eval,[| t |]); mkApp (eval, [|t'|])|]) in - let convert = Proofview.V82.of_tactic (Tactics.convert_concl convert_to Term.VMcast) in - let apply_tac = Proofview.V82.of_tactic (Tactics.apply normalise_thm) in - (Tacticals.tclTHENLIST - [ retype normalise_thm; retype convert_to; - convert ; - apply_tac; - ]) - - ) - -(** A handler tactic, that reifies the goal, and infer the liftings, - and then call its continuation *) -let aac_conclude - (k : Term.constr -> aac_lift -> Theory.Trans.ir -> Matcher.Terms.t -> Matcher.Terms.t -> Proof_type.tactic) = fun goal -> - - let (equation : Term.types) = Tacmach.pf_concl goal in - let envs = Theory.Trans.empty_envs () in - let left, right,r = - match Coq.match_as_equation goal equation with - | None -> Coq.user_error "The goal is not an applied relation" - | Some x -> x in - try infer_lifting r - (fun ~lift goal -> - let eq = Coq.Equivalence.to_relation lift.e in - let tleft,tright, goal = Theory.Trans.t_of_constr goal eq envs (left,right) in - let goal, ir = Theory.Trans.ir_of_envs goal eq envs in - let concl = Tacmach.pf_concl goal in - let _ = pr_constr "concl "concl in - let evar_map = Tacmach.project goal in - Tacticals.tclTHEN - (Refiner.tclEVARS evar_map) - (k left lift ir tleft tright) - goal - )goal - with - | Not_found -> Coq.user_error "No lifting from the goal's relation to an equivalence" - -open Libnames -open Tacexpr -open Tacinterp - -let aac_normalise = fun goal -> - let ids = Tacmach.pf_ids_of_hyps goal in - let loc = Loc.ghost in - let mp = MPfile (DirPath.make (List.map Id.of_string ["AAC"; "AAC_tactics"])) in - let norm_tac = KerName.make2 mp (Label.make "internal_normalize") in - let norm_tac = Misctypes.ArgArg (loc, norm_tac) in - Tacticals.tclTHENLIST - [ - aac_conclude by_aac_normalise; - Proofview.V82.of_tactic (Tacinterp.eval_tactic (TacArg (loc, TacCall (loc, norm_tac, [])))); - Proofview.V82.of_tactic (Tactics.keep ids) - ] goal - -let aac_reflexivity = fun goal -> - aac_conclude - (fun zero lift ir t t' -> - let x,r = Coq.Relation.split (lift.r) in - let r_reflexive = (Coq.Classes.mk_reflexive x r) in - Coq.cps_resolve_one_typeclass ~error:"The goal's relation is not reflexive" - r_reflexive - (fun reflexive -> - let lift_reflexivity = - mkApp (Lazy.force (Theory.Stubs.lift_reflexivity), - [| - x; - r; - lift.e.Coq.Equivalence.eq; - lift.lift; - reflexive - |]) - in - Tacticals.tclTHEN - - (Tacticals.tclTHEN (retype lift_reflexivity) (Proofview.V82.of_tactic (Tactics.apply lift_reflexivity))) - (fun goal -> - let concl = Tacmach.pf_concl goal in - let _ = pr_constr "concl "concl in - by_aac_reflexivity zero lift.e ir t t' goal) - ) - ) goal - -(** A sub-tactic to lift the rewriting using Lift *) -let lift_transitivity in_left (step:constr) preorder lifting (using_eq : Coq.Equivalence.t): tactic = - fun goal -> - (* catch the equation and the two members*) - let concl = Tacmach.pf_concl goal in - let (left, right, _ ) = match Coq.match_as_equation goal concl with - | Some x -> x - | None -> Coq.user_error "The goal is not an equation" - in - let lift_transitivity = - let thm = - if in_left - then - Lazy.force Theory.Stubs.lift_transitivity_left - else - Lazy.force Theory.Stubs.lift_transitivity_right - in - mkApp (thm, - [| - preorder.Coq.Relation.carrier; - preorder.Coq.Relation.r; - using_eq.Coq.Equivalence.eq; - lifting; - step; - left; - right; - |]) - in - Tacticals.tclTHENLIST - [ retype lift_transitivity; - Proofview.V82.of_tactic (Tactics.apply lift_transitivity) - ] goal - - -(** The core tactic for aac_rewrite *) -let core_aac_rewrite ?abort - rewinfo - subst - (by_aac_reflexivity : Matcher.Terms.t -> Matcher.Terms.t -> Proof_type.tactic) - (tr : constr) t left : tactic = - pr_constr "transitivity through" tr; - let tran_tac = - lift_transitivity rewinfo.in_left tr rewinfo.rlt rewinfo.lifting.lift rewinfo.eqt - in - Coq.Rewrite.rewrite ?abort rewinfo.hypinfo subst (fun rew -> - Tacticals.tclTHENSV - (tac_or_exn (tran_tac) Coq.anomaly "Unable to make the transitivity step") - ( - if rewinfo.in_left - then [| by_aac_reflexivity left t ; rew |] - else [| by_aac_reflexivity t left ; rew |] - ) - ) - -exception NoSolutions - - -(** Choose a substitution from a - [(int * Terms.t * Env.env Search_monad.m) Search_monad.m ] *) -(* WARNING: Beware, since the printing function can change the order of the - printed monad, this function has to be updated accordingly *) -let choose_subst subterm sol m= - try - let (depth,pat,envm) = match subterm with - | None -> (* first solution *) - List.nth ( List.rev (Search_monad.to_list m)) 0 - | Some x -> - List.nth ( List.rev (Search_monad.to_list m)) x - in - let env = match sol with - None -> - List.nth ( (Search_monad.to_list envm)) 0 - | Some x -> List.nth ( (Search_monad.to_list envm)) x - in - pat, env - with - | _ -> raise NoSolutions - -(** rewrite the constr modulo AC from left to right in the left member - of the goal *) -let aac_rewrite ?abort rew ?(l2r=true) ?(show = false) ?(in_left=true) ?strict ~occ_subterm ~occ_sol ?extra : Proof_type.tactic = fun goal -> - let envs = Theory.Trans.empty_envs () in - let (concl : Term.types) = Tacmach.pf_concl goal in - let (_,_,rlt) as concl = - match Coq.match_as_equation goal concl with - | None -> Coq.user_error "The goal is not an applied relation" - | Some (left, right, rlt) -> left,right,rlt - in - let check_type x = - Tacmach.pf_conv_x goal x rlt.Coq.Relation.carrier - in - Coq.Rewrite.get_hypinfo rew ~l2r ?check_type:(Some check_type) - (fun hypinfo -> - dispatch in_left concl hypinfo - ( - fun rewinfo -> - let goal = - match extra with - | Some t -> Theory.Trans.add_symbol goal rewinfo.morph_rlt envs t - | None -> goal - in - let pattern, subject, goal = - Theory.Trans.t_of_constr goal rewinfo.morph_rlt envs - (rewinfo.pattern , rewinfo.subject) - in - let goal, ir = Theory.Trans.ir_of_envs goal rewinfo.morph_rlt envs in - - let units = Theory.Trans.ir_to_units ir in - let m = Matcher.subterm ?strict units pattern subject in - (* We sort the monad in increasing size of contet *) - let m = Search_monad.sort (fun (x,_,_) (y,_,_) -> x - y) m in - - if show - then - Print.print rewinfo.morph_rlt ir m (hypinfo.Coq.Rewrite.context) - - else - try - let pat,subst = choose_subst occ_subterm occ_sol m in - let tr_step = Matcher.Subst.instantiate subst pat in - let tr_step_raw = Theory.Trans.raw_constr_of_t ir rewinfo.morph_rlt [] tr_step in - - let conv = (Theory.Trans.raw_constr_of_t ir rewinfo.morph_rlt (hypinfo.Coq.Rewrite.context)) in - let subst = Matcher.Subst.to_list subst in - let subst = List.map (fun (x,y) -> x, conv y) subst in - let by_aac_reflexivity = (by_aac_reflexivity rewinfo.subject rewinfo.eqt ir) in - core_aac_rewrite ?abort rewinfo subst by_aac_reflexivity tr_step_raw tr_step subject - - with - | NoSolutions -> - Tacticals.tclFAIL 0 - (Pp.str (if occ_subterm = None && occ_sol = None - then "No matching occurence modulo AC found" - else "No such solution")) - ) - ) goal - - -open Coq.Rewrite -open Tacmach -open Tacticals -open Tacexpr -open Tacinterp -open Extraargs -open Genarg - -let rec add k x = function - | [] -> [k,x] - | k',_ as ky::q -> - if k'=k then Coq.user_error ("redondant argument ("^k^")") - else ky::add k x q - -let get k l = try Some (List.assoc k l) with Not_found -> None - -let get_lhs l = try List.assoc "in_right" l; false with Not_found -> true - -let aac_rewrite ~args = - aac_rewrite ~occ_subterm:(get "at" args) ~occ_sol:(get "subst" args) ~in_left:(get_lhs args) - - -let pr_aac_args _ _ _ l = - List.fold_left - (fun acc -> function - | ("in_right" as s,_) -> Pp.(++) (Pp.str s) acc - | (k,i) -> Pp.(++) (Pp.(++) (Pp.str k) (Pp.int i)) acc - ) (Pp.str "") l - -ARGUMENT EXTEND aac_args -TYPED AS ((string * int) list ) -PRINTED BY pr_aac_args -| [ "at" integer(n) aac_args(q) ] -> [ add "at" n q ] -| [ "subst" integer(n) aac_args(q) ] -> [ add "subst" n q ] -| [ "in_right" aac_args(q) ] -> [ add "in_right" 0 q ] -| [ ] -> [ [] ] -END - -let pr_constro _ _ _ = fun b -> - match b with - | None -> Pp.str "" - | Some o -> Pp.str "" - -ARGUMENT EXTEND constro -TYPED AS (constr option) -PRINTED BY pr_constro -| [ "[" constr(c) "]" ] -> [ Some c ] -| [ ] -> [ None ] -END - -TACTIC EXTEND _aac_reflexivity_ -| [ "aac_reflexivity" ] -> [ Proofview.V82.tactic aac_reflexivity ] -END - -TACTIC EXTEND _aac_normalise_ -| [ "aac_normalise" ] -> [ Proofview.V82.tactic aac_normalise ] -END - -TACTIC EXTEND _aac_rewrite_ -| [ "aac_rewrite" orient(l2r) constr(c) aac_args(args) constro(extra)] -> - [ Proofview.V82.tactic (fun gl -> aac_rewrite ?extra ~args ~l2r ~strict:true c gl) ] -END - -TACTIC EXTEND _aac_pattern_ -| [ "aac_pattern" orient(l2r) constr(c) aac_args(args) constro(extra)] -> - [ Proofview.V82.tactic (fun gl -> aac_rewrite ?extra ~args ~l2r ~strict:true ~abort:true c gl) ] -END - -TACTIC EXTEND _aac_instances_ -| [ "aac_instances" orient(l2r) constr(c) aac_args(args) constro(extra)] -> - [ Proofview.V82.tactic (fun gl -> aac_rewrite ?extra ~args ~l2r ~strict:true ~show:true c gl) ] -END - -TACTIC EXTEND _aacu_rewrite_ -| [ "aacu_rewrite" orient(l2r) constr(c) aac_args(args) constro(extra)] -> - [ Proofview.V82.tactic (fun gl -> aac_rewrite ?extra ~args ~l2r ~strict:false c gl) ] -END - -TACTIC EXTEND _aacu_pattern_ -| [ "aacu_pattern" orient(l2r) constr(c) aac_args(args) constro(extra)] -> - [ Proofview.V82.tactic (fun gl -> aac_rewrite ?extra ~args ~l2r ~strict:false ~abort:true c gl) ] -END - -TACTIC EXTEND _aacu_instances_ -| [ "aacu_instances" orient(l2r) constr(c) aac_args(args) constro(extra)] -> - [ Proofview.V82.tactic (fun gl -> aac_rewrite ?extra ~args ~l2r ~strict:false ~show:true c gl) ] -END diff --git a/aac_rewrite.mli b/aac_rewrite.mli deleted file mode 100644 index d195075..0000000 --- a/aac_rewrite.mli +++ /dev/null @@ -1,9 +0,0 @@ -(***************************************************************************) -(* This is part of aac_tactics, it is distributed under the terms of the *) -(* GNU Lesser General Public License version 3 *) -(* (see file LICENSE for more details) *) -(* *) -(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) -(***************************************************************************) - -(** Definition of the tactics, and corresponding Coq grammar entries.*) diff --git a/coq.ml b/coq.ml deleted file mode 100755 index eb4b5f1..0000000 --- a/coq.ml +++ /dev/null @@ -1,611 +0,0 @@ -(***************************************************************************) -(* This is part of aac_tactics, it is distributed under the terms of the *) -(* GNU Lesser General Public License version 3 *) -(* (see file LICENSE for more details) *) -(* *) -(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) -(***************************************************************************) - -(** Interface with Coq *) -type constr = Term.constr - -open Term -open Names -open Coqlib -open Sigma.Notations -open Context.Rel.Declaration - -(* The contrib name is used to locate errors when loading constrs *) -let contrib_name = "aac_tactics" - -(* Getting constrs (primitive Coq terms) from existing Coq - libraries. *) -let find_constant contrib dir s = - Universes.constr_of_global (Coqlib.find_reference contrib dir s) - -let init_constant dir s = find_constant contrib_name dir s - -(* A clause specifying that the [let] should not try to fold anything - in the goal *) -let nowhere = - { Locus.onhyps = Some []; - Locus.concl_occs = Locus.NoOccurrences - } - -let retype c gl = - let sigma, _ = Tacmach.pf_apply Typing.type_of gl c in - Refiner.tclEVARS sigma gl - -let cps_mk_letin - (name:string) - (c: constr) - (k : constr -> Proof_type.tactic) -: Proof_type.tactic = - fun goal -> - let name = (Names.id_of_string name) in - let name = Tactics.fresh_id [] name goal in - let letin = (Proofview.V82.of_tactic (Tactics.letin_tac None (Name name) c None nowhere)) in - Tacticals.tclTHENLIST [retype c; letin; (k (mkVar name))] goal - -(** {2 General functions} *) - -type goal_sigma = Proof_type.goal Tacmach.sigma -let goal_update (goal : goal_sigma) evar_map : goal_sigma= - let it = Tacmach.sig_it goal in - Tacmach.re_sig it evar_map - -let fresh_evar goal ty : constr * goal_sigma = - let env = Tacmach.pf_env goal in - let evar_map = Tacmach.project goal in - let evar_map = Sigma.Unsafe.of_evar_map evar_map in - let Sigma (x,em,_) = Evarutil.new_evar env evar_map ty in - let em = Sigma.to_evar_map em in - x,( goal_update goal em) - -let resolve_one_typeclass goal ty : constr*goal_sigma= - let env = Tacmach.pf_env goal in - let evar_map = Tacmach.project goal in - let em,c = Typeclasses.resolve_one_typeclass env evar_map ty in - c, (goal_update goal em) - -let general_error = - "Cannot resolve a typeclass : please report" - -let cps_resolve_one_typeclass ?error : Term.types -> (Term.constr -> Proof_type.tactic) -> Proof_type.tactic = fun t k goal -> - Tacmach.pf_apply - (fun env em -> let em ,c = - try Typeclasses.resolve_one_typeclass env em t - with Not_found -> - begin match error with - | None -> CErrors.anomaly (Pp.str "Cannot resolve a typeclass : please report") - | Some x -> CErrors.error x - end - in - Tacticals.tclTHENLIST [Refiner.tclEVARS em; k c] goal - ) goal - - -let nf_evar goal c : Term.constr= - let evar_map = Tacmach.project goal in - Evarutil.nf_evar evar_map c - -let evar_unit (gl : goal_sigma) (x : constr) : constr * goal_sigma = - let env = Tacmach.pf_env gl in - let evar_map = Tacmach.project gl in - let evar_map = Sigma.Unsafe.of_evar_map evar_map in - let Sigma (x,em,_) = Evarutil.new_evar env evar_map x in - let em = Sigma.to_evar_map em in - x,(goal_update gl em) - -let evar_binary (gl: goal_sigma) (x : constr) = - let env = Tacmach.pf_env gl in - let evar_map = Tacmach.project gl in - let ty = mkArrow x (mkArrow x x) in - let evar_map = Sigma.Unsafe.of_evar_map evar_map in - let Sigma (x,em,_) = Evarutil.new_evar env evar_map ty in - let em = Sigma.to_evar_map em in - x,( goal_update gl em) - -let evar_relation (gl: goal_sigma) (x: constr) = - let env = Tacmach.pf_env gl in - let evar_map = Tacmach.project gl in - let ty = mkArrow x (mkArrow x (mkSort prop_sort)) in - let evar_map = Sigma.Unsafe.of_evar_map evar_map in - let Sigma (r, em, _) = Evarutil.new_evar env evar_map ty in - let em = Sigma.to_evar_map em in - r,( goal_update gl em) - -let cps_evar_relation (x: constr) k = fun goal -> - Tacmach.pf_apply - (fun env em -> - let ty = mkArrow x (mkArrow x (mkSort prop_sort)) in - let em = Sigma.Unsafe.of_evar_map em in - let Sigma (r, em, _) = Evarutil.new_evar env em ty in - let em = Sigma.to_evar_map em in - Tacticals.tclTHENLIST [Refiner.tclEVARS em; k r] goal - ) goal - -(* decomp_term : constr -> (constr, types) kind_of_term *) -let decomp_term c = kind_of_term (strip_outer_cast c) - -let lapp c v = mkApp (Lazy.force c, v) - -(** {2 Bindings with Coq' Standard Library} *) -module Std = struct -(* Here we package the module to be able to use List, later *) - -module Pair = struct - - let path = ["Coq"; "Init"; "Datatypes"] - let typ = lazy (init_constant path "prod") - let pair = lazy (init_constant path "pair") - let of_pair t1 t2 (x,y) = - mkApp (Lazy.force pair, [| t1; t2; x ; y|] ) -end - -module Bool = struct - let path = ["Coq"; "Init"; "Datatypes"] - let typ = lazy (init_constant path "bool") - let ctrue = lazy (init_constant path "true") - let cfalse = lazy (init_constant path "false") - let of_bool b = - if b then Lazy.force ctrue else Lazy.force cfalse -end - -module Comparison = struct - let path = ["Coq"; "Init"; "Datatypes"] - let typ = lazy (init_constant path "comparison") - let eq = lazy (init_constant path "Eq") - let lt = lazy (init_constant path "Lt") - let gt = lazy (init_constant path "Gt") -end - -module Leibniz = struct - let path = ["Coq"; "Init"; "Logic"] - let eq_refl = lazy (init_constant path "eq_refl") - let eq_refl ty x = lapp eq_refl [| ty;x|] -end - -module Option = struct - let path = ["Coq"; "Init"; "Datatypes"] - let typ = lazy (init_constant path "option") - let some = lazy (init_constant path "Some") - let none = lazy (init_constant path "None") - let some t x = mkApp (Lazy.force some, [| t ; x|]) - let none t = mkApp (Lazy.force none, [| t |]) - let of_option t x = match x with - | Some x -> some t x - | None -> none t -end - -module Pos = struct - - let path = ["Coq" ; "PArith"; "BinPos"] - let typ = lazy (init_constant path "positive") - let xI = lazy (init_constant path "xI") - let xO = lazy (init_constant path "xO") - let xH = lazy (init_constant path "xH") - - (* A coq positive from an int *) - let of_int n = - let rec aux n = - begin match n with - | n when n < 1 -> assert false - | 1 -> Lazy.force xH - | n -> mkApp - ( - (if n mod 2 = 0 - then Lazy.force xO - else Lazy.force xI - ), [| aux (n/2)|] - ) - end - in - aux n -end - -module Nat = struct - let path = ["Coq" ; "Init"; "Datatypes"] - let typ = lazy (init_constant path "nat") - let _S = lazy (init_constant path "S") - let _O = lazy (init_constant path "O") - (* A coq nat from an int *) - let of_int n = - let rec aux n = - begin match n with - | n when n < 0 -> assert false - | 0 -> Lazy.force _O - | n -> mkApp - ( - (Lazy.force _S - ), [| aux (n-1)|] - ) - end - in - aux n -end - -(** Lists from the standard library*) -module List = struct - let path = ["Coq"; "Lists"; "List"] - let typ = lazy (init_constant path "list") - let nil = lazy (init_constant path "nil") - let cons = lazy (init_constant path "cons") - let cons ty h t = - mkApp (Lazy.force cons, [| ty; h ; t |]) - let nil ty = - (mkApp (Lazy.force nil, [| ty |])) - let rec of_list ty = function - | [] -> nil ty - | t::q -> cons ty t (of_list ty q) - let type_of_list ty = - mkApp (Lazy.force typ, [|ty|]) -end - -(** Morphisms *) -module Classes = -struct - let classes_path = ["Coq";"Classes"; ] - let morphism = lazy (init_constant (classes_path@["Morphisms"]) "Proper") - let equivalence = lazy (init_constant (classes_path@ ["RelationClasses"]) "Equivalence") - let reflexive = lazy (init_constant (classes_path@ ["RelationClasses"]) "Reflexive") - let transitive = lazy (init_constant (classes_path@ ["RelationClasses"]) "Transitive") - - (** build the type [Equivalence ty rel] *) - let mk_equivalence ty rel = - mkApp (Lazy.force equivalence, [| ty; rel|]) - - - (** build the type [Reflexive ty rel] *) - let mk_reflexive ty rel = - mkApp (Lazy.force reflexive, [| ty; rel|]) - - (** build the type [Proper rel t] *) - let mk_morphism ty rel t = - mkApp (Lazy.force morphism, [| ty; rel; t|]) - - (** build the type [Transitive ty rel] *) - let mk_transitive ty rel = - mkApp (Lazy.force transitive, [| ty; rel|]) -end - -module Relation = struct - type t = - { - carrier : constr; - r : constr; - } - - let make ty r = {carrier = ty; r = r } - let split t = t.carrier, t.r -end - -module Transitive = struct - type t = - { - carrier : constr; - leq : constr; - transitive : constr; - } - let make ty leq transitive = {carrier = ty; leq = leq; transitive = transitive} - let infer goal ty leq = - let ask = Classes.mk_transitive ty leq in - let transitive , goal = resolve_one_typeclass goal ask in - make ty leq transitive, goal - let from_relation goal rlt = - infer goal (rlt.Relation.carrier) (rlt.Relation.r) - let cps_from_relation rlt k = - let ty =rlt.Relation.carrier in - let r = rlt.Relation.r in - let ask = Classes.mk_transitive ty r in - cps_resolve_one_typeclass ask - (fun trans -> k (make ty r trans) ) - let to_relation t = - {Relation.carrier = t.carrier; Relation.r = t.leq} - -end - -module Equivalence = struct - type t = - { - carrier : constr; - eq : constr; - equivalence : constr; - } - let make ty eq equivalence = {carrier = ty; eq = eq; equivalence = equivalence} - let infer goal ty eq = - let ask = Classes.mk_equivalence ty eq in - let equivalence , goal = resolve_one_typeclass goal ask in - make ty eq equivalence, goal - let from_relation goal rlt = - infer goal (rlt.Relation.carrier) (rlt.Relation.r) - let cps_from_relation rlt k = - let ty =rlt.Relation.carrier in - let r = rlt.Relation.r in - let ask = Classes.mk_equivalence ty r in - cps_resolve_one_typeclass ask (fun equiv -> k (make ty r equiv) ) - let to_relation t = - {Relation.carrier = t.carrier; Relation.r = t.eq} - let split t = - t.carrier, t.eq, t.equivalence -end -end -(**[ match_as_equation goal eqt] see [eqt] as an equation. An - optionnal rel-context can be provided to ensure that the term - remains typable*) -let match_as_equation ?(context = Context.Rel.empty) goal equation : (constr*constr* Std.Relation.t) option = - let env = Tacmach.pf_env goal in - let env = Environ.push_rel_context context env in - let evar_map = Tacmach.project goal in - begin - match decomp_term equation with - | App(c,ca) when Array.length ca >= 2 -> - let n = Array.length ca in - let left = ca.(n-2) in - let right = ca.(n-1) in - let r = (mkApp (c, Array.sub ca 0 (n - 2))) in - let carrier = Typing.unsafe_type_of env evar_map left in - let rlt =Std.Relation.make carrier r - in - Some (left, right, rlt ) - | _ -> None - end - - -(** {1 Tacticals} *) - -let tclTIME msg tac = fun gl -> - let u = Sys.time () in - let r = tac gl in - let _ = Feedback.msg_notice (Pp.str (Printf.sprintf "%s:%fs" msg (Sys.time ()-. u))) in - r - -let tclDEBUG msg tac = fun gl -> - let _ = Feedback.msg_debug (Pp.str msg) in - tac gl - -let tclPRINT tac = fun gl -> - let _ = Feedback.msg_notice (Printer.pr_constr (Tacmach.pf_concl gl)) in - tac gl - - -(** {1 Error related mechanisms} *) -(* functions to handle the failures of our tactic. Some should be - reported [anomaly], some are on behalf of the user [user_error]*) -let anomaly msg = - CErrors.anomaly ~label:"[aac_tactics]" (Pp.str msg) - -let user_error msg = - CErrors.error ("[aac_tactics] " ^ msg) - -let warning msg = - Feedback.msg_warning (Pp.str ("[aac_tactics]" ^ msg)) - - -(** {1 Rewriting tactics used in aac_rewrite} *) -module Rewrite = struct -(** Some informations about the hypothesis, with an (informal) - invariant: - - [typeof hyp = hyptype] - - [hyptype = forall context, body] - - [body = rel left right] - -*) - -type hypinfo = - { - hyp : Term.constr; (** the actual constr corresponding to the hypothese *) - hyptype : Term.constr; (** the type of the hypothesis *) - context : Context.Rel.t; (** the quantifications of the hypothese *) - body : Term.constr; (** the body of the type of the hypothesis*) - rel : Std.Relation.t; (** the relation *) - left : Term.constr; (** left hand side *) - right : Term.constr; (** right hand side *) - l2r : bool; (** rewriting from left to right *) - } - -let get_hypinfo c ~l2r ?check_type (k : hypinfo -> Proof_type.tactic) : Proof_type.tactic = fun goal -> - let ctype = Tacmach.pf_unsafe_type_of goal c in - let (rel_context, body_type) = Term.decompose_prod_assum ctype in - let rec check f e = - match decomp_term e with - | Term.Rel i -> f (get_type (Context.Rel.lookup i rel_context)) - | _ -> Term.fold_constr (fun acc x -> acc && check f x) true e - in - begin match check_type with - | None -> () - | Some f -> - if not (check f body_type) - then user_error "Unable to deal with higher-order or heterogeneous patterns"; - end; - begin - match match_as_equation ~context:rel_context goal body_type with - | None -> - user_error "The hypothesis is not an applied relation" - | Some (hleft,hright,hrlt) -> - k { - hyp = c; - hyptype = ctype; - body = body_type; - l2r = l2r; - context = rel_context; - rel = hrlt ; - left =hleft; - right = hright; - } - goal - end - - -(* The problem : Given a term to rewrite of type [H :forall xn ... x1, - t], we have to instanciate the subset of [xi] of type - [carrier]. [subst : (int * constr)] is the mapping the debruijn - indices in [t] to the [constrs]. We need to handle the rest of the - indexes. Two ways : - - - either using fresh evars and rewriting [H tn ?1 tn-2 ?2 ] - - either building a term like [fun 1 2 => H tn 1 tn-2 2] - - Both these terms have the same type. -*) - - -(* Fresh evars for everyone (should be the good way to do this - recompose in Coq v8.4) *) -let recompose_prod - (context : Context.Rel.t) - (subst : (int * Term.constr) list) - env - em - : Evd.evar_map * Term.constr list = - (* the last index of rel relevant for the rewriting *) - let min_n = List.fold_left - (fun acc (x,_) -> min acc x) - (List.length context) subst in - let rec aux context acc em n = - let _ = Printf.printf "%i\n%!" n in - match context with - | [] -> - env, em, acc - | t::q -> - let env, em, acc = aux q acc em (n+1) in - if n >= min_n - then - let em,x = - try em, List.assoc n subst - with | Not_found -> - let em = Sigma.Unsafe.of_evar_map em in - let Sigma (r, em, _) = Evarutil.new_evar env em (Vars.substl acc (get_type t)) in - let em = Sigma.to_evar_map em in - (em, r) - in - (Environ.push_rel t env), em,x::acc - else - env,em,acc - in - let _,em,acc = aux context [] em 1 in - em, acc - -(* no fresh evars : instead, use a lambda abstraction around an - application to handle non-instanciated variables. *) - -let recompose_prod' - (context : Context.Rel.t) - (subst : (int *Term.constr) list) - c - = - let rec popn pop n l = - if n <= 0 then l - else match l with - | [] -> [] - | t::q -> pop t :: (popn pop (n-1) q) - in - let pop_rel_decl = map_type Termops.pop in - let rec aux sign n next app ctxt = - match sign with - | [] -> List.rev app, List.rev ctxt - | decl::sign -> - try let term = (List.assoc n subst) in - aux sign (n+1) next (term::app) (None :: ctxt) - with - | Not_found -> - let term = Term.mkRel next in - aux sign (n+1) (next+1) (term::app) (Some decl :: ctxt) - in - let app,ctxt = aux context 1 1 [] [] in - (* substitutes in the context *) - let rec update ctxt app = - match ctxt,app with - | [],_ -> [] - | _,[] -> [] - | None :: sign, _ :: app -> - None :: update sign (List.map (Termops.pop) app) - | Some decl :: sign, _ :: app -> - Some (Vars.substl_decl app decl)::update sign (List.map (Termops.pop) app) - in - let ctxt = update ctxt app in - (* updates the rel accordingly, taking some care not to go to far - beyond: it is important not to lift indexes homogeneously, we - have to update *) - let rec update ctxt res n = - match ctxt with - | [] -> List.rev res - | None :: sign -> - (update (sign) (popn pop_rel_decl n res) 0) - | Some decl :: sign -> - update sign (decl :: res) (n+1) - in - let ctxt = update ctxt [] 0 in - let c = Term.applistc c (List.rev app) in - let c = Term.it_mkLambda_or_LetIn c (ctxt) in - c - -(* Version de Matthieu -let subst_rel_context k cstr ctx = - let (_, ctx') = - List.fold_left (fun (k, ctx') (id, b, t) -> (succ k, (id, Option.map - (Term.substnl [cstr] k) b, Term.substnl [cstr] k t) :: ctx')) (k, []) - ctx in List.rev ctx' - -let recompose_prod' context subst c = - let len = List.length context in - let rec aux sign n next app ctxt = - match sign with - | [] -> List.rev app, List.rev ctxt - | decl::sign -> - try let term = (List.assoc n subst) in - aux (subst_rel_context 0 term sign) (pred n) (succ next) - (term::List.map (Term.lift (-1)) app) ctxt - with Not_found -> - let term = Term.mkRel (len - next) in - aux sign (pred n) (succ next) (term::app) (decl :: ctxt) - in - let app,ctxt = aux (List.rev context) len 0 [] [] in - Term.it_mkLambda_or_LetIn (Term.applistc c(app)) (List.rev ctxt) -*) - -let build - (h : hypinfo) - (subst : (int *Term.constr) list) - (k :Term.constr -> Proof_type.tactic) - : Proof_type.tactic = fun goal -> - let c = recompose_prod' h.context subst h.hyp in - Tacticals.tclTHENLIST [k c] goal - -let build_with_evar - (h : hypinfo) - (subst : (int *Term.constr) list) - (k :Term.constr -> Proof_type.tactic) - : Proof_type.tactic - = fun goal -> - Tacmach.pf_apply - (fun env em -> - let evar_map, acc = recompose_prod h.context subst env em in - let c = Term.applistc h.hyp (List.rev acc) in - Tacticals.tclTHENLIST [Refiner.tclEVARS evar_map; k c] goal - ) goal - - -let rewrite ?(abort=false)hypinfo subst k = - build hypinfo subst - (fun rew -> - let tac = - if not abort then - Proofview.V82.of_tactic begin - Equality.general_rewrite_bindings - hypinfo.l2r - Locus.AllOccurrences - true (* tell if existing evars must be frozen for instantiation *) - false - (rew,Misctypes.NoBindings) - true - end - else - Tacticals.tclIDTAC - in k tac - ) - - -end - -include Std diff --git a/coq.mli b/coq.mli deleted file mode 100644 index 4d46c7d..0000000 --- a/coq.mli +++ /dev/null @@ -1,232 +0,0 @@ -(***************************************************************************) -(* This is part of aac_tactics, it is distributed under the terms of the *) -(* GNU Lesser General Public License version 3 *) -(* (see file LICENSE for more details) *) -(* *) -(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) -(***************************************************************************) - -(** Interface with Coq where we define some handlers for Coq's API, - and we import several definitions from Coq's standard library. - - This general purpose library could be reused by other plugins. - - Some salient points: - - we use Caml's module system to mimic Coq's one, and avoid cluttering - the namespace; - - we also provide several handlers for standard coq tactics, in - order to provide a unified setting (we replace functions that - modify the evar_map by functions that operate on the whole - goal, and repack the modified evar_map with it). - -*) - -(** {2 Getting Coq terms from the environment} *) - -val init_constant : string list -> string -> Term.constr - -(** {2 General purpose functions} *) - -type goal_sigma = Proof_type.goal Tacmach.sigma -val goal_update : goal_sigma -> Evd.evar_map -> goal_sigma -val resolve_one_typeclass : Proof_type.goal Tacmach.sigma -> Term.types -> Term.constr * goal_sigma -val cps_resolve_one_typeclass: ?error:string -> Term.types -> (Term.constr -> Proof_type.tactic) -> Proof_type.tactic -val nf_evar : goal_sigma -> Term.constr -> Term.constr -val fresh_evar :goal_sigma -> Term.types -> Term.constr* goal_sigma -val evar_unit :goal_sigma ->Term.constr -> Term.constr* goal_sigma -val evar_binary: goal_sigma -> Term.constr -> Term.constr* goal_sigma -val evar_relation: goal_sigma -> Term.constr -> Term.constr* goal_sigma -val cps_evar_relation : Term.constr -> (Term.constr -> Proof_type.tactic) -> Proof_type.tactic -(** [cps_mk_letin name v] binds the constr [v] using a letin tactic *) -val cps_mk_letin : string -> Term.constr -> ( Term.constr -> Proof_type.tactic) -> Proof_type.tactic - -val retype : Term.constr -> Proof_type.tactic - -val decomp_term : Term.constr -> (Term.constr , Term.types) Term.kind_of_term -val lapp : Term.constr lazy_t -> Term.constr array -> Term.constr - -(** {2 Bindings with Coq' Standard Library} *) - -(** Coq lists *) -module List: -sig - (** [of_list ty l] *) - val of_list:Term.constr ->Term.constr list ->Term.constr - - (** [type_of_list ty] *) - val type_of_list:Term.constr ->Term.constr -end - -(** Coq pairs *) -module Pair: -sig - val typ:Term.constr lazy_t - val pair:Term.constr lazy_t - val of_pair : Term.constr -> Term.constr -> Term.constr * Term.constr -> Term.constr -end - -module Bool : sig - val typ : Term.constr lazy_t - val of_bool : bool -> Term.constr -end - - -module Comparison : sig - val typ : Term.constr lazy_t - val eq : Term.constr lazy_t - val lt : Term.constr lazy_t - val gt : Term.constr lazy_t -end - -module Leibniz : sig - val eq_refl : Term.types -> Term.constr -> Term.constr -end - -module Option : sig - val some : Term.constr -> Term.constr -> Term.constr - val none : Term.constr -> Term.constr - val of_option : Term.constr -> Term.constr option -> Term.constr -end - -(** Coq positive numbers (pos) *) -module Pos: -sig - val typ:Term.constr lazy_t - val of_int: int ->Term.constr -end - -(** Coq unary numbers (peano) *) -module Nat: -sig - val typ:Term.constr lazy_t - val of_int: int ->Term.constr -end - -(** Coq typeclasses *) -module Classes: -sig - val mk_morphism: Term.constr -> Term.constr -> Term.constr -> Term.constr - val mk_equivalence: Term.constr -> Term.constr -> Term.constr - val mk_reflexive: Term.constr -> Term.constr -> Term.constr - val mk_transitive: Term.constr -> Term.constr -> Term.constr -end - -module Relation : sig - type t = {carrier : Term.constr; r : Term.constr;} - val make : Term.constr -> Term.constr -> t - val split : t -> Term.constr * Term.constr -end - -module Transitive : sig - type t = - { - carrier : Term.constr; - leq : Term.constr; - transitive : Term.constr; - } - val make : Term.constr -> Term.constr -> Term.constr -> t - val infer : goal_sigma -> Term.constr -> Term.constr -> t * goal_sigma - val from_relation : goal_sigma -> Relation.t -> t * goal_sigma - val cps_from_relation : Relation.t -> (t -> Proof_type.tactic) -> Proof_type.tactic - val to_relation : t -> Relation.t -end - -module Equivalence : sig - type t = - { - carrier : Term.constr; - eq : Term.constr; - equivalence : Term.constr; - } - val make : Term.constr -> Term.constr -> Term.constr -> t - val infer : goal_sigma -> Term.constr -> Term.constr -> t * goal_sigma - val from_relation : goal_sigma -> Relation.t -> t * goal_sigma - val cps_from_relation : Relation.t -> (t -> Proof_type.tactic) -> Proof_type.tactic - val to_relation : t -> Relation.t - val split : t -> Term.constr * Term.constr * Term.constr -end - -(** [match_as_equation ?context goal c] try to decompose c as a - relation applied to two terms. An optionnal rel-context can be - provided to ensure that the term remains typable *) -val match_as_equation : ?context:Context.Rel.t -> goal_sigma -> Term.constr -> (Term.constr * Term.constr * Relation.t) option - -(** {2 Some tacticials} *) - -(** time the execution of a tactic *) -val tclTIME : string -> Proof_type.tactic -> Proof_type.tactic - -(** emit debug messages to see which tactics are failing *) -val tclDEBUG : string -> Proof_type.tactic -> Proof_type.tactic - -(** print the current goal *) -val tclPRINT : Proof_type.tactic -> Proof_type.tactic - - -(** {2 Error related mechanisms} *) - -val anomaly : string -> 'a -val user_error : string -> 'a -val warning : string -> unit - - -(** {2 Rewriting tactics used in aac_rewrite} *) - -module Rewrite : sig - -(** The rewriting tactics used in aac_rewrite, build as handlers of the usual [setoid_rewrite]*) - - -(** {2 Datatypes} *) - -(** We keep some informations about the hypothesis, with an (informal) - invariant: - - [typeof hyp = typ] - - [typ = forall context, body] - - [body = rel left right] - -*) -type hypinfo = - { - hyp : Term.constr; (** the actual constr corresponding to the hypothese *) - hyptype : Term.constr; (** the type of the hypothesis *) - context : Context.Rel.t; (** the quantifications of the hypothese *) - body : Term.constr; (** the body of the hypothese*) - rel : Relation.t; (** the relation *) - left : Term.constr; (** left hand side *) - right : Term.constr; (** right hand side *) - l2r : bool; (** rewriting from left to right *) - } - -(** [get_hypinfo H l2r ?check_type k] analyse the hypothesis H, and - build the related hypinfo, in CPS style. Moreover, an optionnal - function can be provided to check the type of every free - variable of the body of the hypothesis. *) -val get_hypinfo :Term.constr -> l2r:bool -> ?check_type:(Term.types -> bool) -> (hypinfo -> Proof_type.tactic) -> Proof_type.tactic - -(** {2 Rewriting with bindings} - - The problem : Given a term to rewrite of type [H :forall xn ... x1, - t], we have to instanciate the subset of [xi] of type - [carrier]. [subst : (int * constr)] is the mapping the De Bruijn - indices in [t] to the [constrs]. We need to handle the rest of the - indexes. Two ways : - - - either using fresh evars and rewriting [H tn ?1 tn-2 ?2 ] - - either building a term like [fun 1 2 => H tn 1 tn-2 2] - - Both these terms have the same type. -*) - -(** build the constr to rewrite, in CPS style, with lambda abstractions *) -val build : hypinfo -> (int * Term.constr) list -> (Term.constr -> Proof_type.tactic) -> Proof_type.tactic - -(** build the constr to rewrite, in CPS style, with evars *) -val build_with_evar : hypinfo -> (int * Term.constr) list -> (Term.constr -> Proof_type.tactic) -> Proof_type.tactic - -(** [rewrite ?abort hypinfo subst] builds the rewriting tactic - associated with the given [subst] and [hypinfo], and feeds it to - the given continuation. If [abort] is set to [true], we build - [tclIDTAC] instead. *) -val rewrite : ?abort:bool -> hypinfo -> (int * Term.constr) list -> (Proof_type.tactic -> Proof_type.tactic) -> Proof_type.tactic -end diff --git a/default.nix b/default.nix new file mode 100644 index 0000000..8cd0cac --- /dev/null +++ b/default.nix @@ -0,0 +1,28 @@ +{ pkgs ? (import {}), coq-version-or-url, shell ? false }: + +let + coq-version-parts = builtins.match "([0-9]+).([0-9]+)" coq-version-or-url; + coqPackages = + if coq-version-parts == null then + pkgs.mkCoqPackages (import (fetchTarball coq-version-or-url) {}) + else + pkgs."coqPackages_${builtins.concatStringsSep "_" coq-version-parts}"; +in + +with coqPackages; + +pkgs.stdenv.mkDerivation { + + name = "aac-tactics"; + + buildInputs = with coq.ocamlPackages; [ ocaml findlib camlp5 ] + ++ pkgs.lib.optionals shell [ merlin ocp-indent ocp-index ]; + + propagatedBuildInputs = [ + coq + ]; + + src = if shell then null else ./.; + + installFlags = "COQLIB=$(out)/lib/coq/${coq.coq-version}/"; +} diff --git a/description b/description deleted file mode 100644 index 7b7195d..0000000 --- a/description +++ /dev/null @@ -1,15 +0,0 @@ -Name: AACTactics -Title: AAC tactics -Description: This Coq plugin provides tactics for rewriting universally quantified equations, modulo associative (and possibly commutative) operators: -Keywords: reflexive tactic, rewriting, rewriting modulo associativity and commutativity, rewriting modulo AC, reflexive decision procedure -Category: Miscellaneous/Coq Extensions -Author: Thomas Braibant -Email: thomas.braibant@gmail.com -Homepage: http://sardes.inrialpes.fr/~braibant/ -Institution: INRIA/UJF/Ens Lyon -Author: Damien Pous -Email: damien.pous@inria.fr -Homepage: http://sardes.inrialpes.fr/~pous/ -Institution: CNRS -Require: -License: LGPL diff --git a/helper.ml b/helper.ml deleted file mode 100644 index 636b17f..0000000 --- a/helper.ml +++ /dev/null @@ -1,41 +0,0 @@ -(***************************************************************************) -(* This is part of aac_tactics, it is distributed under the terms of the *) -(* GNU Lesser General Public License version 3 *) -(* (see file LICENSE for more details) *) -(* *) -(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) -(***************************************************************************) - -module type CONTROL = sig - val debug : bool - val time : bool - val printing : bool -end - -module Debug (X : CONTROL) = struct - open X - let debug x = - if debug then - Printf.printf "%s\n%!" x - - - let time f x fmt = - if time then - let t = Sys.time () in - let r = f x in - Printf.printf fmt (Sys.time () -. t); - r - else f x - - let pr_constr msg constr = - if printing then - ( Feedback.msg_notice (Pp.str (Printf.sprintf "=====%s====" msg)); - Feedback.msg_notice (Printer.pr_constr constr); - ) - - - let debug_exception msg e = - debug (msg ^ (Printexc.to_string e)) - - -end diff --git a/helper.mli b/helper.mli deleted file mode 100644 index f4e4454..0000000 --- a/helper.mli +++ /dev/null @@ -1,33 +0,0 @@ -(***************************************************************************) -(* This is part of aac_tactics, it is distributed under the terms of the *) -(* GNU Lesser General Public License version 3 *) -(* (see file LICENSE for more details) *) -(* *) -(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) -(***************************************************************************) - -(** Debugging functions, that can be triggered on a per-file base. *) - -module type CONTROL = -sig - val debug : bool - val time : bool - val printing : bool -end -module Debug : - functor (X : CONTROL) -> -sig - (** {!debug} prints the string and end it with a newline *) - val debug : string -> unit - val debug_exception : string -> exn -> unit - - (** {!time} computes the time spent in a function, and then - print it using the given format *) - val time : - ('a -> 'b) -> 'a -> (float -> unit, out_channel, unit) format -> 'b - - (** {!pr_constr} print a Coq constructor, that can be labelled - by a string *) - val pr_constr : string -> Term.constr -> unit - - end diff --git a/matcher.ml b/matcher.ml deleted file mode 100644 index 1dcb1d2..0000000 --- a/matcher.ml +++ /dev/null @@ -1,1293 +0,0 @@ -(***************************************************************************) -(* This is part of aac_tactics, it is distributed under the terms of the *) -(* GNU Lesser General Public License version 3 *) -(* (see file LICENSE for more details) *) -(* *) -(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) -(***************************************************************************) - -(** This module defines our matching functions, modulo associativity - and commutativity (AAC). - - The basic idea is to find a substitution [env] such that the - pattern [p] instantiated by [env] is equal to [t] modulo AAC. - - We proceed by structural decomposition of the pattern, and try all - possible non-deterministic split of the subject when needed. The - function {!matcher} is limited to top-level matching, that is, the - subject must make a perfect match against the pattern ([x+x] do - not match [a+a+b] ). We use a search monad {!Search} to perform - non-deterministic splits in an almost transparent way. We also - provide a function {!subterm} for finding a match that is a - subterm modulo AAC of the subject. Therefore, we are able to solve - the aforementioned case [x+x] against [a+b+a]. - - This file is structured as follows. First, we define the search - monad. Then,we define the two representations of terms (one - representing the AST, and one in normal form ), and environments - from variables to terms. Then, we use these parts to solve - matching problem. Finally, we wrap this function {!matcher} into - {!subterm} -*) - - -module Control = -struct - let debug = false - let time = false - let printing = false -end - -module Debug = Helper.Debug (Control) -open Debug - -module Search = Search_monad (* a handle *) - -type symbol = int -type var = int -type units = (symbol * symbol) list (* from AC/A symbols to the unit *) -type ext_units = - { - unit_for : units; - is_ac : (symbol * bool) list - } - -let print_units units= - List.iter (fun (op,unit) -> Printf.printf "%i %i\t" op unit) units; - Printf.printf "\n%!" - -exception NoUnit - -let get_unit units op = - try List.assoc op units - with - | Not_found -> raise NoUnit - -let is_unit units op unit = List.mem (op,unit) units - - -open Search - -type 'a mset = ('a * int) list -let linear p = - let rec ncons t l = function - | 0 -> l - | n -> t::ncons t l (n-1) - in - let rec aux = function - [ ] -> [] - | (t,n)::q -> let q = aux q in - ncons t q n - in aux p - - - -(** The module {!Terms} defines two different types for expressions. - - - a public type {!Terms.t} that represent abstract syntax trees of - expressions with binary associative (and commutative) operators - - - a private type {!Terms.nf_term} that represent an equivalence - class for terms that are equal modulo AAC. The constructions - functions on this type ensure the property that the term is in - normal form (that is, no sum can appear as a subterm of the same - sum, no trailing units, etc...). - -*) - -module Terms : sig - - (** {1 Abstract syntax tree of terms} - - Terms represented using this datatype are representation of the - AST of an expression. *) - - type t = - Dot of (symbol * t * t) - | Plus of (symbol * t * t) - | Sym of (symbol * t array) - | Var of var - | Unit of symbol - - val equal_aac : units -> t -> t -> bool - val size: t -> int - - (** {1 Terms in normal form} - - A term in normal form is the canonical representative of the - equivalence class of all the terms that are equal modulo - Associativity and Commutativity. Outside the {!Matcher} module, - one does not need to access the actual representation of this - type. *) - - - type nf_term = private - | TAC of symbol * nf_term mset - | TA of symbol * nf_term list - | TSym of symbol * nf_term list - | TUnit of symbol - | TVar of var - - (** {2 Fresh variables: we provide some functions to pick some fresh variables with respect to a term} *) - val fresh_var_term : t -> int - val fresh_var_nfterm : nf_term -> int - - - (** {2 Constructors: we ensure that the terms are always - normalised - - braibant - Fri 27 Aug 2010, 15:11 - Moreover, we assure that we will not build empty sums or empty - products, leaving this task to the caller function. - } - *) - val mk_TAC : units -> symbol -> nf_term mset -> nf_term - val mk_TA : units -> symbol -> nf_term list -> nf_term - val mk_TSym : symbol -> nf_term list -> nf_term - val mk_TVar : var -> nf_term - val mk_TUnit : symbol -> nf_term - - (** {2 Comparisons} *) - val nf_term_compare : nf_term -> nf_term -> int - val nf_equal : nf_term -> nf_term -> bool - - (** {2 Printing function} *) - val sprint_nf_term : nf_term -> string - - (** {2 Conversion functions} *) - val term_of_t : units -> t -> nf_term - val t_of_term : nf_term -> t (* we could return the units here *) -end - = struct - - type t = - Dot of (symbol * t * t) - | Plus of (symbol * t * t) - | Sym of (symbol * t array) - | Var of var - | Unit of symbol - - let rec size = function - | Dot (_,x,y) - | Plus (_,x,y) -> size x+ size y + 1 - | Sym (_,v)-> Array.fold_left (fun acc x -> size x + acc) 1 v - | _ -> 1 - - - type nf_term = - | TAC of symbol * nf_term mset - | TA of symbol * nf_term list - | TSym of symbol * nf_term list - | TUnit of symbol - | TVar of var - - - (** {2 Picking fresh variables} *) - - (** [fresh_var_term] picks a fresh variable with respect to a term *) - let fresh_var_term t = - let rec aux = function - | Dot (_,t1,t2) - | Plus (_,t1,t2) -> max (aux t1) (aux t2) - | Sym (_,v) -> Array.fold_left (fun acc x -> max acc (aux x)) 0 v - | Var v -> assert (v >= 0); v - | Unit _ -> 0 - in - aux t - - (** [fresh_var_nfterm] picks a fresh_variable with respect to a term *) - let fresh_var_nfterm t = - let rec aux = function - | TAC (_,l) -> List.fold_left (fun acc (x,_) -> max acc (aux x)) 0 l - | TA (_,l) - | TSym (_,l) -> List.fold_left (fun acc x -> max acc (aux x)) 0 l - | TVar v -> assert (v >= 0); v - | TUnit _ -> 0 - in - aux t - - (** {2 Constructors: we ensure that the terms are always - normalised} *) - - (** {3 Pre constructors : These constructors ensure that sums and - products are not degenerated (no trailing units)} *) - let mk_TAC' units (s : symbol) l = match l with - | [] -> TUnit (get_unit units s ) - | [_,0] -> assert false - | [t,1] -> t - | _ -> TAC (s,l) - let mk_TA' units (s : symbol) l = match l with - | [] -> TUnit (get_unit units s ) - | [t] -> t - | _ -> TA (s,l) - - - (** {2 Comparison} *) - - let nf_term_compare = Pervasives.compare - let nf_equal a b = - a = b - - (** [merge_ac comp l1 l2] merges two lists of terms with coefficients - into one. Terms that are equal modulo the comparison function - [comp] will see their arities added. *) - - (* This function could be improved by the use of sorted msets *) - let merge_ac (compare : 'a -> 'a -> int) (l : 'a mset) (l' : 'a mset) : 'a mset = - let rec aux l l'= - match l,l' with - | [], _ -> l' - | _, [] -> l - | (t,tar)::q, (t',tar')::q' -> - begin match compare t t' with - | 0 -> ( t,tar+tar'):: aux q q' - | -1 -> (t, tar):: aux q l' - | _ -> (t', tar'):: aux l q' - end - in aux l l' - - (** [merge_map f l] uses the combinator [f] to combine the head of the - list [l] with the merge_maped tail of [l] *) - let rec merge_map (f : 'a -> 'b list -> 'b list) (l : 'a list) : 'b list = - match l with - | [] -> [] - | t::q -> f t (merge_map f q) - - - (** This function has to deal with the arities *) - let rec merge (l : nf_term mset) (l' : nf_term mset) : nf_term mset= - merge_ac nf_term_compare l l' - - let extract_A units s t = - match t with - | TA (s',l) when s' = s -> l - | TUnit u when is_unit units s u -> [] - | _ -> [t] - - let extract_AC units s (t,ar) : nf_term mset = - match t with - | TAC (s',l) when s' = s -> List.map (fun (x,y) -> (x,y*ar)) l - | TUnit u when is_unit units s u -> [] - | _ -> [t,ar] - - - (** {3 Constructors of {!nf_term}}*) - let mk_TAC units (s : symbol) (l : (nf_term *int) list) = - mk_TAC' units s - (merge_map (fun u l -> merge (extract_AC units s u) l) l) - let mk_TA units s l = - mk_TA' units s - (merge_map (fun u l -> (extract_A units s u) @ l) l) - let mk_TSym s l = TSym (s,l) - let mk_TVar v = TVar v - let mk_TUnit s = TUnit s - - - (** {2 Printing function} *) - let print_binary_list (single : 'a -> string) - (unit : string) - (binary : string -> string -> string) (l : 'a list) = - let rec aux l = - match l with - [] -> unit - | [t] -> single t - | t::q -> - let r = aux q in - Printf.sprintf "%s" (binary (single t) r) - in - aux l - - let print_symbol s = - match s with - | s, None -> Printf.sprintf "%i" s - | s , Some u -> Printf.sprintf "%i(unit %i)" s u - - let sprint_ac (single : 'a -> string) (l : 'a mset) = - (print_binary_list - (fun (x,t) -> - if t = 1 - then single x - else Printf.sprintf "%i*%s" t (single x) - ) - "0" - (fun x y -> x ^ " , " ^ y) - l - ) - - let print_symbol single s l = - match l with - [] -> Printf.sprintf "%i" s - | _ -> - Printf.sprintf "%i(%s)" - s - (print_binary_list single "" (fun x y -> x ^ "," ^ y) l) - - - let print_ac single s l = - Printf.sprintf "[%s:AC]{%s}" - (string_of_int s ) - (sprint_ac - single - l - ) - - let print_a single s l = - Printf.sprintf "[%s:A]{%s}" - (string_of_int s) - (print_binary_list single "1" (fun x y -> x ^ " , " ^ y) l) - - let rec sprint_nf_term = function - | TSym (s,l) -> print_symbol sprint_nf_term s l - | TAC (s,l) -> print_ac sprint_nf_term s l - | TA (s,l) -> print_a sprint_nf_term s l - | TVar v -> Printf.sprintf "x%i" v - | TUnit s -> Printf.sprintf "unit%i" s - - - - - (** {2 Conversion functions} *) - - (* rebuilds a tree out of a list, under the assumption that the list is not empty *) - let rec binary_of_list f comb l = - let l = List.rev l in - let rec aux = function - | [] -> assert false - | [t] -> f t - | t::q -> comb (aux q) (f t) - in - aux l - - let term_of_t units : t -> nf_term = - let rec term_of_t = function - | Dot (s,l,r) -> - let l = term_of_t l in - let r = term_of_t r in - mk_TA units s [l;r] - | Plus (s,l,r) -> - let l = term_of_t l in - let r = term_of_t r in - mk_TAC units s [l,1;r,1] - | Unit x -> - mk_TUnit x - | Sym (s,t) -> - let t = Array.to_list t in - let t = List.map term_of_t t in - mk_TSym s t - | Var i -> - mk_TVar ( i) - in - term_of_t - - let rec t_of_term : nf_term -> t = function - | TAC (s,l) -> - (binary_of_list - t_of_term - (fun l r -> Plus ( s,l,r)) - (linear l) - ) - | TA (s,l) -> - (binary_of_list - t_of_term - (fun l r -> Dot ( s,l,r)) - l - ) - | TSym (s,l) -> - let v = Array.of_list l in - let v = Array.map (t_of_term) v in - Sym ( s,v) - | TVar x -> Var x - | TUnit s -> Unit s - - - let equal_aac units x y = - nf_equal (term_of_t units x) (term_of_t units y) - end - - (** Terms environments defined as association lists from variables to - terms in normal form {! Terms.nf_term} *) - module Subst : sig - type t - - val find : t -> var -> Terms.nf_term option - val add : t -> var -> Terms.nf_term -> t - val empty : t - val instantiate : t -> Terms.t -> Terms.t - val sprint : t -> string - val to_list : t -> (var*Terms.t) list - end - = - struct - open Terms - - (** Terms environments, with nf_terms, to avoid costly conversions - of {!Terms.nf_terms} to {!Terms.t}, that will be mostly discarded*) - type t = (var * nf_term) list - - let find : t -> var -> nf_term option = fun t x -> - try Some (List.assoc x t) with | _ -> None - let add t x v = (x,v) :: t - let empty = [] - - let sprint (l : t) = - match l with - | [] -> Printf.sprintf "Empty environment\n" - | _ -> - let s = List.fold_left - (fun acc (x,y) -> - Printf.sprintf "%sX%i -> %s\n" - acc - x - (sprint_nf_term y) - ) - "" - (List.rev l) in - Printf.sprintf "%s\n%!" s - - - - (** [instantiate] is an homomorphism except for the variables*) - let instantiate (t: t) (x:Terms.t) : Terms.t = - let rec aux = function - | Unit _ as x -> x - | Sym (s,t) -> Sym (s,Array.map aux t) - | Plus (s,l,r) -> Plus (s, aux l, aux r) - | Dot (s,l,r) -> Dot (s, aux l, aux r) - | Var i -> - begin match find t i with - | None -> CErrors.error "aac_tactics: instantiate failure" - | Some x -> t_of_term x - end - in aux x - - let to_list t = List.map (fun (x,y) -> x,Terms.t_of_term y) t - end - - (******************) - (* MATCHING UTILS *) - (******************) - - open Terms - - (** Since most of the folowing functions require an extra parameter, - the units, we package them in a module. This functor will then be - applied to a module containing the units, in the exported - functions. *) - module M (X : sig - val units : units - val is_ac : (symbol * bool) list - val strict : bool (* variables cannot be instantiated with units *) - end) = struct - - open X - - let print_units ()= - List.iter (fun (op,unit) -> Printf.printf "%i %i\t" op unit) units; - Printf.printf "\n%!" - - let mk_TAC s l = mk_TAC units s l - let mk_TA s l = mk_TA units s l - let mk_TAC' s l = - try return( mk_TAC s l) - with _ -> fail () - let mk_TA' s l = - try return( mk_TA s l) - with _ -> fail () - - (** First, we need to be able to perform non-deterministic choice of - term splitting to satisfy a pattern. Indeed, we want to show that: - (x+a*b)*c <= a*b*c - *) - let a_nondet_split_raw t : ('a list * 'a list) m = - let rec aux l l' = - match l' with - | [] -> - return ( l,[]) - | t::q -> - return ( l,l' ) - >>| aux (l @ [t]) q - in - aux [] t - - (** Same as the previous [a_nondet_split], but split the list in 3 - parts *) - let a_nondet_middle t : ('a list * 'a list * 'a list) m = - a_nondet_split_raw t >> - (fun (left, right) -> - a_nondet_split_raw left >> - (fun (left, middle) -> return (left, middle, right)) - ) - - (** Non deterministic splits of ac lists *) - let dispatch f n = - let rec aux k = - if k = 0 then return (f n 0) - else return (f (n-k) k) >>| aux (k-1) - in - aux (n ) - - let add_with_arith x ar l = - if ar = 0 then l else (x,ar) ::l - - let ac_nondet_split_raw (l : 'a mset) : ('a mset * 'a mset) m = - let rec aux = function - | [] -> return ([],[]) - | (t,tar)::q -> - aux q - >> - (fun (left,right) -> - dispatch (fun arl arr -> - add_with_arith t arl left, - add_with_arith t arr right - ) - tar - ) - in - aux l - - let a_nondet_split current t : (nf_term * nf_term list) m= - a_nondet_split_raw t - >> - (fun (l,r) -> - if strict && (l=[]) - then fail() - else - mk_TA' current l >> - fun t -> return (t, r) - ) - - let ac_nondet_split current t : (nf_term * nf_term mset) m= - ac_nondet_split_raw t - >> - (fun (l,r) -> - if strict && (l=[]) - then fail() - else - mk_TAC' current l >> - fun t -> return (t, r) - ) - - - (** Try to affect the variable [x] to each left factor of [t]*) - let var_a_nondet_split env current x t = - a_nondet_split current t - >> - (fun (t,res) -> - return ((Subst.add env x t), res) - ) - - (** Try to affect variable [x] to _each_ subset of t. *) - let var_ac_nondet_split (current: symbol) env (x : var) (t : nf_term mset) : (Subst.t * (nf_term mset)) m = - ac_nondet_split current t - >> - (fun (t,res) -> - return ((Subst.add env x t), res) - ) - - (** See the term t as a given AC symbol. Unwrap the first constructor - if necessary *) - let get_AC (s : symbol) (t : nf_term) : (nf_term *int) list = - match t with - | TAC (s',l) when s' = s -> l - | TUnit u when is_unit units s u -> [] - | _ -> [t,1] - - (** See the term t as a given A symbol. Unwrap the first constructor - if necessary *) - let get_A (s : symbol) (t : nf_term) : nf_term list = - match t with - | TA (s',l) when s' = s -> l - | TUnit u when is_unit units s u -> [] - | _ -> [t] - - (** See the term [t] as an symbol [s]. Fail if it is not such - symbol. *) - let get_Sym s t = - match t with - | TSym (s',l) when s' = s -> return l - | _ -> fail () - - (*************) - (* A Removal *) - (*************) - - (** We remove the left factor v in a term list. This function runs - linearly with respect to the size of the first pattern symbol *) - - let left_factor current (v : nf_term) (t : nf_term list) = - let rec aux a b = - match a,b with - | t::q , t' :: q' when nf_equal t t' -> aux q q' - | [], q -> return q - | _, _ -> fail () - in - match v with - | TA (s,l) when s = current -> aux l t - | TUnit u when is_unit units current u -> return t - | _ -> - begin match t with - | [] -> fail () - | t::q -> - if nf_equal v t - then return q - else fail () - end - - - (**************) - (* AC Removal *) - (**************) - - (** {!pick_sym} gather all elements of a list that satisfies a - predicate, and combine them with the residual of the list. That - is, each element of the residual contains exactly one element less - than the original term. - - TODO : This function not as efficient as it could be, using a - proper data-structure *) - - let pick_sym (s : symbol) (t : nf_term mset ) = - let rec aux front back = - match back with - | [] -> fail () - | (t,tar)::q -> - begin match t with - | TSym (s',v') when s = s' -> - let back = - if tar > 1 - then (t,tar -1) ::q - else q - in - return (v' , List.rev_append front back ) - >>| aux ((t,tar)::front) q - | _ -> aux ((t,tar)::front) q - end - in - aux [] t - - - - (** We have to check if we are trying to remove a unit from a term. Could also be located in Terms*) - let is_unit_AC s t = - try nf_equal t (mk_TAC s []) - with | NoUnit -> false - - let is_same_AC s t : nf_term mset option= - match t with - TAC (s',l) when s = s' -> Some l - | _ -> None - - (** We want to remove the term [v] from the term list [t] under an AC - symbol *) - let single_AC_factor (s : symbol) (v : nf_term) v_ar (t : nf_term mset) : (nf_term mset) m = - let rec aux front back = - match back with - | [] -> fail () - | (t,tar)::q -> - begin - if nf_equal v t - then - match () with - | _ when tar < v_ar -> fail () - | _ when tar = v_ar -> return (List.rev_append front q) - | _ -> return (List.rev_append front ((t,tar-v_ar)::q)) - else - aux ((t,tar) :: front) q - end - in - if is_unit_AC s v - then - return t - else - aux [] t - - (* Remove a constant from a mset. If this constant is also a mset for - the same symbol, we remove every elements, one at a time (and we do - care of the arities) *) - let factor_AC (s : symbol) (v: nf_term) (t : nf_term mset) : ( nf_term mset ) m = - match is_same_AC s v with - | None -> single_AC_factor s v 1 t - | Some l -> - (* We are trying to remove an AC factor *) - List.fold_left (fun acc (v,v_ar) -> - acc >> (single_AC_factor s v v_ar) - ) - (return t) - l - - -(** [tri_fold f l acc] folds on the list [l] and give to f the - beginning of the list in reverse order, the considered element, and - the last part of the list - - as an exemple, on the list [1;2;3;4], we get the trace - f () [] 1 [2; 3; 4] - f () [1] 2 [3; 4] - f () [2;1] 3 [ 4] - f () [3;2;1] 4 [] - - it is the duty of the user to reverse the front if needed -*) - -let tri_fold f (l : 'a list) (acc : 'b)= match l with - [] -> acc - | _ -> - let _,_,acc = List.fold_left (fun acc (t : 'a) -> - let l,r,acc = acc in - let r = List.tl r in - t::l,r,f acc l t r - ) ([], l,acc) l - in acc - -(* let test = tri_fold (fun acc l t r -> (l,t,r) :: acc) [1;2;3;4] [] *) - - - - (*****************************) - (* ENUMERATION DES POSITIONS *) - (*****************************) - - -(** The pattern is a unit: we need to try to make it appear at each - position. We do not need to go further with a real matching, since - the match should be trivial. Hence, we proceed recursively to - enumerate all the positions. *) - -module Positions = struct - - - let ac (l: 'a mset) : ('a mset * 'a )m = - let rec aux = function - | [] -> assert false - | [t,1] -> return ([],t) - | [t,tar] -> return ([t,tar -1],t) - | (t,tar) as h :: q -> - (aux q >> (fun (c,x) -> return (h :: c,x))) - >>| (if tar > 1 then return ((t,tar-1) :: q,t) else return (q,t)) - in - aux l - - let ac' current (l: 'a mset) : ('a mset * 'a )m = - ac_nondet_split_raw l >> - (fun (l,r) -> - if l = [] || r = [] - then fail () - else - mk_TAC' current r >> - fun t -> return (l, t) - ) - - let a (l : 'a list) : ('a list * 'a * 'a list) m = - let rec aux left right : ('a list * 'a * 'a list) m = - match right with - | [] -> assert false - | [t] -> return (left,t,[]) - | t::q -> - aux (left@[t]) q - >>| return (left,t,q) - in - aux [] l -end - -let build_ac (current : symbol) (context : nf_term mset) (p : t) : t m= - if context = [] - then return p - else - mk_TAC' current context >> - fun t -> return (Plus (current,t_of_term t,p)) - -let build_a (current : symbol) - (left : nf_term list) (right : nf_term list) (p : t) : t m= - let right_pat p = - if right = [] - then return p - else - mk_TA' current right >> - fun t -> return (Dot (current,p,t_of_term t)) - in - let left_pat p= - if left = [] - then return p - else - mk_TA' current left >> - fun t -> return (Dot (current,t_of_term t,p)) - in - right_pat p >> left_pat >> (fun p -> return p) - - -let conts (hole : t) (l : symbol list) p : t m = - let p = t_of_term p in - (* - aller chercher les symboles ac et les symboles a - - construire pour chaque - * * + - / \ / \ / \ - 1 p p 1 p 1 - *) - let ac,a = List.partition (fun s -> List.assoc s is_ac) l in - let acc = fail () in - let acc = List.fold_left - (fun acc s -> - acc >>| return (Plus (s,p,hole)) - ) acc ac in - let acc = - List.fold_left - (fun acc s -> - acc >>| return (Dot (s,p,hole)) >>| return (Dot (s,hole,p)) - ) acc a - in acc - - -(** - Return the same thing as subterm : - - The size of the context - - The context - - A collection of substitutions (here == return Subst.empty) -*) -let unit_subterm (t : nf_term) (unit : symbol) (hole : t): (int * t * Subst.t m) m = - let symbols = List.fold_left - (fun acc (ac,u) -> if u = unit then ac :: acc else acc ) [] units - in - (* make a unit appear at each strict sub-position of the term*) - let rec positions (t : nf_term) : t m = - match t with - (* with a final unit at the end *) - | TAC (s,l) -> - let symbols' = List.filter (fun x -> x <> s) symbols in - ( - ac_nondet_split_raw l >> - (fun (l,r) -> if l = [] || r = [] then fail () else - ( - match r with - | [p,1] -> - positions p >>| conts hole symbols' p - | _ -> - mk_TAC' s r >> conts hole symbols' - ) >> build_ac s l )) - | TA (s,l) -> - let symbols' = List.filter (fun x -> x <> s) symbols in - ( - (* first the other symbols, and then the more simple case of - this particular symbol *) - a_nondet_middle l >> - (fun (l,m,r) -> - (* meant to break the symmetry *) - if (l = [] && r = []) - then fail () - else - ( - match m with - | [p] -> - positions p >>| conts hole symbols' p - | _ -> - mk_TA' s m >> conts hole symbols' - ) >> build_a s l r )) - >>| - ( - if List.mem s symbols then - begin match l with - | [a] -> assert false - | [a;b] -> build_a s [a] [b] (hole) - | _ -> - (* on ne construit que les elements interieurs, - d'ou la disymetrie *) - (Positions.a l >> - (fun (left,p,right) -> - if left = [] then fail () else - (build_a s left right (Dot (s,hole,t_of_term p))))) - end - else fail () - ) - - | TSym (s,l) -> - tri_fold (fun acc l t r -> - ((positions t) >> - (fun (p) -> - let l = List.map t_of_term l in - let r = List.map t_of_term r in - return (Sym (s, Array.of_list (List.rev_append l (p::r)))) )) - >>| - ( - conts hole symbols t >> - (fun (p) -> - let l = List.map t_of_term l in - let r = List.map t_of_term r in - return (Sym (s, Array.of_list (List.rev_append l (p::r)))) ) - ) - >>| acc - ) l (fail()) - | TVar x -> assert false - | TUnit x when x = unit -> return (hole) - | TUnit x as t -> conts hole symbols t - in - (positions t - >>| - (match t with - | TSym _ -> conts hole symbols t - | TAC (s,l) -> conts hole symbols t - | TA (s,l) -> conts hole symbols t - | _ -> fail()) - ) - >> fun (p) -> return (Terms.size p,p,return Subst.empty) - - - - - (************) - (* Matching *) - (************) - - - -(** {!matching} is the generic matching judgement. Each time a - non-deterministic split is made, we have to come back to this one. - - {!matchingSym} is used to match two applications that have the - same (free) head-symbol. - - {!matchingAC} is used to match two sums (with the subtlety that - [x+y] matches [f a] which is a function application or [a*b] which - is a product). - - {!matchingA} is used to match two products (with the subtlety that - [x*y] matches [f a] which is a function application, or [a+b] - which is a sum). - - -*) - -let matching : Subst.t -> nf_term -> nf_term -> Subst.t Search.m= - let rec matching env (p : nf_term) (t: nf_term) : Subst.t Search.m= - match p with - | TAC (s,l) -> - let l = linear l in - matchingAC env s l (get_AC s t) - | TA (s,l) -> - matchingA env s l (get_A s t) - | TSym (s,l) -> - (get_Sym s t) - >> (fun t -> matchingSym env l t) - | TVar x -> - begin match Subst.find env x with - | None -> return (Subst.add env x t) - | Some v -> if nf_equal v t then return env else fail () - end - | TUnit s -> - if nf_equal p t then return env else fail () - and - matchingAC (env : Subst.t) (current : symbol) (l : nf_term list) (t : (nf_term *int) list) = - match l with - | TSym (s,v):: q -> - pick_sym s t - >> (fun (v',t') -> - matchingSym env v v' - >> (fun env -> matchingAC env current q t')) - - | TAC (s,v)::q when s = current -> - assert false - | TVar x:: q -> (* This is an optimization *) - begin match Subst.find env x with - | None -> - (var_ac_nondet_split current env x t) - >> (fun (env,residual) -> matchingAC env current q residual) - | Some v -> - (factor_AC current v t) - >> (fun residual -> matchingAC env current q residual) - end - | TUnit s as v :: q -> (* this is an optimization *) - (factor_AC current v t) >> - (fun residual -> matchingAC env current q residual) - | h :: q ->(* PAC =/= curent or PA or unit for this symbol*) - (ac_nondet_split current t) - >> - (fun (t,right) -> - matching env h t - >> - ( - fun env -> - matchingAC env current q right - ) - ) - | [] -> if t = [] then return env else fail () - and - matchingA (env : Subst.t) (current : symbol) (l : nf_term list) (t : nf_term list) = - match l with - | TSym (s,v) :: l -> - begin match t with - | TSym (s',v') :: r when s = s' -> - (matchingSym env v v') - >> (fun env -> matchingA env current l r) - | _ -> fail () - end - | TA (s,v) :: l when s = current -> - assert false - | TVar x :: l -> - begin match Subst.find env x with - | None -> - debug (Printf.sprintf "var %i (%s)" x - (let b = Buffer.create 21 in List.iter (fun t -> Buffer.add_string b ( Terms.sprint_nf_term t)) t; Buffer.contents b )); - var_a_nondet_split env current x t - >> (fun (env,residual)-> debug (Printf.sprintf "pl %i %i" x(List.length residual)); matchingA env current l residual) - | Some v -> - (left_factor current v t) - >> (fun residual -> matchingA env current l residual) - end - | TUnit s as v :: q -> (* this is an optimization *) - (left_factor current v t) >> - (fun residual -> matchingA env current q residual) - | h :: l -> - a_nondet_split current t - >> (fun (t,r) -> - matching env h t - >> (fun env -> matchingA env current l r) - ) - | [] -> if t = [] then return env else fail () - and - matchingSym (env : Subst.t) (l : nf_term list) (t : nf_term list) = - List.fold_left2 - (fun acc p t -> acc >> (fun env -> matching env p t)) - (return env) - l - t - - in - fun env l r -> - let _ = debug (Printf.sprintf "pattern :%s\nterm :%s\n%!" (Terms.sprint_nf_term l) (Terms.sprint_nf_term r)) in - let m = matching env l r in - let _ = debug (Printf.sprintf "count %i" (Search.count m)) in - m - - -(** [unitifiable p : Subst.t m] *) -let unitifiable p : (symbol * Subst.t m) m = - let m = List.fold_left - (fun acc (_,unit) -> acc >>| - let m = matching Subst.empty p (mk_TUnit unit) in - if Search.is_empty m then - fail () - else - begin - return (unit,m) - end - ) (fail ()) units - in - m -;; - -let nullifiable p = - let nullable = not strict in - let has_unit s = - try let _ = get_unit units s in true - with NoUnit -> false - in - let rec aux = function - | TA (s,l) -> has_unit s && List.for_all (aux) l - | TAC (s,l) -> has_unit s && List.for_all (fun (x,n) -> aux x) l - | TSym _ -> false - | TVar _ -> nullable - | TUnit _ -> true - in aux p - -let unit_warning p ~nullif ~unitif = - assert ((Search.is_empty unitif) || nullif); - if not (Search.is_empty unitif) - then - begin - Feedback.msg_warning - (Pp.str - "[aac_tactics] This pattern can be instanciated to match units, some solutions can be missing"); - end - -;; - - - - -(***********) -(* Subterm *) -(***********) - - - -(** [subterm] solves a sub-term pattern matching. - - This function is more high-level than {!matcher}, thus takes {!t} - as arguments rather than terms in normal form {!nf_term}. - - We use three mutually recursive functions {!subterm}, - {!subterm_AC}, {!subterm_A} to find the matching subterm, making - non-deterministic choices to split the term into a context and an - intersting sub-term. Intuitively, the only case in which we have to - go in depth is when we are left with a sub-term that is atomic. - - Indeed, rewriting [H: b = c |- a+b+a = a+a+c], we do not want to - find recursively the sub-terms of [a+b] and [b+a], since they will - overlap with the sub-terms of [a+b+a]. - - We rebuild the context on the fly, leaving the variables in the - pattern uninstantiated. We do so in order to allow interaction - with the user, to choose the env. - - Strange patterms like x*y*x can be instanciated by nothing, inside - a product. Therefore, we need to check that all the term is not - going into the context. With proper support for interaction with - the user, we should lift these tests. However, at the moment, they - serve as heuristics to return "interesting" matchings -*) - - let return_non_empty raw_p m = - if is_empty m - then - fail () - else - return (raw_p ,m) - - let subterm (raw_p:t) (raw_t:t): (int * t * Subst.t m) m= - let _ = debug (String.make 40 '=') in - let p = term_of_t units raw_p in - let t = term_of_t units raw_t in - let nullif = nullifiable p in - let unitif = unitifiable p in - let _ = unit_warning p ~nullif ~unitif in - let _ = debug (Printf.sprintf "%s" (Terms.sprint_nf_term p)) in - let _ = debug (Printf.sprintf "%s" (Terms.sprint_nf_term t)) in - let filter_right current right (p,m) = - if right = [] - then return (p,m) - else - mk_TAC' current right >> - fun t -> return (Plus (current,p,t_of_term t),m) - in - let filter_middle current left right (p,m) = - let right_pat p = - if right = [] - then return p - else - mk_TA' current right >> - fun t -> return (Dot (current,p,t_of_term t)) - in - let left_pat p= - if left = [] - then return p - else - mk_TA' current left >> - fun t -> return (Dot (current,t_of_term t,p)) - in - right_pat p >> left_pat >> (fun p -> return (p,m)) - in - let rec subterm (t:nf_term) : (t * Subst.t m) m= - match t with - | TAC (s,l) -> - ((ac_nondet_split_raw l) >> - (fun (left,right) -> - (subterm_AC s left) >> (filter_right s right) - )) - | TA (s,l) -> - (a_nondet_middle l) >> - (fun (left, middle, right) -> - (subterm_A s middle) >> - (filter_middle s left right) - ) - | TSym (s, l) -> - let init = - return_non_empty raw_p (matching Subst.empty p t) - in - tri_fold (fun acc l t r -> - ((subterm t) >> - (fun (p,m) -> - let l = List.map t_of_term l in - let r = List.map t_of_term r in - let p = Sym (s, Array.of_list (List.rev_append l (p::r))) in - return (p,m) - )) >>| acc - ) l init - | TVar x -> assert false - (* this case is superseded by the later disjunction *) - | TUnit s -> fail () - - and subterm_AC s tl = - match tl with - [x,1] -> subterm x - | _ -> - mk_TAC' s tl >> fun t -> - return_non_empty raw_p (matching Subst.empty p t) - and subterm_A s tl = - match tl with - [x] -> subterm x - | _ -> - mk_TA' s tl >> - fun t -> - return_non_empty raw_p (matching Subst.empty p t) - in - match p with - | TUnit unit -> unit_subterm t unit raw_p - | _ when not (Search.is_empty unitif) -> - let unit_matches = - Search.fold - (fun (unit,inst) acc -> - Search.fold - (fun subst acc' -> - let m = unit_subterm t unit (Subst.instantiate subst raw_p) - in - m>>| acc' - ) - inst - acc - ) - unitif - (fail ()) - in - let nullifies (t : Subst.t) = - List.for_all (fun (_,x) -> - List.exists (fun (_,y) -> Unit y = x ) units - ) (Subst.to_list t) - in - let nonunit_matches = - subterm t >> - ( - fun (p,m) -> - let m = Search.filter (fun subst -> not( nullifies subst)) m in - if Search.is_empty m - then fail () - else return (Terms.size p,p,m) - ) - in - unit_matches >>| nonunit_matches - - | _ -> (subterm t >> fun (p,m) -> return (Terms.size p,p,m)) - - - end - - -(* The functions we export, handlers for the previous ones. Some debug - information also *) -let subterm ?(strict = false) units raw t = - let module M = M (struct - let is_ac = units.is_ac - let units = units.unit_for - let strict = strict - end) in - let sols = time (M.subterm raw) t "%fs spent in subterm (including matching)\n" in - debug - (Printf.sprintf "%i possible solution(s)\n" - (Search.fold (fun (_,_,envm) acc -> count envm + acc) sols 0)); - sols - - -let matcher ?(strict = false) units p t = - let module M = M (struct - let is_ac = units.is_ac - let units = units.unit_for - let strict = false - end) in - let units = units.unit_for in - let sols = time - (fun (p,t) -> - let p = (Terms.term_of_t units p) in - let t = (Terms.term_of_t units t) in - M.matching Subst.empty p t) (p,t) - "%fs spent in the matcher\n" - in - debug (Printf.sprintf "%i solutions\n" (count sols)); - sols - diff --git a/matcher.mli b/matcher.mli deleted file mode 100644 index a6b6f46..0000000 --- a/matcher.mli +++ /dev/null @@ -1,189 +0,0 @@ -(***************************************************************************) -(* This is part of aac_tactics, it is distributed under the terms of the *) -(* GNU Lesser General Public License version 3 *) -(* (see file LICENSE for more details) *) -(* *) -(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) -(***************************************************************************) - -(** Standalone module containing the algorithm for matching modulo - associativity and associativity and commutativity - (AAC). Additionnaly, some A or AC operators can have units (U). - - This module could be reused outside of the Coq plugin. - - Matching a pattern [p] against a term [t] modulo AACU boils down - to finding a substitution [env] such that the pattern [p] - instantiated with [env] is equal to [t] modulo AACU. - - We proceed by structural decomposition of the pattern, trying all - possible non-deterministic splittings of the subject, when needed. The - function {!matcher} is limited to top-level matching, that is, the - subject must make a perfect match against the pattern ([x+x] does - not match [a+a+b] ). - - We use a search monad {!Search_monad} to perform non-deterministic - choices in an almost transparent way. - - We also provide a function {!subterm} for finding a match that is - a subterm of the subject modulo AACU. In particular, this function - gives a solution to the aforementioned case ([x+x] against - [a+b+a]). - - On a slightly more involved level : - - it must be noted that we allow several AC/A operators to share - the same units, but that a given AC/A operator can have at most - one unit. - - - if the pattern does not contain "hard" symbols (like constants, - function symbols, AC or A symbols without units), there can be - infinitely many subterms such that the pattern matches: it is - possible to build "subterms" modulo AAC and U that make the size - of the term increase (by making neutral elements appear in a - layered fashion). Hence, in this case, a warning is issued, and - some solutions are omitted. - -*) - -(** {2 Utility functions} *) - -type symbol = int -type var = int - -(** Relationship between units and operators. This is a sparse - representation of a matrix of couples [(op,unit)] where [op] is - the index of the operation, and [unit] the index of the relevant - unit. We make the assumption that any operation has 0 or 1 unit, - and that operations can share a unit). *) - -type units =(symbol * symbol) list (* from AC/A symbols to the unit *) -type ext_units = - { - unit_for : units; (* from AC/A symbols to the unit *) - is_ac : (symbol * bool) list - } - - -(** The arguments of sums (or AC operators) are represented using finite multisets. - (Typically, [a+b+a] corresponds to [2.a+b], i.e. [Sum[a,2;b,1]]) *) -type 'a mset = ('a * int) list - -(** [linear] expands a multiset into a simple list *) -val linear : 'a mset -> 'a list - -(** Representations of expressions - - The module {!Terms} defines two different types for expressions. - - a public type {!Terms.t} that represents abstract syntax trees - of expressions with binary associative and commutative operators - - a private type {!Terms.nf_term}, corresponding to a canonical - representation for the above terms modulo AACU. The construction - functions on this type ensure that these terms are in normal form - (that is, no sum can appear as a subterm of the same sum, no - trailing units, lists corresponding to multisets are sorted, - etc...). - -*) -module Terms : -sig - - (** {2 Abstract syntax tree of terms and patterns} - - We represent both terms and patterns using the following datatype. - - Values of type [symbol] are used to index symbols. Typically, - given two associative operations [(^)] and [( * )], and two - morphisms [f] and [g], the term [f (a^b) (a*g b)] is represented - by the following value - [Sym(0,[| Dot(1, Sym(2,[||]), Sym(3,[||])); - Dot(4, Sym(2,[||]), Sym(5,[|Sym(3,[||])|])) |])] - where the implicit symbol environment associates - [f] to [0], [(^)] to [1], [a] to [2], [b] to [3], [( * )] to [4], and [g] to [5], - - Accordingly, the following value, that contains "variables" - [Sym(0,[| Dot(1, Var x, Unit (1); Dot(4, Var x, - Sym(5,[|Sym(3,[||])|])) |])] represents the pattern [forall x, f - (x^1) (x*g b)]. The relationship between [1] and [( * )] is only - mentionned in the units table. *) - - type t = - Dot of (symbol * t * t) - | Plus of (symbol * t * t) - | Sym of (symbol * t array) - | Var of var - | Unit of symbol - - (** Test for equality of terms modulo AACU (relies on the following - canonical representation of terms). Note than two different - units of a same operator are not considered equal. *) - val equal_aac : units -> t -> t -> bool - - - (** {2 Normalised terms (canonical representation) } - - A term in normal form is the canonical representative of the - equivalence class of all the terms that are equal modulo AACU. - This representation is only used internally; it is exported here - for the sake of completeness *) - - type nf_term - - (** {3 Comparisons} *) - - val nf_term_compare : nf_term -> nf_term -> int - val nf_equal : nf_term -> nf_term -> bool - - (** {3 Printing function} *) - - val sprint_nf_term : nf_term -> string - - (** {3 Conversion functions} *) - - (** we have the following property: [a] and [b] are equal modulo AACU - iif [nf_equal (term_of_t a) (term_of_t b) = true] *) - val term_of_t : units -> t -> nf_term - val t_of_term : nf_term -> t - -end - - -(** Substitutions (or environments) - - The module {!Subst} contains infrastructure to deal with - substitutions, i.e., functions from variables to terms. Only a - restricted subsets of these functions need to be exported. - - As expected, a particular substitution can be used to - instantiate a pattern. -*) -module Subst : -sig - type t - val sprint : t -> string - val instantiate : t -> Terms.t-> Terms.t - val to_list : t -> (var*Terms.t) list -end - - -(** {2 Main functions exported by this module} *) - -(** [matcher p t] computes the set of solutions to the given top-level - matching problem ([p] is the pattern, [t] is the term). If the - [strict] flag is set, solutions where units are used to - instantiate some variables are excluded, unless this unit appears - directly under a function symbol (e.g., f(x) still matches f(1), - while x+x+y does not match a+b+c, since this would require to - assign 1 to x). -*) -val matcher : ?strict:bool -> ext_units -> Terms.t -> Terms.t -> Subst.t Search_monad.m - -(** [subterm p t] computes a set of solutions to the given - subterm-matching problem. - - Return a collection of possible solutions (each with the - associated depth, the context, and the solutions of the matching - problem). The context is actually a {!Terms.t} where the variables - are yet to be instantiated by one of the associated substitutions -*) -val subterm : ?strict:bool -> ext_units -> Terms.t -> Terms.t -> (int * Terms.t * Subst.t Search_monad.m) Search_monad.m - diff --git a/meta.yml b/meta.yml new file mode 100644 index 0000000..bc7f672 --- /dev/null +++ b/meta.yml @@ -0,0 +1,109 @@ +--- +fullname: AAC tactics +shortname: aac-tactics + +description: | + This Coq plugin provides tactics for rewriting universally quantified + equations, modulo associativity and commutativity of some operator. + The tactics can be applied for custom operators by registering the + operators and their properties as type class instances. Many common + operator instances, such as for Z binary arithmetic and booleans, are + provided with the plugin. + +paper: + doi: 10.1007/978-3-642-25379-9_14 + url: https://arxiv.org/abs/1106.4448 + title: Tactics for Reasoning modulo AC in Coq + +authors: +- name: Thomas Braibant + initial: true +- name: Damien Pous + initial: true +- name: Fabian Kunze + initial: false + +maintainers: +- name: Fabian Kunze + nickname: fakusb +- name: Karl Palmskog + nickname: palmskog + +opam-file-maintainer: palmskog@gmail.com + +license: + fullname: GNU Lesser General Public License v3.0 or later + identifier: LGPL-3.0-or-later + +plugin: true + +supported_coq_versions: + text: Coq 8.9 (use the corresponding branch or release for other Coq versions) + opam: '{>= "8.9" & < "8.10~"}' + +tested_coq_versions: +- version_or_url: 8.9 + +tested_coq_opam_version: 8.9 + +namespace: AAC_tactics + +keywords: +- name: reflexive tactic +- name: rewriting +- name: rewriting modulo associativity and commutativity +- name: rewriting modulo ac +- name: decision procedure + +categories: +- name: Miscellaneous/Coq Extensions +- name: Computer Science/Decision Procedures and Certified Algorithms/Decision procedures + +documentation: | + ## Documentation + + The following example shows an application of the tactics for reasoning over Z binary numbers: + ```coq + Require Import AAC_tactics.AAC. + Require AAC_tactics.Instances. + Require Import ZArith. + + Section ZOpp. + Import Instances.Z. + Variables a b c : Z. + Hypothesis H: forall x, x + Z.opp x = 0. + + Goal a + b + c + Z.opp (c + a) = b. + aac_rewrite H. + aac_reflexivity. + Qed. + End ZOpp. + ``` + + The file [Tutorial.v](theories/Tutorial.v) provides a succinct introduction + and more examples of how to use this plugin. + + The file [Instances.v](theories/Instances.v) defines several type class instances + for frequent use-cases of this plugin, that should allow you to use it off-the-shelf. + Namely, it contains instances for: + + - Peano naturals (`Import Instances.Peano.`) + - Z binary numbers (`Import Instances.Z.`) + - N binary numbers (`Import Instances.N.`) + - P binary numbers (`Import Instances.P.`) + - Rational numbers (`Import Instances.Q.`) + - Prop (`Import Instances.Prop_ops.`) + - Booleans (`Import Instances.Bool.`) + - Relations (`Import Instances.Relations.`) + - all of the above (`Import Instances.All.`) + + To understand the inner workings of the tactics, please refer to + the `.mli` files as the main source of information on each `.ml` file. + + ## Acknowledgements + + The initial authors are grateful to Evelyne Contejean, Hugo Herbelin, + Assia Mahboubi, and Matthieu Sozeau for highly instructive discussions. + The plugin took inspiration from the plugin tutorial "constructors" by + Matthieu Sozeau, distributed under the LGPL 2.1. +--- diff --git a/opam b/opam new file mode 100644 index 0000000..cd6aa36 --- /dev/null +++ b/opam @@ -0,0 +1,30 @@ +opam-version: "1.2" +maintainer: "palmskog@gmail.com" + +homepage: "https://github.com/coq-community/aac-tactics" +dev-repo: "https://github.com/coq-community/aac-tactics.git" +bug-reports: "https://github.com/coq-community/aac-tactics/issues" +license: "LGPL-3.0-or-later" + +build: [make "-j%{jobs}%"] +install: [make "install"] +remove: ["rm" "-R" "%{lib}%/coq/user-contrib/AAC_tactics"] +depends: [ + "coq" {>= "8.9" & < "8.10~"} +] + +tags: [ + "category:Miscellaneous/Coq Extensions" + "category:Computer Science/Decision Procedures and Certified Algorithms/Decision procedures" + "keyword:reflexive tactic" + "keyword:rewriting" + "keyword:rewriting modulo associativity and commutativity" + "keyword:rewriting modulo ac" + "keyword:decision procedure" + "logpath:AAC_tactics" +] +authors: [ + "Thomas Braibant" + "Damien Pous" + "Fabian Kunze" +] diff --git a/print.ml b/print.ml deleted file mode 100644 index 427b6dc..0000000 --- a/print.ml +++ /dev/null @@ -1,101 +0,0 @@ -(***************************************************************************) -(* This is part of aac_tactics, it is distributed under the terms of the *) -(* GNU Lesser General Public License version 3 *) -(* (see file LICENSE for more details) *) -(* *) -(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) -(***************************************************************************) - -(* A very basic way to interact with the envs, to choose a possible - solution *) -open Pp -open Matcher -open Context.Rel.Declaration - -type named_env = (Names.name * Terms.t) list - - - -(** {pp_env} prints a substitution mapping names to terms, using the -provided printer *) -let pp_env pt : named_env -> Pp.std_ppcmds = fun env -> - List.fold_left - (fun acc (v,t) -> - begin match v with - | Names.Name s -> str (Printf.sprintf "%s: " (Names.string_of_id s)) - | Names.Anonymous -> str ("_") - end - ++ pt t ++ str "; " ++ acc - ) - (str "") - env - -(** {pp_envm} prints a collection of possible environments, and number -them. This number must remain compatible with the parameters given to -{!aac_rewrite} *) -let pp_envm pt : named_env Search_monad.m -> Pp.std_ppcmds = fun m -> - let _,s = - Search_monad.fold - (fun env (n,acc) -> - n+1, h 0 (str (Printf.sprintf "%i:\t[" n) ++pp_env pt env ++ str "]") ++ fnl () :: acc - ) m (0,[]) - in - List.fold_left (fun acc s -> s ++ acc) (str "") (s) - -let trivial_substitution envm = - match Search_monad.choose envm with - | None -> true (* assert false *) - | Some l -> l=[] - -(** {pp_all} prints a collection of possible contexts and related -possibles substitutions, giving a number to each. This number must -remain compatible with the parameters of {!aac_rewrite} *) -let pp_all pt : (int * Terms.t * named_env Search_monad.m) Search_monad.m -> Pp.std_ppcmds = fun m -> - let _,s = Search_monad.fold - (fun (size,context,envm) (n,acc) -> - let s = str (Printf.sprintf "occurence %i: transitivity through " n) in - let s = s ++ pt context ++ str "\n" in - let s = if trivial_substitution envm then s else - s ++ str (Printf.sprintf "%i possible(s) substitution(s)" (Search_monad.count envm) ) - ++ fnl () ++ pp_envm pt envm - in - n+1, s::acc - ) m (0,[]) in - List.fold_left (fun acc s -> s ++ str "\n" ++ acc) (str "") (s) - -(** The main printing function. {!print} uses the debruijn_env the -rename the variables, and rebuilds raw Coq terms (for the context, and -the terms in the environment). In order to do so, it requires the -information gathered by the {!Theory.Trans} module.*) -let print rlt ir m (context : Context.Rel.t) goal = - if Search_monad.count m = 0 - then - ( - Tacticals.tclFAIL 0 (Pp.str "No subterm modulo AC") goal - ) - else - let _ = Feedback.msg_notice (Pp.str "All solutions:") in - let m = Search_monad.(>>) m - (fun (i,t,envm) -> - let envm = Search_monad.(>>) envm ( fun env -> - let l = Matcher.Subst.to_list env in - let l = List.sort (fun (n,_) (n',_) -> Pervasives.compare n n') l in - let l = - List.map (fun (v,t) -> - get_name (Context.Rel.lookup v context), t - ) l - in - Search_monad.return l - ) - in - Search_monad.return (i,t,envm) - ) - in - let m = Search_monad.sort (fun (x,_,_) (y,_,_) -> x - y) m in - let _ = Feedback.msg_notice - (pp_all - (fun t -> Printer.pr_constr (Theory.Trans.raw_constr_of_t ir rlt context t) ) m - ) - in - Tacticals.tclIDTAC goal - diff --git a/print.mli b/print.mli deleted file mode 100644 index bbb9b20..0000000 --- a/print.mli +++ /dev/null @@ -1,23 +0,0 @@ -(***************************************************************************) -(* This is part of aac_tactics, it is distributed under the terms of the *) -(* GNU Lesser General Public License version 3 *) -(* (see file LICENSE for more details) *) -(* *) -(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) -(***************************************************************************) - -(** Pretty printing functions we use for the aac_instances - tactic. *) - - -(** The main printing function. {!print} uses the rel-context - to rename the variables, and rebuilds raw Coq terms (for the given - context, and the terms in the environment). In order to do so, it - requires the information gathered by the {!Theory.Trans} module.*) -val print : - Coq.Relation.t -> - Theory.Trans.ir -> - (int * Matcher.Terms.t * Matcher.Subst.t Search_monad.m) Search_monad.m -> - Context.Rel.t -> - Proof_type.tactic - diff --git a/search_monad.ml b/search_monad.ml deleted file mode 100644 index 09a6455..0000000 --- a/search_monad.ml +++ /dev/null @@ -1,70 +0,0 @@ -(***************************************************************************) -(* This is part of aac_tactics, it is distributed under the terms of the *) -(* GNU Lesser General Public License version 3 *) -(* (see file LICENSE for more details) *) -(* *) -(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) -(***************************************************************************) - -type 'a m = | F of 'a - | N of 'a m list - -let fold (f : 'a -> 'b -> 'b) (m : 'a m) (acc : 'b) = - let rec aux acc = function - F x -> f x acc - | N l -> - (List.fold_left (fun acc x -> - match x with - | (N []) -> acc - | x -> aux acc x - ) acc l) - in - aux acc m - - -let rec (>>) : 'a m -> ('a -> 'b m) -> 'b m = - fun m f -> - match m with - | F x -> f x - | N l -> - N (List.fold_left (fun acc x -> - match x with - | (N []) -> acc - | x -> (x >> f)::acc - ) [] l) - -let (>>|) (m : 'a m) (n :'a m) : 'a m = match (m,n) with - | N [],_ -> n - | _,N [] -> m - | F x, N l -> N (F x::l) - | N l, F x -> N (F x::l) - | x,y -> N [x;y] - -let return : 'a -> 'a m = fun x -> F x -let fail : unit -> 'a m = fun () -> N [] - -let sprint f m = - fold (fun x acc -> Printf.sprintf "%s\n%s" acc (f x)) m "" -let rec count = function - | F _ -> 1 - | N l -> List.fold_left (fun acc x -> acc+count x) 0 l - -let opt_comb f x y = match x with None -> f y | _ -> x - -let rec choose = function - | F x -> Some x - | N l -> List.fold_left (fun acc x -> - opt_comb choose acc x - ) None l - -let is_empty = fun x -> choose x = None - -let to_list m = (fold (fun x acc -> x::acc) m []) - -let sort f m = - N (List.map (fun x -> F x) (List.sort f (to_list m))) - -(* preserve the structure of the heap *) -let filter f m = - fold (fun x acc -> (if f x then return x else fail ()) >>| acc) m (N []) - diff --git a/search_monad.mli b/search_monad.mli deleted file mode 100644 index 7e2a910..0000000 --- a/search_monad.mli +++ /dev/null @@ -1,42 +0,0 @@ -(***************************************************************************) -(* This is part of aac_tactics, it is distributed under the terms of the *) -(* GNU Lesser General Public License version 3 *) -(* (see file LICENSE for more details) *) -(* *) -(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) -(***************************************************************************) - -(** Search monad that allows to express non-deterministic algorithms - in a legible maner, or programs that solve combinatorial problems. - - @see the - inspiration of this module -*) - -(** A data type that represent a collection of ['a] *) -type 'a m - - (** {2 Monadic operations} *) - -(** bind and return *) -val ( >> ) : 'a m -> ('a -> 'b m) -> 'b m -val return : 'a -> 'a m - -(** non-deterministic choice *) -val ( >>| ) : 'a m -> 'a m -> 'a m - -(** failure *) -val fail : unit -> 'a m - -(** folding through the collection *) -val fold : ('a -> 'b -> 'b) -> 'a m -> 'b -> 'b - -(** {2 Derived facilities } *) - -val sprint : ('a -> string) -> 'a m -> string -val count : 'a m -> int -val choose : 'a m -> 'a option -val to_list : 'a m -> 'a list -val sort : ('a -> 'a -> int) -> 'a m -> 'a m -val is_empty: 'a m -> bool -val filter : ('a -> bool) -> 'a m -> 'a m diff --git a/src/aac.ml4 b/src/aac.ml4 new file mode 100644 index 0000000..e879a69 --- /dev/null +++ b/src/aac.ml4 @@ -0,0 +1,79 @@ +(***************************************************************************) +(* This is part of aac_tactics, it is distributed under the terms of the *) +(* GNU Lesser General Public License version 3 *) +(* (see file LICENSE for more details) *) +(* *) +(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) +(***************************************************************************) + +(** aac -- Reasoning modulo associativity and commutativity *) + +DECLARE PLUGIN "aac_plugin" + + +open Ltac_plugin +open Pcoq.Prim +open Pcoq.Constr +open Stdarg +open Aac_rewrite +open Extraargs +open Genarg + +ARGUMENT EXTEND aac_args +TYPED AS ((string * int) list ) +PRINTED BY pr_aac_args +| [ "at" integer(n) aac_args(q) ] -> [ add "at" n q ] +| [ "subst" integer(n) aac_args(q) ] -> [ add "subst" n q ] +| [ "in_right" aac_args(q) ] -> [ add "in_right" 0 q ] +| [ ] -> [ [] ] +END + +let pr_constro _ _ _ = fun b -> + match b with + | None -> Pp.str "" + | Some o -> Pp.str "" + +ARGUMENT EXTEND constro +TYPED AS (constr option) +PRINTED BY pr_constro +| [ "[" constr(c) "]" ] -> [ Some c ] +| [ ] -> [ None ] +END + +TACTIC EXTEND _aac_reflexivity_ +| [ "aac_reflexivity" ] -> [ Proofview.V82.tactic aac_reflexivity ] +END + +TACTIC EXTEND _aac_normalise_ +| [ "aac_normalise" ] -> [ Proofview.V82.tactic aac_normalise ] +END + +TACTIC EXTEND _aac_rewrite_ +| [ "aac_rewrite" orient(l2r) constr(c) aac_args(args) constro(extra)] -> + [ Proofview.V82.tactic (fun gl -> aac_rewrite ?extra ~args ~l2r ~strict:true c gl) ] +END + +TACTIC EXTEND _aac_pattern_ +| [ "aac_pattern" orient(l2r) constr(c) aac_args(args) constro(extra)] -> + [ Proofview.V82.tactic (fun gl -> aac_rewrite ?extra ~args ~l2r ~strict:true ~abort:true c gl) ] +END + +TACTIC EXTEND _aac_instances_ +| [ "aac_instances" orient(l2r) constr(c) aac_args(args) constro(extra)] -> + [ Proofview.V82.tactic (fun gl -> aac_rewrite ?extra ~args ~l2r ~strict:true ~show:true c gl) ] +END + +TACTIC EXTEND _aacu_rewrite_ +| [ "aacu_rewrite" orient(l2r) constr(c) aac_args(args) constro(extra)] -> + [ Proofview.V82.tactic (fun gl -> aac_rewrite ?extra ~args ~l2r ~strict:false c gl) ] +END + +TACTIC EXTEND _aacu_pattern_ +| [ "aacu_pattern" orient(l2r) constr(c) aac_args(args) constro(extra)] -> + [ Proofview.V82.tactic (fun gl -> aac_rewrite ?extra ~args ~l2r ~strict:false ~abort:true c gl) ] +END + +TACTIC EXTEND _aacu_instances_ +| [ "aacu_instances" orient(l2r) constr(c) aac_args(args) constro(extra)] -> + [ Proofview.V82.tactic (fun gl -> aac_rewrite ?extra ~args ~l2r ~strict:false ~show:true c gl) ] +END diff --git a/src/aac_plugin.mlpack b/src/aac_plugin.mlpack new file mode 100644 index 0000000..ae61b0d --- /dev/null +++ b/src/aac_plugin.mlpack @@ -0,0 +1,8 @@ +Coq +Helper +Search_monad +Matcher +Theory +Print +Aac_rewrite +Aac diff --git a/src/aac_rewrite.ml b/src/aac_rewrite.ml new file mode 100644 index 0000000..697c15c --- /dev/null +++ b/src/aac_rewrite.ml @@ -0,0 +1,425 @@ +(***************************************************************************) +(* This is part of aac_tactics, it is distributed under the terms of the *) +(* GNU Lesser General Public License version 3 *) +(* (see file LICENSE for more details) *) +(* *) +(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) +(***************************************************************************) + +(** aac_rewrite -- rewriting modulo A or AC*) + +open Ltac_plugin + +module Control = struct + let debug = false + let printing = false + let time = false +end + +module Debug = Helper.Debug (Control) +open Debug + +let time_tac msg tac = + if Control.time then Coq.tclTIME msg tac else tac + +let tac_or_exn tac exn msg = fun gl -> + try tac gl with e -> + let env = Tacmach.pf_env gl in + let sigma = Tacmach.project gl in + pr_constr env sigma "last goal" (Tacmach.pf_concl gl); + exn msg e + + +let retype = Coq.retype + +open EConstr +open Names +open Proof_type + + +(** aac_lift : the ideal type beyond AAC_rewrite.v/Lift + + A base relation r, together with an equivalence relation, and the + proof that the former lifts to the later. Howver, we have to + ensure manually the invariant : r.carrier == e.carrier, and that + lift connects the two things *) +type aac_lift = + { + r : Coq.Relation.t; + e : Coq.Equivalence.t; + lift : constr + } + +type rewinfo = + { + hypinfo : Coq.Rewrite.hypinfo; + + in_left : bool; (** are we rewriting in the left hand-sie of the goal *) + pattern : constr; + subject : constr; + morph_rlt : Coq.Relation.t; (** the relation we look for in morphism *) + eqt : Coq.Equivalence.t; (** the equivalence we use as workbase *) + rlt : Coq.Relation.t; (** the relation in the goal *) + lifting: aac_lift + } + +let infer_lifting (rlt: Coq.Relation.t) (k : lift:aac_lift -> Proof_type.tactic) : Proof_type.tactic = + Coq.cps_evar_relation rlt.Coq.Relation.carrier (fun e -> + let lift_ty = + mkApp (Lazy.force Theory.Stubs.lift, + [| + rlt.Coq.Relation.carrier; + rlt.Coq.Relation.r; + e + |] + ) in + Coq.cps_resolve_one_typeclass ~error:(Pp.strbrk "Cannot infer a lifting") + lift_ty (fun lift goal -> + let x = rlt.Coq.Relation.carrier in + let r = rlt.Coq.Relation.r in + let eq = (Coq.nf_evar goal e) in + let equiv = Coq.lapp Theory.Stubs.lift_proj_equivalence [| x;r;eq; lift |] in + let lift = + { + r = rlt; + e = Coq.Equivalence.make x eq equiv; + lift = lift; + } + in + k ~lift:lift goal + )) + +(** Builds a rewinfo, once and for all *) +let dispatch in_left (left,right,rlt) hypinfo (k: rewinfo -> Proof_type.tactic ) : Proof_type.tactic= + let l2r = hypinfo.Coq.Rewrite.l2r in + infer_lifting rlt + (fun ~lift -> + let eq = lift.e in + k { + hypinfo = hypinfo; + in_left = in_left; + pattern = if l2r then hypinfo.Coq.Rewrite.left else hypinfo.Coq.Rewrite.right; + subject = if in_left then left else right; + morph_rlt = Coq.Equivalence.to_relation eq; + eqt = eq; + lifting = lift; + rlt = rlt + } + ) + + + +(** {1 Tactics} *) + + +(** Build the reifiers, the reified terms, and the evaluation fonction *) +let handle eqt zero envs (t : Matcher.Terms.t) (t' : Matcher.Terms.t) k = + + let (x,r,_) = Coq.Equivalence.split eqt in + Theory.Trans.mk_reifier (Coq.Equivalence.to_relation eqt) zero envs + (fun (maps, reifier) -> + (* fold through a term and reify *) + let t = Theory.Trans.reif_constr_of_t reifier t in + let t' = Theory.Trans.reif_constr_of_t reifier t' in + (* Some letins *) + let eval = (mkApp (Lazy.force Theory.Stubs.eval, [|x;r; maps.Theory.Trans.env_sym; maps.Theory.Trans.env_bin; maps.Theory.Trans.env_units|])) in + + Coq.cps_mk_letin "eval" eval (fun eval -> + Coq.cps_mk_letin "left" t (fun t -> + Coq.cps_mk_letin "right" t' (fun t' -> + k maps eval t t')))) + +(** [by_aac_reflexivity] is a sub-tactic that closes a sub-goal that + is merely a proof of equality of two terms modulo AAC *) +let by_aac_reflexivity zero + eqt envs (t : Matcher.Terms.t) (t' : Matcher.Terms.t) : Proof_type.tactic = + handle eqt zero envs t t' + (fun maps eval t t' -> + let (x,r,e) = Coq.Equivalence.split eqt in + let decision_thm = Coq.lapp Theory.Stubs.decide_thm + [|x;r;e; + maps.Theory.Trans.env_sym; + maps.Theory.Trans.env_bin; + maps.Theory.Trans.env_units; + t;t'; + |] + in + (* This convert is required to deal with evars in a proper + way *) + let convert_to = mkApp (r, [| mkApp (eval,[| t |]); mkApp (eval, [|t'|])|]) in + let convert = Proofview.V82.of_tactic (Tactics.convert_concl convert_to Constr.VMcast) in + let apply_tac = Proofview.V82.of_tactic (Tactics.apply decision_thm) in + (Tacticals.tclTHENLIST + [ retype decision_thm; retype convert_to; + convert ; + tac_or_exn apply_tac Coq.user_error (Pp.strbrk "unification failure"); + tac_or_exn (time_tac "vm_norm" (Proofview.V82.of_tactic (Tactics.normalise_in_concl))) Coq.anomaly "vm_compute failure"; + Tacticals.tclORELSE (Proofview.V82.of_tactic Tactics.reflexivity) + (Tacticals.tclFAIL 0 (Pp.str "Not an equality modulo A/AC")) + ]) + ) + +let by_aac_normalise zero lift ir t t' = + let eqt = lift.e in + let rlt = lift.r in + handle eqt zero ir t t' + (fun maps eval t t' -> + let (x,r,e) = Coq.Equivalence.split eqt in + let normalise_thm = Coq.lapp Theory.Stubs.lift_normalise_thm + [|x;r;e; + maps.Theory.Trans.env_sym; + maps.Theory.Trans.env_bin; + maps.Theory.Trans.env_units; + rlt.Coq.Relation.r; + lift.lift; + t;t'; + |] + in + (* This convert is required to deal with evars in a proper + way *) + let convert_to = mkApp (rlt.Coq.Relation.r, [| mkApp (eval,[| t |]); mkApp (eval, [|t'|])|]) in + let convert = Proofview.V82.of_tactic (Tactics.convert_concl convert_to Constr.VMcast) in + let apply_tac = Proofview.V82.of_tactic (Tactics.apply normalise_thm) in + (Tacticals.tclTHENLIST + [ retype normalise_thm; retype convert_to; + convert ; + apply_tac; + ]) + + ) + +(** A handler tactic, that reifies the goal, and infer the liftings, + and then call its continuation *) +let aac_conclude + (k : constr -> aac_lift -> Theory.Trans.ir -> Matcher.Terms.t -> Matcher.Terms.t -> Proof_type.tactic) = fun goal -> + + let (equation : types) = Tacmach.pf_concl goal in + let envs = Theory.Trans.empty_envs () in + let left, right,r = + match Coq.match_as_equation goal equation with + | None -> Coq.user_error @@ Pp.strbrk "The goal is not an applied relation" + | Some x -> x in + try infer_lifting r + (fun ~lift goal -> + let eq = Coq.Equivalence.to_relation lift.e in + let tleft,tright, goal = Theory.Trans.t_of_constr goal eq envs (left,right) in + let goal, ir = Theory.Trans.ir_of_envs goal eq envs in + let concl = Tacmach.pf_concl goal in + let env = Tacmach.pf_env goal in + let sigma = Tacmach.project goal in + let _ = pr_constr env sigma "concl "concl in + let evar_map = Tacmach.project goal in + Tacticals.tclTHEN + (Refiner.tclEVARS evar_map) + (k left lift ir tleft tright) + goal + )goal + with + | Not_found -> Coq.user_error @@ Pp.strbrk "No lifting from the goal's relation to an equivalence" + +open Tacexpr + +let aac_normalise = fun goal -> + let ids = Tacmach.pf_ids_of_hyps goal in + let mp = MPfile (DirPath.make (List.map Id.of_string ["AAC"; "AAC_tactics"])) in + let norm_tac = KerName.make2 mp (Label.make "internal_normalize") in + let norm_tac = Locus.ArgArg (None, norm_tac) in + Tacticals.tclTHENLIST + [ + aac_conclude by_aac_normalise; + Proofview.V82.of_tactic (Tacinterp.eval_tactic (TacArg (None, TacCall (None, (norm_tac, []))))); + Proofview.V82.of_tactic (Tactics.keep ids) + ] goal + +let aac_reflexivity = fun goal -> + aac_conclude + (fun zero lift ir t t' -> + let x,r = Coq.Relation.split (lift.r) in + let r_reflexive = (Coq.Classes.mk_reflexive x r) in + Coq.cps_resolve_one_typeclass ~error:(Pp.strbrk "The goal's relation is not reflexive") + r_reflexive + (fun reflexive -> + let lift_reflexivity = + mkApp (Lazy.force (Theory.Stubs.lift_reflexivity), + [| + x; + r; + lift.e.Coq.Equivalence.eq; + lift.lift; + reflexive + |]) + in + Tacticals.tclTHEN + + (Tacticals.tclTHEN (retype lift_reflexivity) (Proofview.V82.of_tactic (Tactics.apply lift_reflexivity))) + (fun goal -> + let concl = Tacmach.pf_concl goal in + let env = Tacmach.pf_env goal in + let sigma = Tacmach.project goal in + let _ = pr_constr env sigma "concl "concl in + by_aac_reflexivity zero lift.e ir t t' goal) + ) + ) goal + +(** A sub-tactic to lift the rewriting using Lift *) +let lift_transitivity in_left (step:constr) preorder lifting (using_eq : Coq.Equivalence.t): tactic = + fun goal -> + (* catch the equation and the two members*) + let concl = Tacmach.pf_concl goal in + let (left, right, _ ) = match Coq.match_as_equation goal concl with + | Some x -> x + | None -> Coq.user_error @@ Pp.strbrk "The goal is not an equation" + in + let lift_transitivity = + let thm = + if in_left + then + Lazy.force Theory.Stubs.lift_transitivity_left + else + Lazy.force Theory.Stubs.lift_transitivity_right + in + mkApp (thm, + [| + preorder.Coq.Relation.carrier; + preorder.Coq.Relation.r; + using_eq.Coq.Equivalence.eq; + lifting; + step; + left; + right; + |]) + in + Tacticals.tclTHENLIST + [ retype lift_transitivity; + Proofview.V82.of_tactic (Tactics.apply lift_transitivity) + ] goal + + +(** The core tactic for aac_rewrite. Env and sigma are for the constr *) +let core_aac_rewrite ?abort + rewinfo + subst + (by_aac_reflexivity : Matcher.Terms.t -> Matcher.Terms.t -> Proof_type.tactic) + env sigma (tr : constr) t left : tactic = + pr_constr env sigma "transitivity through" tr; + let tran_tac = + lift_transitivity rewinfo.in_left tr rewinfo.rlt rewinfo.lifting.lift rewinfo.eqt + in + Coq.Rewrite.rewrite ?abort rewinfo.hypinfo subst (fun rew -> + Tacticals.tclTHENSV + (tac_or_exn (tran_tac) Coq.anomaly "Unable to make the transitivity step") + ( + if rewinfo.in_left + then [| by_aac_reflexivity left t ; rew |] + else [| by_aac_reflexivity t left ; rew |] + ) + ) + +exception NoSolutions + + +(** Choose a substitution from a + [(int * Terms.t * Env.env Search_monad.m) Search_monad.m ] *) +(* WARNING: Beware, since the printing function can change the order of the + printed monad, this function has to be updated accordingly *) +let choose_subst subterm sol m= + try + let (depth,pat,envm) = match subterm with + | None -> (* first solution *) + List.nth ( List.rev (Search_monad.to_list m)) 0 + | Some x -> + List.nth ( List.rev (Search_monad.to_list m)) x + in + let env = match sol with + None -> + List.nth ( (Search_monad.to_list envm)) 0 + | Some x -> List.nth ( (Search_monad.to_list envm)) x + in + pat, env + with + | _ -> raise NoSolutions + +(** rewrite the constr modulo AC from left to right in the left member + of the goal *) +let aac_rewrite_wrap ?abort rew ?(l2r=true) ?(show = false) ?(in_left=true) ?strict ~occ_subterm ~occ_sol ?extra : Proof_type.tactic = fun goal -> + let envs = Theory.Trans.empty_envs () in + let (concl : types) = Tacmach.pf_concl goal in + let (_,_,rlt) as concl = + match Coq.match_as_equation goal concl with + | None -> Coq.user_error @@ Pp.strbrk "The goal is not an applied relation" + | Some (left, right, rlt) -> left,right,rlt + in + let check_type x = + Tacmach.pf_conv_x goal x rlt.Coq.Relation.carrier + in + Coq.Rewrite.get_hypinfo rew ~l2r ?check_type:(Some check_type) + (fun hypinfo -> + dispatch in_left concl hypinfo + ( + fun rewinfo -> + let goal = + match extra with + | Some t -> Theory.Trans.add_symbol goal rewinfo.morph_rlt envs (EConstr.to_constr (Tacmach.project goal) t) + | None -> goal + in + let pattern, subject, goal = + Theory.Trans.t_of_constr goal rewinfo.morph_rlt envs + (rewinfo.pattern , rewinfo.subject) + in + let goal, ir = Theory.Trans.ir_of_envs goal rewinfo.morph_rlt envs in + + let units = Theory.Trans.ir_to_units ir in + let m = Matcher.subterm ?strict units pattern subject in + (* We sort the monad in increasing size of contet *) + let m = Search_monad.sort (fun (x,_,_) (y,_,_) -> x - y) m in + + if show + then + Print.print rewinfo.morph_rlt ir m (hypinfo.Coq.Rewrite.context) + + else + try + let pat,subst = choose_subst occ_subterm occ_sol m in + let tr_step = Matcher.Subst.instantiate subst pat in + let tr_step_raw = Theory.Trans.raw_constr_of_t ir rewinfo.morph_rlt [] tr_step in + + let conv = (Theory.Trans.raw_constr_of_t ir rewinfo.morph_rlt (hypinfo.Coq.Rewrite.context)) in + let subst = Matcher.Subst.to_list subst in + let subst = List.map (fun (x,y) -> x, conv y) subst in + let by_aac_reflexivity = (by_aac_reflexivity rewinfo.subject rewinfo.eqt ir) in + let env = Tacmach.pf_env goal in + let sigma = Tacmach.project goal in + (* I'm not sure whether this is the right env/sigma for printing tr_step_raw *) + core_aac_rewrite ?abort rewinfo subst by_aac_reflexivity env sigma tr_step_raw tr_step subject + + with + | NoSolutions -> + Tacticals.tclFAIL 0 + (Pp.str (if occ_subterm = None && occ_sol = None + then "No matching occurrence modulo AC found" + else "No such solution")) + ) + ) goal + +let get k l = try Some (List.assoc k l) with Not_found -> None + +let get_lhs l = try ignore (List.assoc "in_right" l); false with Not_found -> true + +let aac_rewrite ~args = + aac_rewrite_wrap ~occ_subterm:(get "at" args) ~occ_sol:(get "subst" args) ~in_left:(get_lhs args) + + + +let rec add k x = function + | [] -> [k,x] + | k',_ as ky::q -> + if k'=k then Coq.user_error @@ Pp.strbrk ("redondant argument ("^k^")") + else ky::add k x q + +let pr_aac_args _ _ _ l = + List.fold_left + (fun acc -> function + | ("in_right" as s,_) -> Pp.(++) (Pp.str s) acc + | (k,i) -> Pp.(++) (Pp.(++) (Pp.str k) (Pp.int i)) acc + ) (Pp.str "") l + diff --git a/src/aac_rewrite.mli b/src/aac_rewrite.mli new file mode 100644 index 0000000..80134af --- /dev/null +++ b/src/aac_rewrite.mli @@ -0,0 +1,24 @@ +(***************************************************************************) +(* This is part of aac_tactics, it is distributed under the terms of the *) +(* GNU Lesser General Public License version 3 *) +(* (see file LICENSE for more details) *) +(* *) +(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) +(***************************************************************************) + +(** aac_rewrite -- rewriting modulo A or AC*) + +val aac_reflexivity : Coq.goal_sigma -> Proof_type.goal list Evd.sigma +val aac_normalise : Coq.goal_sigma -> Proof_type.goal list Evd.sigma + +val aac_rewrite : + args:(string * int) list -> + ?abort:bool -> + EConstr.constr -> + ?l2r:bool -> + ?show:bool -> + ?strict:bool -> ?extra:EConstr.t -> Proof_type.tactic + +val add : string -> 'a -> (string * 'a) list -> (string * 'a) list + +val pr_aac_args : 'a -> 'b -> 'c -> (string * int) list -> Pp.t diff --git a/src/coq.ml b/src/coq.ml new file mode 100644 index 0000000..ae1803a --- /dev/null +++ b/src/coq.ml @@ -0,0 +1,594 @@ +(***************************************************************************) +(* This is part of aac_tactics, it is distributed under the terms of the *) +(* GNU Lesser General Public License version 3 *) +(* (see file LICENSE for more details) *) +(* *) +(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) +(***************************************************************************) + +(** Interface with Coq *) + +open Constr +open EConstr +open Names +open Context.Rel.Declaration + +(* The contrib name is used to locate errors when loading constrs *) +let contrib_name = "aac_tactics" + +(* Getting constrs (primitive Coq terms) from existing Coq + libraries. *) +let find_constant contrib dir s = + UnivGen.constr_of_global (Coqlib.find_reference contrib dir s) + +let init_constant_constr dir s = find_constant contrib_name dir s + +let init_constant dir s = EConstr.of_constr (find_constant contrib_name dir s) + +(* A clause specifying that the [let] should not try to fold anything + in the goal *) +let nowhere = + { Locus.onhyps = Some []; + Locus.concl_occs = Locus.NoOccurrences + } + +let retype c gl = + let sigma, _ = Tacmach.pf_apply Typing.type_of gl c in + Refiner.tclEVARS sigma gl + +let cps_mk_letin + (name:string) + (c: constr) + (k : constr -> Proof_type.tactic) +: Proof_type.tactic = + fun goal -> + let name = (Id.of_string name) in + let name = Tactics.fresh_id Id.Set.empty name goal in + let letin = (Proofview.V82.of_tactic (Tactics.letin_tac None (Name name) c None nowhere)) in + Tacticals.tclTHENLIST [retype c; letin; (k (mkVar name))] goal + +(** {1 General functions} *) + +type goal_sigma = Proof_type.goal Evd.sigma +let goal_update (goal : goal_sigma) evar_map : goal_sigma= + let it = Tacmach.sig_it goal in + Tacmach.re_sig it evar_map + +let resolve_one_typeclass goal ty : constr*goal_sigma= + let env = Tacmach.pf_env goal in + let evar_map = Tacmach.project goal in + let em,c = Typeclasses.resolve_one_typeclass env evar_map ty in + c, (goal_update goal em) + +let cps_resolve_one_typeclass ?error : types -> (constr -> Proof_type.tactic) -> Proof_type.tactic = fun t k goal -> + Tacmach.pf_apply + (fun env em -> let em ,c = + try Typeclasses.resolve_one_typeclass env em t + with Not_found -> + begin match error with + | None -> CErrors.anomaly (Pp.str "Cannot resolve a typeclass : please report") + | Some x -> CErrors.user_err x + end + in + Tacticals.tclTHENLIST [Refiner.tclEVARS em; k c] goal + ) goal + + +let nf_evar goal c : constr= + let evar_map = Tacmach.project goal in + Evarutil.nf_evar evar_map c + + (* TODO: refactor following similar functions*) + +let evar_unit (gl : goal_sigma) (x : constr) : constr * goal_sigma = + let env = Tacmach.pf_env gl in + let evar_map = Tacmach.project gl in + let (em,x) = Evarutil.new_evar env evar_map x in + x,(goal_update gl em) + +let evar_binary (gl: goal_sigma) (x : constr) = + let env = Tacmach.pf_env gl in + let evar_map = Tacmach.project gl in + let ty = mkArrow x (mkArrow x x) in + let (em,x) = Evarutil.new_evar env evar_map ty in + x,( goal_update gl em) + +let evar_relation (gl: goal_sigma) (x: constr) = + let env = Tacmach.pf_env gl in + let evar_map = Tacmach.project gl in + let ty = mkArrow x (mkArrow x (mkSort Sorts.prop)) in + let (em,r) = Evarutil.new_evar env evar_map ty in + r,( goal_update gl em) + +let cps_evar_relation (x: constr) k = fun goal -> + Tacmach.pf_apply + (fun env em -> + let ty = mkArrow x (mkArrow x (mkSort Sorts.prop)) in + let (em, r) = Evarutil.new_evar env em ty in + Tacticals.tclTHENLIST [Refiner.tclEVARS em; k r] goal + ) goal + +let decomp_term sigma c = kind sigma (Termops.strip_outer_cast sigma c) + +let lapp c v = mkApp (Lazy.force c, v) + +(** {2 Bindings with Coq' Standard Library} *) +module Std = struct +(* Here we package the module to be able to use List, later *) + +module Pair = struct + + let path = ["Coq"; "Init"; "Datatypes"] + let typ = lazy (init_constant path "prod") + let pair = lazy (init_constant path "pair") + let of_pair t1 t2 (x,y) = + mkApp (Lazy.force pair, [| t1; t2; x ; y|] ) +end + +module Bool = struct + let path = ["Coq"; "Init"; "Datatypes"] + let typ = lazy (init_constant path "bool") + let ctrue = lazy (init_constant path "true") + let cfalse = lazy (init_constant path "false") + let of_bool b = + if b then Lazy.force ctrue else Lazy.force cfalse +end + +module Comparison = struct + let path = ["Coq"; "Init"; "Datatypes"] + let typ = lazy (init_constant path "comparison") + let eq = lazy (init_constant path "Eq") + let lt = lazy (init_constant path "Lt") + let gt = lazy (init_constant path "Gt") +end + +module Leibniz = struct + let path = ["Coq"; "Init"; "Logic"] + let eq_refl = lazy (init_constant path "eq_refl") + let eq_refl ty x = lapp eq_refl [| ty;x|] +end + +module Option = struct + let path = ["Coq"; "Init"; "Datatypes"] + let typ = lazy (init_constant path "option") + let some = lazy (init_constant path "Some") + let none = lazy (init_constant path "None") + let some t x = mkApp (Lazy.force some, [| t ; x|]) + let none t = mkApp (Lazy.force none, [| t |]) + let of_option t x = match x with + | Some x -> some t x + | None -> none t +end + +module Pos = struct + + let path = ["Coq" ; "Numbers"; "BinNums"] + let typ = lazy (init_constant path "positive") + let xI = lazy (init_constant path "xI") + let xO = lazy (init_constant path "xO") + let xH = lazy (init_constant path "xH") + + (* A coq positive from an int *) + let of_int n = + let rec aux n = + begin match n with + | n when n < 1 -> assert false + | 1 -> Lazy.force xH + | n -> mkApp + ( + (if n mod 2 = 0 + then Lazy.force xO + else Lazy.force xI + ), [| aux (n/2)|] + ) + end + in + aux n +end + +module Nat = struct + let path = ["Coq" ; "Init"; "Datatypes"] + let typ = lazy (init_constant path "nat") + let _S = lazy (init_constant path "S") + let _O = lazy (init_constant path "O") + (* A coq nat from an int *) + let of_int n = + let rec aux n = + begin match n with + | n when n < 0 -> assert false + | 0 -> Lazy.force _O + | n -> mkApp + ( + (Lazy.force _S + ), [| aux (n-1)|] + ) + end + in + aux n +end + +(** Lists from the standard library*) +module List = struct + let path = ["Coq"; "Init"; "Datatypes"] + let typ = lazy (init_constant path "list") + let nil = lazy (init_constant path "nil") + let cons = lazy (init_constant path "cons") + let cons ty h t = + mkApp (Lazy.force cons, [| ty; h ; t |]) + let nil ty = + (mkApp (Lazy.force nil, [| ty |])) + let rec of_list ty = function + | [] -> nil ty + | t::q -> cons ty t (of_list ty q) + let type_of_list ty = + mkApp (Lazy.force typ, [|ty|]) +end + +(** Morphisms *) +module Classes = +struct + let classes_path = ["Coq";"Classes"; ] + let morphism = lazy (init_constant (classes_path@["Morphisms"]) "Proper") + let equivalence = lazy (init_constant (classes_path@ ["RelationClasses"]) "Equivalence") + let reflexive = lazy (init_constant (classes_path@ ["RelationClasses"]) "Reflexive") + let transitive = lazy (init_constant (classes_path@ ["RelationClasses"]) "Transitive") + + (** build the type [Equivalence ty rel] *) + let mk_equivalence ty rel = + mkApp (Lazy.force equivalence, [| ty; rel|]) + + + (** build the type [Reflexive ty rel] *) + let mk_reflexive ty rel = + mkApp (Lazy.force reflexive, [| ty; rel|]) + + (** build the type [Proper rel t] *) + let mk_morphism ty rel t = + mkApp (Lazy.force morphism, [| ty; rel; t|]) + + (** build the type [Transitive ty rel] *) + let mk_transitive ty rel = + mkApp (Lazy.force transitive, [| ty; rel|]) +end + +module Relation = struct + type t = + { + carrier : constr; + r : constr; + } + + let make ty r = {carrier = ty; r = r } + let split t = t.carrier, t.r +end + +module Transitive = struct + type t = + { + carrier : constr; + leq : constr; + transitive : constr; + } + let make ty leq transitive = {carrier = ty; leq = leq; transitive = transitive} + let infer goal ty leq = + let ask = Classes.mk_transitive ty leq in + let transitive , goal = resolve_one_typeclass goal ask in + make ty leq transitive, goal + let from_relation goal rlt = + infer goal (rlt.Relation.carrier) (rlt.Relation.r) + let cps_from_relation rlt k = + let ty =rlt.Relation.carrier in + let r = rlt.Relation.r in + let ask = Classes.mk_transitive ty r in + cps_resolve_one_typeclass ask + (fun trans -> k (make ty r trans) ) + let to_relation t = + {Relation.carrier = t.carrier; Relation.r = t.leq} + +end + +module Equivalence = struct + type t = + { + carrier : constr; + eq : constr; + equivalence : constr; + } + let make ty eq equivalence = {carrier = ty; eq = eq; equivalence = equivalence} + let infer goal ty eq = + let ask = Classes.mk_equivalence ty eq in + let equivalence , goal = resolve_one_typeclass goal ask in + make ty eq equivalence, goal + let from_relation goal rlt = + infer goal (rlt.Relation.carrier) (rlt.Relation.r) + let cps_from_relation rlt k = + let ty =rlt.Relation.carrier in + let r = rlt.Relation.r in + let ask = Classes.mk_equivalence ty r in + cps_resolve_one_typeclass ask (fun equiv -> k (make ty r equiv) ) + let to_relation t = + {Relation.carrier = t.carrier; Relation.r = t.eq} + let split t = + t.carrier, t.eq, t.equivalence +end +end +(**[ match_as_equation goal eqt] see [eqt] as an equation. An + optionnal rel-context can be provided to ensure that the term + remains typable*) +let match_as_equation ?(context = Context.Rel.empty) goal equation : (constr*constr* Std.Relation.t) option = + let env = Tacmach.pf_env goal in + let env = EConstr.push_rel_context context env in + let evar_map = Tacmach.project goal in + begin + match decomp_term evar_map equation with + | App(c,ca) when Array.length ca >= 2 -> + let n = Array.length ca in + let left = ca.(n-2) in + let right = ca.(n-1) in + let r = (mkApp (c, Array.sub ca 0 (n - 2))) in + let carrier = Typing.unsafe_type_of env evar_map left in + let rlt =Std.Relation.make carrier r + in + Some (left, right, rlt ) + | _ -> None + end + + +(** {1 Tacticals} *) + +let tclTIME msg tac = fun gl -> + let u = Sys.time () in + let r = tac gl in + let _ = Feedback.msg_notice (Pp.str (Printf.sprintf "%s:%fs" msg (Sys.time ()-. u))) in + r + +let tclDEBUG msg tac = fun gl -> + let _ = Feedback.msg_debug (Pp.str msg) in + tac gl + +let tclPRINT tac = fun gl -> + let env = Tacmach.pf_env gl in + let sigma = Tacmach.project gl in + let _ = Feedback.msg_notice (Printer.pr_econstr_env env sigma (Tacmach.pf_concl gl)) in + tac gl + + +(** {1 Error related mechanisms} *) +(* functions to handle the failures of our tactic. Some should be + reported [anomaly], some are on behalf of the user [user_error]*) +let anomaly msg = + CErrors.anomaly ~label:"[aac_tactics]" (Pp.str msg) + +let user_error msg = + CErrors.user_err Pp.(str "[aac_tactics] " ++ msg) + +let warning msg = + Feedback.msg_warning (Pp.str ("[aac_tactics]" ^ msg)) + + +(** {1 Rewriting tactics used in aac_rewrite} *) +module Rewrite = struct +(** Some informations about the hypothesis, with an (informal) + invariant: + - [typeof hyp = hyptype] + - [hyptype = forall context, body] + - [body = rel left right] + +*) + +type hypinfo = + { + hyp : constr; (** the actual constr corresponding to the hypothese *) + hyptype : constr; (** the type of the hypothesis *) + context : EConstr.rel_context; (** the quantifications of the hypothese *) + body : constr; (** the body of the type of the hypothesis*) + rel : Std.Relation.t; (** the relation *) + left : constr; (** left hand side *) + right : constr; (** right hand side *) + l2r : bool; (** rewriting from left to right *) + } + +let get_hypinfo c ~l2r ?check_type (k : hypinfo -> Proof_type.tactic) : Proof_type.tactic = fun goal -> + let ctype = Tacmach.pf_unsafe_type_of goal c in + let evar_map = Tacmach.project goal in + let (rel_context, body_type) = decompose_prod_assum evar_map ctype in + let rec check f e = + match decomp_term evar_map e with + | Rel i -> f (get_type (Context.Rel.lookup i rel_context)) + | _ -> fold evar_map (fun acc x -> acc && check f x) true e + in + begin match check_type with + | None -> () + | Some f -> + if not (check f body_type) + then user_error @@ Pp.strbrk "Unable to deal with higher-order or heterogeneous patterns"; + end; + begin + match match_as_equation ~context:rel_context goal body_type with + | None -> + user_error @@ Pp.strbrk "The hypothesis is not an applied relation" + | Some (hleft,hright,hrlt) -> + k { + hyp = c; + hyptype = ctype; + body = body_type; + l2r = l2r; + context = rel_context; + rel = hrlt ; + left =hleft; + right = hright; + } + goal + end + + +(* The problem : Given a term to rewrite of type [H :forall xn ... x1, + t], we have to instanciate the subset of [xi] of type + [carrier]. [subst : (int * constr)] is the mapping the debruijn + indices in [t] to the [constrs]. We need to handle the rest of the + indexes. Two ways : + + - either using fresh evars and rewriting [H tn ?1 tn-2 ?2 ] + - either building a term like [fun 1 2 => H tn 1 tn-2 2] + + Both these terms have the same type. +*) + + +(* Fresh evars for everyone (should be the good way to do this + recompose in Coq v8.4) *) +let recompose_prod + (context : rel_context) + (subst : (int * constr) list) + env + em + : Evd.evar_map * constr list = + (* the last index of rel relevant for the rewriting *) + let min_n = List.fold_left + (fun acc (x,_) -> min acc x) + (List.length context) subst in + let rec aux context acc em n = + let _ = Printf.printf "%i\n%!" n in + match context with + | [] -> + env, em, acc + | t::q -> + let env, em, acc = aux q acc em (n+1) in + if n >= min_n + then + let em,x = + try em, List.assoc n subst + with | Not_found -> + let (em, r) = Evarutil.new_evar env em (Vars.substl acc (get_type t)) in + (em, r) + in + (EConstr.push_rel t env), em,x::acc + else + env,em,acc + in + let _,em,acc = aux context [] em 1 in + em, acc + +(* no fresh evars : instead, use a lambda abstraction around an + application to handle non-instantiated variables. *) + +let recompose_prod' + (context : rel_context) + (subst : (int *constr) list) + c + = + let rec popn pop n l = + if n <= 0 then l + else match l with + | [] -> [] + | t::q -> pop t :: (popn pop (n-1) q) + in + let pop_rel_decl = map_type Termops.pop in + let rec aux sign n next app ctxt = + match sign with + | [] -> List.rev app, List.rev ctxt + | decl::sign -> + try let term = (List.assoc n subst) in + aux sign (n+1) next (term::app) (None :: ctxt) + with + | Not_found -> + let term = mkRel next in + aux sign (n+1) (next+1) (term::app) (Some decl :: ctxt) + in + let app,ctxt = aux context 1 1 [] [] in + (* substitutes in the context *) + let rec update ctxt app = + match ctxt,app with + | [],_ -> [] + | _,[] -> [] + | None :: sign, _ :: app -> + None :: update sign (List.map (Termops.pop) app) + | Some decl :: sign, _ :: app -> + Some (Vars.substl_decl app decl)::update sign (List.map (Termops.pop) app) + in + let ctxt = update ctxt app in + (* updates the rel accordingly, taking some care not to go to far + beyond: it is important not to lift indexes homogeneously, we + have to update *) + let rec update ctxt res n = + match ctxt with + | [] -> List.rev res + | None :: sign -> + (update (sign) (popn pop_rel_decl n res) 0) + | Some decl :: sign -> + update sign (decl :: res) (n+1) + in + let ctxt = update ctxt [] 0 in + let c = applist (c,List.rev app) in + let c = it_mkLambda_or_LetIn c ctxt in + c + +(* Version de Matthieu +let subst_rel_context k cstr ctx = + let (_, ctx') = + List.fold_left (fun (k, ctx') (id, b, t) -> (succ k, (id, Option.map + (Term.substnl [cstr] k) b, Term.substnl [cstr] k t) :: ctx')) (k, []) + ctx in List.rev ctx' + +let recompose_prod' context subst c = + let len = List.length context in + let rec aux sign n next app ctxt = + match sign with + | [] -> List.rev app, List.rev ctxt + | decl::sign -> + try let term = (List.assoc n subst) in + aux (subst_rel_context 0 term sign) (pred n) (succ next) + (term::List.map (Term.lift (-1)) app) ctxt + with Not_found -> + let term = Term.mkRel (len - next) in + aux sign (pred n) (succ next) (term::app) (decl :: ctxt) + in + let app,ctxt = aux (List.rev context) len 0 [] [] in + Term.it_mkLambda_or_LetIn (Term.applistc c(app)) (List.rev ctxt) +*) + +let build + (h : hypinfo) + (subst : (int *constr) list) + (k :constr -> Proof_type.tactic) + : Proof_type.tactic = fun goal -> + let c = recompose_prod' h.context subst h.hyp in + Tacticals.tclTHENLIST [k c] goal + +let build_with_evar + (h : hypinfo) + (subst : (int *constr) list) + (k :constr -> Proof_type.tactic) + : Proof_type.tactic + = fun goal -> + Tacmach.pf_apply + (fun env em -> + let evar_map, acc = recompose_prod h.context subst env em in + let c = applist (h.hyp,List.rev acc) in + Tacticals.tclTHENLIST [Refiner.tclEVARS evar_map; k c] goal + ) goal + + +let rewrite ?(abort=false)hypinfo subst k = + build hypinfo subst + (fun rew -> + let tac = + if not abort then + Proofview.V82.of_tactic begin + Equality.general_rewrite_bindings + hypinfo.l2r + Locus.AllOccurrences + true (* tell if existing evars must be frozen for instantiation *) + false + (rew,Tactypes.NoBindings) + true + end + else + Tacticals.tclIDTAC + in k tac + ) + + +end + +include Std diff --git a/src/coq.mli b/src/coq.mli new file mode 100644 index 0000000..9cf0db7 --- /dev/null +++ b/src/coq.mli @@ -0,0 +1,232 @@ +(***************************************************************************) +(* This is part of aac_tactics, it is distributed under the terms of the *) +(* GNU Lesser General Public License version 3 *) +(* (see file LICENSE for more details) *) +(* *) +(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) +(***************************************************************************) + +(** Interface with Coq where we define some handlers for Coq's API, + and we import several definitions from Coq's standard library. + + This general purpose library could be reused by other plugins. + + Some salient points: + - we use Caml's module system to mimic Coq's one, and avoid cluttering + the namespace; + - we also provide several handlers for standard coq tactics, in + order to provide a unified setting (we replace functions that + modify the evar_map by functions that operate on the whole + goal, and repack the modified evar_map with it). + +*) + +(** {2 Getting Coq terms from the environment} *) + +val init_constant_constr : string list -> string -> Constr.t +val init_constant : string list -> string -> EConstr.constr + +(** {2 General purpose functions} *) + +type goal_sigma = Proof_type.goal Evd.sigma +val resolve_one_typeclass : Proof_type.goal Evd.sigma -> EConstr.types -> EConstr.constr * goal_sigma +val cps_resolve_one_typeclass: ?error:Pp.t -> EConstr.types -> (EConstr.constr -> Proof_type.tactic) -> Proof_type.tactic +val nf_evar : goal_sigma -> EConstr.constr -> EConstr.constr +val evar_unit :goal_sigma ->EConstr.constr -> EConstr.constr* goal_sigma +val evar_binary: goal_sigma -> EConstr.constr -> EConstr.constr* goal_sigma +val evar_relation: goal_sigma -> EConstr.constr -> EConstr.constr* goal_sigma +val cps_evar_relation : EConstr.constr -> (EConstr.constr -> Proof_type.tactic) -> Proof_type.tactic +(** [cps_mk_letin name v] binds the constr [v] using a letin tactic *) +val cps_mk_letin : string -> EConstr.constr -> ( EConstr.constr -> Proof_type.tactic) -> Proof_type.tactic + +val retype : EConstr.constr -> Proof_type.tactic + +val decomp_term : Evd.evar_map -> EConstr.constr -> (EConstr.constr , EConstr.types, EConstr.ESorts.t, EConstr.EInstance.t) Constr.kind_of_term +val lapp : EConstr.constr lazy_t -> EConstr.constr array -> EConstr.constr + +(** {2 Bindings with Coq' Standard Library} *) + +(** Coq lists *) +module List: +sig + (** [of_list ty l] *) + val of_list:EConstr.constr ->EConstr.constr list ->EConstr.constr + + (** [type_of_list ty] *) + val type_of_list:EConstr.constr ->EConstr.constr +end + +(** Coq pairs *) +module Pair: +sig + val typ:EConstr.constr lazy_t + val pair:EConstr.constr lazy_t + val of_pair : EConstr.constr -> EConstr.constr -> EConstr.constr * EConstr.constr -> EConstr.constr +end + +module Bool : sig + val typ : EConstr.constr lazy_t + val of_bool : bool -> EConstr.constr +end + + +module Comparison : sig + val typ : EConstr.constr lazy_t + val eq : EConstr.constr lazy_t + val lt : EConstr.constr lazy_t + val gt : EConstr.constr lazy_t +end + +module Leibniz : sig + val eq_refl : EConstr.types -> EConstr.constr -> EConstr.constr +end + +module Option : sig + val typ : EConstr.constr lazy_t + val some : EConstr.constr -> EConstr.constr -> EConstr.constr + val none : EConstr.constr -> EConstr.constr + val of_option : EConstr.constr -> EConstr.constr option -> EConstr.constr +end + +(** Coq positive numbers (pos) *) +module Pos: +sig + val typ:EConstr.constr lazy_t + val of_int: int ->EConstr.constr +end + +(** Coq unary numbers (peano) *) +module Nat: +sig + val typ:EConstr.constr lazy_t + val of_int: int ->EConstr.constr +end + +(** Coq typeclasses *) +module Classes: +sig + val mk_morphism: EConstr.constr -> EConstr.constr -> EConstr.constr -> EConstr.constr + val mk_equivalence: EConstr.constr -> EConstr.constr -> EConstr.constr + val mk_reflexive: EConstr.constr -> EConstr.constr -> EConstr.constr + val mk_transitive: EConstr.constr -> EConstr.constr -> EConstr.constr +end + +module Relation : sig + type t = {carrier : EConstr.constr; r : EConstr.constr;} + val make : EConstr.constr -> EConstr.constr -> t + val split : t -> EConstr.constr * EConstr.constr +end + +module Transitive : sig + type t = + { + carrier : EConstr.constr; + leq : EConstr.constr; + transitive : EConstr.constr; + } + val make : EConstr.constr -> EConstr.constr -> EConstr.constr -> t + val infer : goal_sigma -> EConstr.constr -> EConstr.constr -> t * goal_sigma + val from_relation : goal_sigma -> Relation.t -> t * goal_sigma + val cps_from_relation : Relation.t -> (t -> Proof_type.tactic) -> Proof_type.tactic + val to_relation : t -> Relation.t +end + +module Equivalence : sig + type t = + { + carrier : EConstr.constr; + eq : EConstr.constr; + equivalence : EConstr.constr; + } + val make : EConstr.constr -> EConstr.constr -> EConstr.constr -> t + val infer : goal_sigma -> EConstr.constr -> EConstr.constr -> t * goal_sigma + val from_relation : goal_sigma -> Relation.t -> t * goal_sigma + val cps_from_relation : Relation.t -> (t -> Proof_type.tactic) -> Proof_type.tactic + val to_relation : t -> Relation.t + val split : t -> EConstr.constr * EConstr.constr * EConstr.constr +end + +(** [match_as_equation ?context goal c] try to decompose c as a + relation applied to two terms. An optionnal rel-context can be + provided to ensure that the term remains typable *) +val match_as_equation : ?context:EConstr.rel_context -> goal_sigma -> EConstr.constr -> (EConstr.constr * EConstr.constr * Relation.t) option + +(** {2 Some tacticials} *) + +(** time the execution of a tactic *) +val tclTIME : string -> Proof_type.tactic -> Proof_type.tactic + +(** emit debug messages to see which tactics are failing *) +val tclDEBUG : string -> Proof_type.tactic -> Proof_type.tactic + +(** print the current goal *) +val tclPRINT : Proof_type.tactic -> Proof_type.tactic + + +(** {2 Error related mechanisms} *) + +val anomaly : string -> 'a +val user_error : Pp.t -> 'a +val warning : string -> unit + + +(** {2 Rewriting tactics used in aac_rewrite} *) + +module Rewrite : sig + +(** The rewriting tactics used in aac_rewrite, build as handlers of the usual [setoid_rewrite]*) + + +(** {2 Datatypes} *) + +(** We keep some informations about the hypothesis, with an (informal) + invariant: + - [typeof hyp = typ] + - [typ = forall context, body] + - [body = rel left right] + +*) +type hypinfo = + { + hyp : EConstr.constr; (** the actual constr corresponding to the hypothese *) + hyptype : EConstr.constr; (** the type of the hypothesis *) + context : EConstr.rel_context; (** the quantifications of the hypothese *) + body : EConstr.constr; (** the body of the hypothese*) + rel : Relation.t; (** the relation *) + left : EConstr.constr; (** left hand side *) + right : EConstr.constr; (** right hand side *) + l2r : bool; (** rewriting from left to right *) + } + +(** [get_hypinfo H l2r ?check_type k] analyse the hypothesis H, and + build the related hypinfo, in CPS style. Moreover, an optionnal + function can be provided to check the type of every free + variable of the body of the hypothesis. *) +val get_hypinfo :EConstr.constr -> l2r:bool -> ?check_type:(EConstr.types -> bool) -> (hypinfo -> Proof_type.tactic) -> Proof_type.tactic + +(** {2 Rewriting with bindings} + + The problem : Given a term to rewrite of type [H :forall xn ... x1, + t], we have to instanciate the subset of [xi] of type + [carrier]. [subst : (int * constr)] is the mapping the De Bruijn + indices in [t] to the [constrs]. We need to handle the rest of the + indexes. Two ways : + + - either using fresh evars and rewriting [H tn ?1 tn-2 ?2 ] + - either building a term like [fun 1 2 => H tn 1 tn-2 2] + + Both these terms have the same type. +*) + +(** build the constr to rewrite, in CPS style, with lambda abstractions *) +val build : hypinfo -> (int * EConstr.constr) list -> (EConstr.constr -> Proof_type.tactic) -> Proof_type.tactic + +(** build the constr to rewrite, in CPS style, with evars *) +val build_with_evar : hypinfo -> (int * EConstr.constr) list -> (EConstr.constr -> Proof_type.tactic) -> Proof_type.tactic + +(** [rewrite ?abort hypinfo subst] builds the rewriting tactic + associated with the given [subst] and [hypinfo], and feeds it to + the given continuation. If [abort] is set to [true], we build + [tclIDTAC] instead. *) +val rewrite : ?abort:bool -> hypinfo -> (int * EConstr.constr) list -> (Proof_type.tactic -> Proof_type.tactic) -> Proof_type.tactic +end diff --git a/src/helper.ml b/src/helper.ml new file mode 100644 index 0000000..c1cae2a --- /dev/null +++ b/src/helper.ml @@ -0,0 +1,41 @@ +(***************************************************************************) +(* This is part of aac_tactics, it is distributed under the terms of the *) +(* GNU Lesser General Public License version 3 *) +(* (see file LICENSE for more details) *) +(* *) +(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) +(***************************************************************************) + +module type CONTROL = sig + val debug : bool + val time : bool + val printing : bool +end + +module Debug (X : CONTROL) = struct + open X + let debug x = + if debug then + Printf.printf "%s\n%!" x + + + let time f x fmt = + if time then + let t = Sys.time () in + let r = f x in + Printf.printf fmt (Sys.time () -. t); + r + else f x + + let pr_constr env evd msg constr = + if printing then + ( Feedback.msg_notice (Pp.str (Printf.sprintf "=====%s====" msg)); + Feedback.msg_notice (Printer.pr_econstr_env env evd constr); + ) + + + let debug_exception msg e = + debug (msg ^ (Printexc.to_string e)) + + +end diff --git a/src/helper.mli b/src/helper.mli new file mode 100644 index 0000000..400e847 --- /dev/null +++ b/src/helper.mli @@ -0,0 +1,33 @@ +(***************************************************************************) +(* This is part of aac_tactics, it is distributed under the terms of the *) +(* GNU Lesser General Public License version 3 *) +(* (see file LICENSE for more details) *) +(* *) +(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) +(***************************************************************************) + +(** Debugging functions, that can be triggered on a per-file base. *) + +module type CONTROL = +sig + val debug : bool + val time : bool + val printing : bool +end +module Debug : + functor (X : CONTROL) -> +sig + (** {!debug} prints the string and end it with a newline *) + val debug : string -> unit + val debug_exception : string -> exn -> unit + + (** {!time} computes the time spent in a function, and then + print it using the given format *) + val time : + ('a -> 'b) -> 'a -> (float -> unit, out_channel, unit) format -> 'b + + (** {!pr_constr} print a Coq constructor, that can be labelled + by a string *) + val pr_constr : Environ.env -> Evd.evar_map -> string -> EConstr.constr -> unit + + end diff --git a/src/matcher.ml b/src/matcher.ml new file mode 100644 index 0000000..cc66961 --- /dev/null +++ b/src/matcher.ml @@ -0,0 +1,1293 @@ +(***************************************************************************) +(* This is part of aac_tactics, it is distributed under the terms of the *) +(* GNU Lesser General Public License version 3 *) +(* (see file LICENSE for more details) *) +(* *) +(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) +(***************************************************************************) + +(** This module defines our matching functions, modulo associativity + and commutativity (AAC). + + The basic idea is to find a substitution [env] such that the + pattern [p] instantiated by [env] is equal to [t] modulo AAC. + + We proceed by structural decomposition of the pattern, and try all + possible non-deterministic split of the subject when needed. The + function {!matcher} is limited to top-level matching, that is, the + subject must make a perfect match against the pattern ([x+x] do + not match [a+a+b] ). We use a search monad {!Search} to perform + non-deterministic splits in an almost transparent way. We also + provide a function {!subterm} for finding a match that is a + subterm modulo AAC of the subject. Therefore, we are able to solve + the aforementioned case [x+x] against [a+b+a]. + + This file is structured as follows. First, we define the search + monad. Then,we define the two representations of terms (one + representing the AST, and one in normal form ), and environments + from variables to terms. Then, we use these parts to solve + matching problem. Finally, we wrap this function {!matcher} into + {!subterm} +*) + + +module Control = +struct + let debug = false + let time = false + let printing = false +end + +module Debug = Helper.Debug (Control) +open Debug + +module Search = Search_monad (* a handle *) + +type symbol = int +type var = int +type units = (symbol * symbol) list (* from AC/A symbols to the unit *) +type ext_units = + { + unit_for : units; + is_ac : (symbol * bool) list + } + +let print_units units= + List.iter (fun (op,unit) -> Printf.printf "%i %i\t" op unit) units; + Printf.printf "\n%!" + +exception NoUnit + +let get_unit units op = + try List.assoc op units + with + | Not_found -> raise NoUnit + +let is_unit units op unit = List.mem (op,unit) units + + +open Search + +type 'a mset = ('a * int) list +let linear p = + let rec ncons t l = function + | 0 -> l + | n -> t::ncons t l (n-1) + in + let rec aux = function + [ ] -> [] + | (t,n)::q -> let q = aux q in + ncons t q n + in aux p + + + +(** The module {!Terms} defines two different types for expressions. + + - a public type {!Terms.t} that represent abstract syntax trees of + expressions with binary associative (and commutative) operators + + - a private type {!Terms.nf_term} that represent an equivalence + class for terms that are equal modulo AAC. The constructions + functions on this type ensure the property that the term is in + normal form (that is, no sum can appear as a subterm of the same + sum, no trailing units, etc...). + +*) + +module Terms : sig + + (** {1 Abstract syntax tree of terms} + + Terms represented using this datatype are representation of the + AST of an expression. *) + + type t = + Dot of (symbol * t * t) + | Plus of (symbol * t * t) + | Sym of (symbol * t array) + | Var of var + | Unit of symbol + + val equal_aac : units -> t -> t -> bool + val size: t -> int + + (** {1 Terms in normal form} + + A term in normal form is the canonical representative of the + equivalence class of all the terms that are equal modulo + Associativity and Commutativity. Outside the {!Matcher} module, + one does not need to access the actual representation of this + type. *) + + + type nf_term = private + | TAC of symbol * nf_term mset + | TA of symbol * nf_term list + | TSym of symbol * nf_term list + | TUnit of symbol + | TVar of var + + (** {2 Fresh variables: we provide some functions to pick some fresh variables with respect to a term} *) + val fresh_var_term : t -> int + val fresh_var_nfterm : nf_term -> int + + + (** {2 Constructors: we ensure that the terms are always + normalised + + braibant - Fri 27 Aug 2010, 15:11 + Moreover, we assure that we will not build empty sums or empty + products, leaving this task to the caller function. + } + *) + val mk_TAC : units -> symbol -> nf_term mset -> nf_term + val mk_TA : units -> symbol -> nf_term list -> nf_term + val mk_TSym : symbol -> nf_term list -> nf_term + val mk_TVar : var -> nf_term + val mk_TUnit : symbol -> nf_term + + (** {2 Comparisons} *) + val nf_term_compare : nf_term -> nf_term -> int + val nf_equal : nf_term -> nf_term -> bool + + (** {2 Printing function} *) + val sprint_nf_term : nf_term -> string + + (** {2 Conversion functions} *) + val term_of_t : units -> t -> nf_term + val t_of_term : nf_term -> t (* we could return the units here *) +end + = struct + + type t = + Dot of (symbol * t * t) + | Plus of (symbol * t * t) + | Sym of (symbol * t array) + | Var of var + | Unit of symbol + + let rec size = function + | Dot (_,x,y) + | Plus (_,x,y) -> size x+ size y + 1 + | Sym (_,v)-> Array.fold_left (fun acc x -> size x + acc) 1 v + | _ -> 1 + + + type nf_term = + | TAC of symbol * nf_term mset + | TA of symbol * nf_term list + | TSym of symbol * nf_term list + | TUnit of symbol + | TVar of var + + + (** {2 Picking fresh variables} *) + + (** [fresh_var_term] picks a fresh variable with respect to a term *) + let fresh_var_term t = + let rec aux = function + | Dot (_,t1,t2) + | Plus (_,t1,t2) -> max (aux t1) (aux t2) + | Sym (_,v) -> Array.fold_left (fun acc x -> max acc (aux x)) 0 v + | Var v -> assert (v >= 0); v + | Unit _ -> 0 + in + aux t + + (** [fresh_var_nfterm] picks a fresh_variable with respect to a term *) + let fresh_var_nfterm t = + let rec aux = function + | TAC (_,l) -> List.fold_left (fun acc (x,_) -> max acc (aux x)) 0 l + | TA (_,l) + | TSym (_,l) -> List.fold_left (fun acc x -> max acc (aux x)) 0 l + | TVar v -> assert (v >= 0); v + | TUnit _ -> 0 + in + aux t + + (** {2 Constructors: we ensure that the terms are always + normalised} *) + + (** {3 Pre constructors : These constructors ensure that sums and + products are not degenerated (no trailing units)} *) + let mk_TAC' units (s : symbol) l = match l with + | [] -> TUnit (get_unit units s ) + | [_,0] -> assert false + | [t,1] -> t + | _ -> TAC (s,l) + let mk_TA' units (s : symbol) l = match l with + | [] -> TUnit (get_unit units s ) + | [t] -> t + | _ -> TA (s,l) + + + (** {2 Comparison} *) + + let nf_term_compare = Pervasives.compare + let nf_equal a b = + a = b + + (** [merge_ac comp l1 l2] merges two lists of terms with coefficients + into one. Terms that are equal modulo the comparison function + [comp] will see their arities added. *) + + (* This function could be improved by the use of sorted msets *) + let merge_ac (compare : 'a -> 'a -> int) (l : 'a mset) (l' : 'a mset) : 'a mset = + let rec aux l l'= + match l,l' with + | [], _ -> l' + | _, [] -> l + | (t,tar)::q, (t',tar')::q' -> + begin match compare t t' with + | 0 -> ( t,tar+tar'):: aux q q' + | -1 -> (t, tar):: aux q l' + | _ -> (t', tar'):: aux l q' + end + in aux l l' + + (** [merge_map f l] uses the combinator [f] to combine the head of the + list [l] with the merge_maped tail of [l] *) + let rec merge_map (f : 'a -> 'b list -> 'b list) (l : 'a list) : 'b list = + match l with + | [] -> [] + | t::q -> f t (merge_map f q) + + + (** This function has to deal with the arities *) + let merge (l : nf_term mset) (l' : nf_term mset) : nf_term mset= + merge_ac nf_term_compare l l' + + let extract_A units s t = + match t with + | TA (s',l) when s' = s -> l + | TUnit u when is_unit units s u -> [] + | _ -> [t] + + let extract_AC units s (t,ar) : nf_term mset = + match t with + | TAC (s',l) when s' = s -> List.map (fun (x,y) -> (x,y*ar)) l + | TUnit u when is_unit units s u -> [] + | _ -> [t,ar] + + + (** {3 Constructors of {!nf_term}}*) + let mk_TAC units (s : symbol) (l : (nf_term *int) list) = + mk_TAC' units s + (merge_map (fun u l -> merge (extract_AC units s u) l) l) + let mk_TA units s l = + mk_TA' units s + (merge_map (fun u l -> (extract_A units s u) @ l) l) + let mk_TSym s l = TSym (s,l) + let mk_TVar v = TVar v + let mk_TUnit s = TUnit s + + + (** {2 Printing function} *) + let print_binary_list (single : 'a -> string) + (unit : string) + (binary : string -> string -> string) (l : 'a list) = + let rec aux l = + match l with + [] -> unit + | [t] -> single t + | t::q -> + let r = aux q in + Printf.sprintf "%s" (binary (single t) r) + in + aux l + + let print_symbol s = + match s with + | s, None -> Printf.sprintf "%i" s + | s , Some u -> Printf.sprintf "%i(unit %i)" s u + + let sprint_ac (single : 'a -> string) (l : 'a mset) = + (print_binary_list + (fun (x,t) -> + if t = 1 + then single x + else Printf.sprintf "%i*%s" t (single x) + ) + "0" + (fun x y -> x ^ " , " ^ y) + l + ) + + let print_symbol single s l = + match l with + [] -> Printf.sprintf "%i" s + | _ -> + Printf.sprintf "%i(%s)" + s + (print_binary_list single "" (fun x y -> x ^ "," ^ y) l) + + + let print_ac single s l = + Printf.sprintf "[%s:AC]{%s}" + (string_of_int s ) + (sprint_ac + single + l + ) + + let print_a single s l = + Printf.sprintf "[%s:A]{%s}" + (string_of_int s) + (print_binary_list single "1" (fun x y -> x ^ " , " ^ y) l) + + let rec sprint_nf_term = function + | TSym (s,l) -> print_symbol sprint_nf_term s l + | TAC (s,l) -> print_ac sprint_nf_term s l + | TA (s,l) -> print_a sprint_nf_term s l + | TVar v -> Printf.sprintf "x%i" v + | TUnit s -> Printf.sprintf "unit%i" s + + + + + (** {2 Conversion functions} *) + + (* rebuilds a tree out of a list, under the assumption that the list is not empty *) + let binary_of_list f comb l = + let l = List.rev l in + let rec aux = function + | [] -> assert false + | [t] -> f t + | t::q -> comb (aux q) (f t) + in + aux l + + let term_of_t units : t -> nf_term = + let rec term_of_t = function + | Dot (s,l,r) -> + let l = term_of_t l in + let r = term_of_t r in + mk_TA units s [l;r] + | Plus (s,l,r) -> + let l = term_of_t l in + let r = term_of_t r in + mk_TAC units s [l,1;r,1] + | Unit x -> + mk_TUnit x + | Sym (s,t) -> + let t = Array.to_list t in + let t = List.map term_of_t t in + mk_TSym s t + | Var i -> + mk_TVar ( i) + in + term_of_t + + let rec t_of_term : nf_term -> t = function + | TAC (s,l) -> + (binary_of_list + t_of_term + (fun l r -> Plus ( s,l,r)) + (linear l) + ) + | TA (s,l) -> + (binary_of_list + t_of_term + (fun l r -> Dot ( s,l,r)) + l + ) + | TSym (s,l) -> + let v = Array.of_list l in + let v = Array.map (t_of_term) v in + Sym ( s,v) + | TVar x -> Var x + | TUnit s -> Unit s + + + let equal_aac units x y = + nf_equal (term_of_t units x) (term_of_t units y) + end + + (** Terms environments defined as association lists from variables to + terms in normal form {! Terms.nf_term} *) + module Subst : sig + type t + + val find : t -> var -> Terms.nf_term option + val add : t -> var -> Terms.nf_term -> t + val empty : t + val instantiate : t -> Terms.t -> Terms.t + val sprint : t -> string + val to_list : t -> (var*Terms.t) list + end + = + struct + open Terms + + (** Terms environments, with nf_terms, to avoid costly conversions + of {!Terms.nf_terms} to {!Terms.t}, that will be mostly discarded*) + type t = (var * nf_term) list + + let find : t -> var -> nf_term option = fun t x -> + try Some (List.assoc x t) with | _ -> None + let add t x v = (x,v) :: t + let empty = [] + + let sprint (l : t) = + match l with + | [] -> Printf.sprintf "Empty environment\n" + | _ -> + let s = List.fold_left + (fun acc (x,y) -> + Printf.sprintf "%sX%i -> %s\n" + acc + x + (sprint_nf_term y) + ) + "" + (List.rev l) in + Printf.sprintf "%s\n%!" s + + + + (** [instantiate] is an homomorphism except for the variables*) + let instantiate (t: t) (x:Terms.t) : Terms.t = + let rec aux = function + | Unit _ as x -> x + | Sym (s,t) -> Sym (s,Array.map aux t) + | Plus (s,l,r) -> Plus (s, aux l, aux r) + | Dot (s,l,r) -> Dot (s, aux l, aux r) + | Var i -> + begin match find t i with + | None -> CErrors.user_err (Pp.strbrk "aac_tactics: instantiate failure") + | Some x -> t_of_term x + end + in aux x + + let to_list t = List.map (fun (x,y) -> x,Terms.t_of_term y) t + end + + (******************) + (* MATCHING UTILS *) + (******************) + + open Terms + + (** Since most of the folowing functions require an extra parameter, + the units, we package them in a module. This functor will then be + applied to a module containing the units, in the exported + functions. *) + module M (X : sig + val units : units + val is_ac : (symbol * bool) list + val strict : bool (* variables cannot be instantiated with units *) + end) = struct + + open X + + let print_units ()= + List.iter (fun (op,unit) -> Printf.printf "%i %i\t" op unit) units; + Printf.printf "\n%!" + + let mk_TAC s l = mk_TAC units s l + let mk_TA s l = mk_TA units s l + let mk_TAC' s l = + try return( mk_TAC s l) + with _ -> fail () + let mk_TA' s l = + try return( mk_TA s l) + with _ -> fail () + + (** First, we need to be able to perform non-deterministic choice of + term splitting to satisfy a pattern. Indeed, we want to show that: + (x+a*b)*c <= a*b*c + *) + let a_nondet_split_raw t : ('a list * 'a list) m = + let rec aux l l' = + match l' with + | [] -> + return ( l,[]) + | t::q -> + return ( l,l' ) + >>| aux (l @ [t]) q + in + aux [] t + + (** Same as the previous [a_nondet_split], but split the list in 3 + parts *) + let a_nondet_middle t : ('a list * 'a list * 'a list) m = + a_nondet_split_raw t >> + (fun (left, right) -> + a_nondet_split_raw left >> + (fun (left, middle) -> return (left, middle, right)) + ) + + (** Non deterministic splits of ac lists *) + let dispatch f n = + let rec aux k = + if k = 0 then return (f n 0) + else return (f (n-k) k) >>| aux (k-1) + in + aux (n ) + + let add_with_arith x ar l = + if ar = 0 then l else (x,ar) ::l + + let ac_nondet_split_raw (l : 'a mset) : ('a mset * 'a mset) m = + let rec aux = function + | [] -> return ([],[]) + | (t,tar)::q -> + aux q + >> + (fun (left,right) -> + dispatch (fun arl arr -> + add_with_arith t arl left, + add_with_arith t arr right + ) + tar + ) + in + aux l + + let a_nondet_split current t : (nf_term * nf_term list) m= + a_nondet_split_raw t + >> + (fun (l,r) -> + if strict && (l=[]) + then fail() + else + mk_TA' current l >> + fun t -> return (t, r) + ) + + let ac_nondet_split current t : (nf_term * nf_term mset) m= + ac_nondet_split_raw t + >> + (fun (l,r) -> + if strict && (l=[]) + then fail() + else + mk_TAC' current l >> + fun t -> return (t, r) + ) + + + (** Try to affect the variable [x] to each left factor of [t]*) + let var_a_nondet_split env current x t = + a_nondet_split current t + >> + (fun (t,res) -> + return ((Subst.add env x t), res) + ) + + (** Try to affect variable [x] to _each_ subset of t. *) + let var_ac_nondet_split (current: symbol) env (x : var) (t : nf_term mset) : (Subst.t * (nf_term mset)) m = + ac_nondet_split current t + >> + (fun (t,res) -> + return ((Subst.add env x t), res) + ) + + (** See the term t as a given AC symbol. Unwrap the first constructor + if necessary *) + let get_AC (s : symbol) (t : nf_term) : (nf_term *int) list = + match t with + | TAC (s',l) when s' = s -> l + | TUnit u when is_unit units s u -> [] + | _ -> [t,1] + + (** See the term t as a given A symbol. Unwrap the first constructor + if necessary *) + let get_A (s : symbol) (t : nf_term) : nf_term list = + match t with + | TA (s',l) when s' = s -> l + | TUnit u when is_unit units s u -> [] + | _ -> [t] + + (** See the term [t] as an symbol [s]. Fail if it is not such + symbol. *) + let get_Sym s t = + match t with + | TSym (s',l) when s' = s -> return l + | _ -> fail () + + (*************) + (* A Removal *) + (*************) + + (** We remove the left factor v in a term list. This function runs + linearly with respect to the size of the first pattern symbol *) + + let left_factor current (v : nf_term) (t : nf_term list) = + let rec aux a b = + match a,b with + | t::q , t' :: q' when nf_equal t t' -> aux q q' + | [], q -> return q + | _, _ -> fail () + in + match v with + | TA (s,l) when s = current -> aux l t + | TUnit u when is_unit units current u -> return t + | _ -> + begin match t with + | [] -> fail () + | t::q -> + if nf_equal v t + then return q + else fail () + end + + + (**************) + (* AC Removal *) + (**************) + + (** {!pick_sym} gather all elements of a list that satisfies a + predicate, and combine them with the residual of the list. That + is, each element of the residual contains exactly one element less + than the original term. + + TODO : This function not as efficient as it could be, using a + proper data-structure *) + + let pick_sym (s : symbol) (t : nf_term mset ) = + let rec aux front back = + match back with + | [] -> fail () + | (t,tar)::q -> + begin match t with + | TSym (s',v') when s = s' -> + let back = + if tar > 1 + then (t,tar -1) ::q + else q + in + return (v' , List.rev_append front back ) + >>| aux ((t,tar)::front) q + | _ -> aux ((t,tar)::front) q + end + in + aux [] t + + + + (** We have to check if we are trying to remove a unit from a term. Could also be located in Terms*) + let is_unit_AC s t = + try nf_equal t (mk_TAC s []) + with | NoUnit -> false + + let is_same_AC s t : nf_term mset option= + match t with + TAC (s',l) when s = s' -> Some l + | _ -> None + + (** We want to remove the term [v] from the term list [t] under an AC + symbol *) + let single_AC_factor (s : symbol) (v : nf_term) v_ar (t : nf_term mset) : (nf_term mset) m = + let rec aux front back = + match back with + | [] -> fail () + | (t,tar)::q -> + begin + if nf_equal v t + then + match () with + | _ when tar < v_ar -> fail () + | _ when tar = v_ar -> return (List.rev_append front q) + | _ -> return (List.rev_append front ((t,tar-v_ar)::q)) + else + aux ((t,tar) :: front) q + end + in + if is_unit_AC s v + then + return t + else + aux [] t + + (* Remove a constant from a mset. If this constant is also a mset for + the same symbol, we remove every elements, one at a time (and we do + care of the arities) *) + let factor_AC (s : symbol) (v: nf_term) (t : nf_term mset) : ( nf_term mset ) m = + match is_same_AC s v with + | None -> single_AC_factor s v 1 t + | Some l -> + (* We are trying to remove an AC factor *) + List.fold_left (fun acc (v,v_ar) -> + acc >> (single_AC_factor s v v_ar) + ) + (return t) + l + + +(** [tri_fold f l acc] folds on the list [l] and give to f the + beginning of the list in reverse order, the considered element, and + the last part of the list + + as an exemple, on the list [1;2;3;4], we get the trace + f () [] 1 [2; 3; 4] + f () [1] 2 [3; 4] + f () [2;1] 3 [ 4] + f () [3;2;1] 4 [] + + it is the duty of the user to reverse the front if needed +*) + +let tri_fold f (l : 'a list) (acc : 'b)= match l with + [] -> acc + | _ -> + let _,_,acc = List.fold_left (fun acc (t : 'a) -> + let l,r,acc = acc in + let r = List.tl r in + t::l,r,f acc l t r + ) ([], l,acc) l + in acc + +(* let test = tri_fold (fun acc l t r -> (l,t,r) :: acc) [1;2;3;4] [] *) + + + + (*****************************) + (* ENUMERATION DES POSITIONS *) + (*****************************) + + +(** The pattern is a unit: we need to try to make it appear at each + position. We do not need to go further with a real matching, since + the match should be trivial. Hence, we proceed recursively to + enumerate all the positions. *) + +module Positions = struct + + + let ac (l: 'a mset) : ('a mset * 'a )m = + let rec aux = function + | [] -> assert false + | [t,1] -> return ([],t) + | [t,tar] -> return ([t,tar -1],t) + | (t,tar) as h :: q -> + (aux q >> (fun (c,x) -> return (h :: c,x))) + >>| (if tar > 1 then return ((t,tar-1) :: q,t) else return (q,t)) + in + aux l + + let ac' current (l: 'a mset) : ('a mset * 'a )m = + ac_nondet_split_raw l >> + (fun (l,r) -> + if l = [] || r = [] + then fail () + else + mk_TAC' current r >> + fun t -> return (l, t) + ) + + let a (l : 'a list) : ('a list * 'a * 'a list) m = + let rec aux left right : ('a list * 'a * 'a list) m = + match right with + | [] -> assert false + | [t] -> return (left,t,[]) + | t::q -> + aux (left@[t]) q + >>| return (left,t,q) + in + aux [] l +end + +let build_ac (current : symbol) (context : nf_term mset) (p : t) : t m= + if context = [] + then return p + else + mk_TAC' current context >> + fun t -> return (Plus (current,t_of_term t,p)) + +let build_a (current : symbol) + (left : nf_term list) (right : nf_term list) (p : t) : t m= + let right_pat p = + if right = [] + then return p + else + mk_TA' current right >> + fun t -> return (Dot (current,p,t_of_term t)) + in + let left_pat p= + if left = [] + then return p + else + mk_TA' current left >> + fun t -> return (Dot (current,t_of_term t,p)) + in + right_pat p >> left_pat >> (fun p -> return p) + + +let conts (hole : t) (l : symbol list) p : t m = + let p = t_of_term p in + (* - aller chercher les symboles ac et les symboles a + - construire pour chaque + * * + + / \ / \ / \ + 1 p p 1 p 1 + *) + let ac,a = List.partition (fun s -> List.assoc s is_ac) l in + let acc = fail () in + let acc = List.fold_left + (fun acc s -> + acc >>| return (Plus (s,p,hole)) + ) acc ac in + let acc = + List.fold_left + (fun acc s -> + acc >>| return (Dot (s,p,hole)) >>| return (Dot (s,hole,p)) + ) acc a + in acc + + +(** + Return the same thing as subterm : + - The size of the context + - The context + - A collection of substitutions (here == return Subst.empty) +*) +let unit_subterm (t : nf_term) (unit : symbol) (hole : t): (int * t * Subst.t m) m = + let symbols = List.fold_left + (fun acc (ac,u) -> if u = unit then ac :: acc else acc ) [] units + in + (* make a unit appear at each strict sub-position of the term*) + let rec positions (t : nf_term) : t m = + match t with + (* with a final unit at the end *) + | TAC (s,l) -> + let symbols' = List.filter (fun x -> x <> s) symbols in + ( + ac_nondet_split_raw l >> + (fun (l,r) -> if l = [] || r = [] then fail () else + ( + match r with + | [p,1] -> + positions p >>| conts hole symbols' p + | _ -> + mk_TAC' s r >> conts hole symbols' + ) >> build_ac s l )) + | TA (s,l) -> + let symbols' = List.filter (fun x -> x <> s) symbols in + ( + (* first the other symbols, and then the more simple case of + this particular symbol *) + a_nondet_middle l >> + (fun (l,m,r) -> + (* meant to break the symmetry *) + if (l = [] && r = []) + then fail () + else + ( + match m with + | [p] -> + positions p >>| conts hole symbols' p + | _ -> + mk_TA' s m >> conts hole symbols' + ) >> build_a s l r )) + >>| + ( + if List.mem s symbols then + begin match l with + | [a] -> assert false + | [a;b] -> build_a s [a] [b] (hole) + | _ -> + (* on ne construit que les elements interieurs, + d'ou la disymetrie *) + (Positions.a l >> + (fun (left,p,right) -> + if left = [] then fail () else + (build_a s left right (Dot (s,hole,t_of_term p))))) + end + else fail () + ) + + | TSym (s,l) -> + tri_fold (fun acc l t r -> + ((positions t) >> + (fun (p) -> + let l = List.map t_of_term l in + let r = List.map t_of_term r in + return (Sym (s, Array.of_list (List.rev_append l (p::r)))) )) + >>| + ( + conts hole symbols t >> + (fun (p) -> + let l = List.map t_of_term l in + let r = List.map t_of_term r in + return (Sym (s, Array.of_list (List.rev_append l (p::r)))) ) + ) + >>| acc + ) l (fail()) + | TVar x -> assert false + | TUnit x when x = unit -> return (hole) + | TUnit x as t -> conts hole symbols t + in + (positions t + >>| + (match t with + | TSym _ -> conts hole symbols t + | TAC (s,l) -> conts hole symbols t + | TA (s,l) -> conts hole symbols t + | _ -> fail()) + ) + >> fun (p) -> return (Terms.size p,p,return Subst.empty) + + + + + (************) + (* Matching *) + (************) + + + +(** {!matching} is the generic matching judgement. Each time a + non-deterministic split is made, we have to come back to this one. + + {!matchingSym} is used to match two applications that have the + same (free) head-symbol. + + {!matchingAC} is used to match two sums (with the subtlety that + [x+y] matches [f a] which is a function application or [a*b] which + is a product). + + {!matchingA} is used to match two products (with the subtlety that + [x*y] matches [f a] which is a function application, or [a+b] + which is a sum). + + +*) + +let matching : Subst.t -> nf_term -> nf_term -> Subst.t Search.m= + let rec matching env (p : nf_term) (t: nf_term) : Subst.t Search.m= + match p with + | TAC (s,l) -> + let l = linear l in + matchingAC env s l (get_AC s t) + | TA (s,l) -> + matchingA env s l (get_A s t) + | TSym (s,l) -> + (get_Sym s t) + >> (fun t -> matchingSym env l t) + | TVar x -> + begin match Subst.find env x with + | None -> return (Subst.add env x t) + | Some v -> if nf_equal v t then return env else fail () + end + | TUnit s -> + if nf_equal p t then return env else fail () + and + matchingAC (env : Subst.t) (current : symbol) (l : nf_term list) (t : (nf_term *int) list) = + match l with + | TSym (s,v):: q -> + pick_sym s t + >> (fun (v',t') -> + matchingSym env v v' + >> (fun env -> matchingAC env current q t')) + + | TAC (s,v)::q when s = current -> + assert false + | TVar x:: q -> (* This is an optimization *) + begin match Subst.find env x with + | None -> + (var_ac_nondet_split current env x t) + >> (fun (env,residual) -> matchingAC env current q residual) + | Some v -> + (factor_AC current v t) + >> (fun residual -> matchingAC env current q residual) + end + | TUnit s as v :: q -> (* this is an optimization *) + (factor_AC current v t) >> + (fun residual -> matchingAC env current q residual) + | h :: q ->(* PAC =/= curent or PA or unit for this symbol*) + (ac_nondet_split current t) + >> + (fun (t,right) -> + matching env h t + >> + ( + fun env -> + matchingAC env current q right + ) + ) + | [] -> if t = [] then return env else fail () + and + matchingA (env : Subst.t) (current : symbol) (l : nf_term list) (t : nf_term list) = + match l with + | TSym (s,v) :: l -> + begin match t with + | TSym (s',v') :: r when s = s' -> + (matchingSym env v v') + >> (fun env -> matchingA env current l r) + | _ -> fail () + end + | TA (s,v) :: l when s = current -> + assert false + | TVar x :: l -> + begin match Subst.find env x with + | None -> + debug (Printf.sprintf "var %i (%s)" x + (let b = Buffer.create 21 in List.iter (fun t -> Buffer.add_string b ( Terms.sprint_nf_term t)) t; Buffer.contents b )); + var_a_nondet_split env current x t + >> (fun (env,residual)-> debug (Printf.sprintf "pl %i %i" x(List.length residual)); matchingA env current l residual) + | Some v -> + (left_factor current v t) + >> (fun residual -> matchingA env current l residual) + end + | TUnit s as v :: q -> (* this is an optimization *) + (left_factor current v t) >> + (fun residual -> matchingA env current q residual) + | h :: l -> + a_nondet_split current t + >> (fun (t,r) -> + matching env h t + >> (fun env -> matchingA env current l r) + ) + | [] -> if t = [] then return env else fail () + and + matchingSym (env : Subst.t) (l : nf_term list) (t : nf_term list) = + List.fold_left2 + (fun acc p t -> acc >> (fun env -> matching env p t)) + (return env) + l + t + + in + fun env l r -> + let _ = debug (Printf.sprintf "pattern :%s\nterm :%s\n%!" (Terms.sprint_nf_term l) (Terms.sprint_nf_term r)) in + let m = matching env l r in + let _ = debug (Printf.sprintf "count %i" (Search.count m)) in + m + + +(** [unitifiable p : Subst.t m] *) +let unitifiable p : (symbol * Subst.t m) m = + let m = List.fold_left + (fun acc (_,unit) -> acc >>| + let m = matching Subst.empty p (mk_TUnit unit) in + if Search.is_empty m then + fail () + else + begin + return (unit,m) + end + ) (fail ()) units + in + m +;; + +let nullifiable p = + let nullable = not strict in + let has_unit s = + try let _ = get_unit units s in true + with NoUnit -> false + in + let rec aux = function + | TA (s,l) -> has_unit s && List.for_all (aux) l + | TAC (s,l) -> has_unit s && List.for_all (fun (x,n) -> aux x) l + | TSym _ -> false + | TVar _ -> nullable + | TUnit _ -> true + in aux p + +let unit_warning p ~nullif ~unitif = + assert ((Search.is_empty unitif) || nullif); + if not (Search.is_empty unitif) + then + begin + Feedback.msg_warning + (Pp.str + "[aac_tactics] This pattern can be instantiated to match units, some solutions can be missing"); + end + +;; + + + + +(***********) +(* Subterm *) +(***********) + + + +(** [subterm] solves a sub-term pattern matching. + + This function is more high-level than {!matcher}, thus takes {!t} + as arguments rather than terms in normal form {!nf_term}. + + We use three mutually recursive functions {!subterm}, + {!subterm_AC}, {!subterm_A} to find the matching subterm, making + non-deterministic choices to split the term into a context and an + intersting sub-term. Intuitively, the only case in which we have to + go in depth is when we are left with a sub-term that is atomic. + + Indeed, rewriting [H: b = c |- a+b+a = a+a+c], we do not want to + find recursively the sub-terms of [a+b] and [b+a], since they will + overlap with the sub-terms of [a+b+a]. + + We rebuild the context on the fly, leaving the variables in the + pattern uninstantiated. We do so in order to allow interaction + with the user, to choose the env. + + Strange patterms like x*y*x can be instantiated by nothing, inside + a product. Therefore, we need to check that all the term is not + going into the context. With proper support for interaction with + the user, we should lift these tests. However, at the moment, they + serve as heuristics to return "interesting" matchings +*) + + let return_non_empty raw_p m = + if is_empty m + then + fail () + else + return (raw_p ,m) + + let subterm (raw_p:t) (raw_t:t): (int * t * Subst.t m) m= + let _ = debug (String.make 40 '=') in + let p = term_of_t units raw_p in + let t = term_of_t units raw_t in + let nullif = nullifiable p in + let unitif = unitifiable p in + let _ = unit_warning p ~nullif ~unitif in + let _ = debug (Printf.sprintf "%s" (Terms.sprint_nf_term p)) in + let _ = debug (Printf.sprintf "%s" (Terms.sprint_nf_term t)) in + let filter_right current right (p,m) = + if right = [] + then return (p,m) + else + mk_TAC' current right >> + fun t -> return (Plus (current,p,t_of_term t),m) + in + let filter_middle current left right (p,m) = + let right_pat p = + if right = [] + then return p + else + mk_TA' current right >> + fun t -> return (Dot (current,p,t_of_term t)) + in + let left_pat p= + if left = [] + then return p + else + mk_TA' current left >> + fun t -> return (Dot (current,t_of_term t,p)) + in + right_pat p >> left_pat >> (fun p -> return (p,m)) + in + let rec subterm (t:nf_term) : (t * Subst.t m) m= + match t with + | TAC (s,l) -> + ((ac_nondet_split_raw l) >> + (fun (left,right) -> + (subterm_AC s left) >> (filter_right s right) + )) + | TA (s,l) -> + (a_nondet_middle l) >> + (fun (left, middle, right) -> + (subterm_A s middle) >> + (filter_middle s left right) + ) + | TSym (s, l) -> + let init = + return_non_empty raw_p (matching Subst.empty p t) + in + tri_fold (fun acc l t r -> + ((subterm t) >> + (fun (p,m) -> + let l = List.map t_of_term l in + let r = List.map t_of_term r in + let p = Sym (s, Array.of_list (List.rev_append l (p::r))) in + return (p,m) + )) >>| acc + ) l init + | TVar x -> assert false + (* this case is superseded by the later disjunction *) + | TUnit s -> fail () + + and subterm_AC s tl = + match tl with + [x,1] -> subterm x + | _ -> + mk_TAC' s tl >> fun t -> + return_non_empty raw_p (matching Subst.empty p t) + and subterm_A s tl = + match tl with + [x] -> subterm x + | _ -> + mk_TA' s tl >> + fun t -> + return_non_empty raw_p (matching Subst.empty p t) + in + match p with + | TUnit unit -> unit_subterm t unit raw_p + | _ when not (Search.is_empty unitif) -> + let unit_matches = + Search.fold + (fun (unit,inst) acc -> + Search.fold + (fun subst acc' -> + let m = unit_subterm t unit (Subst.instantiate subst raw_p) + in + m>>| acc' + ) + inst + acc + ) + unitif + (fail ()) + in + let nullifies (t : Subst.t) = + List.for_all (fun (_,x) -> + List.exists (fun (_,y) -> Unit y = x ) units + ) (Subst.to_list t) + in + let nonunit_matches = + subterm t >> + ( + fun (p,m) -> + let m = Search.filter (fun subst -> not( nullifies subst)) m in + if Search.is_empty m + then fail () + else return (Terms.size p,p,m) + ) + in + unit_matches >>| nonunit_matches + + | _ -> (subterm t >> fun (p,m) -> return (Terms.size p,p,m)) + + + end + + +(* The functions we export, handlers for the previous ones. Some debug + information also *) +let subterm ?(strict = false) units raw t = + let module M = M (struct + let is_ac = units.is_ac + let units = units.unit_for + let strict = strict + end) in + let sols = time (M.subterm raw) t "%fs spent in subterm (including matching)\n" in + debug + (Printf.sprintf "%i possible solution(s)\n" + (Search.fold (fun (_,_,envm) acc -> count envm + acc) sols 0)); + sols + + +let matcher ?(strict = false) units p t = + let module M = M (struct + let is_ac = units.is_ac + let units = units.unit_for + let strict = false + end) in + let units = units.unit_for in + let sols = time + (fun (p,t) -> + let p = (Terms.term_of_t units p) in + let t = (Terms.term_of_t units t) in + M.matching Subst.empty p t) (p,t) + "%fs spent in the matcher\n" + in + debug (Printf.sprintf "%i solutions\n" (count sols)); + sols + diff --git a/src/matcher.mli b/src/matcher.mli new file mode 100644 index 0000000..a6b6f46 --- /dev/null +++ b/src/matcher.mli @@ -0,0 +1,189 @@ +(***************************************************************************) +(* This is part of aac_tactics, it is distributed under the terms of the *) +(* GNU Lesser General Public License version 3 *) +(* (see file LICENSE for more details) *) +(* *) +(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) +(***************************************************************************) + +(** Standalone module containing the algorithm for matching modulo + associativity and associativity and commutativity + (AAC). Additionnaly, some A or AC operators can have units (U). + + This module could be reused outside of the Coq plugin. + + Matching a pattern [p] against a term [t] modulo AACU boils down + to finding a substitution [env] such that the pattern [p] + instantiated with [env] is equal to [t] modulo AACU. + + We proceed by structural decomposition of the pattern, trying all + possible non-deterministic splittings of the subject, when needed. The + function {!matcher} is limited to top-level matching, that is, the + subject must make a perfect match against the pattern ([x+x] does + not match [a+a+b] ). + + We use a search monad {!Search_monad} to perform non-deterministic + choices in an almost transparent way. + + We also provide a function {!subterm} for finding a match that is + a subterm of the subject modulo AACU. In particular, this function + gives a solution to the aforementioned case ([x+x] against + [a+b+a]). + + On a slightly more involved level : + - it must be noted that we allow several AC/A operators to share + the same units, but that a given AC/A operator can have at most + one unit. + + - if the pattern does not contain "hard" symbols (like constants, + function symbols, AC or A symbols without units), there can be + infinitely many subterms such that the pattern matches: it is + possible to build "subterms" modulo AAC and U that make the size + of the term increase (by making neutral elements appear in a + layered fashion). Hence, in this case, a warning is issued, and + some solutions are omitted. + +*) + +(** {2 Utility functions} *) + +type symbol = int +type var = int + +(** Relationship between units and operators. This is a sparse + representation of a matrix of couples [(op,unit)] where [op] is + the index of the operation, and [unit] the index of the relevant + unit. We make the assumption that any operation has 0 or 1 unit, + and that operations can share a unit). *) + +type units =(symbol * symbol) list (* from AC/A symbols to the unit *) +type ext_units = + { + unit_for : units; (* from AC/A symbols to the unit *) + is_ac : (symbol * bool) list + } + + +(** The arguments of sums (or AC operators) are represented using finite multisets. + (Typically, [a+b+a] corresponds to [2.a+b], i.e. [Sum[a,2;b,1]]) *) +type 'a mset = ('a * int) list + +(** [linear] expands a multiset into a simple list *) +val linear : 'a mset -> 'a list + +(** Representations of expressions + + The module {!Terms} defines two different types for expressions. + - a public type {!Terms.t} that represents abstract syntax trees + of expressions with binary associative and commutative operators + - a private type {!Terms.nf_term}, corresponding to a canonical + representation for the above terms modulo AACU. The construction + functions on this type ensure that these terms are in normal form + (that is, no sum can appear as a subterm of the same sum, no + trailing units, lists corresponding to multisets are sorted, + etc...). + +*) +module Terms : +sig + + (** {2 Abstract syntax tree of terms and patterns} + + We represent both terms and patterns using the following datatype. + + Values of type [symbol] are used to index symbols. Typically, + given two associative operations [(^)] and [( * )], and two + morphisms [f] and [g], the term [f (a^b) (a*g b)] is represented + by the following value + [Sym(0,[| Dot(1, Sym(2,[||]), Sym(3,[||])); + Dot(4, Sym(2,[||]), Sym(5,[|Sym(3,[||])|])) |])] + where the implicit symbol environment associates + [f] to [0], [(^)] to [1], [a] to [2], [b] to [3], [( * )] to [4], and [g] to [5], + + Accordingly, the following value, that contains "variables" + [Sym(0,[| Dot(1, Var x, Unit (1); Dot(4, Var x, + Sym(5,[|Sym(3,[||])|])) |])] represents the pattern [forall x, f + (x^1) (x*g b)]. The relationship between [1] and [( * )] is only + mentionned in the units table. *) + + type t = + Dot of (symbol * t * t) + | Plus of (symbol * t * t) + | Sym of (symbol * t array) + | Var of var + | Unit of symbol + + (** Test for equality of terms modulo AACU (relies on the following + canonical representation of terms). Note than two different + units of a same operator are not considered equal. *) + val equal_aac : units -> t -> t -> bool + + + (** {2 Normalised terms (canonical representation) } + + A term in normal form is the canonical representative of the + equivalence class of all the terms that are equal modulo AACU. + This representation is only used internally; it is exported here + for the sake of completeness *) + + type nf_term + + (** {3 Comparisons} *) + + val nf_term_compare : nf_term -> nf_term -> int + val nf_equal : nf_term -> nf_term -> bool + + (** {3 Printing function} *) + + val sprint_nf_term : nf_term -> string + + (** {3 Conversion functions} *) + + (** we have the following property: [a] and [b] are equal modulo AACU + iif [nf_equal (term_of_t a) (term_of_t b) = true] *) + val term_of_t : units -> t -> nf_term + val t_of_term : nf_term -> t + +end + + +(** Substitutions (or environments) + + The module {!Subst} contains infrastructure to deal with + substitutions, i.e., functions from variables to terms. Only a + restricted subsets of these functions need to be exported. + + As expected, a particular substitution can be used to + instantiate a pattern. +*) +module Subst : +sig + type t + val sprint : t -> string + val instantiate : t -> Terms.t-> Terms.t + val to_list : t -> (var*Terms.t) list +end + + +(** {2 Main functions exported by this module} *) + +(** [matcher p t] computes the set of solutions to the given top-level + matching problem ([p] is the pattern, [t] is the term). If the + [strict] flag is set, solutions where units are used to + instantiate some variables are excluded, unless this unit appears + directly under a function symbol (e.g., f(x) still matches f(1), + while x+x+y does not match a+b+c, since this would require to + assign 1 to x). +*) +val matcher : ?strict:bool -> ext_units -> Terms.t -> Terms.t -> Subst.t Search_monad.m + +(** [subterm p t] computes a set of solutions to the given + subterm-matching problem. + + Return a collection of possible solutions (each with the + associated depth, the context, and the solutions of the matching + problem). The context is actually a {!Terms.t} where the variables + are yet to be instantiated by one of the associated substitutions +*) +val subterm : ?strict:bool -> ext_units -> Terms.t -> Terms.t -> (int * Terms.t * Subst.t Search_monad.m) Search_monad.m + diff --git a/src/print.ml b/src/print.ml new file mode 100644 index 0000000..6800200 --- /dev/null +++ b/src/print.ml @@ -0,0 +1,104 @@ +(***************************************************************************) +(* This is part of aac_tactics, it is distributed under the terms of the *) +(* GNU Lesser General Public License version 3 *) +(* (see file LICENSE for more details) *) +(* *) +(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) +(***************************************************************************) + +(* A very basic way to interact with the envs, to choose a possible + solution *) +open Pp +open Matcher +open Context.Rel.Declaration +open Names + +type named_env = (Name.t * Terms.t) list + + + +(** {pp_env} prints a substitution mapping names to terms, using the +provided printer *) +let pp_env pt : named_env -> Pp.t = fun env -> + List.fold_left + (fun acc (v,t) -> + begin match v with + | Names.Name s -> str (Printf.sprintf "%s: " (Id.to_string s)) + | Names.Anonymous -> str ("_") + end + ++ pt t ++ str "; " ++ acc + ) + (str "") + env + +(** {pp_envm} prints a collection of possible environments, and number +them. This number must remain compatible with the parameters given to +{!aac_rewrite} *) +let pp_envm pt : named_env Search_monad.m -> Pp.t = fun m -> + let _,s = + Search_monad.fold + (fun env (n,acc) -> + n+1, h 0 (str (Printf.sprintf "%i:\t[" n) ++pp_env pt env ++ str "]") ++ fnl () :: acc + ) m (0,[]) + in + List.fold_left (fun acc s -> s ++ acc) (str "") (s) + +let trivial_substitution envm = + match Search_monad.choose envm with + | None -> true (* assert false *) + | Some l -> l=[] + +(** {pp_all} prints a collection of possible contexts and related +possibles substitutions, giving a number to each. This number must +remain compatible with the parameters of {!aac_rewrite} *) +let pp_all pt : (int * Terms.t * named_env Search_monad.m) Search_monad.m -> Pp.t = fun m -> + let _,s = Search_monad.fold + (fun (size,context,envm) (n,acc) -> + let s = str (Printf.sprintf "occurrence %i: transitivity through " n) in + let s = s ++ pt context ++ str "\n" in + let s = if trivial_substitution envm then s else + s ++ str (Printf.sprintf "%i possible(s) substitution(s)" (Search_monad.count envm) ) + ++ fnl () ++ pp_envm pt envm + in + n+1, s::acc + ) m (0,[]) in + List.fold_left (fun acc s -> s ++ str "\n" ++ acc) (str "") (s) + +(** The main printing function. {!print} uses the debruijn_env the +rename the variables, and rebuilds raw Coq terms (for the context, and +the terms in the environment). In order to do so, it requires the +information gathered by the {!Theory.Trans} module.*) +let print rlt ir m (context : EConstr.rel_context) goal = + if Search_monad.count m = 0 + then + ( + Tacticals.tclFAIL 0 (Pp.str "No subterm modulo AC") goal + ) + else + let _ = Feedback.msg_notice (Pp.str "All solutions:") in + let m = Search_monad.(>>) m + (fun (i,t,envm) -> + let envm = Search_monad.(>>) envm ( fun env -> + let l = Matcher.Subst.to_list env in + let l = List.sort (fun (n,_) (n',_) -> Pervasives.compare n n') l in + let l = + List.map (fun (v,t) -> + get_name (Context.Rel.lookup v context), t + ) l + in + Search_monad.return l + ) + in + Search_monad.return (i,t,envm) + ) + in + let m = Search_monad.sort (fun (x,_,_) (y,_,_) -> x - y) m in + let env = Tacmach.pf_env goal in + let sigma = Tacmach.project goal in + let _ = Feedback.msg_notice + (pp_all + (fun t -> Printer.pr_letype_env env sigma (Theory.Trans.raw_constr_of_t ir rlt context t)) m + ) + in + Tacticals.tclIDTAC goal + diff --git a/src/print.mli b/src/print.mli new file mode 100644 index 0000000..5ca6e27 --- /dev/null +++ b/src/print.mli @@ -0,0 +1,23 @@ +(***************************************************************************) +(* This is part of aac_tactics, it is distributed under the terms of the *) +(* GNU Lesser General Public License version 3 *) +(* (see file LICENSE for more details) *) +(* *) +(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) +(***************************************************************************) + +(** Pretty printing functions we use for the aac_instances + tactic. *) + + +(** The main printing function. {!print} uses the rel-context + to rename the variables, and rebuilds raw Coq terms (for the given + context, and the terms in the environment). In order to do so, it + requires the information gathered by the {!Theory.Trans} module.*) +val print : + Coq.Relation.t -> + Theory.Trans.ir -> + (int * Matcher.Terms.t * Matcher.Subst.t Search_monad.m) Search_monad.m -> + EConstr.rel_context -> + Proof_type.tactic + diff --git a/src/search_monad.ml b/src/search_monad.ml new file mode 100644 index 0000000..09a6455 --- /dev/null +++ b/src/search_monad.ml @@ -0,0 +1,70 @@ +(***************************************************************************) +(* This is part of aac_tactics, it is distributed under the terms of the *) +(* GNU Lesser General Public License version 3 *) +(* (see file LICENSE for more details) *) +(* *) +(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) +(***************************************************************************) + +type 'a m = | F of 'a + | N of 'a m list + +let fold (f : 'a -> 'b -> 'b) (m : 'a m) (acc : 'b) = + let rec aux acc = function + F x -> f x acc + | N l -> + (List.fold_left (fun acc x -> + match x with + | (N []) -> acc + | x -> aux acc x + ) acc l) + in + aux acc m + + +let rec (>>) : 'a m -> ('a -> 'b m) -> 'b m = + fun m f -> + match m with + | F x -> f x + | N l -> + N (List.fold_left (fun acc x -> + match x with + | (N []) -> acc + | x -> (x >> f)::acc + ) [] l) + +let (>>|) (m : 'a m) (n :'a m) : 'a m = match (m,n) with + | N [],_ -> n + | _,N [] -> m + | F x, N l -> N (F x::l) + | N l, F x -> N (F x::l) + | x,y -> N [x;y] + +let return : 'a -> 'a m = fun x -> F x +let fail : unit -> 'a m = fun () -> N [] + +let sprint f m = + fold (fun x acc -> Printf.sprintf "%s\n%s" acc (f x)) m "" +let rec count = function + | F _ -> 1 + | N l -> List.fold_left (fun acc x -> acc+count x) 0 l + +let opt_comb f x y = match x with None -> f y | _ -> x + +let rec choose = function + | F x -> Some x + | N l -> List.fold_left (fun acc x -> + opt_comb choose acc x + ) None l + +let is_empty = fun x -> choose x = None + +let to_list m = (fold (fun x acc -> x::acc) m []) + +let sort f m = + N (List.map (fun x -> F x) (List.sort f (to_list m))) + +(* preserve the structure of the heap *) +let filter f m = + fold (fun x acc -> (if f x then return x else fail ()) >>| acc) m (N []) + diff --git a/src/search_monad.mli b/src/search_monad.mli new file mode 100644 index 0000000..7e2a910 --- /dev/null +++ b/src/search_monad.mli @@ -0,0 +1,42 @@ +(***************************************************************************) +(* This is part of aac_tactics, it is distributed under the terms of the *) +(* GNU Lesser General Public License version 3 *) +(* (see file LICENSE for more details) *) +(* *) +(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) +(***************************************************************************) + +(** Search monad that allows to express non-deterministic algorithms + in a legible maner, or programs that solve combinatorial problems. + + @see the + inspiration of this module +*) + +(** A data type that represent a collection of ['a] *) +type 'a m + + (** {2 Monadic operations} *) + +(** bind and return *) +val ( >> ) : 'a m -> ('a -> 'b m) -> 'b m +val return : 'a -> 'a m + +(** non-deterministic choice *) +val ( >>| ) : 'a m -> 'a m -> 'a m + +(** failure *) +val fail : unit -> 'a m + +(** folding through the collection *) +val fold : ('a -> 'b -> 'b) -> 'a m -> 'b -> 'b + +(** {2 Derived facilities } *) + +val sprint : ('a -> string) -> 'a m -> string +val count : 'a m -> int +val choose : 'a m -> 'a option +val to_list : 'a m -> 'a list +val sort : ('a -> 'a -> int) -> 'a m -> 'a m +val is_empty: 'a m -> bool +val filter : ('a -> bool) -> 'a m -> 'a m diff --git a/src/theory.ml b/src/theory.ml new file mode 100644 index 0000000..7871fe4 --- /dev/null +++ b/src/theory.ml @@ -0,0 +1,1153 @@ +(***************************************************************************) +(* This is part of aac_tactics, it is distributed under the terms of the *) +(* GNU Lesser General Public License version 3 *) +(* (see file LICENSE for more details) *) +(* *) +(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) +(***************************************************************************) + +(** Constr from the theory of this particular development + + The inner-working of this module is highly correlated with the + particular structure of {b AAC_rewrite.v}, therefore, it should + be of little interest to most readers. +*) +open EConstr + +module Control = struct + let printing = true + let debug = false + let time = false +end + +module Debug = Helper.Debug (Control) +open Debug + +(** {1 HMap : Specialized Hashtables based on constr} *) + + + (* TODO module HMap = Hashtbl, du coup ? *) +module HMap = Hashtbl.Make(Constr) + +let ac_theory_path = ["AAC_tactics"; "AAC"] +let ac_util_path = ["AAC_tactics"; "Utils"] + +module Stubs = struct + let path = ac_theory_path@["Internal"] + + (** The constants from the inductive type *) + let _Tty = lazy (Coq.init_constant path "T") + let vTty = lazy (Coq.init_constant path "vT") + + let rsum = lazy (Coq.init_constant path "sum") + let rprd = lazy (Coq.init_constant path "prd") + let rsym = lazy (Coq.init_constant path "sym") + let runit = lazy (Coq.init_constant path "unit") + + let vnil = lazy (Coq.init_constant path "vnil") + let vcons = lazy (Coq.init_constant path "vcons") + let eval = lazy (Coq.init_constant path "eval") + + + let decide_thm = lazy (Coq.init_constant path "decide") + let lift_normalise_thm = lazy (Coq.init_constant path "lift_normalise") + + let lift = + lazy (Coq.init_constant ac_theory_path "AAC_lift") + let lift_proj_equivalence= + lazy (Coq.init_constant ac_theory_path "aac_lift_equivalence") + let lift_transitivity_left = + lazy(Coq.init_constant ac_theory_path "lift_transitivity_left") + let lift_transitivity_right = + lazy(Coq.init_constant ac_theory_path "lift_transitivity_right") + let lift_reflexivity = + lazy(Coq.init_constant ac_theory_path "lift_reflexivity") +end + +module Classes = struct + module Associative = struct + let path = ac_theory_path + let typ = lazy (Coq.init_constant path "Associative") + let ty (rlt : Coq.Relation.t) (value : constr) = + mkApp (Lazy.force typ, [| rlt.Coq.Relation.carrier; + rlt.Coq.Relation.r; + value + |] ) + let infer goal rlt value = + let ty = ty rlt value in + Coq.resolve_one_typeclass goal ty + end + + module Commutative = struct + let path = ac_theory_path + let typ = lazy (Coq.init_constant path "Commutative") + let ty (rlt : Coq.Relation.t) (value : constr) = + mkApp (Lazy.force typ, [| rlt.Coq.Relation.carrier; + rlt.Coq.Relation.r; + value + |] ) + + end + + module Proper = struct + let path = ac_theory_path @ ["Internal";"Sym"] + let typeof = lazy (Coq.init_constant path "type_of") + let relof = lazy (Coq.init_constant path "rel_of") + let mk_typeof : Coq.Relation.t -> int -> constr = fun rlt n -> + let x = rlt.Coq.Relation.carrier in + mkApp (Lazy.force typeof, [| x ; Coq.Nat.of_int n |]) + let mk_relof : Coq.Relation.t -> int -> constr = fun rlt n -> + let (x,r) = Coq.Relation.split rlt in + mkApp (Lazy.force relof, [| x;r ; Coq.Nat.of_int n |]) + + let ty rlt op ar = + let typeof = mk_typeof rlt ar in + let relof = mk_relof rlt ar in + Coq.Classes.mk_morphism typeof relof op + let infer goal rlt op ar = + let ty = ty rlt op ar in + Coq.resolve_one_typeclass goal ty + end + + module Unit = struct + let path = ac_theory_path + let typ = lazy (Coq.init_constant path "Unit") + let ty (rlt : Coq.Relation.t) (value : constr) (unit : constr)= + mkApp (Lazy.force typ, [| rlt.Coq.Relation.carrier; + rlt.Coq.Relation.r; + value; + unit + |] ) + end + +end + +(* Non empty lists *) +module NEList = struct + let path = ac_util_path + let typ = lazy (Coq.init_constant path "list") + let nil = lazy (Coq.init_constant path "nil") + let cons = lazy (Coq.init_constant path "cons") + let cons ty h t = + mkApp (Lazy.force cons, [| ty; h ; t |]) + let nil ty x = + (mkApp (Lazy.force nil, [| ty ; x|])) + let rec of_list ty = function + | [] -> invalid_arg "NELIST" + | [x] -> nil ty x + | t::q -> cons ty t (of_list ty q) + + let type_of_list ty = + mkApp (Lazy.force typ, [|ty|]) +end + +(** a [mset] is a ('a * pos) list *) +let mk_mset ty (l : (constr * int) list) = + let pos = Lazy.force Coq.Pos.typ in + let pair (x : constr) (ar : int) = + Coq.Pair.of_pair ty pos (x, Coq.Pos.of_int ar) + in + let pair_ty = Coq.lapp Coq.Pair.typ [| ty ; pos|] in + let rec aux = function + | [ ] -> assert false + | [x,ar] -> NEList.nil pair_ty (pair x ar) + | (t,ar)::q -> NEList.cons pair_ty (pair t ar) (aux q) + in + aux l + +module Sigma = struct + let sigma = lazy (Coq.init_constant ac_theory_path "sigma") + let sigma_empty = lazy (Coq.init_constant ac_theory_path "sigma_empty") + let sigma_add = lazy (Coq.init_constant ac_theory_path "sigma_add") + let sigma_get = lazy (Coq.init_constant ac_theory_path "sigma_get") + + let add ty n x map = + mkApp (Lazy.force sigma_add,[|ty; n; x ; map|]) + let empty ty = + mkApp (Lazy.force sigma_empty,[|ty |]) + let typ ty = + mkApp (Lazy.force sigma, [|ty|]) + + let to_fun ty null map = + mkApp (Lazy.force sigma_get, [|ty;null;map|]) + + let of_list ty null l = + match l with + | (_,t)::q -> + let map = + List.fold_left + (fun acc (i,t) -> + assert (i > 0); + add ty (Coq.Pos.of_int i) ( t) acc) + (empty ty) + q + in to_fun ty (t) map + | [] -> debug "of_list empty" ; to_fun ty (null) (empty ty) + + +end + + +module Sym = struct + type pack = {ar: Constr.t; value: Constr.t ; morph: Constr.t} + let path = ac_theory_path @ ["Internal";"Sym"] + let typ = lazy (Coq.init_constant path "pack") + let mkPack = lazy (Coq.init_constant path "mkPack") + let value = lazy (Coq.init_constant path "value") + let null = lazy (Coq.init_constant path "null") + let mk_pack (rlt: Coq.Relation.t) s = + let (x,r) = Coq.Relation.split rlt in + mkApp (Lazy.force mkPack, [|x;r; EConstr.of_constr s.ar;EConstr.of_constr s.value;EConstr.of_constr s.morph|]) + let null rlt = + let x = rlt.Coq.Relation.carrier in + let r = rlt.Coq.Relation.r in + mkApp (Lazy.force null, [| x;r;|]) + + let mk_ty : Coq.Relation.t -> constr = fun rlt -> + let (x,r) = Coq.Relation.split rlt in + mkApp (Lazy.force typ, [| x; r|] ) +end + +module Bin =struct + type pack = {value : Constr.t; + compat : Constr.t; + assoc : Constr.t; + comm : Constr.t option; + } + + let path = ac_theory_path @ ["Internal";"Bin"] + let typ = lazy (Coq.init_constant path "pack") + let mkPack = lazy (Coq.init_constant path "mk_pack") + + let mk_pack: Coq.Relation.t -> pack -> constr = fun (rlt) s -> + let (x,r) = Coq.Relation.split rlt in + let comm_ty = Classes.Commutative.ty rlt (EConstr.of_constr s.value) in + mkApp (Lazy.force mkPack , [| x ; r; + EConstr.of_constr s.value; + EConstr.of_constr s.compat ; + EConstr.of_constr s.assoc; + Coq.Option.of_option comm_ty (Option.map EConstr.of_constr s.comm) + |]) + let mk_ty : Coq.Relation.t -> constr = fun rlt -> + let (x,r) = Coq.Relation.split rlt in + mkApp (Lazy.force typ, [| x; r|] ) +end + +module Unit = struct + let path = ac_theory_path @ ["Internal"] + let unit_of_ty = lazy (Coq.init_constant path "unit_of") + let unit_pack_ty = lazy (Coq.init_constant path "unit_pack") + let mk_unit_of = lazy (Coq.init_constant path "mk_unit_for") + let mk_unit_pack = lazy (Coq.init_constant path "mk_unit_pack") + + type unit_of = + { + uf_u : Constr.t; (* u *) + uf_idx : Constr.t; + uf_desc : Constr.t; (* Unit R (e_bin uf_idx) u *) + } + + type pack = { + u_value : constr; (* X *) + u_desc : (unit_of) list (* unit_of u_value *) + } + + let ty_unit_of rlt e_bin u = + let (x,r) = Coq.Relation.split rlt in + mkApp ( Lazy.force unit_of_ty, [| x; r; e_bin; u |]) + + let ty_unit_pack rlt e_bin = + let (x,r) = Coq.Relation.split rlt in + mkApp (Lazy.force unit_pack_ty, [| x; r; e_bin |]) + + let mk_unit_of rlt e_bin u unit_of = + let (x,r) = Coq.Relation.split rlt in + mkApp (Lazy.force mk_unit_of , [| x; + r; + e_bin ; + u; + EConstr.of_constr unit_of.uf_idx; + EConstr.of_constr unit_of.uf_desc + |]) + + let mk_pack rlt e_bin pack : constr = + let (x,r) = Coq.Relation.split rlt in + let ty = ty_unit_of rlt e_bin pack.u_value in + let mk_unit_of = mk_unit_of rlt e_bin pack.u_value in + let u_desc =Coq.List.of_list ( ty ) (List.map mk_unit_of pack.u_desc) in + mkApp (Lazy.force mk_unit_pack, [|x;r; + e_bin ; + pack.u_value; + u_desc + |]) + + let default x : pack = + { u_value = x; + u_desc = [] + } + +end + +let anomaly msg = + CErrors.anomaly ~label:"aac_tactics" (Pp.str msg) + +let user_error msg = + CErrors.user_err Pp.(strbrk "aac_tactics: " ++ msg) + +module Trans = struct + + (** {1 From Coq to Abstract Syntax Trees (AST)} + + This module provides facilities to interpret a Coq term with + arbitrary operators as an abstract syntax tree. Considering an + application, we try to infer instances of our classes. + + We consider that [A] operators are coarser than [AC] operators, + which in turn are coarser than raw morphisms. That means that + [List.append], of type [(A : Type) -> list A -> list A -> list + A] can be understood as an [A] operator. + + During this reification, we gather some informations that will + be used to rebuild Coq terms from AST ( type {!envs}) + + We use a main hash-table from [constr] to [pack], in order to + discriminate the various constructors. All these are mixed in + order to improve on the number of comparisons in the tables *) + + (* 'a * (unit, op_unit) option *) + type 'a with_unit = 'a * (Unit.unit_of) option + type pack = + (* used to infer the AC/A structure in the first pass {!Gather} *) + | Bin of Bin.pack with_unit + (* will only be used in the second pass : {!Parse}*) + | Sym of Sym.pack + | Unit of Constr.t (* to avoid confusion in bloom *) + + module PackHash = + struct + open Hashset.Combine + + type t = pack + + let eq_sym_pack p1 p2 = + let open Sym in + Constr.equal p1.ar p2.ar && + Constr.equal p1.value p2.value && + Constr.equal p1.morph p2.morph + + let hash_sym_pack p = + let open Sym in + combine3 (Constr.hash p.ar) (Constr.hash p.value) (Constr.hash p.morph) + + let eq_bin_pack p1 p2 = + let open Bin in + Constr.equal p1.value p2.value && + Constr.equal p1.compat p2.compat && + Constr.equal p1.assoc p2.assoc && + Option.equal Constr.equal p1.comm p2.comm + + let hash_bin_pack p = + let open Bin in + combine4 (Constr.hash p.value) (Constr.hash p.compat) + (Constr.hash p.assoc) (Option.hash Constr.hash p.comm) + + let eq_unit_of u1 u2 = + let open Unit in + Constr.equal u1.uf_u u2.uf_u && + Constr.equal u1.uf_idx u2.uf_idx && + Constr.equal u1.uf_desc u2.uf_desc + + let hash_unit_of u = + let open Unit in + combine3 (Constr.hash u.uf_u) (Constr.hash u.uf_idx) + (Constr.hash u.uf_desc) + + let equal p1 p2 = match p1, p2 with + | Bin (p1, o1), Bin (p2, o2) -> + eq_bin_pack p1 p2 && Option.equal eq_unit_of o1 o2 + | Sym p1, Sym p2 -> eq_sym_pack p1 p2 + | Unit c1, Unit c2 -> Constr.equal c1 c2 + | _ -> false + + let hash p = match p with + | Bin (p, o) -> + combinesmall 1 (combine (hash_bin_pack p) (Option.hash hash_unit_of o)) + | Sym p -> + combinesmall 2 (hash_sym_pack p) + | Unit c -> + combinesmall 3 (Constr.hash c) + + end + + module PackTable = Hashtbl.Make(PackHash) + + (** discr is a map from {!constr} to {!pack}. + + [None] means that it is not possible to instantiate this partial + application to an interesting class. + + [Some x] means that we found something in the past. This means + in particular that a single [constr] cannot be two things at the + same time. + + The field [bloom] allows to give a unique number to each class we + found. *) + type envs = + { + discr : (pack option) HMap.t ; + bloom : int PackTable.t; + bloom_back : (int, pack ) Hashtbl.t; + bloom_next : int ref; + } + + let empty_envs () = + { + discr = HMap.create 17; + bloom = PackTable.create 17; + bloom_back = Hashtbl.create 17; + bloom_next = ref 1; + } + + + + let add_bloom envs pack = + if PackTable.mem envs.bloom pack + then () + else + let x = ! (envs.bloom_next) in + PackTable.add envs.bloom pack x; + Hashtbl.add envs.bloom_back x pack; + incr (envs.bloom_next) + + let find_bloom envs pack = + try PackTable.find envs.bloom pack + with Not_found -> assert false + + (*********************************************) + (* (\* Gather the occurring AC/A symbols *\) *) + (*********************************************) + + (** This modules exhibit a function that memoize in the environment + all the AC/A operators as well as the morphism that occur. This + staging process allows us to prefer AC/A operators over raw + morphisms. Moreover, for each AC/A operators, we need to try to + infer units. Otherwise, we do not have [x * y * x <= a * a] since 1 + does not occur. + + But, do we also need to check whether constants are + units. Otherwise, we do not have the ability to rewrite [0 = a + + a] in [a = ...]*) + module Gather : sig + val gather : Coq.goal_sigma -> Coq.Relation.t -> envs -> constr -> Coq.goal_sigma + end + = struct + + let memoize envs t pack : unit = + begin + HMap.add envs.discr t (Some pack); + add_bloom envs pack; + match pack with + | Bin (_, None) | Sym _ -> () + | Bin (_, Some (unit_of)) -> + let unit = unit_of.Unit.uf_u in + HMap.add envs.discr unit (Some (Unit unit)); + add_bloom envs (Unit unit); + | Unit _ -> assert false + end + + + let get_unit (rlt : Coq.Relation.t) op goal : + (Coq.goal_sigma * constr * constr ) option= + let x = (rlt.Coq.Relation.carrier) in + let unit, goal = Coq.evar_unit goal x in + let ty =Classes.Unit.ty rlt op unit in + let result = + try + let t,goal = Coq.resolve_one_typeclass goal ty in + Some (goal,t,unit) + with Not_found -> None + in + match result with + | None -> None + | Some (goal,s,unit) -> + let unit = Coq.nf_evar goal unit in + Some (goal, unit, s) + + + + (** gives back the class and the operator *) + let is_bin (rlt: Coq.Relation.t) (op: constr) ( goal: Coq.goal_sigma) + : (Coq.goal_sigma * Bin.pack) option = + let assoc_ty = Classes.Associative.ty rlt op in + let comm_ty = Classes.Commutative.ty rlt op in + let proper_ty = Classes.Proper.ty rlt op 2 in + try + let proper , goal = Coq.resolve_one_typeclass goal proper_ty in + let assoc, goal = Coq.resolve_one_typeclass goal assoc_ty in + let comm , goal = + try + let comm, goal = Coq.resolve_one_typeclass goal comm_ty in + Some comm, goal + with Not_found -> + None, goal + in + let bin = + {Bin.value = EConstr.to_constr (Tacmach.project goal) op; + Bin.compat = EConstr.to_constr (Tacmach.project goal) proper; + Bin.assoc = EConstr.to_constr (Tacmach.project goal) assoc; + Bin.comm = Option.map (EConstr.to_constr (Tacmach.project goal)) comm } + in + Some (goal,bin) + with |Not_found -> None + + let is_bin (rlt : Coq.Relation.t) (op : constr) (goal : Coq.goal_sigma)= + match is_bin rlt op goal with + | None -> None + | Some (goal, bin_pack) -> + match get_unit rlt op goal with + | None -> Some (goal, Bin (bin_pack, None)) + | Some (gl, unit,s) -> + let unit_of = + { + Unit.uf_u = EConstr.to_constr (Tacmach.project goal) unit; + (* TRICK : this term is not well-typed by itself, + but it is okay the way we use it in the other + functions *) + Unit.uf_idx = EConstr.to_constr (Tacmach.project goal) op; + Unit.uf_desc = EConstr.to_constr (Tacmach.project goal) s; + } + in Some (gl,Bin (bin_pack, Some (unit_of))) + + + (** {is_morphism} try to infer the kind of operator we are + dealing with *) + let is_morphism goal (rlt : Coq.Relation.t) (papp : constr) (ar : int) : (Coq.goal_sigma * pack ) option = + let typeof = Classes.Proper.mk_typeof rlt ar in + let relof = Classes.Proper.mk_relof rlt ar in + let m = Coq.Classes.mk_morphism typeof relof papp in + try + let m,goal = Coq.resolve_one_typeclass goal m in + let pack = {Sym.ar = EConstr.to_constr (Tacmach.project goal) (Coq.Nat.of_int ar); + Sym.value= EConstr.to_constr (Tacmach.project goal) papp; + Sym.morph= EConstr.to_constr (Tacmach.project goal) m} in + Some (goal, Sym pack) + with + | Not_found -> None + + + (* [crop_app f [| a_0 ; ... ; a_n |]] + returns Some (f a_0 ... a_(n-2), [|a_(n-1); a_n |] ) + *) + let crop_app t ca : (constr * constr array) option= + let n = Array.length ca in + if n <= 1 + then None + else + let papp = mkApp (t, Array.sub ca 0 (n-2) ) in + let args = Array.sub ca (n-2) 2 in + Some (papp, args ) + + let fold goal (rlt : Coq.Relation.t) envs t ca cont = + let fold_morphism t ca = + let nb_params = Array.length ca in + let rec aux n = + assert (n < nb_params && 0 < nb_params ); + let papp = mkApp (t, Array.sub ca 0 n) in + match is_morphism goal rlt papp (nb_params - n) with + | None -> + (* here we have to memoize the failures *) + HMap.add envs.discr (EConstr.to_constr (Tacmach.project goal) papp) None; + if n < nb_params - 1 then aux (n+1) else goal + | Some (goal, pack) -> + let args = Array.sub ca (n) (nb_params -n)in + let goal = Array.fold_left cont goal args in + memoize envs (EConstr.to_constr (Tacmach.project goal) papp) pack; + goal + in + if nb_params = 0 then goal else aux 0 + in + let is_aac t = is_bin rlt t in + match crop_app t ca with + | None -> + fold_morphism t ca + | Some (papp, args) -> + begin match is_aac papp goal with + | None -> fold_morphism t ca + | Some (goal, pack) -> + memoize envs (EConstr.to_constr (Tacmach.project goal) papp) pack; + Array.fold_left cont goal args + end + + (* update in place the envs of known stuff, using memoization. We + have to memoize failures, here. *) + let gather goal (rlt : Coq.Relation.t ) envs t : Coq.goal_sigma = + let rec aux goal x = + match Coq.decomp_term (Tacmach.project goal) x with + | Constr.App (t,ca) -> + fold goal rlt envs t ca (aux ) + | _ -> goal + in + aux goal t + end + + (** We build a term out of a constr, updating in place the + environment if needed (the only kind of such updates are the + constants). *) + module Parse : + sig + val t_of_constr : Coq.goal_sigma -> Coq.Relation.t -> envs -> constr -> Matcher.Terms.t * Coq.goal_sigma + end + = struct + + (** [discriminates goal envs rlt t ca] infer : + + - its {! pack } (is it an AC operator, or an A operator, or a + Symbol ?), + + - the relevant partial application, + + - the vector of the remaining arguments + + We use an expansion to handle the special case of units, + before going back to the general discrimination + procedure. That means that a unit is more coarse than a raw + morphism + + This functions is prevented to go through [ar < 0] by the fact + that a constant is a morphism. But not an eva. *) + + let is_morphism goal (rlt : Coq.Relation.t) (papp : constr) (ar : int) : (Coq.goal_sigma * pack ) option = + let typeof = Classes.Proper.mk_typeof rlt ar in + let relof = Classes.Proper.mk_relof rlt ar in + let m = Coq.Classes.mk_morphism typeof relof papp in + try + let m,goal = Coq.resolve_one_typeclass goal m in + let pack = {Sym.ar = EConstr.to_constr ~abort_on_undefined_evars:(false)(Tacmach.project goal) (Coq.Nat.of_int ar); + Sym.value= EConstr.to_constr ~abort_on_undefined_evars:(false)(Tacmach.project goal) papp; + Sym.morph= EConstr.to_constr ~abort_on_undefined_evars:(false)(Tacmach.project goal) m} in + Some (goal, Sym pack) + with + | e -> None + + exception NotReflexive + let discriminate goal envs (rlt : Coq.Relation.t) t ca : Coq.goal_sigma * pack * constr * constr array = + let nb_params = Array.length ca in + let rec fold goal ar :Coq.goal_sigma * pack * constr * constr array = + begin + assert (0 <= ar && ar <= nb_params); + let p_app = mkApp (t, Array.sub ca 0 (nb_params - ar)) in + begin + try + begin match HMap.find envs.discr (EConstr.to_constr ~abort_on_undefined_evars:(false) (Tacmach.project goal) p_app) with + | None -> + fold goal (ar-1) + | Some pack -> + (goal, pack, p_app, Array.sub ca (nb_params -ar ) ar) + end + with + Not_found -> (* Could not find this constr *) + memoize (is_morphism goal rlt p_app ar) p_app ar + end + end + and memoize (x) p_app ar = + assert (0 <= ar && ar <= nb_params); + match x with + | Some (goal, pack) -> + HMap.add envs.discr (EConstr.to_constr ~abort_on_undefined_evars:(false) (Tacmach.project goal) p_app) (Some pack); + add_bloom envs pack; + (goal, pack, p_app, Array.sub ca (nb_params-ar) ar) + | None -> + + if ar = 0 then raise NotReflexive; + begin + (* to memoise the failures *) + HMap.add envs.discr (EConstr.to_constr ~abort_on_undefined_evars:(false) (Tacmach.project goal) p_app) None; + (* will terminate, since [const] is capped, and it is + easy to find an instance of a constant *) + fold goal (ar-1) + end + in + try match HMap.find envs.discr (EConstr.to_constr ~abort_on_undefined_evars:(false) (Tacmach.project goal) (mkApp (t,ca))) with + | None -> fold goal (nb_params) + | Some pack -> goal, pack, (mkApp (t,ca)), [| |] + with Not_found -> fold goal (nb_params) + + let discriminate goal envs rlt x = + try + match Coq.decomp_term (Tacmach.project goal) x with + | Constr.App (t,ca) -> + discriminate goal envs rlt t ca + | _ -> discriminate goal envs rlt x [| |] + with + | NotReflexive -> user_error @@ Pp.strbrk "The relation to which the goal was lifted is not Reflexive" + (* TODO: is it the only source of invalid use that fall + into this catch_all ? *) + | e -> + user_error @@ Pp.strbrk "Cannot handle this kind of hypotheses (variables occurring under a function symbol which is not a proper morphism)." + + (** [t_of_constr goal rlt envs cstr] builds the abstract syntax tree + of the term [cstr]. Doing so, it modifies the environment of + known stuff [envs], and eventually creates fresh + evars. Therefore, we give back the goal updated accordingly *) + let t_of_constr goal (rlt: Coq.Relation.t ) envs : constr -> Matcher.Terms.t * Coq.goal_sigma = + let r_goal = ref (goal) in + let rec aux x = + match Coq.decomp_term (Tacmach.project goal) x with + | Constr.Rel i -> Matcher.Terms.Var i + | _ -> + let goal, pack , p_app, ca = discriminate (!r_goal) envs rlt x in + r_goal := goal; + let k = find_bloom envs pack + in + match pack with + | Bin (pack,_) -> + begin match pack.Bin.comm with + | Some _ -> + assert (Array.length ca = 2); + Matcher.Terms.Plus ( k, aux ca.(0), aux ca.(1)) + | None -> + assert (Array.length ca = 2); + Matcher.Terms.Dot ( k, aux ca.(0), aux ca.(1)) + end + | Unit _ -> + assert (Array.length ca = 0); + Matcher.Terms.Unit ( k) + | Sym _ -> + Matcher.Terms.Sym ( k, Array.map aux ca) + in + ( + fun x -> let r = aux x in r, !r_goal + ) + + end (* Parse *) + + let add_symbol goal rlt envs l = + let goal = Gather.gather goal rlt envs (EConstr.of_constr (Constr.mkApp (l, [| Constr.mkRel 0;Constr.mkRel 0|]))) in + goal + + (* [t_of_constr] buils a the abstract syntax tree of a constr, + updating in place the environment. Doing so, we infer all the + morphisms and the AC/A operators. It is mandatory to do so both + for the pattern and the term, since AC symbols can occur in one + and not the other *) + let t_of_constr goal rlt envs (l,r) = + let goal = Gather.gather goal rlt envs l in + let goal = Gather.gather goal rlt envs r in + let l,goal = Parse.t_of_constr goal rlt envs l in + let r, goal = Parse.t_of_constr goal rlt envs r in + l, r, goal + + (* An intermediate representation of the environment, with association lists for AC/A operators, and also the raw [packer] information *) + + type ir = + { + packer : (int, pack) Hashtbl.t; (* = bloom_back *) + bin : (int * Bin.pack) list ; + units : (int * Unit.pack) list; + sym : (int * constr) list ; + matcher_units : Matcher.ext_units + } + + let ir_to_units ir = ir.matcher_units + + let ir_of_envs goal (rlt : Coq.Relation.t) envs = + let add x y l = (x,y)::l in + let nil = [] in + let sym , + (bin : (int * Bin.pack with_unit) list), + (units : (int * Constr.t) list) = + Hashtbl.fold + (fun int pack (sym,bin,units) -> + match pack with + | Bin s -> + sym, add (int) s bin, units + | Sym s -> + add (int) s sym, bin, units + | Unit s -> + sym, bin, add int s units + ) + envs.bloom_back + (nil,nil,nil) + in + let matcher_units = + let unit_for , is_ac = + List.fold_left + (fun ((unit_for,is_ac) as acc) (n,(bp,wu)) -> + match wu with + | None -> acc + | Some (unit_of) -> + let unit_n = PackTable.find envs.bloom + (Unit unit_of.Unit.uf_u) + in + ( n, unit_n)::unit_for, + (n, bp.Bin.comm <> None )::is_ac + + ) + ([],[]) bin + in + {Matcher.unit_for = unit_for; Matcher.is_ac = is_ac} + + in + let units : (int * Unit.pack) list = + List.fold_left + (fun acc (n,u) -> + (* first, gather all bins with this unit *) + let all_bin = + List.fold_left + ( fun acc (nop,(bp,wu)) -> + match wu with + | None -> acc + | Some unit_of -> + if Constr.equal (unit_of.Unit.uf_u) u + then + {unit_of with + Unit.uf_idx = EConstr.to_constr (Tacmach.project goal) (Coq.Pos.of_int nop)} :: acc + else + acc + ) + [] bin + in + (n,{ + Unit.u_value = EConstr.of_constr u; + Unit.u_desc = all_bin + })::acc + ) + [] units + + in + goal, { + packer = envs.bloom_back; + bin = (List.map (fun (n,(s,_)) -> n, s) bin); + units = units; + sym = (List.map (fun (n,s) -> n,(Sym.mk_pack rlt s)) sym); + matcher_units = matcher_units + } + + + + (** {1 From AST back to Coq } + + The next functions allow one to map OCaml abstract syntax trees + to Coq terms *) + + (** {2 Building raw, natural, terms} *) + + (** [raw_constr_of_t_debruijn] rebuilds a term in the raw + representation, without products on top, and maybe escaping free + debruijn indices (in the case of a pattern for example). *) + let raw_constr_of_t_debruijn ir (t : Matcher.Terms.t) : constr * int list = + let add_set,get = + let r = ref [] in + let rec add x = function + [ ] -> [x] + | t::q when t = x -> t::q + | t::q -> t:: (add x q) + in + (fun x -> r := add x !r),(fun () -> !r) + in + (* Here, we rely on the invariant that the maps are well formed: + it is meanigless to fail to find a symbol in the maps, or to + find the wrong kind of pack in the maps *) + let rec aux t = + match t with + | Matcher.Terms.Plus (s,l,r) -> + begin match Hashtbl.find ir.packer s with + | Bin (s,_) -> + mkApp (EConstr.of_constr s.Bin.value , [|(aux l);(aux r)|]) + | _ -> Printf.printf "erreur:%i\n%!"s; + assert false + end + | Matcher.Terms.Dot (s,l,r) -> + begin match Hashtbl.find ir.packer s with + | Bin (s,_) -> + mkApp (EConstr.of_constr s.Bin.value, [|(aux l);(aux r)|]) + | _ -> assert false + end + | Matcher.Terms.Sym (s,t) -> + begin match Hashtbl.find ir.packer s with + | Sym s -> + mkApp (EConstr.of_constr s.Sym.value, Array.map aux t) + | _ -> assert false + end + | Matcher.Terms.Unit x -> + begin match Hashtbl.find ir.packer x with + | Unit s -> EConstr.of_constr s + | _ -> assert false + end + | Matcher.Terms.Var i -> add_set i; + mkRel (i) + in + let t = aux t in + t , get ( ) + + (** [raw_constr_of_t] rebuilds a term in the raw representation *) + let raw_constr_of_t ir rlt (context:rel_context) t = + (** cap rebuilds the products in front of the constr *) + let rec cap c = function [] -> c + | t::q -> + let i = Context.Rel.lookup t context in + cap (mkProd_or_LetIn i c) q + in + let t,indices = raw_constr_of_t_debruijn ir t in + cap t (List.sort (Pervasives.compare) indices) + + + (** {2 Building reified terms} *) + + (* Some informations to be able to build the maps *) + type reif_params = + { + bin_null : Bin.pack; (* the default A operator *) + sym_null : constr; + unit_null : Unit.pack; + sym_ty : constr; (* the type, as it appears in e_sym *) + bin_ty : constr + } + + + (** A record containing the environments that will be needed by the + decision procedure, as a Coq constr. Contains the functions + from the symbols (as ints) to indexes (as constr) *) + + type sigmas = { + env_sym : constr; + env_bin : constr; + env_units : constr; (* the [idx -> X:constr] *) + } + + + type sigma_maps = { + sym_to_pos : int -> constr; + bin_to_pos : int -> constr; + units_to_pos : int -> constr; + } + + + (** infers some stuff that will be required when we will build + environments (our environments need a default case, so we need + an Op_AC, an Op_A, and a symbol) *) + (* Note : this function can fail if the user is using the wrong + relation, like proving a = b, while the classes are defined with + another relation (==) *) + let build_reif_params goal (rlt : Coq.Relation.t) (zero) : + reif_params * Coq.goal_sigma = + let carrier = rlt.Coq.Relation.carrier in + let bin_null = + try + let op,goal = Coq.evar_binary goal carrier in + let assoc,goal = Classes.Associative.infer goal rlt op in + let compat,goal = Classes.Proper.infer goal rlt op 2 in + let op = Coq.nf_evar goal op in + { + Bin.value = EConstr.to_constr (Tacmach.project goal) op; + Bin.compat = EConstr.to_constr (Tacmach.project goal) compat; + Bin.assoc = EConstr.to_constr (Tacmach.project goal) assoc; + Bin.comm = None + } + with Not_found -> user_error @@ Pp.strbrk "Cannot infer a default A operator (required at least to be Proper and Associative)" + in + let zero, goal = + try + let evar_op,goal = Coq.evar_binary goal carrier in + let evar_unit, goal = Coq.evar_unit goal carrier in + let query = Classes.Unit.ty rlt evar_op evar_unit in + let _, goal = Coq.resolve_one_typeclass goal query in + Coq.nf_evar goal evar_unit, goal + with _ -> zero, goal in + let sym_null = Sym.null rlt in + let unit_null = Unit.default zero in + let record = + { + bin_null = bin_null; + sym_null = sym_null; + unit_null = unit_null; + sym_ty = Sym.mk_ty rlt ; + bin_ty = Bin.mk_ty rlt + } + in + record, goal + + (* We want to lift down the indexes of symbols. *) + let renumber (l: (int * 'a) list ) = + let _, l = List.fold_left (fun (next,acc) (glob,x) -> + (next+1, (next,glob,x)::acc) + ) (1,[]) l + in + let rec to_global loc = function + | [] -> assert false + | (local, global,x)::q when local = loc -> global + | _::q -> to_global loc q + in + let rec to_local glob = function + | [] -> assert false + | (local, global,x)::q when global = glob -> local + | _::q -> to_local glob q + in + let locals = List.map (fun (local,global,x) -> local,x) l in + locals, (fun x -> to_local x l) , (fun x -> to_global x l) + + (** [build_sigma_maps] given a envs and some reif_params, we are + able to build the sigmas *) + let build_sigma_maps (rlt : Coq.Relation.t) zero ir (k : sigmas * sigma_maps -> Proof_type.tactic ) : Proof_type.tactic = fun goal -> + let rp,goal = build_reif_params goal rlt zero in + let renumbered_sym, to_local, to_global = renumber ir.sym in + let env_sym = Sigma.of_list + rp.sym_ty + (rp.sym_null) + renumbered_sym + in + Coq.cps_mk_letin "env_sym" env_sym + (fun env_sym -> + let bin = (List.map ( fun (n,s) -> n, Bin.mk_pack rlt s) ir.bin) in + let env_bin = + Sigma.of_list + rp.bin_ty + (Bin.mk_pack rlt rp.bin_null) + bin + in + Coq.cps_mk_letin "env_bin" env_bin + (fun env_bin -> + let units = (List.map (fun (n,s) -> n, Unit.mk_pack rlt env_bin s)ir.units) in + let env_units = + Sigma.of_list + (Unit.ty_unit_pack rlt env_bin) + (Unit.mk_pack rlt env_bin rp.unit_null ) + units + in + + Coq.cps_mk_letin "env_units" env_units + (fun env_units -> + let sigmas = + { + env_sym = env_sym ; + env_bin = env_bin ; + env_units = env_units; + } in + let f = List.map (fun (x,_) -> (x,Coq.Pos.of_int x)) in + let sigma_maps = + { + sym_to_pos = (let sym = f renumbered_sym in fun x -> (List.assoc (to_local x) sym)); + bin_to_pos = (let bin = f bin in fun x -> (List.assoc x bin)); + units_to_pos = (let units = f units in fun x -> (List.assoc x units)); + } + in + k (sigmas, sigma_maps ) + ) + ) + ) goal + + (** builders for the reification *) + type reif_builders = + { + rsum: constr -> constr -> constr -> constr; + rprd: constr -> constr -> constr -> constr; + rsym: constr -> constr array -> constr; + runit : constr -> constr + } + + (* donne moi une tactique, je rajoute ma part. Potentiellement, il + est possible d'utiliser la notation 'do' a la Haskell: + http://www.cas.mcmaster.ca/~carette/pa_monad/ *) + let (>>) : 'a * Proof_type.tactic -> ('a -> 'b * Proof_type.tactic) -> 'b * Proof_type.tactic = + fun (x,t) f -> + let b,t' = f x in + b, Tacticals.tclTHEN t t' + + let return x = x, Tacticals.tclIDTAC + + let mk_vect vnil vcons v = + let ar = Array.length v in + let rec aux = function + | 0 -> vnil + | n -> let n = n-1 in + mkApp( vcons, + [| + (Coq.Nat.of_int n); + v.(ar - 1 - n); + (aux (n)) + |] + ) + in aux ar + + (* TODO: use a do notation *) + let mk_reif_builders (rlt: Coq.Relation.t) (env_sym:constr) (k: reif_builders -> Proof_type.tactic) = + let x = (rlt.Coq.Relation.carrier) in + let r = (rlt.Coq.Relation.r) in + + let x_r_env = [|x;r;env_sym|] in + let tty = mkApp (Lazy.force Stubs._Tty, x_r_env) in + let rsum = mkApp (Lazy.force Stubs.rsum, x_r_env) in + let rprd = mkApp (Lazy.force Stubs.rprd, x_r_env) in + let rsym = mkApp (Lazy.force Stubs.rsym, x_r_env) in + let vnil = mkApp (Lazy.force Stubs.vnil, x_r_env) in + let vcons = mkApp (Lazy.force Stubs.vcons, x_r_env) in + Coq.cps_mk_letin "tty" tty + (fun tty -> + Coq.cps_mk_letin "rsum" rsum + (fun rsum -> + Coq.cps_mk_letin "rprd" rprd + (fun rprd -> + Coq.cps_mk_letin "rsym" rsym + (fun rsym -> + Coq.cps_mk_letin "vnil" vnil + (fun vnil -> + Coq.cps_mk_letin "vcons" vcons + (fun vcons -> + let r ={ + rsum = + begin fun idx l r -> + mkApp (rsum, [| idx ; mk_mset tty [l,1 ; r,1]|]) + end; + rprd = + begin fun idx l r -> + let lst = NEList.of_list tty [l;r] in + mkApp (rprd, [| idx; lst|]) + end; + rsym = + begin fun idx v -> + let vect = mk_vect vnil vcons v in + mkApp (rsym, [| idx; vect|]) + end; + runit = fun idx -> (* could benefit of a letin *) + mkApp (Lazy.force Stubs.runit , [|x;r;env_sym;idx; |]) + } + in k r + )))))) + + + + type reifier = sigma_maps * reif_builders + + + let mk_reifier rlt zero envs (k : sigmas *reifier -> Proof_type.tactic) = + build_sigma_maps rlt zero envs + (fun (s,sm) -> + mk_reif_builders rlt s.env_sym + (fun rb ->k (s,(sm,rb)) ) + + ) + + (** [reif_constr_of_t reifier t] rebuilds the term [t] in the + reified form. We use the [reifier] to minimise the size of the + terms (we make as much lets as possible)*) + let reif_constr_of_t (sm,rb) (t:Matcher.Terms.t) : constr = + let rec aux = function + | Matcher.Terms.Plus (s,l,r) -> + let idx = sm.bin_to_pos s in + rb.rsum idx (aux l) (aux r) + | Matcher.Terms.Dot (s,l,r) -> + let idx = sm.bin_to_pos s in + rb.rprd idx (aux l) (aux r) + | Matcher.Terms.Sym (s,t) -> + let idx = sm.sym_to_pos s in + rb.rsym idx (Array.map aux t) + | Matcher.Terms.Unit s -> + let idx = sm.units_to_pos s in + rb.runit idx + | Matcher.Terms.Var i -> + anomaly "call to reif_constr_of_t on a term with variables." + in aux t +end + + + diff --git a/src/theory.mli b/src/theory.mli new file mode 100644 index 0000000..e7bfbfe --- /dev/null +++ b/src/theory.mli @@ -0,0 +1,197 @@ +(***************************************************************************) +(* This is part of aac_tactics, it is distributed under the terms of the *) +(* GNU Lesser General Public License version 3 *) +(* (see file LICENSE for more details) *) +(* *) +(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) +(***************************************************************************) + +(** Bindings for Coq constants that are specific to the plugin; + reification and translation functions. + + Note: this module is highly correlated with the definitions of {i + AAC_rewrite.v}. + + This module interfaces with the above Coq module; it provides + facilities to interpret a term with arbitrary operators as an + abstract syntax tree, and to convert an AST into a Coq term + (either using the Coq "raw" terms, as written in the starting + goal, or using the reified Coq datatypes we define in {i + AAC_rewrite.v}). +*) + +(** Both in OCaml and Coq, we represent finite multisets using + weighted lists ([('a*int) list]), see {!Matcher.mset}. + + [mk_mset ty l] constructs a Coq multiset from an OCaml multiset + [l] of Coq terms of type [ty] *) + +val mk_mset:EConstr.constr -> (EConstr.constr * int) list ->EConstr.constr + +(** {2 Packaging modules} *) + +(** Environments *) +module Sigma: +sig + (** [add ty n x map] adds the value [x] of type [ty] with key [n] in [map] *) + val add: EConstr.constr ->EConstr.constr ->EConstr.constr ->EConstr.constr ->EConstr.constr + + (** [empty ty] create an empty map of type [ty] *) + val empty: EConstr.constr ->EConstr.constr + + (** [of_list ty null l] translates an OCaml association list into a Coq one *) + val of_list: EConstr.constr -> EConstr.constr -> (int * EConstr.constr ) list -> EConstr.constr + + (** [to_fun ty null map] converts a Coq association list into a Coq function (with default value [null]) *) + val to_fun: EConstr.constr ->EConstr.constr ->EConstr.constr ->EConstr.constr +end + + +(** Dynamically typed morphisms *) +module Sym: +sig + (** mimics the Coq record [Sym.pack] *) + type pack = {ar: Constr.t; value: Constr.t ; morph: Constr.t} + + val typ: EConstr.constr lazy_t + + + (** [mk_pack rlt (ar,value,morph)] *) + val mk_pack: Coq.Relation.t -> pack -> EConstr.constr + + (** [null] builds a dummy (identity) symbol, given an {!Coq.Relation.t} *) + val null: Coq.Relation.t -> EConstr.constr + +end + + +(** We need to export some Coq stubs out of this module. They are used + by the main tactic, see {!Rewrite} *) +module Stubs : sig + val lift : EConstr.constr Lazy.t + val lift_proj_equivalence : EConstr.constr Lazy.t + val lift_transitivity_left : EConstr.constr Lazy.t + val lift_transitivity_right : EConstr.constr Lazy.t + val lift_reflexivity : EConstr.constr Lazy.t + (** The evaluation fonction, used to convert a reified coq term to a + raw coq term *) + val eval: EConstr.constr lazy_t + + (** The main lemma of our theory, that is + [compare (norm u) (norm v) = Eq -> eval u == eval v] *) + val decide_thm:EConstr.constr lazy_t + + val lift_normalise_thm : EConstr.constr lazy_t +end + +(** {2 Building reified terms} + + We define a bundle of functions to build reified versions of the + terms (those that will be given to the reflexive decision + procedure). In particular, each field takes as first argument the + index of the symbol rather than the symbol itself. *) + +(** Tranlations between Coq and OCaml *) +module Trans : sig + + (** This module provides facilities to interpret a term with + arbitrary operators as an instance of an abstract syntax tree + {!Matcher.Terms.t}. + + For each Coq application [f x_1 ... x_n], this amounts to + deciding whether one of the partial applications [f x_1 + ... x_i], [i<=n] is a proper morphism, whether the partial + application with [i=n-2] yields an A or AC binary operator, and + whether the whole term is the unit for some A or AC operator. We + use typeclass resolution to test each of these possibilities. + + Note that there are ambiguous terms: + - a term like [f x y] might yield a unary morphism ([f x]) and a + binary one ([f]); we select the latter one (unless [f] is A or + AC, in which case we declare it accordingly); + - a term like [S O] can be considered as a morphism ([S]) + applied to a unit for [(+)], or as a unit for [( * )]; we + chose to give priority to units, so that the latter + interpretation is selected in this case; + - an element might be the unit for several operations + *) + + (** To achieve this reification, one need to record informations + about the collected operators (symbols, binary operators, + units). We use the following imperative internal data-structure to + this end. *) + + type envs + val empty_envs : unit -> envs + + + (** {2 Reification: from Coq terms to AST {!Matcher.Terms.t} } *) + + + (** [t_of_constr goal rlt envs (left,right)] builds the abstract + syntax tree of the terms [left] and [right]. We rely on the [goal] + to perform typeclasses resolutions to find morphisms compatible + with the relation [rlt]. Doing so, it modifies the reification + environment [envs]. Moreover, we need to create fresh + evars; this is why we give back the [goal], accordingly + updated. *) + + val t_of_constr : Coq.goal_sigma -> Coq.Relation.t -> envs -> (EConstr.constr * EConstr.constr) -> Matcher.Terms.t * Matcher.Terms.t * Coq.goal_sigma + + (** [add_symbol] adds a given binary symbol to the environment of + known stuff. *) + val add_symbol : Coq.goal_sigma -> Coq.Relation.t -> envs -> Constr.t -> Coq.goal_sigma + + (** {2 Reconstruction: from AST back to Coq terms } + + The next functions allow one to map OCaml abstract syntax trees + to Coq terms. We need two functions to rebuild different kind of + terms: first, raw terms, like the one encountered by + {!t_of_constr}; second, reified Coq terms, that are required for + the reflexive decision procedure. *) + + type ir + val ir_of_envs : Coq.goal_sigma -> Coq.Relation.t -> envs -> Coq.goal_sigma * ir + val ir_to_units : ir -> Matcher.ext_units + + (** {2 Building raw, natural, terms} *) + + (** [raw_constr_of_t] rebuilds a term in the raw representation, and + reconstruct the named products on top of it. In particular, this + allow us to print the context put around the left (or right) + hand side of a pattern. *) + val raw_constr_of_t : ir -> Coq.Relation.t -> EConstr.rel_context -> Matcher.Terms.t -> EConstr.constr + + (** {2 Building reified terms} *) + + (** The reification environments, as Coq constrs *) + + type sigmas = { + env_sym : EConstr.constr; + env_bin : EConstr.constr; + env_units : EConstr.constr; (* the [idx -> X:constr] *) + } + + + + (** We need to reify two terms (left and right members of a goal) + that share the same reification envirnoment. Therefore, we need + to add letins to the proof context in order to ensure some + sharing in the proof terms we produce. + + Moreover, in order to have as much sharing as possible, we also + add letins for various partial applications that are used + throughout the terms. + + To achieve this, we decompose the reconstruction function into + two steps: first, we build the reification environment and then + reify each term successively.*) + type reifier + + val mk_reifier : Coq.Relation.t -> EConstr.constr -> ir -> (sigmas * reifier -> Proof_type.tactic) -> Proof_type.tactic + + (** [reif_constr_of_t reifier t] rebuilds the term [t] in the + reified form. *) + val reif_constr_of_t : reifier -> Matcher.Terms.t -> EConstr.constr + +end diff --git a/theories/AAC.v b/theories/AAC.v new file mode 100644 index 0000000..d7cc7a2 --- /dev/null +++ b/theories/AAC.v @@ -0,0 +1,901 @@ +(***************************************************************************) +(* This is part of aac_tactics, it is distributed under the terms of the *) +(* GNU Lesser General Public License version 3 *) +(* (see file LICENSE for more details) *) +(* *) +(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) +(***************************************************************************) + +(** * Theory file for the aac_rewrite tactic + + We define several base classes to package associative and possibly + commutative operators, and define a data-type for reified (or + quoted) expressions (with morphisms). + + We then define a reflexive decision procedure to decide the + equality of reified terms: first normalise reified terms, then + compare them. This allows us to close transitivity steps + automatically, in the [aac_rewrite] tactic. + + We restrict ourselves to the case where all symbols operate on a + single fixed type. In particular, this means that we cannot handle + situations like + + [H: forall x y, nat_of_pos (pos_of_nat (x) + y) + x = ....] + + where one occurrence of [+] operates on nat while the other one + operates on positive. *) + +Require Import Arith NArith. +Require Import List. +Require Import FMapPositive FMapFacts. +Require Import RelationClasses Equality. +Require Export Morphisms. + +From AAC_tactics +Require Import Utils. + +Set Implicit Arguments. +Set Asymmetric Patterns. + +Local Open Scope signature_scope. + +(** * Environments for the reification process: we use positive maps to index elements *) + +Section sigma. + Definition sigma := PositiveMap.t. + Definition sigma_get A (null : A) (map : sigma A) (n : positive) : A := + match PositiveMap.find n map with + | None => null + | Some x => x + end. + Definition sigma_add := @PositiveMap.add. + Definition sigma_empty := @PositiveMap.empty. +End sigma. + + +(** * Classes for properties of operators *) + +Class Associative (X:Type) (R:relation X) (dot: X -> X -> X) := + law_assoc : forall x y z, R (dot x (dot y z)) (dot (dot x y) z). +Class Commutative (X:Type) (R: relation X) (plus: X -> X -> X) := + law_comm: forall x y, R (plus x y) (plus y x). +Class Unit (X:Type) (R:relation X) (op : X -> X -> X) (unit:X) := { + law_neutral_left: forall x, R (op unit x) x; + law_neutral_right: forall x, R (op x unit) x +}. + + +(** Class used to find the equivalence relation on which operations + are A or AC, starting from the relation appearing in the goal *) + +Class AAC_lift X (R: relation X) (E : relation X) := { + aac_lift_equivalence : Equivalence E; + aac_list_proper : Proper (E ==> E ==> iff) R +}. + +(** simple instances, when we have a subrelation, or an equivalence *) + +Instance aac_lift_subrelation {X} {R} {E} {HE: Equivalence E} + {HR: @Transitive X R} {HER: subrelation E R}: AAC_lift R E | 3. +Proof. + constructor; trivial. + intros ? ? H ? ? H'. split; intro G. + rewrite <- H, G. apply HER, H'. + rewrite H, G. apply HER. symmetry. apply H'. +Qed. + +Instance aac_lift_proper {X} {R : relation X} {E} {HE: Equivalence E} + {HR: Proper (E==>E==>iff) R}: AAC_lift R E | 4 := {}. + + + +Module Internal. +(** * Utilities for the evaluation function *) + +Section copy. + + Context {X} {R} {HR: @Equivalence X R} {plus} + (op: Associative R plus) (op': Commutative R plus) (po: Proper (R ==> R ==> R) plus). + + (* copy n x = x+...+x (n times) *) + Fixpoint copy' n x := match n with + | xH => x + | xI n => let xn := copy' n x in plus (plus xn xn) x + | xO n => let xn := copy' n x in (plus xn xn) + end. + Definition copy n x := Prect (fun _ => X) x (fun _ xn => plus x xn) n. + + Lemma copy_plus : forall n m x, R (copy (n+m) x) (plus (copy n x) (copy m x)). + Proof. + unfold copy. + induction n using Pind; intros m x. + rewrite Prect_base. rewrite <- Pplus_one_succ_l. rewrite Prect_succ. reflexivity. + rewrite Pplus_succ_permute_l. rewrite 2Prect_succ. rewrite IHn. apply op. + Qed. + Lemma copy_xH : forall x, R (copy 1 x) x. + Proof. intros; unfold copy; rewrite Prect_base. reflexivity. Qed. + Lemma copy_Psucc : forall n x, R (copy (Pos.succ n) x) (plus x (copy n x)). + Proof. intros; unfold copy; rewrite Prect_succ. reflexivity. Qed. + + Global Instance copy_compat n: Proper (R ==> R) (copy n). + Proof. + unfold copy. + induction n using Pind; intros x y H. + rewrite 2Prect_base. assumption. + rewrite 2Prect_succ. apply po; auto. + Qed. + +End copy. + +(** * Packaging structures *) + +(** ** free symbols *) + +Module Sym. + Section t. + Context {X} {R : relation X} . + + (** type of an arity *) + Fixpoint type_of (n: nat) := + match n with + | O => X + | S n => X -> type_of n + end. + + (** relation to be preserved at an arity *) + Fixpoint rel_of n : relation (type_of n) := + match n with + | O => R + | S n => respectful R (rel_of n) + end. + + (** a symbol package contains an arity, + a value of the corresponding type, + and a proof that the value is a proper morphism *) + Record pack : Type := mkPack { + ar : nat; + value :> type_of ar; + morph : Proper (rel_of ar) value + }. + + (** helper to build default values, when filling reification environments *) + Definition null: pack := mkPack 1 (fun x => x) (fun _ _ H => H). + + End t. + +End Sym. + +(** ** binary operations *) + +Module Bin. + Section t. + Context {X} {R: relation X}. + + Record pack := mk_pack { + value:> X -> X -> X; + compat: Proper (R ==> R ==> R) value; + assoc: Associative R value; + comm: option (Commutative R value) + }. + End t. + (* See #Instances.v# for concrete instances of these classes. *) + +End Bin. + + +(** * Reification, normalisation, and decision *) + +Section s. + Context {X} {R: relation X} {E: @Equivalence X R}. + Infix "==" := R (at level 80). + + (* We use environments to store the various operators and the + morphisms.*) + + Variable e_sym: idx -> @Sym.pack X R. + Variable e_bin: idx -> @Bin.pack X R. + + + (** packaging units (depends on e_bin) *) + + Record unit_of u := mk_unit_for { + uf_idx: idx; + uf_desc: Unit R (Bin.value (e_bin uf_idx)) u + }. + + Record unit_pack := mk_unit_pack { + u_value:> X; + u_desc: list (unit_of u_value) + }. + Variable e_unit: positive -> unit_pack. + + Hint Resolve e_bin e_unit: typeclass_instances. + + (** ** Almost normalised syntax + a term in [T] is in normal form if: + - sums do not contain sums + - products do not contain products + - there are no unary sums or products + - lists and msets are lexicographically sorted according to the order we define below + + [vT n] denotes the set of term vectors of size [n] (the mutual dependency could be removed), + + Note that [T] and [vT] depend on the [e_sym] environment (which + contains, among other things, the arity of symbols) + *) + + Inductive T: Type := + | sum: idx -> mset T -> T + | prd: idx -> nelist T -> T + | sym: forall i, vT (Sym.ar (e_sym i)) -> T + | unit : idx -> T + with vT: nat -> Type := + | vnil: vT O + | vcons: forall n, T -> vT n -> vT (S n). + + + (** lexicographic rpo over the normalised syntax *) + Fixpoint compare (u v: T) := + match u,v with + | sum i l, sum j vs => lex (idx_compare i j) (mset_compare compare l vs) + | prd i l, prd j vs => lex (idx_compare i j) (list_compare compare l vs) + | sym i l, sym j vs => lex (idx_compare i j) (vcompare l vs) + | unit i , unit j => idx_compare i j + | unit _ , _ => Lt + | _ , unit _ => Gt + | sum _ _, _ => Lt + | _ , sum _ _ => Gt + | prd _ _, _ => Lt + | _ , prd _ _ => Gt + + end + with vcompare i j (us: vT i) (vs: vT j) := + match us,vs with + | vnil, vnil => Eq + | vnil, _ => Lt + | _, vnil => Gt + | vcons _ u us, vcons _ v vs => lex (compare u v) (vcompare us vs) + end. + + + + (** ** Evaluation from syntax to the abstract domain *) + + Fixpoint eval u: X := + match u with + | sum i l => let o := Bin.value (e_bin i) in + fold_map (fun un => let '(u,n):=un in @copy _ o n (eval u)) o l + | prd i l => fold_map eval (Bin.value (e_bin i)) l + | sym i v => eval_aux v (Sym.value (e_sym i)) + | unit i => e_unit i + end + with eval_aux i (v: vT i): Sym.type_of i -> X := + match v with + | vnil => fun f => f + | vcons _ u v => fun f => eval_aux v (f (eval u)) + end. + + (** we need to show that compare reflects equality (this is because + we work with msets rather than lists with arities) *) + Lemma tcompare_weak_spec: forall (u v : T), compare_weak_spec u v (compare u v) + with vcompare_reflect_eqdep: forall i us j vs (H: i=j), vcompare us vs = Eq -> cast vT H us = vs. + Proof. + induction u. + destruct v; simpl; try constructor. + case (pos_compare_weak_spec p p0); intros; try constructor. + case (mset_compare_weak_spec compare tcompare_weak_spec m m0); intros; try constructor. + destruct v; simpl; try constructor. + case (pos_compare_weak_spec p p0); intros; try constructor. + case (list_compare_weak_spec compare tcompare_weak_spec n n0); intros; try constructor. + destruct v0; simpl; try constructor. + case_eq (idx_compare i i0); intro Hi; try constructor. + apply idx_compare_reflect_eq in Hi. symmetry in Hi. subst. (* the [symmetry] is required ! *) + case_eq (vcompare v v0); intro Hv; try constructor. + rewrite <- (vcompare_reflect_eqdep _ _ _ _ eq_refl Hv). constructor. + destruct v; simpl; try constructor. + case_eq (idx_compare p p0); intro Hi; try constructor. + apply idx_compare_reflect_eq in Hi. symmetry in Hi. subst. constructor. + + induction us; destruct vs; simpl; intros H Huv; try discriminate. + apply cast_eq, eq_nat_dec. + injection H; intro Hn. + revert Huv; case (tcompare_weak_spec t t0); intros; try discriminate. + symmetry in Hn. subst. (* symmetry required *) + rewrite <- (IHus _ _ eq_refl Huv). + apply cast_eq, eq_nat_dec. + Qed. + + Instance eval_aux_compat i (l: vT i): Proper (@Sym.rel_of X R i ==> R) (eval_aux l). + Proof. + induction l; simpl; repeat intro. + assumption. + apply IHl, H. reflexivity. + Qed. + + + (* is [i] a unit for [j] ? *) + Definition is_unit_of j i := + List.existsb (fun p => eq_idx_bool j (uf_idx p)) (u_desc (e_unit i)). + + (* is [i] commutative ? *) + Definition is_commutative i := + match Bin.comm (e_bin i) with Some _ => true | None => false end. + + + (** ** Normalisation *) + + Inductive discr {A} : Type := + | Is_op : A -> discr + | Is_unit : idx -> discr + | Is_nothing : discr . + + (* This is called sum in the std lib *) + Inductive m {A} {B} := + | left : A -> m + | right : B -> m. + + Definition comp A B (merge : B -> B -> B) (l : B) (l' : @m A B) : @m A B := + match l' with + | left _ => right l + | right l' => right (merge l l') + end. + + (** auxiliary functions, to clean up sums *) + + Section sums. + Variable i : idx. + Variable is_unit : idx -> bool. + + Definition sum' (u: mset T): T := + match u with + | nil (u,xH) => u + | _ => sum i u + end. + + Definition is_sum (u: T) : @discr (mset T) := + match u with + | sum j l => if eq_idx_bool j i then Is_op l else Is_nothing + | unit j => if is_unit j then Is_unit j else Is_nothing + | u => Is_nothing + end. + + Definition copy_mset n (l: mset T): mset T := + match n with + | xH => l + | _ => nelist_map (fun vm => let '(v,m):=vm in (v,Pmult n m)) l + end. + + Definition return_sum u n := + match is_sum u with + | Is_nothing => right (nil (u,n)) + | Is_op l' => right (copy_mset n l') + | Is_unit j => left j + end. + + Definition add_to_sum u n (l : @m idx (mset T)) := + match is_sum u with + | Is_nothing => comp (merge_msets compare) (nil (u,n)) l + | Is_op l' => comp (merge_msets compare) (copy_mset n l') l + | Is_unit _ => l + end. + + + Definition norm_msets_ norm (l: mset T) := + fold_map' + (fun un => let '(u,n) := un in return_sum (norm u) n) + (fun un l => let '(u,n) := un in add_to_sum (norm u) n l) l. + + + End sums. + + (** similar functions for products *) + + Section prds. + + Variable i : idx. + Variable is_unit : idx -> bool. + Definition prd' (u: nelist T): T := + match u with + | nil u => u + | _ => prd i u + end. + + Definition is_prd (u: T) : @discr (nelist T) := + match u with + | prd j l => if eq_idx_bool j i then Is_op l else Is_nothing + | unit j => if is_unit j then Is_unit j else Is_nothing + | u => Is_nothing + end. + + + Definition return_prd u := + match is_prd u with + | Is_nothing => right (nil (u)) + | Is_op l' => right (l') + | Is_unit j => left j + end. + + Definition add_to_prd u (l : @m idx (nelist T)) := + match is_prd u with + | Is_nothing => comp (@appne T) (nil (u)) l + | Is_op l' => comp (@appne T) (l') l + | Is_unit _ => l + end. + + Definition norm_lists_ norm (l : nelist T) := + fold_map' + (fun u => return_prd (norm u)) + (fun u l => add_to_prd (norm u) l) l. + + + End prds. + + + Definition run_list x := match x with + | left n => nil (unit n) + | right l => l + end. + + Definition norm_lists norm i l := + let is_unit := is_unit_of i in + run_list (norm_lists_ i is_unit norm l). + + Definition run_msets x := match x with + | left n => nil (unit n, xH) + | right l => l + end. + + Definition norm_msets norm i l := + let is_unit := is_unit_of i in + run_msets (norm_msets_ i is_unit norm l). + + Fixpoint norm u {struct u}:= + match u with + | sum i l => if is_commutative i then sum' i (norm_msets norm i l) else u + | prd i l => prd' i (norm_lists norm i l) + | sym i l => sym i (vnorm l) + | unit i => unit i + end + with vnorm i (l: vT i): vT i := + match l with + | vnil => vnil + | vcons _ u l => vcons (norm u) (vnorm l) + end. + + (** ** Correctness *) + + Lemma is_unit_of_Unit : forall i j : idx, + is_unit_of i j = true -> Unit R (Bin.value (e_bin i)) (eval (unit j)). + Proof. + intros. unfold is_unit_of in H. + rewrite existsb_exists in H. + destruct H as [x [H H']]. + revert H' ; case (eq_idx_spec); [intros H' _ ; subst| intros _ H'; discriminate]. + simpl. destruct x. simpl. auto. + Qed. + + Instance Binvalue_Commutative i (H : is_commutative i = true) : Commutative R (@Bin.value _ _ (e_bin i) ). + Proof. + unfold is_commutative in H. + destruct (Bin.comm (e_bin i)); auto. + discriminate. + Qed. + + Instance Binvalue_Associative i :Associative R (@Bin.value _ _ (e_bin i) ). + Proof. + destruct ((e_bin i)); auto. + Qed. + + Instance Binvalue_Proper i : Proper (R ==> R ==> R) (@Bin.value _ _ (e_bin i) ). + Proof. + destruct ((e_bin i)); auto. + Qed. + Hint Resolve Binvalue_Proper Binvalue_Associative Binvalue_Commutative. + + (** auxiliary lemmas about sums *) + + Hint Resolve is_unit_of_Unit. + Section sum_correctness. + Variable i : idx. + Variable is_unit : idx -> bool. + Hypothesis is_unit_sum_Unit : forall j, is_unit j = true-> @Unit X R (Bin.value (e_bin i)) (eval (unit j)). + + Inductive is_sum_spec_ind : T -> @discr (mset T) -> Prop := + | is_sum_spec_op : forall j l, j = i -> is_sum_spec_ind (sum j l) (Is_op l) + | is_sum_spec_unit : forall j, is_unit j = true -> is_sum_spec_ind (unit j) (Is_unit j) + | is_sum_spec_nothing : forall u, is_sum_spec_ind u (Is_nothing). + + Lemma is_sum_spec u : is_sum_spec_ind u (is_sum i is_unit u). + Proof. + unfold is_sum; case u; intros; try constructor. + case_eq (eq_idx_bool p i); intros; subst; try constructor; auto. + revert H. case eq_idx_spec; try discriminate. auto. + case_eq (is_unit p); intros; try constructor. auto. + Qed. + + Instance assoc : @Associative X R (Bin.value (e_bin i)). + Proof. + destruct (e_bin i). simpl. assumption. + Qed. + Instance proper : Proper (R ==> R ==> R)(Bin.value (e_bin i)). + Proof. + destruct (e_bin i). simpl. assumption. + Qed. + Hypothesis comm : @Commutative X R (Bin.value (e_bin i)). + + Lemma sum'_sum : forall (l: mset T), eval (sum' i l) ==eval (sum i l) . + Proof. + intros [[a n] | [a n] l]; destruct n; simpl; reflexivity. + Qed. + + Lemma eval_sum_nil x: + eval (sum i (nil (x,xH))) == (eval x). + Proof. rewrite <- sum'_sum. reflexivity. Qed. + + Lemma eval_sum_cons : forall n a (l: mset T), + (eval (sum i ((a,n)::l))) == (@Bin.value _ _ (e_bin i) (@copy _ (@Bin.value _ _ (e_bin i)) n (eval a)) (eval (sum i l))). + Proof. + intros n a [[? ? ]|[b m] l]; simpl; reflexivity. + Qed. + + Inductive compat_sum_unit : @m idx (mset T) -> Prop := + | csu_left : forall x, is_unit x = true-> compat_sum_unit (left x) + | csu_right : forall m, compat_sum_unit (right m) + . + + Lemma compat_sum_unit_return x n : compat_sum_unit (return_sum i is_unit x n). + Proof. + unfold return_sum. + case is_sum_spec; intros; try constructor; auto. + Qed. + + Lemma compat_sum_unit_add : forall x n h, + compat_sum_unit h + -> + compat_sum_unit + (add_to_sum i (is_unit_of i) x n + h). + Proof. + unfold add_to_sum;intros; inversion H; + case_eq (is_sum i (is_unit_of i) x); + intros; simpl; try constructor || eauto. apply H0. + Qed. + + (* Hint Resolve copy_plus. : this lags because of the inference of the implicit arguments *) + Hint Extern 5 (copy (?n + ?m) (eval ?a) == Bin.value (copy ?n (eval ?a)) (copy ?m (eval ?a))) => apply copy_plus. + Hint Extern 5 (?x == ?x) => reflexivity. + Hint Extern 5 ( Bin.value ?x ?y == Bin.value ?y ?x) => apply Bin.comm. + + Lemma eval_merge_bin : forall (h k: mset T), + eval (sum i (merge_msets compare h k)) == @Bin.value _ _ (e_bin i) (eval (sum i h)) (eval (sum i k)). + Proof. + induction h as [[a n]|[a n] h IHh]; intro k. + simpl. + induction k as [[b m]|[b m] k IHk]; simpl. + destruct (tcompare_weak_spec a b) as [a|a b|a b]; simpl; auto. apply copy_plus; auto. + + destruct (tcompare_weak_spec a b) as [a|a b|a b]; simpl; auto. + rewrite copy_plus,law_assoc; auto. + rewrite IHk; clear IHk. rewrite 2 law_assoc. apply proper. apply law_comm. reflexivity. + + induction k as [[b m]|[b m] k IHk]; simpl; simpl in IHh. + destruct (tcompare_weak_spec a b) as [a|a b|a b]; simpl. + rewrite (law_comm _ (copy m (eval a))), law_assoc, <- copy_plus, Pplus_comm; auto. + rewrite <- law_assoc, IHh. reflexivity. + rewrite law_comm. reflexivity. + + simpl in IHk. + destruct (tcompare_weak_spec a b) as [a|a b|a b]; simpl. + rewrite IHh; clear IHh. rewrite 2 law_assoc. rewrite (law_comm _ (copy m (eval a))). rewrite law_assoc, <- copy_plus, Pplus_comm; auto. + rewrite IHh; clear IHh. simpl. rewrite law_assoc. reflexivity. + rewrite 2 (law_comm (copy m (eval b))). rewrite law_assoc. apply proper; [ | reflexivity]. + rewrite <- IHk. reflexivity. + Qed. + + + Lemma copy_mset' n (l: mset T): copy_mset n l = nelist_map (fun vm => let '(v,m):=vm in (v,Pmult n m)) l. + Proof. + unfold copy_mset. destruct n; try reflexivity. + + simpl. induction l as [|[a l] IHl]; simpl; try congruence. destruct a. reflexivity. + Qed. + + + Lemma copy_mset_succ n (l: mset T): eval (sum i (copy_mset (Pos.succ n) l)) == @Bin.value _ _ (e_bin i) (eval (sum i l)) (eval (sum i (copy_mset n l))). + Proof. + rewrite 2 copy_mset'. + + induction l as [[a m]|[a m] l IHl]. + simpl eval. rewrite <- copy_plus; auto. rewrite Pmult_Sn_m. reflexivity. + + simpl nelist_map. rewrite ! eval_sum_cons. rewrite IHl. clear IHl. + rewrite Pmult_Sn_m. rewrite copy_plus; auto. rewrite <- !law_assoc. + apply Binvalue_Proper; try reflexivity. + rewrite law_comm . rewrite <- !law_assoc. apply proper; try reflexivity. + apply law_comm. + Qed. + + Lemma copy_mset_copy : forall n (m : mset T), eval (sum i (copy_mset n m)) == @copy _ (@Bin.value _ _ (e_bin i)) n (eval (sum i m)). + Proof. + induction n using Pind; intros. + + unfold copy_mset. rewrite copy_xH. reflexivity. + rewrite copy_mset_succ. rewrite copy_Psucc. rewrite IHn. reflexivity. + Qed. + + Instance compat_sum_unit_Unit : forall p, compat_sum_unit (left p) -> + @Unit X R (Bin.value (e_bin i)) (eval (unit p)). + Proof. + intros. + inversion H. subst. auto. + Qed. + + Lemma copy_n_unit : forall j n, is_unit j = true -> + eval (unit j) == @copy _ (Bin.value (e_bin i)) n (eval (unit j)). + Proof. + intros. + induction n using Prect. + rewrite copy_xH. reflexivity. + rewrite copy_Psucc. rewrite <- IHn. apply is_unit_sum_Unit in H. rewrite law_neutral_left. reflexivity. + Qed. + + + Lemma z0 l r (H : compat_sum_unit r): + eval (sum i (run_msets (comp (merge_msets compare) l r))) == eval (sum i ((merge_msets compare) (l) (run_msets r))). + Proof. + unfold comp. unfold run_msets. + case_eq r; intros; subst. + rewrite eval_merge_bin; auto. + rewrite eval_sum_nil. + apply compat_sum_unit_Unit in H. rewrite law_neutral_right. reflexivity. + reflexivity. + Qed. + + Lemma z1 : forall n x, + eval (sum i (run_msets (return_sum i (is_unit) x n ))) + == @copy _ (@Bin.value _ _ (e_bin i)) n (eval x). + Proof. + intros. unfold return_sum. unfold run_msets. + case (is_sum_spec); intros; subst. + rewrite copy_mset_copy. + reflexivity. + + rewrite eval_sum_nil. apply copy_n_unit. auto. + reflexivity. + Qed. + Lemma z2 : forall u n x, compat_sum_unit x -> + eval (sum i ( run_msets + (add_to_sum i (is_unit) u n x))) + == + @Bin.value _ _ (e_bin i) (@copy _ (@Bin.value _ _ (e_bin i)) n (eval u)) (eval (sum i (run_msets x))). + Proof. + intros u n x Hix . + unfold add_to_sum. + case is_sum_spec; intros; subst. + + rewrite z0 by auto. rewrite eval_merge_bin. rewrite copy_mset_copy. reflexivity. + rewrite <- copy_n_unit by assumption. + apply is_unit_sum_Unit in H. + rewrite law_neutral_left. reflexivity. + + + rewrite z0 by auto. rewrite eval_merge_bin. reflexivity. + Qed. + End sum_correctness. + Lemma eval_norm_msets i norm + (Comm : Commutative R (Bin.value (e_bin i))) + (Hnorm: forall u, eval (norm u) == eval u) : forall h, eval (sum i (norm_msets norm i h)) == eval (sum i h). + Proof. + unfold norm_msets. + assert (H : + forall h : mset T, + eval (sum i (run_msets (norm_msets_ i (is_unit_of i) norm h))) == eval (sum i h) /\ compat_sum_unit (is_unit_of i) (norm_msets_ i (is_unit_of i) norm h)). + + induction h as [[a n] | [a n] h [IHh IHh']]; simpl norm_msets_; split. + rewrite z1 by auto. rewrite Hnorm . reflexivity. auto. + apply compat_sum_unit_return. + + rewrite z2 by auto. rewrite IHh. + rewrite eval_sum_cons. rewrite Hnorm. reflexivity. apply compat_sum_unit_add, IHh'. + + apply H. + Defined. + + (** auxiliary lemmas about products *) + + Section prd_correctness. + Variable i : idx. + Variable is_unit : idx -> bool. + Hypothesis is_unit_prd_Unit : forall j, is_unit j = true-> @Unit X R (Bin.value (e_bin i)) (eval (unit j)). + + Inductive is_prd_spec_ind : T -> @discr (nelist T) -> Prop := + | is_prd_spec_op : + forall j l, j = i -> is_prd_spec_ind (prd j l) (Is_op l) + | is_prd_spec_unit : + forall j, is_unit j = true -> is_prd_spec_ind (unit j) (Is_unit j) + | is_prd_spec_nothing : + forall u, is_prd_spec_ind u (Is_nothing). + + Lemma is_prd_spec u : is_prd_spec_ind u (is_prd i is_unit u). + Proof. + unfold is_prd; case u; intros; try constructor. + case (eq_idx_spec); intros; subst; try constructor; auto. + case_eq (is_unit p); intros; try constructor; auto. + Qed. + + Lemma prd'_prd : forall (l: nelist T), eval (prd' i l) == eval (prd i l). + Proof. + intros [?|? [|? ?]]; simpl; reflexivity. + Qed. + + + Lemma eval_prd_nil x: eval (prd i (nil x)) == eval x. + Proof. + rewrite <- prd'_prd. simpl. reflexivity. + Qed. + Lemma eval_prd_cons a : forall (l: nelist T), eval (prd i (a::l)) == @Bin.value _ _ (e_bin i) (eval a) (eval (prd i l)). + Proof. + intros [|b l]; simpl; reflexivity. + Qed. + Lemma eval_prd_app : forall (h k: nelist T), eval (prd i (h++k)) == @Bin.value _ _ (e_bin i) (eval (prd i h)) (eval (prd i k)). + Proof. + induction h; intro k. simpl; try reflexivity. + simpl appne. rewrite 2 eval_prd_cons, IHh, law_assoc. reflexivity. + Qed. + + Inductive compat_prd_unit : @m idx (nelist T) -> Prop := + | cpu_left : forall x, is_unit x = true -> compat_prd_unit (left x) + | cpu_right : forall m, compat_prd_unit (right m) + . + + Lemma compat_prd_unit_return x: compat_prd_unit (return_prd i is_unit x). + Proof. + unfold return_prd. + case (is_prd_spec); intros; try constructor; auto. + Qed. + + Lemma compat_prd_unit_add : forall x h, + compat_prd_unit h + -> + compat_prd_unit + (add_to_prd i is_unit x + h). + Proof. + intros. + unfold add_to_prd. + unfold comp. + case (is_prd_spec); intros; try constructor; auto. + unfold comp; case h; try constructor. + unfold comp; case h; try constructor. + Qed. + + + Instance compat_prd_Unit : forall p, compat_prd_unit (left p) -> + @Unit X R (Bin.value (e_bin i)) (eval (unit p)). + Proof. + intros. + inversion H; subst. apply is_unit_prd_Unit. assumption. + Qed. + + Lemma z0' : forall l (r: @m idx (nelist T)), compat_prd_unit r -> + eval (prd i (run_list (comp (@appne T) l r))) == eval (prd i ((appne (l) (run_list r)))). + Proof. + intros. + unfold comp. unfold run_list. case_eq r; intros; auto; subst. + rewrite eval_prd_app. + rewrite eval_prd_nil. + apply compat_prd_Unit in H. rewrite law_neutral_right. reflexivity. + reflexivity. + Qed. + + Lemma z1' a : eval (prd i (run_list (return_prd i is_unit a))) == eval (prd i (nil a)). + Proof. + intros. unfold return_prd. unfold run_list. + case (is_prd_spec); intros; subst; reflexivity. + Qed. + Lemma z2' : forall u x, compat_prd_unit x -> + eval (prd i ( run_list + (add_to_prd i is_unit u x))) == @Bin.value _ _ (e_bin i) (eval u) (eval (prd i (run_list x))). + Proof. + intros u x Hix. + unfold add_to_prd. + case (is_prd_spec); intros; subst. + rewrite z0' by auto. rewrite eval_prd_app. reflexivity. + apply is_unit_prd_Unit in H. rewrite law_neutral_left. reflexivity. + rewrite z0' by auto. rewrite eval_prd_app. reflexivity. + Qed. + + End prd_correctness. + + + + + Lemma eval_norm_lists i (Hnorm: forall u, eval (norm u) == eval u) : forall h, eval (prd i (norm_lists norm i h)) == eval (prd i h). + Proof. + unfold norm_lists. + assert (H : forall h : nelist T, + eval (prd i (run_list (norm_lists_ i (is_unit_of i) norm h))) == + eval (prd i h) + /\ compat_prd_unit (is_unit_of i) (norm_lists_ i (is_unit_of i) norm h)). + + + induction h as [a | a h [IHh IHh']]; simpl norm_lists_; split. + rewrite z1'. simpl. apply Hnorm. + + apply compat_prd_unit_return. + + rewrite z2'. rewrite IHh. rewrite eval_prd_cons. rewrite Hnorm. reflexivity. apply is_unit_of_Unit. + auto. + + apply compat_prd_unit_add. auto. + apply H. + Defined. + + (** correctness of the normalisation function *) + + Theorem eval_norm: forall u, eval (norm u) == eval u + with eval_norm_aux: forall i (l: vT i) (f: Sym.type_of i), + Proper (@Sym.rel_of X R i) f -> eval_aux (vnorm l) f == eval_aux l f. + Proof. + induction u as [ p m | p l | ? | ?]; simpl norm. + case_eq (is_commutative p); intros. + rewrite sum'_sum. + apply eval_norm_msets; auto. + reflexivity. + + rewrite prd'_prd. + apply eval_norm_lists; auto. + + apply eval_norm_aux, Sym.morph. + + reflexivity. + + induction l; simpl; intros f Hf. reflexivity. + rewrite eval_norm. apply IHl, Hf; reflexivity. + Qed. + + + (** corollaries, for goal normalisation or decision *) + + Lemma normalise : forall (u v: T), eval (norm u) == eval (norm v) -> eval u == eval v. + Proof. intros u v. rewrite 2 eval_norm. trivial. Qed. + + Lemma compare_reflect_eq: forall u v, compare u v = Eq -> eval u == eval v. + Proof. intros u v. case (tcompare_weak_spec u v); intros; try congruence. reflexivity. Qed. + + Lemma decide: forall (u v: T), compare (norm u) (norm v) = Eq -> eval u == eval v. + Proof. intros u v H. apply normalise. apply compare_reflect_eq. apply H. Qed. + + Lemma lift_normalise {S} {H : AAC_lift S R}: + forall (u v: T), (let x := norm u in let y := norm v in + S (eval x) (eval y)) -> S (eval u) (eval v). + Proof. destruct H. intros u v; simpl; rewrite 2 eval_norm. trivial. Qed. + +End s. +End Internal. + +Local Ltac internal_normalize := + let x := fresh in let y := fresh in + intro x; intro y; vm_compute in x; vm_compute in y; unfold x; unfold y; + compute [Internal.eval Utils.fold_map Internal.copy Prect]; simpl. + + +(** * Lemmas for performing transitivity steps + given an instance of AAC_lift *) + +Section t. + Context `{AAC_lift}. + + Lemma lift_transitivity_left (y x z : X): E x y -> R y z -> R x z. + Proof. destruct H as [Hequiv Hproper]; intros G;rewrite G. trivial. Qed. + + Lemma lift_transitivity_right (y x z : X): E y z -> R x y -> R x z. + Proof. destruct H as [Hequiv Hproper]; intros G. rewrite G. trivial. Qed. + + Lemma lift_reflexivity {HR :Reflexive R}: forall x y, E x y -> R x y. + Proof. destruct H. intros ? ? G. rewrite G. reflexivity. Qed. + +End t. + +Declare ML Module "aac_plugin". diff --git a/theories/Caveats.v b/theories/Caveats.v new file mode 100644 index 0000000..d7824cd --- /dev/null +++ b/theories/Caveats.v @@ -0,0 +1,376 @@ +(***************************************************************************) +(* This is part of aac_tactics, it is distributed under the terms of the *) +(* GNU Lesser General Public License version 3 *) +(* (see file LICENSE for more details) *) +(* *) +(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) +(***************************************************************************) + +(** * Currently known caveats and limitations of the [aac_tactics] library. + + Depending on your installation, either uncomment the following two + lines, or add them to your .coqrc files, replacing "." + with the path to the [aac_tactics] library +*) + +Require NArith Minus. + +From AAC_tactics +Require Import AAC. +From AAC_tactics +Require Instances. + +(** ** Limitations *) + +(** *** 1. Dependent parameters + The type of the rewriting hypothesis must be of the form + + [forall (x_1: T_1) ... (x_n: T_n), R l r], + + where [R] is a relation over some type [T] and such that for all + variable [x_i] appearing in the left-hand side ([l]), we actually + have [T_i]=[T]. The goal should be of the form [S g d], where [S] + is a relation on [T]. + + In other words, we cannot instantiate arguments of an exogeneous + type. *) + +Section parameters. + + Context {X} {R} {E: @Equivalence X R} + {plus} {plus_A: Associative R plus} {plus_C: Commutative R plus} + {plus_Proper: Proper (R ==> R ==> R) plus} + {zero} {Zero: Unit R plus zero}. + + Notation "x == y" := (R x y) (at level 70). + Notation "x + y" := (plus x y) (at level 50, left associativity). + Notation "0" := (zero). + + Variable f: nat -> X -> X. + + (** in [Hf], the parameter [n] has type [nat], it cannot be instantiated automatically *) + Hypothesis Hf: forall n x, f n x + x == x. + Hypothesis Hf': forall n, Proper (R ==> R) (f n). + + Goal forall a b k, a + f k (b+a) + b == a+b. + intros. + Fail aac_rewrite Hf. (** [aac_rewrite] does not instantiate [n] automatically *) + aac_rewrite (Hf k). (** of course, this argument can be given explicitly *) + aac_reflexivity. + Qed. + + (** for the same reason, we cannot handle higher-order parameters (here, [g])*) + Hypothesis H : forall g x y, g x + g y == g (x + y). + Variable g : X -> X. + Hypothesis Hg : Proper (R ==> R) g. + Goal forall a b c, g a + g b + g c == g (a + b + c). + intros. + Fail aac_rewrite H. + do 2 aac_rewrite (H g). aac_reflexivity. + Qed. + +End parameters. + + +(** *** 2. Exogeneous morphisms + + We do not handle `exogeneous' morphisms: morphisms that move from + type [T] to some other type [T']. *) + +Section morphism. + Import NArith Minus. + Open Scope nat_scope. + + (** Typically, although [N_of_nat] is a proper morphism from + [@eq nat] to [@eq N], we cannot rewrite under [N_of_nat] *) + Goal forall a b: nat, N_of_nat (a+b-(b+a)) = 0%N. + intros. + Fail aac_rewrite minus_diag. + Abort. + + + (* More generally, this prevents us from rewriting under + propositional contexts *) + Context {P} {HP : Proper (@eq nat ==> iff) P}. + Hypothesis H : P 0. + + Goal forall a b, P (a + b - (b + a)). + intros a b. + Fail aac_rewrite minus_diag. + (** a solution is to introduce an evar to replace the part to be + rewritten. This tiresome process should be improved in the + future. Here, it can be done using eapply and the morphism. *) + eapply HP. + aac_rewrite minus_diag. + reflexivity. + exact H. + Qed. + + Goal forall a b, a+b-(b+a) = 0 /\ b-b = 0. + intros. + (** similarly, we need to bring equations to the toplevel before + being able to rewrite *) + Fail aac_rewrite minus_diag. + split; aac_rewrite minus_diag; reflexivity. + Qed. + +End morphism. + + +(** *** 3. Treatment of variance with inequations. + + We do not take variance into account when we compute the set of + solutions to a matching problem modulo AC. As a consequence, + [aac_instances] may propose solutions for which [aac_rewrite] will + fail, due to the lack of adequate morphisms *) + +Section ineq. + + Import ZArith. + Import Instances.Z. + Open Scope Z_scope. + + Instance Zplus_incr: Proper (Z.le ==> Z.le ==> Z.le) Zplus. + Proof. intros ? ? H ? ? H'. apply Zplus_le_compat; assumption. Qed. + + Hypothesis H: forall x, x+x <= x. + Goal forall a b c, c + - (a + a) + b + b <= c. + intros. + (** this fails because the first solution is not valid ([Zopp] is not increasing), *) + Fail aac_rewrite H. + aac_instances H. + (** on the contrary, the second solution is valid: *) + aac_rewrite H at 1. + (** Currently, we cannot filter out such invalid solutions in an easy way; + this should be fixed in the future *) + Abort. + +End ineq. + + + +(** ** Caveats *) + +(** *** 1. Special treatment for units. + [S O] is considered as a unit for multiplication whenever a [Peano.mult] + appears in the goal. The downside is that [S x] does not match [1], + and [1] does not match [S(0+0)] whenever [Peano.mult] appears in + the goal. *) + +Section Peano. + Import Instances.Peano. + + Hypothesis H : forall x, x + S x = S (x+x). + + Goal 1 = 1. + (** ok (no multiplication around), [x] is instantiated with [O] *) + aacu_rewrite H. + Abort. + + Goal 1*1 = 1. + (** fails since 1 is seen as a unit, not the application of the + morphism [S] to the constant [O] *) + Fail aacu_rewrite H. + Abort. + + Hypothesis H': forall x, x+1 = 1+x. + + Goal forall a, a+S(0+0) = 1+a. + (** ok (no multiplication around), [x] is instantiated with [a]*) + intro. aac_rewrite H'. + Abort. + + Goal forall a, a*a+S(0+0) = 1+a*a. + (** fails: although [S(0+0)] is understood as the application of + the morphism [S] to the constant [O], it is not recognised + as the unit [S O] of multiplication *) + intro. Fail aac_rewrite H'. + Abort. + + (** More generally, similar counter-intuitive behaviours can appear + when declaring an applied morphism as an unit. *) + +End Peano. + + + +(** *** 2. Existential variables. +We implemented an algorithm for _matching_ modulo AC, not for +_unifying_ modulo AC. As a consequence, existential variables +appearing in a goal are considered as constants, they will not be +instantiated. *) + +Section evars. + Import ZArith. + Import Instances.Z. + + Variable P: Prop. + Hypothesis H: forall x y, x+y+x = x -> P. + Hypothesis idem: forall x, x+x = x. + Goal P. + eapply H. + aac_rewrite idem. (** this works: [x] is instantiated with an evar *) + instantiate (2 := 0). + symmetry. aac_reflexivity. (** this does work but there are remaining evars in the end *) + Abort. + + Hypothesis H': forall x, 3+x = x -> P. + Goal P. + eapply H'. + Fail aac_rewrite idem. (** this fails since we do not instantiate evars *) + Abort. +End evars. + + +(** *** 3. Distinction between [aac_rewrite] and [aacu_rewrite] *) + +Section U. + Context {X} {R} {E: @Equivalence X R} + {dot} {dot_A: Associative R dot} {dot_Proper: Proper (R ==> R ==> R) dot} + {one} {One: Unit R dot one}. + + Infix "==" := R (at level 70). + Infix "*" := dot. + Notation "1" := one. + + (** In some situations, the [aac_rewrite] tactic allows + instantiations of a variable with a unit, when the variable occurs + directly under a function symbol: *) + + Variable f : X -> X. + Hypothesis Hf : Proper (R ==> R) f. + Hypothesis dot_inv_left : forall x, f x*x == x. + Goal f 1 == 1. + aac_rewrite dot_inv_left. reflexivity. + Qed. + + (** This behaviour seems desirable in most situations: these + solutions with units are less peculiar than the other ones, since + the unit comes from the goal. However, this policy is not properly + enforced for now (hard to do with the current algorithm): *) + + Hypothesis dot_inv_right : forall x, x*f x == x. + Goal f 1 == 1. + Fail aac_rewrite dot_inv_right. + aacu_rewrite dot_inv_right. reflexivity. + Qed. + +End U. + +(** *** 4. Rewriting units *) +Section V. + Context {X} {R} {E: @Equivalence X R} + {dot} {dot_A: Associative R dot} {dot_Proper: Proper (R ==> R ==> R) dot} + {one} {One: Unit R dot one}. + + Infix "==" := R (at level 70). + Infix "*" := dot. + Notation "1" := one. + + (** [aac_rewrite] uses the symbols appearing in the goal and the + hypothesis to infer the AC and A operations. In the following + example, [dot] appears neither in the left-hand-side of the goal, + nor in the right-hand side of the hypothesis. Hence, 1 is not + recognised as a unit. To circumvent this problem, we can force + [aac_rewrite] to take into account a given operation, by giving + it an extra argument. This extra argument seems useful only in + this peculiar case. *) + + Lemma inv_unique: forall x y y', x*y == 1 -> y'*x == 1 -> y==y'. + Proof. + intros x y y' Hxy Hy'x. + aac_instances <- Hy'x [dot]. + aac_rewrite <- Hy'x at 1 [dot]. + aac_rewrite Hxy. aac_reflexivity. + Qed. +End V. + +(** *** 5. Rewriting too much things. *) +Section W. + Variables a b c: nat. + Hypothesis H: 0 = c. + + Goal b*(a+a) <= b*(c+a+a+1). + + (** [aac_rewrite] finds a pattern modulo AC that matches a given + hypothesis, and then makes a call to [setoid_rewrite]. This + [setoid_rewrite] can unfortunately make several rewrites (in a + non-intuitive way: below, the [1] in the right-hand side is + rewritten into [S c]) *) + aac_rewrite H. + + (** To this end, we provide a companion tactic to [aac_rewrite] + and [aacu_rewrite], that makes the transitivity step, but not the + setoid_rewrite: + + This allows the user to select the relevant occurrences in which + to rewrite. *) + aac_pattern H at 2. setoid_rewrite H at 1. + Abort. + +End W. + +(** *** 6. Rewriting nullifiable patterns. *) +Section Z. + +(** If the pattern of the rewritten hypothesis does not contain "hard" +symbols (like constants, function symbols, AC or A symbols without +units), there can be infinitely many subterms such that the pattern +matches: it is possible to build "subterms" modulo ACU that make the +size of the term increase (by making neutral elements appear in a +layered fashion). Hence, we settled with heuristics to propose only +"some" of these solutions. In such cases, the tactic displays a +(conservative) warning. *) + +Variables a b c: nat. +Variable f: nat -> nat. +Hypothesis H0: forall x, 0 = x - x. +Hypothesis H1: forall x, 1 = x * x. + +Goal a+b*c = c. + aac_instances H0. + (** In this case, only three solutions are proposed, while there are + infinitely many solutions. E.g. + - a+b*c*(1+[]) + - a+b*c*(1+0*(1+ [])) + - ... + *) +Abort. + +(** **** If the pattern is a unit or can be instantiated to be equal + to a unit: + + The heuristic is to make the unit appear at each possible position + in the term, e.g. transforming [a] into [1*a] and [a*1], but this + process is not recursive (we will not transform [1*a]) into + [(1+0*1)*a] *) + +Goal a+b+c = c. + + aac_instances H0 [mult]. + (** 1 solution, we miss solutions like [(a+b+c*(1+0*(1+[])))] and so on *) + + aac_instances H1 [mult]. + (** 7 solutions, we miss solutions like [(a+b+c+0*(1+0*[]))]*) +Abort. + +(** *** Another example of the former case is the following, where the hypothesis can be instantiated to be equal to [1] *) +Hypothesis H : forall x y, (x+y)*x = x*x + y *x. +Goal a*a+b*a + c = c. + (** Here, only one solution if we use the aac_instance tactic *) + aac_instances <- H. + + (** There are 8 solutions using aacu_instances (but, here, + there are infinitely many different solutions). We miss e.g. [a*a +b*a + + (x*x + y*x)*c], which seems to be more peculiar. *) + aacu_instances <- H. + + (** The 7 last solutions are the same as if we were matching [1] *) + aacu_instances H1. Abort. + +(** The behavior of the tactic is not satisfying in this case. It is +still unclear how to handle properly this kind of situation : we plan +to investigate on this in the future *) + +End Z. + diff --git a/theories/Instances.v b/theories/Instances.v new file mode 100644 index 0000000..d007ceb --- /dev/null +++ b/theories/Instances.v @@ -0,0 +1,261 @@ +(***************************************************************************) +(* This is part of aac_tactics, it is distributed under the terms of the *) +(* GNU Lesser General Public License version 3 *) +(* (see file LICENSE for more details) *) +(* *) +(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) +(***************************************************************************) + +Require List. +Require Arith NArith Max Min. +Require ZArith Zminmax. +Require QArith Qminmax. +Require Relations. + +From AAC_tactics +Require Export AAC. + +(** Instances for aac_rewrite.*) + + +(* This one is not declared as an instance: this interferes badly with setoid_rewrite *) +Lemma eq_subr {X} {R} `{@Reflexive X R}: subrelation eq R. +Proof. intros x y ->. reflexivity. Qed. + +(* At the moment, all the instances are exported even if they are packaged into modules. Even using LocalInstances in fact*) + +Module Peano. + Import Arith NArith Max Min. + Instance aac_plus_Assoc : Associative eq plus := plus_assoc. + Instance aac_plus_Comm : Commutative eq plus := plus_comm. + + Instance aac_mult_Comm : Commutative eq mult := mult_comm. + Instance aac_mult_Assoc : Associative eq mult := mult_assoc. + + Instance aac_min_Comm : Commutative eq min := min_comm. + Instance aac_min_Assoc : Associative eq min := min_assoc. + + Instance aac_max_Comm : Commutative eq max := max_comm. + Instance aac_max_Assoc : Associative eq max := max_assoc. + + Instance aac_one : Unit eq mult 1 := Build_Unit eq mult 1 mult_1_l mult_1_r. + Instance aac_zero_plus : Unit eq plus O := Build_Unit eq plus (O) plus_0_l plus_0_r. + Instance aac_zero_max : Unit eq max O := Build_Unit eq max 0 max_0_l max_0_r. + + + (* We also provide liftings from le to eq *) + Instance preorder_le : PreOrder le := Build_PreOrder _ le_refl le_trans. + Instance lift_le_eq : AAC_lift le eq := Build_AAC_lift eq_equivalence _. + +End Peano. + + +Module Z. + Import ZArith Zminmax. + Open Scope Z_scope. + Instance aac_Zplus_Assoc : Associative eq Zplus := Zplus_assoc. + Instance aac_Zplus_Comm : Commutative eq Zplus := Zplus_comm. + + Instance aac_Zmult_Comm : Commutative eq Zmult := Zmult_comm. + Instance aac_Zmult_Assoc : Associative eq Zmult := Zmult_assoc. + + Instance aac_Zmin_Comm : Commutative eq Z.min := Z.min_comm. + Instance aac_Zmin_Assoc : Associative eq Z.min := Z.min_assoc. + + Instance aac_Zmax_Comm : Commutative eq Z.max := Z.max_comm. + Instance aac_Zmax_Assoc : Associative eq Z.max := Z.max_assoc. + + Instance aac_one : Unit eq Zmult 1 := Build_Unit eq Zmult 1 Zmult_1_l Zmult_1_r. + Instance aac_zero_Zplus : Unit eq Zplus 0 := Build_Unit eq Zplus 0 Zplus_0_l Zplus_0_r. + + (* We also provide liftings from le to eq *) + Instance preorder_Zle : PreOrder Z.le := Build_PreOrder _ Z.le_refl Z.le_trans. + Instance lift_le_eq : AAC_lift Z.le eq := Build_AAC_lift eq_equivalence _. + +End Z. + +Module Lists. + Import List. + Instance aac_append_Assoc {A} : Associative eq (@app A) := @app_assoc A. + Instance aac_nil_append {A} : @Unit (list A) eq (@app A) (@nil A) := Build_Unit _ (@app A) (@nil A) (@app_nil_l A) (@app_nil_r A). + Instance aac_append_Proper {A} : Proper (eq ==> eq ==> eq) (@app A). + Proof. + repeat intro. + subst. + reflexivity. + Qed. +End Lists. + + +Module N. + Import NArith. + Open Scope N_scope. + Instance aac_Nplus_Assoc : Associative eq Nplus := Nplus_assoc. + Instance aac_Nplus_Comm : Commutative eq Nplus := Nplus_comm. + + Instance aac_Nmult_Comm : Commutative eq Nmult := Nmult_comm. + Instance aac_Nmult_Assoc : Associative eq Nmult := Nmult_assoc. + + Instance aac_Nmin_Comm : Commutative eq N.min := N.min_comm. + Instance aac_Nmin_Assoc : Associative eq N.min := N.min_assoc. + + Instance aac_Nmax_Comm : Commutative eq N.max := N.max_comm. + Instance aac_Nmax_Assoc : Associative eq N.max := N.max_assoc. + + Instance aac_one : Unit eq Nmult (1)%N := Build_Unit eq Nmult (1)%N Nmult_1_l Nmult_1_r. + Instance aac_zero : Unit eq Nplus (0)%N := Build_Unit eq Nplus (0)%N Nplus_0_l Nplus_0_r. + Instance aac_zero_max : Unit eq N.max 0 := Build_Unit eq N.max 0 N.max_0_l N.max_0_r. + + (* We also provide liftings from le to eq *) + Instance preorder_le : PreOrder N.le := Build_PreOrder N.le N.le_refl N.le_trans. + Instance lift_le_eq : AAC_lift N.le eq := Build_AAC_lift eq_equivalence _. + +End N. + +Module P. + Import NArith. + Open Scope positive_scope. + Instance aac_Pplus_Assoc : Associative eq Pplus := Pplus_assoc. + Instance aac_Pplus_Comm : Commutative eq Pplus := Pplus_comm. + + Instance aac_Pmult_Comm : Commutative eq Pmult := Pmult_comm. + Instance aac_Pmult_Assoc : Associative eq Pmult := Pmult_assoc. + + Instance aac_Pmin_Comm : Commutative eq Pos.min := Pos.min_comm. + Instance aac_Pmin_Assoc : Associative eq Pos.min := Pos.min_assoc. + + Instance aac_Pmax_Comm : Commutative eq Pos.max := Pos.max_comm. + Instance aac_Pmax_Assoc : Associative eq Pos.max := Pos.max_assoc. + + Instance aac_one : Unit eq Pmult 1 := Build_Unit eq Pmult 1 _ Pmult_1_r. + intros; reflexivity. Qed. (* TODO : add this lemma in the stdlib *) + Instance aac_one_max : Unit eq Pos.max 1 := Build_Unit eq Pos.max 1 Pos.max_1_l Pos.max_1_r. + + (* We also provide liftings from le to eq *) + Instance preorder_le : PreOrder Pos.le := Build_PreOrder Pos.le Pos.le_refl Pos.le_trans. + Instance lift_le_eq : AAC_lift Pos.le eq := Build_AAC_lift eq_equivalence _. +End P. + +Module Q. + Import QArith Qminmax. + Instance aac_Qplus_Assoc : Associative Qeq Qplus := Qplus_assoc. + Instance aac_Qplus_Comm : Commutative Qeq Qplus := Qplus_comm. + + Instance aac_Qmult_Comm : Commutative Qeq Qmult := Qmult_comm. + Instance aac_Qmult_Assoc : Associative Qeq Qmult := Qmult_assoc. + + Instance aac_Qmin_Comm : Commutative Qeq Qmin := Q.min_comm. + Instance aac_Qmin_Assoc : Associative Qeq Qmin := Q.min_assoc. + + Instance aac_Qmax_Comm : Commutative Qeq Qmax := Q.max_comm. + Instance aac_Qmax_Assoc : Associative Qeq Qmax := Q.max_assoc. + + Instance aac_one : Unit Qeq Qmult 1 := Build_Unit Qeq Qmult 1 Qmult_1_l Qmult_1_r. + Instance aac_zero_Qplus : Unit Qeq Qplus 0 := Build_Unit Qeq Qplus 0 Qplus_0_l Qplus_0_r. + + (* We also provide liftings from le to eq *) + Instance preorder_le : PreOrder Qle := Build_PreOrder Qle Qle_refl Qle_trans. + Instance lift_le_eq : AAC_lift Qle Qeq := Build_AAC_lift QOrderedType.QOrder.TO.eq_equiv _. + +End Q. + +Module Prop_ops. + Instance aac_or_Assoc : Associative iff or. Proof. unfold Associative; tauto. Qed. + Instance aac_or_Comm : Commutative iff or. Proof. unfold Commutative; tauto. Qed. + Instance aac_and_Assoc : Associative iff and. Proof. unfold Associative; tauto. Qed. + Instance aac_and_Comm : Commutative iff and. Proof. unfold Commutative; tauto. Qed. + Instance aac_True : Unit iff or False. Proof. constructor; firstorder. Qed. + Instance aac_False : Unit iff and True. Proof. constructor; firstorder. Qed. + + Program Instance aac_not_compat : Proper (iff ==> iff) not. + Solve All Obligations with firstorder. + + Instance lift_impl_iff : AAC_lift Basics.impl iff := Build_AAC_lift _ _. +End Prop_ops. + +Module Bool. + Instance aac_orb_Assoc : Associative eq orb. Proof. unfold Associative; firstorder. Qed. + Instance aac_orb_Comm : Commutative eq orb. Proof. unfold Commutative; firstorder. Qed. + Instance aac_andb_Assoc : Associative eq andb. Proof. unfold Associative; firstorder. Qed. + Instance aac_andb_Comm : Commutative eq andb. Proof. unfold Commutative; firstorder. Qed. + Instance aac_true : Unit eq orb false. Proof. constructor; firstorder. Qed. + Instance aac_false : Unit eq andb true. Proof. constructor; intros [|];firstorder. Qed. + + Instance negb_compat : Proper (eq ==> eq) negb. Proof. intros [|] [|]; auto. Qed. +End Bool. + +Module Relations. + Import Relations.Relations. + Section defs. + Variable T : Type. + Variables R S: relation T. + Definition inter : relation T := fun x y => R x y /\ S x y. + Definition compo : relation T := fun x y => exists z : T, R x z /\ S z y. + Definition negr : relation T := fun x y => ~ R x y. + (* union and converse are already defined in the standard library *) + + Definition bot : relation T := fun _ _ => False. + Definition top : relation T := fun _ _ => True. + End defs. + + Instance eq_same_relation T : Equivalence (same_relation T). Proof. firstorder. Qed. + + Instance aac_union_Comm T : Commutative (same_relation T) (union T). Proof. unfold Commutative; compute; intuition. Qed. + Instance aac_union_Assoc T : Associative (same_relation T) (union T). Proof. unfold Associative; compute; intuition. Qed. + Instance aac_bot T : Unit (same_relation T) (union T) (bot T). Proof. constructor; compute; intuition. Qed. + + Instance aac_inter_Comm T : Commutative (same_relation T) (inter T). Proof. unfold Commutative; compute; intuition. Qed. + Instance aac_inter_Assoc T : Associative (same_relation T) (inter T). Proof. unfold Associative; compute; intuition. Qed. + Instance aac_top T : Unit (same_relation T) (inter T) (top T). Proof. constructor; compute; intuition. Qed. + + (* note that we use [eq] directly as a neutral element for composition *) + Instance aac_compo T : Associative (same_relation T) (compo T). Proof. unfold Associative; compute; firstorder. Qed. + Instance aac_eq T : Unit (same_relation T) (compo T) (eq). Proof. compute; firstorder subst; trivial. Qed. + + Instance negr_compat T : Proper (same_relation T ==> same_relation T) (negr T). + Proof. compute. firstorder. Qed. + + Instance transp_compat T : Proper (same_relation T ==> same_relation T) (transp T). + Proof. compute. firstorder. Qed. + + Instance clos_trans_incr T : Proper (inclusion T ==> inclusion T) (clos_trans T). + Proof. + intros R S H x y Hxy. induction Hxy. + constructor 1. apply H. assumption. + econstructor 2; eauto 3. + Qed. + Instance clos_trans_compat T: Proper (same_relation T ==> same_relation T) (clos_trans T). + Proof. intros R S H; split; apply clos_trans_incr, H. Qed. + + Instance clos_refl_trans_incr T : Proper (inclusion T ==> inclusion T) (clos_refl_trans T). + Proof. + intros R S H x y Hxy. induction Hxy. + constructor 1. apply H. assumption. + constructor 2. + econstructor 3; eauto 3. + Qed. + Instance clos_refl_trans_compat T : Proper (same_relation T ==> same_relation T) (clos_refl_trans T). + Proof. intros R S H; split; apply clos_refl_trans_incr, H. Qed. + + Instance preorder_inclusion T : PreOrder (inclusion T). + Proof. constructor; unfold Reflexive, Transitive, inclusion; intuition. Qed. + + Instance lift_inclusion_same_relation T: AAC_lift (inclusion T) (same_relation T) := + Build_AAC_lift (eq_same_relation T) _. + Proof. firstorder. Qed. + +End Relations. + +Module All. + Export Peano. + Export Z. + Export P. + Export N. + Export Prop_ops. + Export Bool. + Export Relations. +End All. + +(* Here, we should not see any instance of our classes. + Print HintDb typeclass_instances. +*) diff --git a/theories/Tutorial.v b/theories/Tutorial.v new file mode 100644 index 0000000..e403d92 --- /dev/null +++ b/theories/Tutorial.v @@ -0,0 +1,401 @@ +(***************************************************************************) +(* This is part of aac_tactics, it is distributed under the terms of the *) +(* GNU Lesser General Public License version 3 *) +(* (see file LICENSE for more details) *) +(* *) +(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) +(***************************************************************************) + +(** * Tutorial for using the [aac_tactics] library. + + Depending on your installation, either modify the following two + lines, or add them to your .coqrc files, replacing "." with the + path to the [aac_tactics] library. *) + +Require Arith ZArith. + +From AAC_tactics +Require Import AAC. +From AAC_tactics +Require Instances. + +(** ** Introductory example + + Here is a first example with relative numbers ([Z]): one can + rewrite an universally quantified hypothesis modulo the + associativity and commutativity of [Zplus]. *) + +Section introduction. + + Import ZArith. + Import Instances.Z. + + Variables a b c : Z. + Hypothesis H: forall x, x + Z.opp x = 0. + Goal a + b + c + Z.opp (c + a) = b. + aac_rewrite H. + aac_reflexivity. + Qed. + Goal a + c + Z.opp (b + a + Z.opp b) = c. + do 2 aac_rewrite H. + reflexivity. + Qed. + + (** Notes: + - the tactic handles arbitrary function symbols like [Zopp] (as + long as they are proper morphisms w.r.t. the considered + equivalence relation); + - here, ring would have done the job. + *) + +End introduction. + + +(** ** Usage + + One can also work in an abstract context, with arbitrary + associative and commutative operators. (Note that one can declare + several operations of each kind.) *) + +Section base. + Context {X} {R} {E: Equivalence R} + {plus} + {dot} + {zero} + {one} + {dot_A: @Associative X R dot } + {plus_A: @Associative X R plus } + {plus_C: @Commutative X R plus } + {dot_Proper :Proper (R ==> R ==> R) dot} + {plus_Proper :Proper (R ==> R ==> R) plus} + {Zero : Unit R plus zero} + {One : Unit R dot one} + . + + Notation "x == y" := (R x y) (at level 70). + Notation "x * y" := (dot x y) (at level 40, left associativity). + Notation "1" := (one). + Notation "x + y" := (plus x y) (at level 50, left associativity). + Notation "0" := (zero). + + (** In the very first example, [ring] would have solved the + goal. Here, since [dot] does not necessarily distribute over [plus], + it is not possible to rely on it. *) + + Section reminder. + Hypothesis H : forall x, x * x == x. + Variables a b c : X. + + Goal (a+b+c)*(c+a+b) == a+b+c. + aac_rewrite H. + aac_reflexivity. + Qed. + + (** The tactic starts by normalising terms, so that trailing units + are always eliminated. *) + + Goal ((a+b)+0+c)*((c+a)+b*1) == a+b+c. + aac_rewrite H. + aac_reflexivity. + Qed. + End reminder. + + (** The tactic can deal with "proper" morphisms of arbitrary arity + (here [f] and [g], or [Zopp] earlier): it rewrites under such + morphisms ([g]), and, more importantly, it is able to reorder + terms modulo AC under these morphisms ([f]). *) + + Section morphisms. + Variable f : X -> X -> X. + Hypothesis Hf : Proper (R ==> R ==> R) f. + Variable g : X -> X. + Hypothesis Hg : Proper (R ==> R) g. + + Variable a b: X. + Hypothesis H : forall x y, x+f (b+y) x == y+x. + Goal g ((f (a+b) a) + a) == g (a+a). + aac_rewrite H. + reflexivity. + Qed. + End morphisms. + + (** *** Selecting what and where to rewrite + + There are sometimes several solutions to the matching problem. We + now show how to interact with the tactic to select the desired + one. *) + + Section occurrence. + Variable f : X -> X. + Variable a : X. + Hypothesis Hf : Proper (R ==> R) f. + Hypothesis H : forall x, x + x == x. + + Goal f(a+a)+f(a+a) == f a. + (** In case there are several possible solutions, one can print + the different solutions using the [aac_instances] tactic (in + proof-general, look at buffer *coq* ): *) + aac_instances H. + (** the default choice is the occurrence with the smallest + possible context (number 0), but one can choose the desired + one; *) + aac_rewrite H at 1. + (** now the goal is [f a + f a == f a], there is only one solution. *) + aac_rewrite H. + reflexivity. + Qed. + + End occurrence. + + Section subst. + Variables a b c d : X. + Hypothesis H: forall x y, a*x*y*b == a*(x+y)*b. + Hypothesis H': forall x, x + x == x. + + Goal a*c*d*c*d*b == a*c*d*b. + (** Here, there is only one possible occurrence, but several substitutions; *) + aac_instances H. + (** one can select them with the proper keyword. *) + aac_rewrite H subst 1. + aac_rewrite H'. + aac_reflexivity. + Qed. + End subst. + + (** As expected, one can use both keywords together to select the + occurrence and the substitution. We also provide a keyword to + specify that the rewrite should be done in the right-hand side of + the equation. *) + + Section both. + Variables a b c d : X. + Hypothesis H: forall x y, a*x*y*b == a*(x+y)*b. + Hypothesis H': forall x, x + x == x. + + Goal a*c*d*c*d*b*b == a*(c*d*b)*b. + aac_instances H. + aac_rewrite H at 1 subst 1. + aac_instances H. + aac_rewrite H. + aac_rewrite H'. + aac_rewrite H at 0 subst 1 in_right. + aac_reflexivity. + Qed. + + End both. + + (** *** Distinction between [aac_rewrite] and [aacu_rewrite]: + + [aac_rewrite] rejects solutions in which variables are instantiated + by units, while the companion tactic, [aacu_rewrite] allows such + solutions. *) + + Section dealing_with_units. + Variables a b c: X. + Hypothesis H: forall x, a*x*a == x. + Goal a*a == 1. + (** Here, [x] must be instantiated with [1], so that the [aac_*] + tactics give no solutions; *) + try aac_instances H. + (** while we get solutions with the [aacu_*] tactics. *) + aacu_instances H. + aacu_rewrite H. + reflexivity. + Qed. + + (** We introduced this distinction because it allows us to rule + out dummy cases in common situations: *) + + Hypothesis H': forall x y z, x*y + x*z == x*(y+z). + Goal a*b*c + a*c + a*b == a*(c+b*(1+c)). + (** 6 solutions without units, *) + aac_instances H'. + aac_rewrite H' at 0. + (** more than 52 with units. *) + aacu_instances H'. + Abort. + + End dealing_with_units. +End base. + +(** *** Declaring instances + + To use one's own operations: it suffices to declare them as + instances of our classes. (Note that the following instances are + already declared in file [Instances.v].) *) + +Section Peano. + Import Arith. + + Instance aac_plus_Assoc : Associative eq plus := plus_assoc. + Instance aac_plus_Comm : Commutative eq plus := plus_comm. + + Instance aac_one : Unit eq mult 1 := + Build_Unit eq mult 1 mult_1_l mult_1_r. + Instance aac_zero_plus : Unit eq plus O := + Build_Unit eq plus (O) plus_0_l plus_0_r. + + + (** Two (or more) operations may share the same units: in the + following example, [0] is understood as the unit of [max] as well as + the unit of [plus]. *) + + Instance aac_max_Comm : Commutative eq Max.max := Max.max_comm. + Instance aac_max_Assoc : Associative eq Max.max := Max.max_assoc. + + Instance aac_zero_max : Unit eq Max.max O := + Build_Unit eq Max.max 0 Max.max_0_l Max.max_0_r. + + Variable a b c : nat. + Goal Max.max (a + 0) 0 = a. + aac_reflexivity. + Qed. + + (** Furthermore, several operators can be mixed: *) + + Hypothesis H : forall x y z, Max.max (x + y) (x + z) = x+ Max.max y z. + + Goal Max.max (a + b) (c + (a * 1)) = Max.max c b + a. + aac_instances H. aac_rewrite H. aac_reflexivity. + Qed. + Goal Max.max (a + b) (c + Max.max (a*1+0) 0) = a + Max.max b c. + aac_instances H. aac_rewrite H. aac_reflexivity. + Qed. + + + (** *** Working with inequations + + To be able to use the tactics, the goal must be a relation [R] + applied to two arguments, and the rewritten hypothesis must end + with a relation [Q] applied to two arguments. These relations are + not necessarily equivalences, but they should be related + according to the occurrence where the rewrite takes place; we + leave this check to the underlying call to [setoid_rewrite]. *) + + (** One can rewrite equations in the left member of inequations, *) + Goal (forall x, x + x = x) -> a + b + b + a <= a + b. + intro Hx. + aac_rewrite Hx. + reflexivity. + Qed. + + (** or in the right member of inequations, using the [in_right] keyword *) + Goal (forall x, x + x = x) -> a + b <= a + b + b + a. + intro Hx. + aac_rewrite Hx in_right. + reflexivity. + Qed. + + (** Similarly, one can rewrite inequations in inequations, *) + Goal (forall x, x + x <= x) -> a + b + b + a <= a + b. + intro Hx. + aac_rewrite Hx. + reflexivity. + Qed. + + (** possibly in the right-hand side. *) + Goal (forall x, x <= x + x) -> a + b <= a + b + b + a. + intro Hx. + aac_rewrite <- Hx in_right. + reflexivity. + Qed. + + (** [aac_reflexivity] deals with "trivial" inequations too *) + Goal Max.max (a + b) (c + a) <= Max.max (b + a) (c + 1*a). + aac_reflexivity. + Qed. + + (** In the last three examples, there were no equivalence relation + involved in the goal. However, we actually had to guess the + equivalence relation with respect to which the operators + ([plus,max,0]) were AC. In this case, it was Leibniz equality + [eq] so that it was automatically inferred; more generally, one + can specify which equivalence relation to use by declaring + instances of the [AAC_lift] type class: *) + + Instance lift_le_eq : AAC_lift le eq := {}. + (** (This instance is automatically inferred because [eq] is always a + valid candidate, here for [le].) *) + + +End Peano. + + +(** *** Normalising goals + + We also provide a tactic to normalise terms modulo AC. This + normalisation is the one we use internally. *) + +Section AAC_normalise. + + Import Instances.Z. + Import ZArith. + Open Scope Z_scope. + + Variable a b c d : Z. + Goal a + (b + c*c*d) + a + 0 + d*1 = a. + aac_normalise. + Abort. + +End AAC_normalise. + + + + +(** ** Examples from the web page *) +Section Examples. + + Import Instances.Z. + Import ZArith. + Open Scope Z_scope. + + (** *** Reverse triangle inequality *) + + Lemma Zabs_triangle : forall x y, Z.abs (x + y) <= Z.abs x + Z.abs y . + Proof Z.abs_triangle. + + Lemma Zplus_opp_r : forall x, x + -x = 0. + Proof Zplus_opp_r. + + (** The following morphisms are required to perform the required rewrites *) + Instance Zminus_compat : Proper (Z.ge ==> Z.le) Z.opp. + Proof. intros x y. omega. Qed. + + Instance Proper_Zplus : Proper (Z.le ==> Z.le ==> Z.le) Zplus. + Proof. firstorder. Qed. + + Goal forall a b, Z.abs a - Z.abs b <= Z.abs (a - b). + intros. unfold Zminus. + aac_instances <- (Zminus_diag b). + aac_rewrite <- (Zminus_diag b) at 3. + unfold Zminus. + aac_rewrite Z.abs_triangle. + aac_rewrite Zplus_opp_r. + aac_reflexivity. + Qed. + + + (** *** Pythagorean triples *) + + Notation "x ^2" := (x*x) (at level 40). + Notation "2 ⋅ x" := (x+x) (at level 41). + + Lemma Hbin1: forall x y, (x+y)^2 = x^2 + y^2 + 2⋅x*y. Proof. intros; ring. Qed. + Lemma Hbin2: forall x y, x^2 + y^2 = (x+y)^2 + -(2⋅x*y). Proof. intros; ring. Qed. + Lemma Hopp : forall x, x + -x = 0. Proof Zplus_opp_r. + + Variables a b c : Z. + Hypothesis H : c^2 + 2⋅(a+1)*b = (a+1+b)^2. + Goal a^2 + b^2 + 2⋅a + 1 = c^2. + aacu_rewrite <- Hbin1. + rewrite Hbin2. + aac_rewrite <- H. + aac_rewrite Hopp. + aac_reflexivity. + Qed. + + (** Note: after the [aac_rewrite <- H], one could use [ring] to close the proof.*) + +End Examples. + + diff --git a/theories/Utils.v b/theories/Utils.v new file mode 100644 index 0000000..0f37ae1 --- /dev/null +++ b/theories/Utils.v @@ -0,0 +1,257 @@ +(***************************************************************************) +(* This is part of aac_tactics, it is distributed under the terms of the *) +(* GNU Lesser General Public License version 3 *) +(* (see file LICENSE for more details) *) +(* *) +(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) +(***************************************************************************) + + +Require Import Arith NArith. +Require Import List. +Require Import FMapPositive FMapFacts. +Require Import RelationClasses Equality. + +Set Implicit Arguments. +Set Asymmetric Patterns. + +(** * Utilities for positive numbers + which we use as: + - indices for morphisms and symbols + - multiplicity of terms in sums *) + +Notation idx := positive. + +Fixpoint eq_idx_bool i j := + match i,j with + | xH, xH => true + | xO i, xO j => eq_idx_bool i j + | xI i, xI j => eq_idx_bool i j + | _, _ => false + end. + +Fixpoint idx_compare i j := + match i,j with + | xH, xH => Eq + | xH, _ => Lt + | _, xH => Gt + | xO i, xO j => idx_compare i j + | xI i, xI j => idx_compare i j + | xI _, xO _ => Gt + | xO _, xI _ => Lt + end. + +Notation pos_compare := idx_compare (only parsing). + +(** Specification predicate for boolean binary functions *) +Inductive decide_spec {A} {B} (R : A -> B -> Prop) (x : A) (y : B) : bool -> Prop := +| decide_true : R x y -> decide_spec R x y true +| decide_false : ~(R x y) -> decide_spec R x y false. + +Lemma eq_idx_spec : forall i j, decide_spec (@eq _) i j (eq_idx_bool i j). +Proof. + induction i; destruct j; simpl; try (constructor; congruence). + case (IHi j); constructor; congruence. + case (IHi j); constructor; congruence. +Qed. + +(** weak specification predicate for comparison functions: only the 'Eq' case is specified *) +Inductive compare_weak_spec A: A -> A -> comparison -> Prop := +| pcws_eq: forall i, compare_weak_spec i i Eq +| pcws_lt: forall i j, compare_weak_spec i j Lt +| pcws_gt: forall i j, compare_weak_spec i j Gt. + +Lemma pos_compare_weak_spec: forall i j, compare_weak_spec i j (pos_compare i j). +Proof. induction i; destruct j; simpl; try constructor; case (IHi j); intros; constructor. Qed. + +Lemma idx_compare_reflect_eq: forall i j, idx_compare i j = Eq -> i=j. +Proof. intros i j. case (pos_compare_weak_spec i j); intros; congruence. Qed. + +(** * Dependent types utilities *) + +Notation cast T H u := (eq_rect _ T u _ H). + +Section dep. + Variable U: Type. + Variable T: U -> Type. + + Lemma cast_eq: (forall u v: U, {u=v}+{u<>v}) -> + forall A (H: A=A) (u: T A), cast T H u = u. + Proof. intros. rewrite <- Eqdep_dec.eq_rect_eq_dec; trivial. Qed. + + Variable f: forall A B, T A -> T B -> comparison. + Definition reflect_eqdep := forall A u B v (H: A=B), @f A B u v = Eq -> cast T H u = v. + + (* these lemmas have to remain transparent to get structural recursion + in the lemma [tcompare_weak_spec] below *) + Lemma reflect_eqdep_eq: reflect_eqdep -> + forall A u v, @f A A u v = Eq -> u = v. + Proof. intros H A u v He. apply (H _ _ _ _ eq_refl He). Defined. + + Lemma reflect_eqdep_weak_spec: reflect_eqdep -> + forall A u v, compare_weak_spec u v (@f A A u v). + Proof. + intros. case_eq (f u v); try constructor. + intro H'. apply reflect_eqdep_eq in H'. subst. constructor. assumption. + Defined. +End dep. + + +(** * Utilities about (non-empty) lists and multisets *) + +Inductive nelist (A : Type) : Type := +| nil : A -> nelist A +| cons : A -> nelist A -> nelist A. + +Notation "x :: y" := (cons x y). + +Fixpoint nelist_map (A B: Type) (f: A -> B) l := + match l with + | nil x => nil ( f x) + | cons x l => cons ( f x) (nelist_map f l) + end. + +Fixpoint appne A l l' : nelist A := + match l with + nil x => cons x l' + | cons t q => cons t (appne A q l') + end. + +Notation "x ++ y" := (appne x y). + +(** finite multisets are represented with ordered lists with multiplicities *) +Definition mset A := nelist (A*positive). + +(** lexicographic composition of comparisons (this is a notation to keep it lazy) *) +Notation lex e f := (match e with Eq => f | _ => e end). + + +Section lists. + + (** comparison functions *) + + Section c. + Variables A B: Type. + Variable compare: A -> B -> comparison. + Fixpoint list_compare h k := + match h,k with + | nil x, nil y => compare x y + | nil x, _ => Lt + | _, nil x => Gt + | u::h, v::k => lex (compare u v) (list_compare h k) + end. + End c. + Definition mset_compare A B compare: mset A -> mset B -> comparison := + list_compare (fun un vm => + let '(u,n) := un in + let '(v,m) := vm in + lex (compare u v) (pos_compare n m)). + + Section list_compare_weak_spec. + Variable A: Type. + Variable compare: A -> A -> comparison. + Hypothesis Hcompare: forall u v, compare_weak_spec u v (compare u v). + (* this lemma has to remain transparent to get structural recursion + in the lemma [tcompare_weak_spec] below *) + Lemma list_compare_weak_spec: forall h k, + compare_weak_spec h k (list_compare compare h k). + Proof. + induction h as [|u h IHh]; destruct k as [|v k]; simpl; try constructor. + + case (Hcompare a a0 ); try constructor. + case (Hcompare u v ); try constructor. + case (IHh k); intros; constructor. + Defined. + End list_compare_weak_spec. + + Section mset_compare_weak_spec. + Variable A: Type. + Variable compare: A -> A -> comparison. + Hypothesis Hcompare: forall u v, compare_weak_spec u v (compare u v). + (* this lemma has to remain transparent to get structural recursion + in the lemma [tcompare_weak_spec] below *) + Lemma mset_compare_weak_spec: forall h k, + compare_weak_spec h k (mset_compare compare h k). + Proof. + apply list_compare_weak_spec. + intros [u n] [v m]. + case (Hcompare u v); try constructor. + case (pos_compare_weak_spec n m); try constructor. + Defined. + End mset_compare_weak_spec. + + (** (sorted) merging functions *) + + Section m. + Variable A: Type. + Variable compare: A -> A -> comparison. + Definition insert n1 h1 := + let fix insert_aux l2 := + match l2 with + | nil (h2,n2) => + match compare h1 h2 with + | Eq => nil (h1,Pplus n1 n2) + | Lt => (h1,n1):: nil (h2,n2) + | Gt => (h2,n2):: nil (h1,n1) + end + | (h2,n2)::t2 => + match compare h1 h2 with + | Eq => (h1,Pplus n1 n2):: t2 + | Lt => (h1,n1)::l2 + | Gt => (h2,n2)::insert_aux t2 + end + end + in insert_aux. + + Fixpoint merge_msets l1 := + match l1 with + | nil (h1,n1) => fun l2 => insert n1 h1 l2 + | (h1,n1)::t1 => + let fix merge_aux l2 := + match l2 with + | nil (h2,n2) => + match compare h1 h2 with + | Eq => (h1,Pplus n1 n2) :: t1 + | Lt => (h1,n1):: merge_msets t1 l2 + | Gt => (h2,n2):: l1 + end + | (h2,n2)::t2 => + match compare h1 h2 with + | Eq => (h1,Pplus n1 n2)::merge_msets t1 t2 + | Lt => (h1,n1)::merge_msets t1 l2 + | Gt => (h2,n2)::merge_aux t2 + end + end + in merge_aux + end. + + (** interpretation of a list with a constant and a binary operation *) + + Variable B: Type. + Variable map: A -> B. + Variable b2: B -> B -> B. + Fixpoint fold_map l := + match l with + | nil x => map x + | u::l => b2 (map u) (fold_map l) + end. + + (** mapping and merging *) + + Variable merge: A -> nelist B -> nelist B. + Fixpoint merge_map (l: nelist A): nelist B := + match l with + | nil x => nil (map x) + | u::l => merge u (merge_map l) + end. + + Variable ret : A -> B. + Variable bind : A -> B -> B. + Fixpoint fold_map' (l : nelist A) : B := + match l with + | nil x => ret x + | u::l => bind u (fold_map' l) + end. + + End m. +End lists. diff --git a/theory.ml b/theory.ml deleted file mode 100644 index 20cb299..0000000 --- a/theory.ml +++ /dev/null @@ -1,1149 +0,0 @@ -(***************************************************************************) -(* This is part of aac_tactics, it is distributed under the terms of the *) -(* GNU Lesser General Public License version 3 *) -(* (see file LICENSE for more details) *) -(* *) -(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) -(***************************************************************************) - -(** Constr from the theory of this particular development - - The inner-working of this module is highly correlated with the - particular structure of {b AAC.v}, therefore, it should - be of little interest to most readers. -*) -open Term -open Context - -module Control = struct - let printing = true - let debug = false - let time = false -end - -module Debug = Helper.Debug (Control) -open Debug - -(** {1 HMap : Specialized Hashtables based on constr} *) - - - (* TODO module HMap = Hashtbl, du coup ? *) -module HMap = Hashtbl.Make(Constr) - -let ac_theory_path = ["AAC_tactics"; "AAC"] - -module Stubs = struct - let path = ac_theory_path@["Internal"] - - (** The constants from the inductive type *) - let _Tty = lazy (Coq.init_constant path "T") - let vTty = lazy (Coq.init_constant path "vT") - - let rsum = lazy (Coq.init_constant path "sum") - let rprd = lazy (Coq.init_constant path "prd") - let rsym = lazy (Coq.init_constant path "sym") - let runit = lazy (Coq.init_constant path "unit") - - let vnil = lazy (Coq.init_constant path "vnil") - let vcons = lazy (Coq.init_constant path "vcons") - let eval = lazy (Coq.init_constant path "eval") - - - let decide_thm = lazy (Coq.init_constant path "decide") - let lift_normalise_thm = lazy (Coq.init_constant path "lift_normalise") - - let lift = - lazy (Coq.init_constant ac_theory_path "AAC_lift") - let lift_proj_equivalence= - lazy (Coq.init_constant ac_theory_path "aac_lift_equivalence") - let lift_transitivity_left = - lazy(Coq.init_constant ac_theory_path "lift_transitivity_left") - let lift_transitivity_right = - lazy(Coq.init_constant ac_theory_path "lift_transitivity_right") - let lift_reflexivity = - lazy(Coq.init_constant ac_theory_path "lift_reflexivity") -end - -module Classes = struct - module Associative = struct - let path = ac_theory_path - let typ = lazy (Coq.init_constant path "Associative") - let ty (rlt : Coq.Relation.t) (value : Term.constr) = - mkApp (Lazy.force typ, [| rlt.Coq.Relation.carrier; - rlt.Coq.Relation.r; - value - |] ) - let infer goal rlt value = - let ty = ty rlt value in - Coq.resolve_one_typeclass goal ty - end - - module Commutative = struct - let path = ac_theory_path - let typ = lazy (Coq.init_constant path "Commutative") - let ty (rlt : Coq.Relation.t) (value : Term.constr) = - mkApp (Lazy.force typ, [| rlt.Coq.Relation.carrier; - rlt.Coq.Relation.r; - value - |] ) - - end - - module Proper = struct - let path = ac_theory_path @ ["Internal";"Sym"] - let typeof = lazy (Coq.init_constant path "type_of") - let relof = lazy (Coq.init_constant path "rel_of") - let mk_typeof : Coq.Relation.t -> int -> constr = fun rlt n -> - let x = rlt.Coq.Relation.carrier in - mkApp (Lazy.force typeof, [| x ; Coq.Nat.of_int n |]) - let mk_relof : Coq.Relation.t -> int -> constr = fun rlt n -> - let (x,r) = Coq.Relation.split rlt in - mkApp (Lazy.force relof, [| x;r ; Coq.Nat.of_int n |]) - - let ty rlt op ar = - let typeof = mk_typeof rlt ar in - let relof = mk_relof rlt ar in - Coq.Classes.mk_morphism typeof relof op - let infer goal rlt op ar = - let ty = ty rlt op ar in - Coq.resolve_one_typeclass goal ty - end - - module Unit = struct - let path = ac_theory_path - let typ = lazy (Coq.init_constant path "Unit") - let ty (rlt : Coq.Relation.t) (value : Term.constr) (unit : Term.constr)= - mkApp (Lazy.force typ, [| rlt.Coq.Relation.carrier; - rlt.Coq.Relation.r; - value; - unit - |] ) - end - -end - -(* Non empty lists *) -module NEList = struct - let path = ac_theory_path @ ["Internal"] - let typ = lazy (Coq.init_constant path "list") - let nil = lazy (Coq.init_constant path "nil") - let cons = lazy (Coq.init_constant path "cons") - let cons ty h t = - mkApp (Lazy.force cons, [| ty; h ; t |]) - let nil ty x = - (mkApp (Lazy.force nil, [| ty ; x|])) - let rec of_list ty = function - | [] -> invalid_arg "NELIST" - | [x] -> nil ty x - | t::q -> cons ty t (of_list ty q) - - let type_of_list ty = - mkApp (Lazy.force typ, [|ty|]) -end - -(** a [mset] is a ('a * pos) list *) -let mk_mset ty (l : (constr * int) list) = - let pos = Lazy.force Coq.Pos.typ in - let pair (x : constr) (ar : int) = - Coq.Pair.of_pair ty pos (x, Coq.Pos.of_int ar) - in - let pair_ty = Coq.lapp Coq.Pair.typ [| ty ; pos|] in - let rec aux = function - | [ ] -> assert false - | [x,ar] -> NEList.nil pair_ty (pair x ar) - | (t,ar)::q -> NEList.cons pair_ty (pair t ar) (aux q) - in - aux l - -module Sigma = struct - let sigma = lazy (Coq.init_constant ac_theory_path "sigma") - let sigma_empty = lazy (Coq.init_constant ac_theory_path "sigma_empty") - let sigma_add = lazy (Coq.init_constant ac_theory_path "sigma_add") - let sigma_get = lazy (Coq.init_constant ac_theory_path "sigma_get") - - let add ty n x map = - mkApp (Lazy.force sigma_add,[|ty; n; x ; map|]) - let empty ty = - mkApp (Lazy.force sigma_empty,[|ty |]) - let typ ty = - mkApp (Lazy.force sigma, [|ty|]) - - let to_fun ty null map = - mkApp (Lazy.force sigma_get, [|ty;null;map|]) - - let of_list ty null l = - match l with - | (_,t)::q -> - let map = - List.fold_left - (fun acc (i,t) -> - assert (i > 0); - add ty (Coq.Pos.of_int i) ( t) acc) - (empty ty) - q - in to_fun ty (t) map - | [] -> debug "of_list empty" ; to_fun ty (null) (empty ty) - - -end - - -module Sym = struct - type pack = {ar: Term.constr; value: Term.constr ; morph: Term.constr} - let path = ac_theory_path @ ["Internal";"Sym"] - let typ = lazy (Coq.init_constant path "pack") - let mkPack = lazy (Coq.init_constant path "mkPack") - let value = lazy (Coq.init_constant path "value") - let null = lazy (Coq.init_constant path "null") - let mk_pack (rlt: Coq.Relation.t) s = - let (x,r) = Coq.Relation.split rlt in - mkApp (Lazy.force mkPack, [|x;r; s.ar;s.value;s.morph|]) - let null rlt = - let x = rlt.Coq.Relation.carrier in - let r = rlt.Coq.Relation.r in - mkApp (Lazy.force null, [| x;r;|]) - - let mk_ty : Coq.Relation.t -> constr = fun rlt -> - let (x,r) = Coq.Relation.split rlt in - mkApp (Lazy.force typ, [| x; r|] ) -end - -module Bin =struct - type pack = {value : Term.constr; - compat : Term.constr; - assoc : Term.constr; - comm : Term.constr option; - } - - let path = ac_theory_path @ ["Internal";"Bin"] - let typ = lazy (Coq.init_constant path "pack") - let mkPack = lazy (Coq.init_constant path "mk_pack") - - let mk_pack: Coq.Relation.t -> pack -> Term.constr = fun (rlt) s -> - let (x,r) = Coq.Relation.split rlt in - let comm_ty = Classes.Commutative.ty rlt s.value in - mkApp (Lazy.force mkPack , [| x ; r; - s.value; - s.compat ; - s.assoc; - Coq.Option.of_option comm_ty s.comm - |]) - let mk_ty : Coq.Relation.t -> constr = fun rlt -> - let (x,r) = Coq.Relation.split rlt in - mkApp (Lazy.force typ, [| x; r|] ) -end - -module Unit = struct - let path = ac_theory_path @ ["Internal"] - let unit_of_ty = lazy (Coq.init_constant path "unit_of") - let unit_pack_ty = lazy (Coq.init_constant path "unit_pack") - let mk_unit_of = lazy (Coq.init_constant path "mk_unit_for") - let mk_unit_pack = lazy (Coq.init_constant path "mk_unit_pack") - - type unit_of = - { - uf_u : Term.constr; (* u *) - uf_idx : Term.constr; - uf_desc : Term.constr; (* Unit R (e_bin uf_idx) u *) - } - - type pack = { - u_value : Term.constr; (* X *) - u_desc : (unit_of) list (* unit_of u_value *) - } - - let ty_unit_of rlt e_bin u = - let (x,r) = Coq.Relation.split rlt in - mkApp ( Lazy.force unit_of_ty, [| x; r; e_bin; u |]) - - let ty_unit_pack rlt e_bin = - let (x,r) = Coq.Relation.split rlt in - mkApp (Lazy.force unit_pack_ty, [| x; r; e_bin |]) - - let mk_unit_of rlt e_bin u unit_of = - let (x,r) = Coq.Relation.split rlt in - mkApp (Lazy.force mk_unit_of , [| x; - r; - e_bin ; - u; - unit_of.uf_idx; - unit_of.uf_desc - |]) - - let mk_pack rlt e_bin pack : Term.constr = - let (x,r) = Coq.Relation.split rlt in - let ty = ty_unit_of rlt e_bin pack.u_value in - let mk_unit_of = mk_unit_of rlt e_bin pack.u_value in - let u_desc =Coq.List.of_list ( ty ) (List.map mk_unit_of pack.u_desc) in - mkApp (Lazy.force mk_unit_pack, [|x;r; - e_bin ; - pack.u_value; - u_desc - |]) - - let default x : pack = - { u_value = x; - u_desc = [] - } - -end - -let anomaly msg = - CErrors.anomaly ~label:"aac_tactics" (Pp.str msg) - -let user_error msg = - CErrors.error ("aac_tactics: " ^ msg) - -module Trans = struct - - (** {1 From Coq to Abstract Syntax Trees (AST)} - - This module provides facilities to interpret a Coq term with - arbitrary operators as an abstract syntax tree. Considering an - application, we try to infer instances of our classes. - - We consider that [A] operators are coarser than [AC] operators, - which in turn are coarser than raw morphisms. That means that - [List.append], of type [(A : Type) -> list A -> list A -> list - A] can be understood as an [A] operator. - - During this reification, we gather some informations that will - be used to rebuild Coq terms from AST ( type {!envs}) - - We use a main hash-table from [constr] to [pack], in order to - discriminate the various constructors. All these are mixed in - order to improve on the number of comparisons in the tables *) - - (* 'a * (unit, op_unit) option *) - type 'a with_unit = 'a * (Unit.unit_of) option - type pack = - (* used to infer the AC/A structure in the first pass {!Gather} *) - | Bin of Bin.pack with_unit - (* will only be used in the second pass : {!Parse}*) - | Sym of Sym.pack - | Unit of constr (* to avoid confusion in bloom *) - - module PackHash = - struct - open Hashset.Combine - - type t = pack - - let eq_sym_pack p1 p2 = - let open Sym in - Constr.equal p1.ar p2.ar && - Constr.equal p1.value p2.value && - Constr.equal p1.morph p2.morph - - let hash_sym_pack p = - let open Sym in - combine3 (Constr.hash p.ar) (Constr.hash p.value) (Constr.hash p.morph) - - let eq_bin_pack p1 p2 = - let open Bin in - Constr.equal p1.value p2.value && - Constr.equal p1.compat p2.compat && - Constr.equal p1.assoc p2.assoc && - Option.equal Constr.equal p1.comm p2.comm - - let hash_bin_pack p = - let open Bin in - combine4 (Constr.hash p.value) (Constr.hash p.compat) - (Constr.hash p.assoc) (Option.hash Constr.hash p.comm) - - let eq_unit_of u1 u2 = - let open Unit in - Constr.equal u1.uf_u u2.uf_u && - Constr.equal u1.uf_idx u2.uf_idx && - Constr.equal u1.uf_desc u2.uf_desc - - let hash_unit_of u = - let open Unit in - combine3 (Constr.hash u.uf_u) (Constr.hash u.uf_idx) - (Constr.hash u.uf_desc) - - let equal p1 p2 = match p1, p2 with - | Bin (p1, o1), Bin (p2, o2) -> - eq_bin_pack p1 p2 && Option.equal eq_unit_of o1 o2 - | Sym p1, Sym p2 -> eq_sym_pack p1 p2 - | Unit c1, Unit c2 -> Constr.equal c1 c2 - | _ -> false - - let hash p = match p with - | Bin (p, o) -> - combinesmall 1 (combine (hash_bin_pack p) (Option.hash hash_unit_of o)) - | Sym p -> - combinesmall 2 (hash_sym_pack p) - | Unit c -> - combinesmall 3 (Constr.hash c) - - end - - module PackTable = Hashtbl.Make(PackHash) - - (** discr is a map from {!Term.constr} to {!pack}. - - [None] means that it is not possible to instantiate this partial - application to an interesting class. - - [Some x] means that we found something in the past. This means - in particular that a single [constr] cannot be two things at the - same time. - - The field [bloom] allows to give a unique number to each class we - found. *) - type envs = - { - discr : (pack option) HMap.t ; - bloom : int PackTable.t; - bloom_back : (int, pack ) Hashtbl.t; - bloom_next : int ref; - } - - let empty_envs () = - { - discr = HMap.create 17; - bloom = PackTable.create 17; - bloom_back = Hashtbl.create 17; - bloom_next = ref 1; - } - - - - let add_bloom envs pack = - if PackTable.mem envs.bloom pack - then () - else - let x = ! (envs.bloom_next) in - PackTable.add envs.bloom pack x; - Hashtbl.add envs.bloom_back x pack; - incr (envs.bloom_next) - - let find_bloom envs pack = - try PackTable.find envs.bloom pack - with Not_found -> assert false - - (********************************************) - (* (\* Gather the occuring AC/A symbols *\) *) - (********************************************) - - (** This modules exhibit a function that memoize in the environment - all the AC/A operators as well as the morphism that occur. This - staging process allows us to prefer AC/A operators over raw - morphisms. Moreover, for each AC/A operators, we need to try to - infer units. Otherwise, we do not have [x * y * x <= a * a] since 1 - does not occur. - - But, do we also need to check whether constants are - units. Otherwise, we do not have the ability to rewrite [0 = a + - a] in [a = ...]*) - module Gather : sig - val gather : Coq.goal_sigma -> Coq.Relation.t -> envs -> Term.constr -> Coq.goal_sigma - end - = struct - - let memoize envs t pack : unit = - begin - HMap.add envs.discr t (Some pack); - add_bloom envs pack; - match pack with - | Bin (_, None) | Sym _ -> () - | Bin (_, Some (unit_of)) -> - let unit = unit_of.Unit.uf_u in - HMap.add envs.discr unit (Some (Unit unit)); - add_bloom envs (Unit unit); - | Unit _ -> assert false - end - - - let get_unit (rlt : Coq.Relation.t) op goal : - (Coq.goal_sigma * constr * constr ) option= - let x = (rlt.Coq.Relation.carrier) in - let unit, goal = Coq.evar_unit goal x in - let ty =Classes.Unit.ty rlt op unit in - let result = - try - let t,goal = Coq.resolve_one_typeclass goal ty in - Some (goal,t,unit) - with Not_found -> None - in - match result with - | None -> None - | Some (goal,s,unit) -> - let unit = Coq.nf_evar goal unit in - Some (goal, unit, s) - - - - (** gives back the class and the operator *) - let is_bin (rlt: Coq.Relation.t) (op: constr) ( goal: Coq.goal_sigma) - : (Coq.goal_sigma * Bin.pack) option = - let assoc_ty = Classes.Associative.ty rlt op in - let comm_ty = Classes.Commutative.ty rlt op in - let proper_ty = Classes.Proper.ty rlt op 2 in - try - let proper , goal = Coq.resolve_one_typeclass goal proper_ty in - let assoc, goal = Coq.resolve_one_typeclass goal assoc_ty in - let comm , goal = - try - let comm, goal = Coq.resolve_one_typeclass goal comm_ty in - Some comm, goal - with Not_found -> - None, goal - in - let bin = - {Bin.value = op; - Bin.compat = proper; - Bin.assoc = assoc; - Bin.comm = comm } - in - Some (goal,bin) - with |Not_found -> None - - let is_bin (rlt : Coq.Relation.t) (op : constr) (goal : Coq.goal_sigma)= - match is_bin rlt op goal with - | None -> None - | Some (goal, bin_pack) -> - match get_unit rlt op goal with - | None -> Some (goal, Bin (bin_pack, None)) - | Some (gl, unit,s) -> - let unit_of = - { - Unit.uf_u = unit; - (* TRICK : this term is not well-typed by itself, - but it is okay the way we use it in the other - functions *) - Unit.uf_idx = op; - Unit.uf_desc = s; - } - in Some (gl,Bin (bin_pack, Some (unit_of))) - - - (** {is_morphism} try to infer the kind of operator we are - dealing with *) - let is_morphism goal (rlt : Coq.Relation.t) (papp : Term.constr) (ar : int) : (Coq.goal_sigma * pack ) option = - let typeof = Classes.Proper.mk_typeof rlt ar in - let relof = Classes.Proper.mk_relof rlt ar in - let m = Coq.Classes.mk_morphism typeof relof papp in - try - let m,goal = Coq.resolve_one_typeclass goal m in - let pack = {Sym.ar = (Coq.Nat.of_int ar); Sym.value= papp; Sym.morph= m} in - Some (goal, Sym pack) - with - | Not_found -> None - - - (* [crop_app f [| a_0 ; ... ; a_n |]] - returns Some (f a_0 ... a_(n-2), [|a_(n-1); a_n |] ) - *) - let crop_app t ca : (constr * constr array) option= - let n = Array.length ca in - if n <= 1 - then None - else - let papp = Term.mkApp (t, Array.sub ca 0 (n-2) ) in - let args = Array.sub ca (n-2) 2 in - Some (papp, args ) - - let fold goal (rlt : Coq.Relation.t) envs t ca cont = - let fold_morphism t ca = - let nb_params = Array.length ca in - let rec aux n = - assert (n < nb_params && 0 < nb_params ); - let papp = Term.mkApp (t, Array.sub ca 0 n) in - match is_morphism goal rlt papp (nb_params - n) with - | None -> - (* here we have to memoize the failures *) - HMap.add envs.discr papp None; - if n < nb_params - 1 then aux (n+1) else goal - | Some (goal, pack) -> - let args = Array.sub ca (n) (nb_params -n)in - let goal = Array.fold_left cont goal args in - memoize envs papp pack; - goal - in - if nb_params = 0 then goal else aux 0 - in - let is_aac t = is_bin rlt t in - match crop_app t ca with - | None -> - fold_morphism t ca - | Some (papp, args) -> - begin match is_aac papp goal with - | None -> fold_morphism t ca - | Some (goal, pack) -> - memoize envs papp pack; - Array.fold_left cont goal args - end - - (* update in place the envs of known stuff, using memoization. We - have to memoize failures, here. *) - let gather goal (rlt : Coq.Relation.t ) envs t : Coq.goal_sigma = - let rec aux goal x = - match Coq.decomp_term x with - | App (t,ca) -> - fold goal rlt envs t ca (aux ) - | _ -> goal - in - aux goal t - end - - (** We build a term out of a constr, updating in place the - environment if needed (the only kind of such updates are the - constants). *) - module Parse : - sig - val t_of_constr : Coq.goal_sigma -> Coq.Relation.t -> envs -> constr -> Matcher.Terms.t * Coq.goal_sigma - end - = struct - - (** [discriminates goal envs rlt t ca] infer : - - - its {! pack } (is it an AC operator, or an A operator, or a - Symbol ?), - - - the relevant partial application, - - - the vector of the remaining arguments - - We use an expansion to handle the special case of units, - before going back to the general discrimination - procedure. That means that a unit is more coarse than a raw - morphism - - This functions is prevented to go through [ar < 0] by the fact - that a constant is a morphism. But not an eva. *) - - let is_morphism goal (rlt : Coq.Relation.t) (papp : Term.constr) (ar : int) : (Coq.goal_sigma * pack ) option = - let typeof = Classes.Proper.mk_typeof rlt ar in - let relof = Classes.Proper.mk_relof rlt ar in - let m = Coq.Classes.mk_morphism typeof relof papp in - try - let m,goal = Coq.resolve_one_typeclass goal m in - let pack = {Sym.ar = (Coq.Nat.of_int ar); Sym.value= papp; Sym.morph= m} in - Some (goal, Sym pack) - with - | e -> None - - exception NotReflexive - let discriminate goal envs (rlt : Coq.Relation.t) t ca : Coq.goal_sigma * pack * constr * constr array = - let nb_params = Array.length ca in - let rec fold goal ar :Coq.goal_sigma * pack * constr * constr array = - begin - assert (0 <= ar && ar <= nb_params); - let p_app = mkApp (t, Array.sub ca 0 (nb_params - ar)) in - begin - try - begin match HMap.find envs.discr p_app with - | None -> - fold goal (ar-1) - | Some pack -> - (goal, pack, p_app, Array.sub ca (nb_params -ar ) ar) - end - with - Not_found -> (* Could not find this constr *) - memoize (is_morphism goal rlt p_app ar) p_app ar - end - end - and memoize (x) p_app ar = - assert (0 <= ar && ar <= nb_params); - match x with - | Some (goal, pack) -> - HMap.add envs.discr p_app (Some pack); - add_bloom envs pack; - (goal, pack, p_app, Array.sub ca (nb_params-ar) ar) - | None -> - - if ar = 0 then raise NotReflexive; - begin - (* to memoise the failures *) - HMap.add envs.discr p_app None; - (* will terminate, since [const] is capped, and it is - easy to find an instance of a constant *) - fold goal (ar-1) - end - in - try match HMap.find envs.discr (mkApp (t,ca))with - | None -> fold goal (nb_params) - | Some pack -> goal, pack, (mkApp (t,ca)), [| |] - with Not_found -> fold goal (nb_params) - - let discriminate goal envs rlt x = - try - match Coq.decomp_term x with - | App (t,ca) -> - discriminate goal envs rlt t ca - | _ -> discriminate goal envs rlt x [| |] - with - | NotReflexive -> user_error "The relation to which the goal was lifted is not Reflexive" - (* TODO: is it the only source of invalid use that fall - into this catch_all ? *) - | e -> - user_error "Cannot handle this kind of hypotheses (variables occuring under a function symbol which is not a proper morphism)." - - (** [t_of_constr goal rlt envs cstr] builds the abstract syntax tree - of the term [cstr]. Doing so, it modifies the environment of - known stuff [envs], and eventually creates fresh - evars. Therefore, we give back the goal updated accordingly *) - let t_of_constr goal (rlt: Coq.Relation.t ) envs : constr -> Matcher.Terms.t * Coq.goal_sigma = - let r_goal = ref (goal) in - let rec aux x = - match Coq.decomp_term x with - | Rel i -> Matcher.Terms.Var i - | _ -> - let goal, pack , p_app, ca = discriminate (!r_goal) envs rlt x in - r_goal := goal; - let k = find_bloom envs pack - in - match pack with - | Bin (pack,_) -> - begin match pack.Bin.comm with - | Some _ -> - assert (Array.length ca = 2); - Matcher.Terms.Plus ( k, aux ca.(0), aux ca.(1)) - | None -> - assert (Array.length ca = 2); - Matcher.Terms.Dot ( k, aux ca.(0), aux ca.(1)) - end - | Unit _ -> - assert (Array.length ca = 0); - Matcher.Terms.Unit ( k) - | Sym _ -> - Matcher.Terms.Sym ( k, Array.map aux ca) - in - ( - fun x -> let r = aux x in r, !r_goal - ) - - end (* Parse *) - - let add_symbol goal rlt envs l = - let goal = Gather.gather goal rlt envs (Term.mkApp (l, [| Term.mkRel 0;Term.mkRel 0|])) in - goal - - (* [t_of_constr] buils a the abstract syntax tree of a constr, - updating in place the environment. Doing so, we infer all the - morphisms and the AC/A operators. It is mandatory to do so both - for the pattern and the term, since AC symbols can occur in one - and not the other *) - let t_of_constr goal rlt envs (l,r) = - let goal = Gather.gather goal rlt envs l in - let goal = Gather.gather goal rlt envs r in - let l,goal = Parse.t_of_constr goal rlt envs l in - let r, goal = Parse.t_of_constr goal rlt envs r in - l, r, goal - - (* An intermediate representation of the environment, with association lists for AC/A operators, and also the raw [packer] information *) - - type ir = - { - packer : (int, pack) Hashtbl.t; (* = bloom_back *) - bin : (int * Bin.pack) list ; - units : (int * Unit.pack) list; - sym : (int * Term.constr) list ; - matcher_units : Matcher.ext_units - } - - let ir_to_units ir = ir.matcher_units - - let ir_of_envs goal (rlt : Coq.Relation.t) envs = - let add x y l = (x,y)::l in - let nil = [] in - let sym , - (bin : (int * Bin.pack with_unit) list), - (units : (int * constr) list) = - Hashtbl.fold - (fun int pack (sym,bin,units) -> - match pack with - | Bin s -> - sym, add (int) s bin, units - | Sym s -> - add (int) s sym, bin, units - | Unit s -> - sym, bin, add int s units - ) - envs.bloom_back - (nil,nil,nil) - in - let matcher_units = - let unit_for , is_ac = - List.fold_left - (fun ((unit_for,is_ac) as acc) (n,(bp,wu)) -> - match wu with - | None -> acc - | Some (unit_of) -> - let unit_n = PackTable.find envs.bloom - (Unit unit_of.Unit.uf_u) - in - ( n, unit_n)::unit_for, - (n, bp.Bin.comm <> None )::is_ac - - ) - ([],[]) bin - in - {Matcher.unit_for = unit_for; Matcher.is_ac = is_ac} - - in - let units : (int * Unit.pack) list = - List.fold_left - (fun acc (n,u) -> - (* first, gather all bins with this unit *) - let all_bin = - List.fold_left - ( fun acc (nop,(bp,wu)) -> - match wu with - | None -> acc - | Some unit_of -> - if Constr.equal (unit_of.Unit.uf_u) u - then - {unit_of with - Unit.uf_idx = (Coq.Pos.of_int nop)} :: acc - else - acc - ) - [] bin - in - (n,{ - Unit.u_value = u; - Unit.u_desc = all_bin - })::acc - ) - [] units - - in - goal, { - packer = envs.bloom_back; - bin = (List.map (fun (n,(s,_)) -> n, s) bin); - units = units; - sym = (List.map (fun (n,s) -> n,(Sym.mk_pack rlt s)) sym); - matcher_units = matcher_units - } - - - - (** {1 From AST back to Coq } - - The next functions allow one to map OCaml abstract syntax trees - to Coq terms *) - - (** {2 Building raw, natural, terms} *) - - (** [raw_constr_of_t_debruijn] rebuilds a term in the raw - representation, without products on top, and maybe escaping free - debruijn indices (in the case of a pattern for example). *) - let raw_constr_of_t_debruijn ir (t : Matcher.Terms.t) : Term.constr * int list = - let add_set,get = - let r = ref [] in - let rec add x = function - [ ] -> [x] - | t::q when t = x -> t::q - | t::q -> t:: (add x q) - in - (fun x -> r := add x !r),(fun () -> !r) - in - (* Here, we rely on the invariant that the maps are well formed: - it is meanigless to fail to find a symbol in the maps, or to - find the wrong kind of pack in the maps *) - let rec aux t = - match t with - | Matcher.Terms.Plus (s,l,r) -> - begin match Hashtbl.find ir.packer s with - | Bin (s,_) -> - mkApp (s.Bin.value , [|(aux l);(aux r)|]) - | _ -> Printf.printf "erreur:%i\n%!"s; - assert false - end - | Matcher.Terms.Dot (s,l,r) -> - begin match Hashtbl.find ir.packer s with - | Bin (s,_) -> - mkApp (s.Bin.value, [|(aux l);(aux r)|]) - | _ -> assert false - end - | Matcher.Terms.Sym (s,t) -> - begin match Hashtbl.find ir.packer s with - | Sym s -> - mkApp (s.Sym.value, Array.map aux t) - | _ -> assert false - end - | Matcher.Terms.Unit x -> - begin match Hashtbl.find ir.packer x with - | Unit s -> s - | _ -> assert false - end - | Matcher.Terms.Var i -> add_set i; - mkRel (i) - in - let t = aux t in - t , get ( ) - - (** [raw_constr_of_t] rebuilds a term in the raw representation *) - let raw_constr_of_t ir rlt (context:Context.Rel.t) t = - (** cap rebuilds the products in front of the constr *) - let rec cap c = function [] -> c - | t::q -> - let i = Context.Rel.lookup t context in - cap (mkProd_or_LetIn i c) q - in - let t,indices = raw_constr_of_t_debruijn ir t in - cap t (List.sort (Pervasives.compare) indices) - - - (** {2 Building reified terms} *) - - (* Some informations to be able to build the maps *) - type reif_params = - { - bin_null : Bin.pack; (* the default A operator *) - sym_null : constr; - unit_null : Unit.pack; - sym_ty : constr; (* the type, as it appears in e_sym *) - bin_ty : constr - } - - - (** A record containing the environments that will be needed by the - decision procedure, as a Coq constr. Contains the functions - from the symbols (as ints) to indexes (as constr) *) - - type sigmas = { - env_sym : Term.constr; - env_bin : Term.constr; - env_units : Term.constr; (* the [idx -> X:constr] *) - } - - - type sigma_maps = { - sym_to_pos : int -> Term.constr; - bin_to_pos : int -> Term.constr; - units_to_pos : int -> Term.constr; - } - - - (** infers some stuff that will be required when we will build - environments (our environments need a default case, so we need - an Op_AC, an Op_A, and a symbol) *) - (* Note : this function can fail if the user is using the wrong - relation, like proving a = b, while the classes are defined with - another relation (==) *) - let build_reif_params goal (rlt : Coq.Relation.t) (zero) : - reif_params * Coq.goal_sigma = - let carrier = rlt.Coq.Relation.carrier in - let bin_null = - try - let op,goal = Coq.evar_binary goal carrier in - let assoc,goal = Classes.Associative.infer goal rlt op in - let compat,goal = Classes.Proper.infer goal rlt op 2 in - let op = Coq.nf_evar goal op in - { - Bin.value = op; - Bin.compat = compat; - Bin.assoc = assoc; - Bin.comm = None - } - with Not_found -> user_error "Cannot infer a default A operator (required at least to be Proper and Associative)" - in - let zero, goal = - try - let evar_op,goal = Coq.evar_binary goal carrier in - let evar_unit, goal = Coq.evar_unit goal carrier in - let query = Classes.Unit.ty rlt evar_op evar_unit in - let _, goal = Coq.resolve_one_typeclass goal query in - Coq.nf_evar goal evar_unit, goal - with _ -> zero, goal in - let sym_null = Sym.null rlt in - let unit_null = Unit.default zero in - let record = - { - bin_null = bin_null; - sym_null = sym_null; - unit_null = unit_null; - sym_ty = Sym.mk_ty rlt ; - bin_ty = Bin.mk_ty rlt - } - in - record, goal - - (* We want to lift down the indexes of symbols. *) - let renumber (l: (int * 'a) list ) = - let _, l = List.fold_left (fun (next,acc) (glob,x) -> - (next+1, (next,glob,x)::acc) - ) (1,[]) l - in - let rec to_global loc = function - | [] -> assert false - | (local, global,x)::q when local = loc -> global - | _::q -> to_global loc q - in - let rec to_local glob = function - | [] -> assert false - | (local, global,x)::q when global = glob -> local - | _::q -> to_local glob q - in - let locals = List.map (fun (local,global,x) -> local,x) l in - locals, (fun x -> to_local x l) , (fun x -> to_global x l) - - (** [build_sigma_maps] given a envs and some reif_params, we are - able to build the sigmas *) - let build_sigma_maps (rlt : Coq.Relation.t) zero ir (k : sigmas * sigma_maps -> Proof_type.tactic ) : Proof_type.tactic = fun goal -> - let rp,goal = build_reif_params goal rlt zero in - let renumbered_sym, to_local, to_global = renumber ir.sym in - let env_sym = Sigma.of_list - rp.sym_ty - (rp.sym_null) - renumbered_sym - in - Coq.cps_mk_letin "env_sym" env_sym - (fun env_sym -> - let bin = (List.map ( fun (n,s) -> n, Bin.mk_pack rlt s) ir.bin) in - let env_bin = - Sigma.of_list - rp.bin_ty - (Bin.mk_pack rlt rp.bin_null) - bin - in - Coq.cps_mk_letin "env_bin" env_bin - (fun env_bin -> - let units = (List.map (fun (n,s) -> n, Unit.mk_pack rlt env_bin s)ir.units) in - let env_units = - Sigma.of_list - (Unit.ty_unit_pack rlt env_bin) - (Unit.mk_pack rlt env_bin rp.unit_null ) - units - in - - Coq.cps_mk_letin "env_units" env_units - (fun env_units -> - let sigmas = - { - env_sym = env_sym ; - env_bin = env_bin ; - env_units = env_units; - } in - let f = List.map (fun (x,_) -> (x,Coq.Pos.of_int x)) in - let sigma_maps = - { - sym_to_pos = (let sym = f renumbered_sym in fun x -> (List.assoc (to_local x) sym)); - bin_to_pos = (let bin = f bin in fun x -> (List.assoc x bin)); - units_to_pos = (let units = f units in fun x -> (List.assoc x units)); - } - in - k (sigmas, sigma_maps ) - ) - ) - ) goal - - (** builders for the reification *) - type reif_builders = - { - rsum: constr -> constr -> constr -> constr; - rprd: constr -> constr -> constr -> constr; - rsym: constr -> constr array -> constr; - runit : constr -> constr - } - - (* donne moi une tactique, je rajoute ma part. Potentiellement, il - est possible d'utiliser la notation 'do' a la Haskell: - http://www.cas.mcmaster.ca/~carette/pa_monad/ *) - let (>>) : 'a * Proof_type.tactic -> ('a -> 'b * Proof_type.tactic) -> 'b * Proof_type.tactic = - fun (x,t) f -> - let b,t' = f x in - b, Tacticals.tclTHEN t t' - - let return x = x, Tacticals.tclIDTAC - - let mk_vect vnil vcons v = - let ar = Array.length v in - let rec aux = function - | 0 -> vnil - | n -> let n = n-1 in - mkApp( vcons, - [| - (Coq.Nat.of_int n); - v.(ar - 1 - n); - (aux (n)) - |] - ) - in aux ar - - (* TODO: use a do notation *) - let mk_reif_builders (rlt: Coq.Relation.t) (env_sym:constr) (k: reif_builders -> Proof_type.tactic) = - let x = (rlt.Coq.Relation.carrier) in - let r = (rlt.Coq.Relation.r) in - - let x_r_env = [|x;r;env_sym|] in - let tty = mkApp (Lazy.force Stubs._Tty, x_r_env) in - let rsum = mkApp (Lazy.force Stubs.rsum, x_r_env) in - let rprd = mkApp (Lazy.force Stubs.rprd, x_r_env) in - let rsym = mkApp (Lazy.force Stubs.rsym, x_r_env) in - let vnil = mkApp (Lazy.force Stubs.vnil, x_r_env) in - let vcons = mkApp (Lazy.force Stubs.vcons, x_r_env) in - Coq.cps_mk_letin "tty" tty - (fun tty -> - Coq.cps_mk_letin "rsum" rsum - (fun rsum -> - Coq.cps_mk_letin "rprd" rprd - (fun rprd -> - Coq.cps_mk_letin "rsym" rsym - (fun rsym -> - Coq.cps_mk_letin "vnil" vnil - (fun vnil -> - Coq.cps_mk_letin "vcons" vcons - (fun vcons -> - let r ={ - rsum = - begin fun idx l r -> - mkApp (rsum, [| idx ; mk_mset tty [l,1 ; r,1]|]) - end; - rprd = - begin fun idx l r -> - let lst = NEList.of_list tty [l;r] in - mkApp (rprd, [| idx; lst|]) - end; - rsym = - begin fun idx v -> - let vect = mk_vect vnil vcons v in - mkApp (rsym, [| idx; vect|]) - end; - runit = fun idx -> (* could benefit of a letin *) - mkApp (Lazy.force Stubs.runit , [|x;r;env_sym;idx; |]) - } - in k r - )))))) - - - - type reifier = sigma_maps * reif_builders - - - let mk_reifier rlt zero envs (k : sigmas *reifier -> Proof_type.tactic) = - build_sigma_maps rlt zero envs - (fun (s,sm) -> - mk_reif_builders rlt s.env_sym - (fun rb ->k (s,(sm,rb)) ) - - ) - - (** [reif_constr_of_t reifier t] rebuilds the term [t] in the - reified form. We use the [reifier] to minimise the size of the - terms (we make as much lets as possible)*) - let reif_constr_of_t (sm,rb) (t:Matcher.Terms.t) : constr = - let rec aux = function - | Matcher.Terms.Plus (s,l,r) -> - let idx = sm.bin_to_pos s in - rb.rsum idx (aux l) (aux r) - | Matcher.Terms.Dot (s,l,r) -> - let idx = sm.bin_to_pos s in - rb.rprd idx (aux l) (aux r) - | Matcher.Terms.Sym (s,t) -> - let idx = sm.sym_to_pos s in - rb.rsym idx (Array.map aux t) - | Matcher.Terms.Unit s -> - let idx = sm.units_to_pos s in - rb.runit idx - | Matcher.Terms.Var i -> - anomaly "call to reif_constr_of_t on a term with variables." - in aux t -end - - - diff --git a/theory.mli b/theory.mli deleted file mode 100644 index fe79a11..0000000 --- a/theory.mli +++ /dev/null @@ -1,197 +0,0 @@ -(***************************************************************************) -(* This is part of aac_tactics, it is distributed under the terms of the *) -(* GNU Lesser General Public License version 3 *) -(* (see file LICENSE for more details) *) -(* *) -(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *) -(***************************************************************************) - -(** Bindings for Coq constants that are specific to the plugin; - reification and translation functions. - - Note: this module is highly correlated with the definitions of {i - AAC.v}. - - This module interfaces with the above Coq module; it provides - facilities to interpret a term with arbitrary operators as an - abstract syntax tree, and to convert an AST into a Coq term - (either using the Coq "raw" terms, as written in the starting - goal, or using the reified Coq datatypes we define in {i - AAC.v}). -*) - -(** Both in OCaml and Coq, we represent finite multisets using - weighted lists ([('a*int) list]), see {!Matcher.mset}. - - [mk_mset ty l] constructs a Coq multiset from an OCaml multiset - [l] of Coq terms of type [ty] *) - -val mk_mset:Term.constr -> (Term.constr * int) list ->Term.constr - -(** {2 Packaging modules} *) - -(** Environments *) -module Sigma: -sig - (** [add ty n x map] adds the value [x] of type [ty] with key [n] in [map] *) - val add: Term.constr ->Term.constr ->Term.constr ->Term.constr ->Term.constr - - (** [empty ty] create an empty map of type [ty] *) - val empty: Term.constr ->Term.constr - - (** [of_list ty null l] translates an OCaml association list into a Coq one *) - val of_list: Term.constr -> Term.constr -> (int * Term.constr ) list -> Term.constr - - (** [to_fun ty null map] converts a Coq association list into a Coq function (with default value [null]) *) - val to_fun: Term.constr ->Term.constr ->Term.constr ->Term.constr -end - - -(** Dynamically typed morphisms *) -module Sym: -sig - (** mimics the Coq record [Sym.pack] *) - type pack = {ar: Term.constr; value: Term.constr ; morph: Term.constr} - - val typ: Term.constr lazy_t - - - (** [mk_pack rlt (ar,value,morph)] *) - val mk_pack: Coq.Relation.t -> pack -> Term.constr - - (** [null] builds a dummy (identity) symbol, given an {!Coq.Relation.t} *) - val null: Coq.Relation.t -> Term.constr - -end - - -(** We need to export some Coq stubs out of this module. They are used - by the main tactic, see {!Rewrite} *) -module Stubs : sig - val lift : Term.constr Lazy.t - val lift_proj_equivalence : Term.constr Lazy.t - val lift_transitivity_left : Term.constr Lazy.t - val lift_transitivity_right : Term.constr Lazy.t - val lift_reflexivity : Term.constr Lazy.t - (** The evaluation fonction, used to convert a reified coq term to a - raw coq term *) - val eval: Term.constr lazy_t - - (** The main lemma of our theory, that is - [compare (norm u) (norm v) = Eq -> eval u == eval v] *) - val decide_thm:Term.constr lazy_t - - val lift_normalise_thm : Term.constr lazy_t -end - -(** {2 Building reified terms} - - We define a bundle of functions to build reified versions of the - terms (those that will be given to the reflexive decision - procedure). In particular, each field takes as first argument the - index of the symbol rather than the symbol itself. *) - -(** Tranlations between Coq and OCaml *) -module Trans : sig - - (** This module provides facilities to interpret a term with - arbitrary operators as an instance of an abstract syntax tree - {!Matcher.Terms.t}. - - For each Coq application [f x_1 ... x_n], this amounts to - deciding whether one of the partial applications [f x_1 - ... x_i], [i<=n] is a proper morphism, whether the partial - application with [i=n-2] yields an A or AC binary operator, and - whether the whole term is the unit for some A or AC operator. We - use typeclass resolution to test each of these possibilities. - - Note that there are ambiguous terms: - - a term like [f x y] might yield a unary morphism ([f x]) and a - binary one ([f]); we select the latter one (unless [f] is A or - AC, in which case we declare it accordingly); - - a term like [S O] can be considered as a morphism ([S]) - applied to a unit for [(+)], or as a unit for [( * )]; we - chose to give priority to units, so that the latter - interpretation is selected in this case; - - an element might be the unit for several operations - *) - - (** To achieve this reification, one need to record informations - about the collected operators (symbols, binary operators, - units). We use the following imperative internal data-structure to - this end. *) - - type envs - val empty_envs : unit -> envs - - - (** {2 Reification: from Coq terms to AST {!Matcher.Terms.t} } *) - - - (** [t_of_constr goal rlt envs (left,right)] builds the abstract - syntax tree of the terms [left] and [right]. We rely on the [goal] - to perform typeclasses resolutions to find morphisms compatible - with the relation [rlt]. Doing so, it modifies the reification - environment [envs]. Moreover, we need to create fresh - evars; this is why we give back the [goal], accordingly - updated. *) - - val t_of_constr : Coq.goal_sigma -> Coq.Relation.t -> envs -> (Term.constr * Term.constr) -> Matcher.Terms.t * Matcher.Terms.t * Coq.goal_sigma - - (** [add_symbol] adds a given binary symbol to the environment of - known stuff. *) - val add_symbol : Coq.goal_sigma -> Coq.Relation.t -> envs -> Term.constr -> Coq.goal_sigma - - (** {2 Reconstruction: from AST back to Coq terms } - - The next functions allow one to map OCaml abstract syntax trees - to Coq terms. We need two functions to rebuild different kind of - terms: first, raw terms, like the one encountered by - {!t_of_constr}; second, reified Coq terms, that are required for - the reflexive decision procedure. *) - - type ir - val ir_of_envs : Coq.goal_sigma -> Coq.Relation.t -> envs -> Coq.goal_sigma * ir - val ir_to_units : ir -> Matcher.ext_units - - (** {2 Building raw, natural, terms} *) - - (** [raw_constr_of_t] rebuilds a term in the raw representation, and - reconstruct the named products on top of it. In particular, this - allow us to print the context put around the left (or right) - hand side of a pattern. *) - val raw_constr_of_t : ir -> Coq.Relation.t -> (Context.Rel.t) ->Matcher.Terms.t -> Term.constr - - (** {2 Building reified terms} *) - - (** The reification environments, as Coq constrs *) - - type sigmas = { - env_sym : Term.constr; - env_bin : Term.constr; - env_units : Term.constr; (* the [idx -> X:constr] *) - } - - - - (** We need to reify two terms (left and right members of a goal) - that share the same reification envirnoment. Therefore, we need - to add letins to the proof context in order to ensure some - sharing in the proof terms we produce. - - Moreover, in order to have as much sharing as possible, we also - add letins for various partial applications that are used - throughout the terms. - - To achieve this, we decompose the reconstruction function into - two steps: first, we build the reification environment and then - reify each term successively.*) - type reifier - - val mk_reifier : Coq.Relation.t -> Term.constr -> ir -> (sigmas * reifier -> Proof_type.tactic) -> Proof_type.tactic - - (** [reif_constr_of_t reifier t] rebuilds the term [t] in the - reified form. *) - val reif_constr_of_t : reifier -> Matcher.Terms.t -> Term.constr - -end -- cgit v1.2.3