From 1aa8b6f6a876af22f538c869f022bc4ca5986b40 Mon Sep 17 00:00:00 2001 From: Stephane Glondu Date: Mon, 29 Nov 2010 23:30:48 +0100 Subject: Imported Upstream version 0.1-r13244 --- AAC.v | 689 +++++++++++++++++++++++++++++++++++++++ COPYING | 674 ++++++++++++++++++++++++++++++++++++++ COPYING.LESSER | 165 ++++++++++ Instances.v | 150 +++++++++ LICENSE | 13 + README.txt | 65 ++++ Tests.v | 373 +++++++++++++++++++++ Tutorial.v | 321 +++++++++++++++++++ aac_rewrite.ml | 308 ++++++++++++++++++ aac_rewrite.mli | 59 ++++ coq.ml | 212 ++++++++++++ coq.mli | 117 +++++++ files.txt | 7 + magic.txt | 4 + make_makefile | 49 +++ matcher.ml | 980 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++ matcher.mli | 192 +++++++++++ theory.ml | 708 ++++++++++++++++++++++++++++++++++++++++ theory.mli | 219 +++++++++++++ 19 files changed, 5305 insertions(+) create mode 100644 AAC.v create mode 100644 COPYING create mode 100644 COPYING.LESSER create mode 100644 Instances.v create mode 100644 LICENSE create mode 100644 README.txt create mode 100644 Tests.v create mode 100644 Tutorial.v create mode 100644 aac_rewrite.ml create mode 100644 aac_rewrite.mli create mode 100644 coq.ml create mode 100644 coq.mli create mode 100644 files.txt create mode 100644 magic.txt create mode 100755 make_makefile create mode 100644 matcher.ml create mode 100644 matcher.mli create mode 100644 theory.ml create mode 100644 theory.mli diff --git a/AAC.v b/AAC.v new file mode 100644 index 0000000..bdbe1b0 --- /dev/null +++ b/AAC.v @@ -0,0 +1,689 @@ +(***************************************************************************) +(* 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 normalize 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 occurence 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. +Unset Automatic Introduction. +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. + +(** * 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 *) + Program Definition null (H: Equivalence R) (x : X) := mkPack 0 x _. + + End t. + +End Sym. + +(** ** Associative commutative operations (commutative monoids) + + See #Instances.v# for concrete instances of these classes. + +*) + +Class Op_AC (X:Type) (R: relation X) (plus: X -> X -> X) (zero:X) := { + plus_compat:> Proper (R ==> R ==> R) plus; + plus_neutral_left: forall x, R (plus zero x) x; + plus_assoc: forall x y z, R (plus x (plus y z)) (plus (plus x y) z); + plus_com: forall x y, R (plus x y) (plus y x) +}. + +Module Sum. + Section t. + Context {X : Type} {R: relation X}. + Class pack : Type := mkPack { + plus : X -> X -> X; + zero : X; + opac :> @Op_AC X R plus zero + }. + End t. +End Sum. + + + +(** ** Associative operations (monoids) *) +Class Op_A (X:Type) (R:relation X) (dot: X -> X -> X) (one: X) := { + dot_compat:> Proper (R ==> R ==> R) dot; + dot_neutral_left: forall x, R (dot one x) x; + dot_neutral_right: forall x, R (dot x one) x; + dot_assoc: forall x y z, R (dot x (dot y z)) (dot (dot x y) z) +}. + + +Module Prd. + Section t. + Context {X : Type} {R: relation X}. + Class pack : Type := mkPack { + dot : X -> X -> X; + one : X; + opa :> @Op_A X R dot one + }. + End t. +End Prd. + + +(** an helper lemma to derive an A class out of an AC one (not + declared as an instance to avoid useless branches in typeclass + resolution) *) +Lemma AC_A {X} {R:relation X} {EQ: Equivalence R} {plus} {zero} (op: Op_AC R plus zero): Op_A R plus zero. +Proof. + split; try apply op. + intros. rewrite plus_com. apply op. +Qed. + + + +(** * Utilities for the evaluation function *) +Section copy. + + Context {X} {R: relation X} {HR : @Equivalence X R} {plus} {zero} {op: @Op_AC X R plus zero}. + + (* copy n x = x+...+x (n times) *) + 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 plus_assoc. + 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 plus_compat; auto. + Qed. + +End copy. + +(** * 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 lists and multisets *) + +(** finite multisets are represented with ordered lists with multiplicities *) +Definition mset A := list (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 function *) + Section c. + Variables A B: Type. + Variable compare: A -> B -> comparison. + Fixpoint list_compare h k := + match h,k with + | nil, nil => Eq + | nil, _ => Lt + | _, nil => 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 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. + + Fixpoint merge_msets l1 := + match l1 with + | nil => fun l2 => l2 + | (h1,n1)::t1 => + let fix merge_aux l2 := + match l2 with + | nil => l1 + | (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. + Variable b0: B. + Fixpoint fold_map l := + match l with + | nil => b0 + | u::nil => map u + | u::l => b2 (map u) (fold_map l) + end. + + (** mapping and merging *) + Variable merge: A -> list B -> list B. + Fixpoint merge_map (l: list A): list B := + match l with + | nil => nil + | u::l => merge u (merge_map l) + end. + + End m. + +End lists. + + + +(** * 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_prd: idx -> @Prd.pack X R. + Variable e_sum: idx -> @Sum.pack X R. + Hint Resolve e_sum e_prd: 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 -> list T -> T + | sym: forall i, vT (Sym.ar (e_sym i)) -> 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) + | 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. + + + (** 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 l l0); 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. + + 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. + + (** ** Evaluation from syntax to the abstract domain *) + Fixpoint eval u: X := + match u with + | sum i l => fold_map (fun un => let '(u,n):=un in @copy _ (@Sum.plus _ _ (e_sum i)) n (eval u)) + (@Sum.plus _ _ (e_sum i)) (@Sum.zero _ _ (e_sum i)) l + | prd i l => fold_map eval + (@Prd.dot _ _ (e_prd i)) (@Prd.one _ _ (e_prd i)) l + | sym i v => @eval_aux _ v (Sym.value (e_sym 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. + + + 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. + + + (** ** Normalisation *) + + (** auxiliary functions, to clean up sums and products *) + Definition sum' i (u: mset T): T := + match u with + | (u,xH)::nil => u + | _ => sum i u + end. + Definition is_sum i (u: T): option (mset T) := + match u with + | sum j l => if eq_idx_bool j i then Some l else None + | u => None + end. + Definition copy_mset n (l: mset T): mset T := + match n with + | xH => l + | _ => List.map (fun vm => let '(v,m):=vm in (v,Pmult n m)) l + end. + Definition sum_to_mset i (u: T) n := + match is_sum i u with + | Some l' => copy_mset n l' + | None => (u,n)::nil + end. + + Definition prd' i (u: list T): T := + match u with + | u::nil => u + | _ => prd i u + end. + Definition is_prd i (u: T): option (list T) := + match u with + | prd j l => if eq_idx_bool j i then Some l else None + | u => None + end. + Definition prd_to_list i (u: T) := + match is_prd i u with + | Some l' => l' + | None => u::nil + end. + + + (** we deduce the normalisation function *) + Fixpoint norm u := + match u with + | sum i l => sum' i (merge_map (fun un l => let '(u,n):=un in + merge_msets compare (sum_to_mset i (norm u) n) l) l) + | prd i l => prd' i (merge_map (fun u l => prd_to_list i (norm u) ++ l) l) + | sym i l => sym i (vnorm l) + end + with vnorm i (l: vT i): vT i := + match l with + | vnil => vnil + | vcons _ u l => vcons (norm u) (vnorm l) + end. + + + (** ** Correctness *) + + (** auxiliary lemmas about sums *) + Lemma sum'_sum i (l: mset T): eval (sum' i l) ==eval (sum i l) . + Proof. + intros i [|? [|? ?]]. + reflexivity. + destruct p; destruct p; simpl; reflexivity. + destruct p; destruct p; simpl; reflexivity. + Qed. + + Lemma eval_sum_cons i n a (l: mset T): + (eval (sum i ((a,n)::l))) == (@Sum.plus _ _ (e_sum i) (@copy _ (@Sum.plus _ _ (e_sum i)) n (eval a)) (eval (sum i l))). + Proof. + intros i n a [|[b m] l]; simpl. + rewrite plus_com, plus_neutral_left. reflexivity. + reflexivity. + Qed. + + Lemma eval_sum_app i (h k: mset T): eval (sum i (h++k)) == @Sum.plus _ _ (e_sum i) (eval (sum i h)) (eval (sum i k)). + Proof. + induction h; intro k. + simpl. rewrite plus_neutral_left. reflexivity. + simpl app. destruct a. rewrite 2 eval_sum_cons, IHh, plus_assoc. reflexivity. + Qed. + + Lemma eval_merge_sum i (h k: mset T): + eval (sum i (merge_msets compare h k)) == @Sum.plus _ _ (e_sum i) (eval (sum i h)) (eval (sum i k)). + Proof. + induction h as [|[a n] h IHh]; intro k. + simpl. rewrite plus_neutral_left. reflexivity. + induction k as [|[b m] k IHk]. + simpl. setoid_rewrite plus_com. rewrite plus_neutral_left. reflexivity. + simpl merge_msets. rewrite eval_sum_cons. destruct (tcompare_weak_spec a b) as [a|a b|a b]. + rewrite 2eval_sum_cons. rewrite IHh. rewrite <- plus_assoc. setoid_rewrite plus_assoc at 3. + setoid_rewrite plus_com at 5. rewrite !plus_assoc. rewrite copy_plus. reflexivity. + + rewrite eval_sum_cons. rewrite IHh. apply plus_assoc. + + rewrite <- eval_sum_cons. setoid_rewrite eval_sum_cons at 3. + rewrite plus_com, <- plus_assoc. rewrite plus_com in IHk. rewrite <- IHk. + rewrite eval_sum_cons. apply plus_compat; reflexivity. + Qed. + + Inductive sum_spec: T -> option (mset T) -> Prop := + | ss_sum1: forall i l, sum_spec (sum i l) (Some l) + | ss_sum2: forall j i l, i<>j -> sum_spec (sum i l) None + | ss_prd: forall i l, sum_spec (prd i l) None + | ss_sym: forall i l, sum_spec (@sym i l) None. + Lemma is_sum_spec: forall i (a: T), sum_spec a (is_sum i a). + Proof. + intros i [j l|j l|j l]; simpl; try constructor. + case eq_idx_spec; intro. + subst. constructor. + econstructor. eassumption. + Qed. + + Lemma eval_is_sum: forall i (a: T) k, is_sum i a = Some k -> eval (sum i k) = eval a. + Proof. + intros i [j l|j l|j l] k; simpl is_sum; try (intro; discriminate). + case eq_idx_spec; intros ? H. + injection H. clear H. intros <-. congruence. + discriminate. + Qed. + + Lemma copy_mset' n (l: mset T): + copy_mset n l = List.map (fun vm => let '(v,m):=vm in (v,Pmult n m)) l. + Proof. + intros. destruct n; try reflexivity. + simpl. induction l as [|[a l] IHl]; simpl; congruence. + Qed. + + + Lemma copy_mset_succ i n (l: mset T): + eval (sum i (copy_mset (Psucc n) l)) == @Sum.plus _ _ (e_sum i) (eval (sum i l)) (eval (sum i (copy_mset n l))). + Proof. + intros. rewrite 2copy_mset'. + induction l as [|[a m] l IHl]. + simpl eval at 2. rewrite plus_neutral_left. destruct n; reflexivity. + simpl map. rewrite ! eval_sum_cons. rewrite IHl. + rewrite Pmult_Sn_m. rewrite copy_plus. rewrite <- !plus_assoc. apply plus_compat; try reflexivity. + setoid_rewrite plus_com at 3. rewrite <- !plus_assoc. apply plus_compat; try reflexivity. apply plus_com. + Qed. + + Lemma eval_sum_to_mset: forall i n a, eval (sum i (sum_to_mset i a n)) == @copy _ (@Sum.plus _ _ (e_sum i)) n (eval a). + Proof. + intros. unfold sum_to_mset. + case_eq (is_sum i a). + intros k Hk. induction n using Pind. + simpl copy_mset. rewrite (eval_is_sum _ _ Hk). reflexivity. + rewrite copy_mset_succ. rewrite IHn. rewrite (eval_is_sum _ _ Hk). + unfold copy at 2. rewrite Prect_succ. reflexivity. + reflexivity. + Qed. + + + (** auxiliary lemmas about products *) + Lemma prd'_prd i (l: list T): eval (prd' i l) == eval (prd i l). + Proof. intros i [|? [|? ?]]; reflexivity. Qed. + + Lemma eval_prd_cons i a (l: list T): eval (prd i (a::l)) == @Prd.dot _ _ (e_prd i) (eval a) (eval (prd i l)). + Proof. + intros i a [|b l]; simpl. + rewrite dot_neutral_right. reflexivity. + reflexivity. + Qed. + + Lemma eval_prd_app i (h k: list T): eval (prd i (h++k)) == @Prd.dot _ _ (e_prd i) (eval (prd i h)) (eval (prd i k)). + Proof. + induction h; intro k. + simpl. rewrite dot_neutral_left. reflexivity. + simpl app. rewrite 2 eval_prd_cons, IHh, dot_assoc. reflexivity. + Qed. + + Lemma eval_is_prd: forall i (a: T) k, is_prd i a = Some k -> eval (prd i k) = eval a. + Proof. + intros i [j l|j l|j l] k; simpl is_prd; try (intro; discriminate). + case eq_idx_spec; intros ? H. + injection H. clear H. intros <-. subst. congruence. + discriminate. + Qed. + + Lemma eval_prd_to_list: forall i a, eval (prd i (prd_to_list i a)) = eval a. + Proof. + intros. unfold prd_to_list. + case_eq (is_prd i a). + intros k Hk. apply eval_is_prd. assumption. + reflexivity. + Qed. + + + (** 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. + simpl norm. rewrite sum'_sum. induction m as [|[u n] m IHm]. + reflexivity. + simpl merge_map. rewrite eval_merge_sum. + rewrite eval_sum_to_mset, IHm, eval_norm. + rewrite eval_sum_cons. reflexivity. + + simpl norm. rewrite prd'_prd. induction l. + reflexivity. + simpl merge_map. + rewrite eval_prd_app. + rewrite eval_prd_to_list, IHl, eval_norm. + rewrite eval_prd_cons. reflexivity. + + simpl. apply eval_norm_aux, Sym.morph. + + + 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. + +End s. +Declare ML Module "aac_tactics". diff --git a/COPYING b/COPYING new file mode 100644 index 0000000..94a9ed0 --- /dev/null +++ b/COPYING @@ -0,0 +1,674 @@ + 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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Such new +versions will be similar in spirit to the present version, but may +differ in detail to address new problems or concerns. + + Each version is given a distinguishing version number. If the +Library as you received it specifies that a certain numbered version +of the GNU Lesser General Public License "or any later version" +applies to it, you have the option of following the terms and +conditions either of that published version or of any later version +published by the Free Software Foundation. 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/Instances.v b/Instances.v new file mode 100644 index 0000000..1a2e08f --- /dev/null +++ b/Instances.v @@ -0,0 +1,150 @@ +(***************************************************************************) +(* 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. *) +(***************************************************************************) + +(** Instances for aac_rewrite.*) + +(* At the moment, all the instances are exported even if they are packaged into modules. *) + +Require Export AAC. + +Module Peano. + Require Import Arith NArith Max. + Program Instance aac_plus : @Op_AC nat eq plus 0 := @Build_Op_AC nat (@eq nat) plus 0 _ plus_0_l plus_assoc plus_comm. + + + Program Instance aac_mult : Op_AC eq mult 1 := Build_Op_AC _ _ _ mult_assoc mult_comm. + (* We also declare a default associative operation: this is currently + required to fill reification environments *) + Definition default_a := AC_A aac_plus. Existing Instance default_a. +End Peano. + +Module Z. + Require Import ZArith. + Open Scope Z_scope. + Program Instance aac_plus : Op_AC eq Zplus 0 := Build_Op_AC _ _ _ Zplus_assoc Zplus_comm. + Program Instance aac_mult : Op_AC eq Zmult 1 := Build_Op_AC _ _ Zmult_1_l Zmult_assoc Zmult_comm. + Definition default_a := AC_A aac_plus. Existing Instance default_a. +End Z. + +Module Q. + Require Import QArith. + Program Instance aac_plus : Op_AC Qeq Qplus 0 := Build_Op_AC _ _ Qplus_0_l Qplus_assoc Qplus_comm. + Program Instance aac_mult : Op_AC Qeq Qmult 1 := Build_Op_AC _ _ Qmult_1_l Qmult_assoc Qmult_comm. + Definition default_a := AC_A aac_plus. Existing Instance default_a. +End Q. + +Module Prop_ops. + Program Instance aac_or : Op_AC iff or False. Solve All Obligations using tauto. + + Program Instance aac_and : Op_AC iff and True. Solve All Obligations using tauto. + + Definition default_a := AC_A aac_and. Existing Instance default_a. + + Program Instance aac_not_compat : Proper (iff ==> iff) not. + Solve All Obligations using firstorder. +End Prop_ops. + +Module Bool. + Program Instance aac_orb : Op_AC eq orb false. + Solve All Obligations using firstorder. + + Program Instance aac_andb : Op_AC eq andb true. + Solve All Obligations using firstorder. + + Definition default_a := AC_A aac_andb. Existing Instance default_a. + + Instance negb_compat : Proper (eq ==> eq) negb. + Proof. intros [|] [|]; auto. Qed. +End Bool. + +Module Relations. + Require Import 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. + + Program Instance aac_union T : Op_AC (same_relation T) (union T) (bot T). + Solve All Obligations using compute; [tauto || intuition]. + + Program Instance aac_inter T : Op_AC (same_relation T) (inter T) (top T). + Solve All Obligations using compute; firstorder. + + (* note that we use [eq] directly as a neutral element for composition *) + Program Instance aac_compo T : Op_A (same_relation T) (compo T) eq. + Solve All Obligations using compute; firstorder. + Solve All Obligations using compute; firstorder subst; trivial. + + 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. + +End Relations. + +Module All. + Export Peano. + Export Z. + Export Prop_ops. + Export Bool. + Export Relations. +End All. + +(* A small test to ensure that everything can live together *) +(* Section test. + Import All. + Require Import ZArith. + Open Scope Z_scope. + Notation "x ^2" := (x*x) (at level 40). + Hypothesis H : forall x y, x^2 + y^2 + x*y + x* y = (x+y)^2. + Goal forall a b c, a*1*(a ^2)*a + c + (b+0)*1*b + a^2*b + a*b*a = (a^2+b)^2+c. + intros. + aac_rewrite H. + aac_rewrite <- H . + symmetry. + aac_rewrite <- H . + aac_reflexivity. + Qed. + Open Scope nat_scope. + Notation "x ^^2" := (x * x) (at level 40). + Hypothesis H' : forall (x y : nat), x^^2 + y^^2 + x*y + x* y = (x+y)^^2. + + Goal forall (a b c : nat), a*1*(a ^^2)*a + c + (b+0)*1*b + a^^2*b + a*b*a = (a^^2+b)^^2+c. + intros. aac_rewrite H'. aac_reflexivity. + Qed. +End test. + +*) + diff --git a/LICENSE b/LICENSE new file mode 100644 index 0000000..1be0ca5 --- /dev/null +++ b/LICENSE @@ -0,0 +1,13 @@ +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. + +This library is distributed in the hope that it will be useful, but +WITHOUT ANY WARRANTY; without even the implied warranty of +MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU +Lesser General Public License for more details. + +You should have received a copy of the GNU Lesser General Public +License along with this library. If not, see +. diff --git a/README.txt b/README.txt new file mode 100644 index 0000000..ad3b5c9 --- /dev/null +++ b/README.txt @@ -0,0 +1,65 @@ + + aac_tactics + =========== + + Thomas Braibant & Damien Pous + +Laboratoire d'Informatique de Grenoble (UMR 5217), INRIA, CNRS, France + +Webpage of the project: http://sardes.inrialpes.fr/~braibant/aac_tactics/ + + +FOREWORD +======== + +This plugin provides tactics for rewriting universally quantified +equations, modulo associativity and commutativity of some operators. + +INSTALL +======= + +- run [./make_makefile] in the top-level directory to generate a Makefile +- run [make world] to build the plugin and the documentation + +This should work with Coq v8.3 `beta' : -r 13027, -r 13060 and hopefully the +following ones. + +To be able to import the plugin from anywhere, add the following to +your ~/.coqrc + +Add Rec LoadPath "path_to_aac_tactics". +Add ML Path "path_to_aac_tactics". + + +DOCUMENTATION +============= + +Please refer to Tutorial.v for a succint 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) +- 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/Tests.v b/Tests.v new file mode 100644 index 0000000..87cc121 --- /dev/null +++ b/Tests.v @@ -0,0 +1,373 @@ +(***************************************************************************) +(* 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 AAC. + +Section sanity. + + Context {X} {R} {E: Equivalence R} {plus} {zero} + {dot} {one} {A: @Op_A X R dot one} {AC: Op_AC R plus zero}. + + + 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). + + Section t0. + Hypothesis H : forall x, x+x == x. + Goal forall a b, a+b +a == a+b. + intros a b; aacu_rewrite H; aac_reflexivity. + Qed. + End t0. + + Section t1. + Variables a b : X. + + Variable P : X -> Prop. + Hypothesis H : forall x y, P y -> x+y+x == x. + Hypothesis Hb : P b. + Goal a+b +a == a. + aacu_rewrite H. + aac_reflexivity. + auto. + Qed. + End t1. + + Section t. + Variable f : X -> X. + Hypothesis Hf : Proper (R ==> R) f. + Hypothesis H : forall x, x+x == x. + Goal forall y, f(y+y) == f (y). + intros y. + aacu_instances H. + aacu_rewrite H. + aac_reflexivity. + Qed. + + Goal forall y z, f(y+z+z+y) == f (y+z). + intros y z. + aacu_instances H. + aacu_rewrite H. + aac_reflexivity. + Qed. + + Hypothesis H' : forall x, x*x == x. + Goal forall y z, f(y*z*y*z) == f (y*z). + intros y z. + aacu_instances H'. + aacu_rewrite H'. + aac_reflexivity. + Qed. + End t. + + Goal forall x y, (plus x y) == (plus y x). intros. + aac_reflexivity. + Qed. + + Goal forall x y, R (plus (dot x one) y) (plus y x). intros. + aac_reflexivity. + Qed. + + Goal (forall f y, f y + f y == y) -> forall a b g (Hg: Proper (R ==> R) g), g (a+b) + g b + g (b+a) == a + b + g b. + intros. + try aacu_rewrite H. + aacu_rewrite (H g). + Restart. + intros. + set (H' := H g). + aacu_rewrite H'. + aac_reflexivity. + Qed. + + Section t2. + Variables a b: X. + Variables g : X ->X. + Hypothesis Hg:Proper (R ==> R) g. + Hypothesis H: (forall y, (g y) + y == y). + Goal g (a+b) + b +a+a == a + b +a. + intros. + aacu_rewrite H. + aac_reflexivity. + Qed. + End t2. + Section t3. + Variables a b: X. + Variables g : X ->X. + Hypothesis Hg:Proper (R ==> R) g. + Hypothesis H: (forall f y, f y + y == y). + + Goal g (a+b) + b +a+a == a + b +a. + intros. + aacu_rewrite (H g). + aac_reflexivity. + Qed. +End t3. + +Section null. + Hypothesis dot_ann : forall x, 0 * x == 0. + Goal forall a, 0*a == 0. + intros a. aac_rewrite dot_ann. reflexivity. + Qed. +End null. + +End sanity. + +Section abs. + Context + {X} {E: Equivalence (@eq X)} + {plus} {zero} {AC: @Op_AC X eq plus zero} + {plus'} {zero'} {AC': @Op_AC X eq plus' zero'} + {dot} {one} {A: @Op_A X eq dot one} + {dot'} {one'} {A': @Op_A X eq dot' one'}. + + Notation "x * y" := (dot x y) (at level 40, left associativity). + Notation "x @ y" := (dot' x y) (at level 20, left associativity). + Notation "1" := (one). + Notation "1'" := (one'). + Notation "x + y" := (plus x y) (at level 50, left associativity). + Notation "x ^ y" := (plus' x y) (at level 30, right associativity). + Notation "0" := (zero). + Notation "0'" := (zero'). + + Variables a b c d e k: X. + Variables f g: X -> X. + Hypothesis Hf: Proper (eq ==> eq) f. + Hypothesis Hg: Proper (eq ==> eq) g. + + Variables h: X -> X -> X. + Hypothesis Hh: Proper (eq ==> eq ==> eq) h. + + Variables q: X -> X -> X -> X. + Hypothesis Hq: Proper (eq ==> eq ==> eq ==> eq) q. + + + + + Hypothesis dot_ann_left: forall x, 0*x = 0. + + Goal f (a*0*b*c*d) = k. + aac_rewrite dot_ann_left. + Restart. + aac_rewrite dot_ann_left subterm 1 subst 0. + Abort. + + + + + Hypothesis distr_dp: forall x y z, x*(y+z) = x*y+x*z. + Hypothesis distr_dp': forall x y z, x*y+x*z = x*(y+z). + + Hypothesis distr_pp: forall x y z, x^(y+z) = x^y+x^z. + Hypothesis distr_pp': forall x y z, x^y+x^z = x^(y+z). + + Hypothesis distr_dd: forall x y z, x@(y*z) = x@y * x@z. + Hypothesis distr_dd': forall x y z, x@y * x@z = x@(y*z). + + + Goal a*b*(c+d+e) = k. + aac_instances distr_dp. + aacu_instances distr_dp. + aac_rewrite distr_dp subterm 0 subst 4. + + aac_instances distr_dp'. + aac_rewrite distr_dp'. + Abort. + + Goal a@(b*c)@d@(e*e) = k. + aacu_instances distr_dd. + aac_instances distr_dd. + aac_rewrite distr_dd. + Abort. + + Goal a^(b+c)^d^(e+e) = k. + aacu_instances distr_dd. + aac_instances distr_dd. + aacu_instances distr_pp. + aac_instances distr_pp. + aac_rewrite distr_pp subterm 9. + Abort. + + + + + Hypothesis plus_idem: forall x, x+x = x. + Hypothesis plus_idem3: forall x, x+x+x = x. + Hypothesis dot_idem: forall x, x*x = x. + Hypothesis dot_idem3: forall x, x*x*x = x. + + Goal a*b*a*b = k. + aac_instances dot_idem. + aacu_instances dot_idem. + aac_rewrite dot_idem. + aac_instances distr_dp. (* TODO: no solution -> fail ? *) + aacu_instances distr_dp. + Abort. + + Goal a+a+a+a = k. + aac_instances plus_idem3. + aac_rewrite plus_idem3. + Abort. + + Goal a*a*a*a = k. + aac_instances dot_idem3. + aac_rewrite dot_idem3. + Abort. + + Goal a*a*a*a*a*a = k. + aac_instances dot_idem3. + aac_rewrite dot_idem3. + Abort. + + Goal f(a*a)*f(a*a) = k. + aac_instances dot_idem. + (* TODO: pas clair qu'on doive réécrire les deux petits + sous-termes quand on nous demande de n'en réécrire qu'un... + d'autant plus que le comportement est nécessairement + imprévisible (cf. les deux exemples en dessous) + ceci dit, je suis conscient que c'est relou à empêcher... *) + aac_rewrite dot_idem subterm 1. + Abort. + + Goal f(a*a*b)*f(a*a*b) = k. + aac_instances dot_idem. + aac_rewrite dot_idem subterm 2. (* une seule instance réécrite *) + Abort. + + Goal f(b*a*a)*f(b*a*a) = k. + aac_instances dot_idem. + aac_rewrite dot_idem subterm 2. (* deux instances réécrites *) + Abort. + + Goal h (a*a) (a*b*c*d*e) = k. + aac_rewrite dot_idem. + Abort. + + Goal h (a+a) (a+b+c+d+e) = k. + aac_rewrite plus_idem. + Abort. + + Goal (a*a+a*a+b)*c = k. + aac_rewrite plus_idem. + Restart. + aac_rewrite dot_idem. + Abort. + + + + + Hypothesis dot_inv: forall x, x*f x = 1. + Hypothesis plus_inv: forall x, x+f x = 0. + Hypothesis dot_inv': forall x, f x*x = 1. + + Goal a*f(1)*b = k. + try aac_rewrite dot_inv. (* TODO: à terme, j'imagine qu'on souhaite accepter ça, même en aac *) + aacu_rewrite dot_inv. + Abort. + + Goal f(1) = k. + aacu_rewrite dot_inv. + Restart. + aacu_rewrite dot_inv'. + Abort. + + Goal b*f(b) = k. + aac_rewrite dot_inv. + Abort. + + Goal f(b)*b = k. + aac_rewrite dot_inv'. + Abort. + + Goal a*f(a*b)*a*b*c = k. + aac_rewrite dot_inv'. + Abort. + + Goal g (a*f(a*b)*a*b*c+d) = k. + aac_rewrite dot_inv'. + Abort. + + Goal g (a+f(a+b)+c+b+b) = k. + aac_rewrite plus_inv. + Abort. + + Goal g (a+f(0)+c) = k. + aac_rewrite plus_inv. + Abort. + + + + + + Goal forall R S, Equivalence R -> Equivalence S -> R a k -> S a k. + intros R S HR HS H. + try (aac_rewrite H ; fail 1 "ne doit pas passer"). + try (aac_instances H; fail 1 "ne doit pas passer"). + Abort. + + Goal forall R S, Equivalence R -> Equivalence S -> (forall x, R (f x) k) -> S (f a) k. + intros R S HR HS H. + try (aac_instances H; fail 1 "ne doit pas passer"). + try (aac_rewrite H ; fail 1 "ne doit pas passer"). + Abort. + + + + + + Hypothesis star_f: forall x y, x*f(y*x) = f(x*y)*x. + + Goal g (a+b*c*d*f(a*b*c*d)+b) = k. + aac_instances star_f. + aac_rewrite star_f. + Abort. + + Goal b*f(a*b)*f(b*b*f(a*b)) = k. + aac_instances star_f. + aac_rewrite star_f. + aac_rewrite star_f. + Abort. + + + + + + Hypothesis dot_select: forall x y, f x * y * g x = g y*x*f y. + + Goal f(a+b)*c*g(b+a) = k. + aac_rewrite dot_select. + Abort. + + Goal f(a+b)*g(b+a) = k. + try (aac_rewrite dot_select; fail 1 "ne doit pas passer"). + aacu_rewrite dot_select. + Abort. + + Goal f b*f(a+b)*g(b+a)*g b = k. + aacu_instances dot_select. + aac_instances dot_select. + aac_rewrite dot_select. + aacu_rewrite dot_select. + aacu_rewrite dot_select. + Abort. + + + + + Hypothesis galois_dot: forall x y z, h x (y*z) = h (x*y) z. + Hypothesis galois_plus: forall x y z, h x (y+z) = h (x+y) z. + + Hypothesis select_dot: forall x y, h x (y*x) = h (x*y) x. + Hypothesis select_plus: forall x y, h x (y+x) = h (x+y) x. + + Hypothesis h_dot: forall x y, h (x*y) (y*x) = h x y. + Hypothesis h_plus: forall x y, h (x+y) (y+x) = h x y. + +End abs. + diff --git a/Tutorial.v b/Tutorial.v new file mode 100644 index 0000000..84d40fc --- /dev/null +++ b/Tutorial.v @@ -0,0 +1,321 @@ +(***************************************************************************) +(* 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. *) +(***************************************************************************) + +(** * Examples about uses of the aac_rewrite library *) + +(* + Depending on your installation, either uncomment the following two + lines, or add them to your .coqrc files, replacing path_to_aac_tactics + with the correct path for your installation: + + Add Rec LoadPath "path_to_aac_tactics". + Add ML Path "path_to_aac_tactics". +*) +Require Export AAC. +Require Instances. + +(** ** Introductory examples *) + +(** First, we settle in the context of Z, and show an usage of our +tactics: we rewrite an universally quantified hypothesis modulo +associativity and commutativity. *) + +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); + - here, ring would have done the job. + *) + +End introduction. + +(** Second, we show how to exploit binomial identities to prove a goal +about pythagorean triples, without breaking a sweat. By comparison, +even if the goal and the hypothesis are both in normal form, making +the rewrites using standard tools is difficult. +*) + +Section binomials. + + Import ZArith. + Import Instances.Z. + + 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. + aac_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 binomials. + +(** ** 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; + however, to be able to use this plugin, one currently needs at least + one associative operator, and one associative-commutative + operator.) *) + +Section base. + + Context {X} {R} {E: Equivalence R} + {plus} {zero} + {dot} {one} + {A: @Op_A X R dot one} + {AC: Op_AC R plus zero}. + + 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. + + (** Note: the tactic starts by normalizing 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. + + (** We can deal with "proper" morphisms of arbitrary arity (here [f], + or [Zopp] earlier), and rewrite under morphisms (here [g]). *) + + 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 possible rewriting. 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 + proofgeneral, look at buffer *coq* ): *) + aac_instances H. + (** the default choice is the smallest possible context (number + 0), but one can choose the desired context; *) + aac_rewrite H subterm 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 context, but several substitutions; *) + aac_instances H. + (** we 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 keyword together to select the + correct subterm and the correct substitution. *) + + 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 subterm 1 subst 1. + aac_rewrite H. + aac_rewrite H'. + aac_reflexivity. + Qed. + + End both. + + (** We now turn on explaining the distinction between [aac_rewrite] + and [aacu_rewrite]: [aac_rewrite] rejects solutions in which + variables are instantiated by units, 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, hence no solutions; *) + aac_instances H. + (** while we get solutions with the "aacu" tactic. *) + 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'. + (** more than 52 with units. *) + aacu_instances H'. + Abort. + + End dealing_with_units. +End base. + +(** ** Declaring instances *) + +(** One can use one's own operations: it suffices to declare them as + instances of our classes. (Note that these instances are already + declared in file [Instances.v].) *) + +Section Peano. + Require Import Arith. + + Program Instance nat_plus : Op_AC eq plus O. + Solve All Obligations using firstorder. + + Program Instance nat_dot : Op_AC eq mult 1. + Solve All Obligations using firstorder. + + + (** Caveat: we need at least an instance of an operator that is AC + and one that is A for a given relation. However, one can reuse an + AC operator as an A operator. *) + + Definition default_a := AC_A nat_dot. Existing Instance default_a. +End Peano. + +(** ** Caveats *) + +Section Peano'. + Import Instances.Peano. + + (** 1. We have a special treatment for units, thus, [S x + x] does not + match [S 0], which is considered as a unit (one) of the mult + operation. *) + + Section caveat_one. + Definition double_plus_one x := 2*x + 1. + Hypothesis H : forall x, S x + x = double_plus_one x. + Goal S O = double_plus_one O. + try aac_rewrite H. (** 0 solutions (normal since it would use 0 to instantiate x) *) + try aacu_rewrite H. (** 0 solutions (abnormal) *) + Abort. + End caveat_one. + + (** 2. We cannot at the moment have two classes with the same + units: in the following example, [0] is understood as the unit of + [max] rather than as the unit of [plus]. *) + + Section max. + Program Instance aac_max : Op_AC eq Max.max O := Build_Op_AC _ _ _ Max.max_assoc Max.max_comm. + Variable a : nat. + Goal 0 + a = a. + try aac_reflexivity. + Abort. + End max. +End Peano'. + +(** 3. If some computations are possible in the goal or in the +hypothesis, the inference mechanism we use will make the +conversion. However, it seems that in most cases, these conversions +can be done by hand using simpl rather than rewrite. *) + +Section Z. + + (* The order of the following lines is important in order to get the right scope afterwards. *) + Import ZArith. + Open Scope Z_scope. + Opaque Zmult. + + Hypothesis dot_ann_left :forall x, x * 0 = 0. + Hypothesis dot_ann_right :forall x, 0 * x = 0. + + Goal forall a, a*0 = 0. + intros. aacu_rewrite dot_ann_left. reflexivity. + Qed. + + (** Here the tactic fails, since the 0*a is converted to 0, and no + rewrite can occur (even though Zmult is opaque). *) + Goal forall a, 0*a = 0. + intros. try aacu_rewrite dot_ann_left. + Abort. + + (** Here the tactic fails, since the 0*x is converted to 0 in the hypothesis. *) + Goal forall a, a*0 = 0. + intros. try aacu_rewrite dot_ann_right. + Abort. + + +End Z. + diff --git a/aac_rewrite.ml b/aac_rewrite.ml new file mode 100644 index 0000000..c4c88e0 --- /dev/null +++ b/aac_rewrite.ml @@ -0,0 +1,308 @@ +(***************************************************************************) +(* 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 *) + +(* 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 = + Util.anomaly ("aac_tactics: " ^ msg) + +let user_error msg = + Util.error ("aac_tactics: " ^ msg) + +(* debug and timing functions *) +let debug = false +let printing = false +let time = false + +let debug x = + if debug then + Printf.printf "%s\n%!" x + +let pr_constr msg constr = + if printing then + ( Pp.msgnl (Pp.str (Printf.sprintf "=====%s====" msg)); + Pp.msgnl (Printer.pr_constr constr); + ) + +let time msg tac = + if time then Coq.tclTIME msg tac + else tac + +let tac_or_exn tac exn msg = + fun gl -> try tac gl with e -> exn msg e + +(* 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 + | Util.Anomaly _ as e -> raise e + | _ -> anomaly msg + +module M = Matcher + +open Term +open Names +open Coqlib +open Proof_type + + +(** [find_applied_equivalence goal eq] try to ensure that the goal is + an applied equivalence relation, with two operands of the same type. + + This function is transitionnal, as we plan to integrate with the + new rewrite features of Coq 8.3, that seems to handle this kind of + things. +*) +let find_applied_equivalence goal : Term.constr -> Coq.eqtype *Term.constr *Term.constr* Coq.goal_sigma = fun equation -> + let env = Tacmach.pf_env goal in + let evar_map = Tacmach.project goal in + match Coq.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 left_type = Tacmach.pf_type_of goal left in + (* let right_type = Tacmach.pf_type_of goal right in *) + let partial_app = mkApp (c, Array.sub ca 0 (n-2) ) in + let eq_type = Coq.Classes.mk_equivalence left_type partial_app in + begin + try + let evar_map, equivalence = Typeclasses.resolve_one_typeclass env evar_map eq_type in + let goal = Coq.goal_update goal evar_map in + ((left_type, partial_app), equivalence),left, right, goal + with + | Not_found -> user_error "The goal is not an applied equivalence" + end + | _ -> user_error "The goal is not an equation" + + +(* [find_applied_equivalence_force] brutally decomposes the given + equation, and fail if we find a relation that is not the same as + the one we look for [r] *) +let find_applied_equivalence_force rlt goal : Term.constr -> Term.constr *Term.constr* Coq.goal_sigma = fun equation -> + match Coq.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 partial_app = mkApp (c, Array.sub ca 0 (n-2) ) in + let _ = if snd rlt = partial_app then () else user_error "The goal and the hypothesis have different top-level relations" in + + left, right, goal + | _ -> user_error "The hypothesis is not an equation" + + + +(** Build a couple of [t] from an hypothesis (variable names are not + relevant) *) +let rec t_of_hyp goal rlt envs e = match Coq.decomp_term e with + | Prod (name,ty,c) -> let x, y = t_of_hyp goal rlt envs c + in x,y+1 + | _ -> + let l,r ,goal = find_applied_equivalence_force rlt goal e in + let l,goal = Theory.Trans.t_of_constr goal rlt envs l in + let r,goal = Theory.Trans.t_of_constr goal rlt envs r in + (l,r),0 + +(** @deprecated: [option_rip] will be removed as soon as a proper interface is + given to the user *) +let option_rip = function | Some x -> x | None -> raise Not_found + +let mk_hyp_app conv c env = + let l = Matcher.Subst.to_list env in + let l = List.sort (fun (x,_) (y,_) -> compare y x) l in + let l = List.map (fun x -> conv (snd x)) l in + let v = Array.of_list l in + mkApp (c,v) + + + +(** {1 Tactics} *) + +(** [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 eqt envs (t : Matcher.Terms.t) (t' : Matcher.Terms.t) : Proof_type.tactic = fun goal -> + let maps, reifier,reif_letins = Theory.Trans.mk_reifier eqt envs goal in + let ((x,r),e) = eqt in + + (* 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, eval_letin = Coq.mk_letin' "eval_name" (mkApp (Lazy.force Theory.eval, [|x;r; maps.Theory.Trans.env_sym; maps.Theory.Trans.env_prd; maps.Theory.Trans.env_sum|])) in + let _ = pr_constr "left" t in + let _ = pr_constr "right" t' in + let t , t_letin = Coq.mk_letin "left" t in + let t', t_letin'= Coq.mk_letin "right" t' in + let apply_tac = Tactics.apply ( + mkApp (Lazy.force Theory.decide_thm, [|x;r;e; maps.Theory.Trans.env_sym; maps.Theory.Trans.env_prd; maps.Theory.Trans.env_sum;t;t'|])) + in + Tactics.tclABSTRACT None + (Tacticals.tclTHENLIST + [ + tac_or_exn reif_letins anomaly "invalid letin (1)"; + tac_or_exn eval_letin anomaly "invalid letin (2)"; + tac_or_exn t_letin anomaly "invalid letin (3)"; + tac_or_exn t_letin' anomaly "invalid letin (4)"; + tac_or_exn apply_tac anomaly "could not apply the decision theorem"; + tac_or_exn (time "vm_norm" (Tactics.normalise_vm_in_concl)) anomaly "vm_compute failure"; + Tacticals.tclORELSE Tactics.reflexivity + (Tacticals.tclFAIL 0 (Pp.str "Not an equality modulo A/AC")) + ]) + goal + + +(** The core tactic for aac_rewrite *) +let core_aac_rewrite ?(l2r = true) envs eqt rew p subst left = fun goal -> + let rlt = fst eqt in + let t = Matcher.Subst.instantiate subst p in + let tr = Theory.Trans.raw_constr_of_t envs rlt t in + let tac_rew_l2r = + if l2r then Equality.rewriteLR rew else Equality.rewriteRL rew + in + pr_constr "transitivity through" tr; + Tacticals.tclTHENSV + (tac_or_exn (Tactics.transitivity tr) anomaly "Unable to make the transitivity step") + [| + (* si by_aac_reflexivity foire (pas un theoreme), il fait tclFAIL, et on rattrape ici *) + tac_or_exn (time "by_aac_refl" (by_aac_reflexivity eqt envs left t)) push_anomaly "Invalid transitivity step"; + tac_or_exn (tac_rew_l2r) anomaly "Unable to rewrite"; + |] goal + +exception NoSolutions + +(** Choose a substitution from a + [(int * Terms.t * Env.env Search.m) Search.m ] *) +let choose_subst subterm sol m= + try + let (depth,pat,envm) = match subterm with + | None -> (* first solution *) + option_rip (Matcher.Search.choose m) + | Some x -> + List.nth (List.rev (Matcher.Search.to_list m)) x + in + let env = match sol with + None -> option_rip (Matcher.Search.choose envm) + | Some x -> List.nth (List.rev (Matcher.Search.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 rew ?(l2r=true) ?(show = false) ?strict ?occ_subterm ?occ_sol : Proof_type.tactic = fun goal -> + let envs = Theory.Trans.empty_envs () in + let rew_type = Tacmach.pf_type_of goal rew in + let (gl : Term.types) = Tacmach.pf_concl goal in + let _ = pr_constr "h" rew_type in + let _ = pr_constr "gl" gl in + let eqt, left, right, goal = find_applied_equivalence goal gl in + let rlt = fst eqt in + let left,goal = Theory.Trans.t_of_constr goal rlt envs left in + let right,goal = Theory.Trans.t_of_constr goal rlt envs right in + let (hleft,hright),hn = t_of_hyp goal rlt envs rew_type in + let pattern = if l2r then hleft else hright in + let m = Matcher.subterm ?strict pattern left in + + let m = Matcher.Search.sort (fun (x,_,_) (y,_,_) -> x - y) m in + if show + then + let _ = Pp.msgnl (Pp.str "All solutions:") in + let _ = Pp.msgnl (Matcher.pp_all (fun t -> Printer.pr_constr (Theory.Trans.raw_constr_of_t envs rlt t) ) m) in + Tacticals.tclIDTAC goal + else + try + let pat,env = choose_subst occ_subterm occ_sol m in + let rew = mk_hyp_app (Theory.Trans.raw_constr_of_t envs rlt )rew env in + core_aac_rewrite ~l2r envs eqt rew pat env left goal + with + | NoSolutions -> Tacticals.tclFAIL 0 + (Pp.str (if occ_subterm = None && occ_sol = None + then "No matching subterm found" + else "No such solution")) goal + + +let aac_reflexivity : Proof_type.tactic = fun goal -> + let (gl : Term.types) = Tacmach.pf_concl goal in + let envs = Theory.Trans.empty_envs() in + let eqt, left, right, evar_map = find_applied_equivalence goal gl in + let rlt = fst eqt in + let left,goal = Theory.Trans.t_of_constr goal rlt envs left in + let right,goal = Theory.Trans.t_of_constr goal rlt envs right in + by_aac_reflexivity eqt envs left right goal + + +open Tacmach +open Tacticals +open Tacexpr +open Tacinterp +open Extraargs + + (* Adding entries to the grammar of tactics *) +TACTIC EXTEND _aac1_ +| [ "aac_reflexivity" ] -> [ aac_reflexivity ] +END + + +TACTIC EXTEND _aac2_ +| [ "aac_rewrite" orient(o) constr(c) "subterm" integer(i) "subst" integer(j)] -> + [ fun gl -> aac_rewrite ~l2r:o ~strict:true ~occ_subterm:i ~occ_sol:j c gl ] +END + +TACTIC EXTEND _aac3_ +| [ "aac_rewrite" orient(o) constr(c) "subst" integer(j)] -> + [ fun gl -> aac_rewrite ~l2r:o ~strict:true ~occ_subterm:0 ~occ_sol:j c gl ] +END + +TACTIC EXTEND _aac4_ +| [ "aac_rewrite" orient(o) constr(c) "subterm" integer(i)] -> + [ fun gl -> aac_rewrite ~l2r:o ~strict:true ~occ_subterm:i ~occ_sol:0 c gl ] +END + +TACTIC EXTEND _aac5_ +| [ "aac_rewrite" orient(o) constr(c) ] -> + [ fun gl -> aac_rewrite ~l2r:o ~strict:true c gl ] +END + +TACTIC EXTEND _aac6_ +| [ "aac_instances" orient(o) constr(c)] -> + [ fun gl -> aac_rewrite ~l2r:o ~strict:true ~show:true c gl ] +END + + + + +TACTIC EXTEND _aacu2_ +| [ "aacu_rewrite" orient(o) constr(c) "subterm" integer(i) "subst" integer(j)] -> + [ fun gl -> aac_rewrite ~l2r:o ~occ_subterm:i ~occ_sol:j c gl ] +END + +TACTIC EXTEND _aacu3_ +| [ "aacu_rewrite" orient(o) constr(c) "subst" integer(j)] -> + [ fun gl -> aac_rewrite ~l2r:o ~occ_subterm:0 ~occ_sol:j c gl ] +END + +TACTIC EXTEND _aacu4_ +| [ "aacu_rewrite" orient(o) constr(c) "subterm" integer(i)] -> + [ fun gl -> aac_rewrite ~l2r:o ~occ_subterm:i ~occ_sol:0 c gl ] +END + +TACTIC EXTEND _aacu5_ +| [ "aacu_rewrite" orient(o) constr(c) ] -> + [ fun gl -> aac_rewrite ~l2r:o c gl ] +END + +TACTIC EXTEND _aacu6_ +| [ "aacu_instances" orient(o) constr(c)] -> + [ fun gl -> aac_rewrite ~l2r:o ~show:true c gl ] +END diff --git a/aac_rewrite.mli b/aac_rewrite.mli new file mode 100644 index 0000000..81818b6 --- /dev/null +++ b/aac_rewrite.mli @@ -0,0 +1,59 @@ +(***************************************************************************) +(* 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. *) +(***************************************************************************) + +(** Main module for the Coq plug-in ; provides the [aac_rewrite] and + [aac_reflexivity] tactics. + + This file defines the entry point for the tactic aac_rewrite. It + does Joe-the-plumbing, given a goal, reifies the interesting + subterms (rewrited hypothesis, goal) into abstract syntax tree + {!Matcher.Terms} (see {!Theory.Trans.t_of_constr}). Then, we use the + results from the matcher to rebuild terms and make a transitivity + step toward a term in which the hypothesis can be rewritten using + the standard rewrite. + + Doing so, we generate a sub-goal which we solve using a reflexive + decision procedure for the equality of terms modulo + AAC. Therefore, we also need to reflect the goal into a concrete + data-structure. See {i AAC.v} for more informations, + especially the data-type {b T} and the {b decide} theorem. + +*) + +(** {2 Transitional functions} + + We define some plumbing functions that will be removed when we + integrate the new rewrite features of Coq 8.3 +*) + +(** [find_applied_equivalence goal eq] checks that the goal is + an applied equivalence relation, with two operands of the same + type. + +*) +val find_applied_equivalence : Proof_type.goal Tacmach.sigma -> Term.constr -> Coq.eqtype * Term.constr * Term.constr * Proof_type.goal Tacmach.sigma + +(** Build a couple of [t] from an hypothesis (variable names are not + relevant) *) +val t_of_hyp : Proof_type.goal Tacmach.sigma -> Coq.reltype -> Theory.Trans.envs -> Term.types -> (Matcher.Terms.t * Matcher.Terms.t) * int + +(** {2 Tactics} *) + +(** the [aac_reflexivity] tactic solves equalities modulo AAC, by + reflection: it reifies the goal to apply theorem [decide], from + file {i AAC.v}, and then concludes using [vm_compute] + and [reflexivity] +*) +val aac_reflexivity : Proof_type.tactic + +(** [aac_rewrite] is the tactic for in-depth reqwriting modulo AAC +with some options to choose the orientation of the rewriting and a +solution (first the subterm, then the solution)*) + +val aac_rewrite : Term.constr -> ?l2r:bool -> ?show:bool -> ?strict: bool -> ?occ_subterm:int -> ?occ_sol:int -> Proof_type.tactic + diff --git a/coq.ml b/coq.ml new file mode 100644 index 0000000..731204c --- /dev/null +++ b/coq.ml @@ -0,0 +1,212 @@ +(***************************************************************************) +(* 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 + + +(* The contrib name is used to locate errors when loading constrs *) +let contrib_name = "aac_tactics" + +(* Getting constrs (primitive Coq terms) from exisiting Coq + libraries. *) +let find_constant contrib dir s = + Libnames.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 = + { Tacexpr.onhyps = Some []; + Tacexpr.concl_occs = Rawterm.no_occurrences_expr + } + + +let mk_letin (name: string) (constr: constr) : constr * Proof_type.tactic = + let name = (Names.id_of_string name) in + let letin = + (Tactics.letin_tac + None + (Name name) + constr + None (* or Some ty *) + nowhere + ) in + mkVar name, letin + +let mk_letin' (name: string) (constr: constr) : constr * Proof_type.tactic + = + constr, Tacticals.tclIDTAC + + + +(** {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 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 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) (rlt : constr * constr) : constr * goal_sigma = + let env = Tacmach.pf_env gl in + let evar_map = Tacmach.project gl in + let (em,x) = Evarutil.new_evar evar_map env (fst rlt) in + x,(goal_update gl em) + +let evar_binary (gl: goal_sigma) (rlt: constr * constr) = + let env = Tacmach.pf_env gl in + let evar_map = Tacmach.project gl in + let x = fst rlt in + let ty = mkArrow x (mkArrow x x) in + let (em,x) = Evarutil.new_evar evar_map env ty in + x,( goal_update gl em) + +(* decomp_term : constr -> (constr, types) kind_of_term *) +let decomp_term c = kind_of_term (strip_outer_cast c) + + +(** {2 Bindings with Coq' Standard Library} *) + +module Pair = struct + + let path = ["Coq"; "Init"; "Datatypes"] + let typ = lazy (init_constant path "prod") + let pair = lazy (init_constant path "pair") +end + +module Pos = struct + + let path = ["Coq" ; "NArith"; "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") + + (** build the constr [Proper rel t] *) + let mk_morphism ty rel t = + mkApp (Lazy.force morphism, [| ty; rel; t|]) + + let equivalence = lazy (init_constant (classes_path@ ["RelationClasses"]) "Equivalence") + + (** build the constr [Equivalence ty rel] *) + let mk_equivalence ty rel = + mkApp (Lazy.force equivalence, [| ty; rel|]) +end + +(** a [mset] is a ('a * pos) list *) +let mk_mset ty (l : (constr * int) list) = + let pos = Lazy.force Pos.typ in + let pair_ty = mkApp (Lazy.force Pair.typ , [| ty; pos|]) in + let pair (x : constr) (ar : int) = + mkApp (Lazy.force Pair.pair, [| ty ; pos ; x ; Pos.of_int ar|]) in + let rec aux = function + | [] -> List.nil pair_ty + | (t,ar)::q -> List.cons pair_ty (pair t ar) (aux q) + in + aux l + +(** x r *) +type reltype = Term.constr * Term.constr +(** x r e *) +type eqtype = reltype * Term.constr + + +(** {1 Tacticals} *) + +let tclTIME msg tac = fun gl -> + let u = Sys.time () in + let r = tac gl in + let _ = Pp.msgnl (Pp.str (Printf.sprintf "%s:%fs" msg (Sys.time ()-. u))) in + r + +let tclDEBUG msg tac = fun gl -> + let _ = Pp.msgnl (Pp.str msg) in + tac gl + +let tclPRINT tac = fun gl -> + let _ = Pp.msgnl (Printer.pr_constr (Tacmach.pf_concl gl)) in + tac gl + diff --git a/coq.mli b/coq.mli new file mode 100644 index 0000000..588a922 --- /dev/null +++ b/coq.mli @@ -0,0 +1,117 @@ +(***************************************************************************) +(* 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 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 nf_evar : goal_sigma -> Term.constr -> Term.constr +val evar_unit :goal_sigma ->Term.constr*Term.constr-> Term.constr* goal_sigma +val evar_binary: goal_sigma -> Term.constr*Term.constr -> Term.constr* goal_sigma + + +(** [mk_letin name v] binds the constr [v] using a letin tactic *) +val mk_letin : string ->Term.constr ->Term.constr * Proof_type.tactic + +(** [mk_letin' name v] is a drop-in replacement for [mk_letin' name v] + that does not make a binding (useful to test whether using lets is + efficient) *) +val mk_letin' : string ->Term.constr ->Term.constr * Proof_type.tactic + +(* decomp_term : constr -> (constr, types) kind_of_term *) +val decomp_term : Term.constr -> (Term.constr , Term.types) Term.kind_of_term + + +(** {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 +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 +end + +(** 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 + +(** pairs [(x,r)] such that [r: Relation x] *) +type reltype = Term.constr * Term.constr + +(** triples [((x,r),e)] such that [e : @Equivalence x r]*) +type eqtype = reltype * Term.constr + + +(** {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 + + diff --git a/files.txt b/files.txt new file mode 100644 index 0000000..7fa1801 --- /dev/null +++ b/files.txt @@ -0,0 +1,7 @@ +matcher.ml +coq.ml +theory.ml +aac_rewrite.ml +AAC.v +Instances.v +Tutorial.v diff --git a/magic.txt b/magic.txt new file mode 100644 index 0000000..d0272c3 --- /dev/null +++ b/magic.txt @@ -0,0 +1,4 @@ +-custom "mkdir -p doc; $(CAMLBIN)ocamldoc -html -rectypes -d doc -m A $(ZDEBUG) $(ZFLAGS) $^ && touch doc" "$(MLIFILES)" doc +-custom "$(CAMLOPTLINK) $(ZDEBUG) $(ZFLAGS) -shared -linkall -o aac_tactics.cmxs $(CMXFILES)" "$(CMXFILES)" aac_tactics.cmxs +-custom "$(CAMLLINK) -g -a -o aac_tactics.cma $(CMOFILES)" "$(CMOFILES)" aac_tactics.cma +-custom "$(CAMLBIN)ocamldep -slash $(OCAMLLIBS) $^ > .depend" "$(MLIFILES)" .depend diff --git a/make_makefile b/make_makefile new file mode 100755 index 0000000..3cfc096 --- /dev/null +++ b/make_makefile @@ -0,0 +1,49 @@ +# - set variable MLIFILES so that it can be used in custom targets to +# build documentation and dependencies for .mli files +# - remove useless ' -I . .../plugins/ ' lines so that we can read something in the compilation log +# - rename the rule for Makefile auto-regeneration, and replace it with an appropriate command +# - patch the rule for cleaning doc +# - include the .depend custom target, to take .mli dependencies into account + +( +coq_makefile -no-install -opt MLIFILES = '$(MLFILES:.ml=.mli)' $(cat files.txt) -f magic.txt \ + | sed 's|.*/plugins/.*||g;s|Makefile:|\.dummy:|g;s|rm -f doc|rm -rf doc|g' +cat < Makefile + + + +# TOTHINK: coq_makefile est bête, [make all] compile systématiquement +# les .cmxs associés à chaque module, alors que ça ne sert à rien, +# seul aac_rewrite.cmxs compte + +# problemes lorsqu'un module porte un meme nom qu'un module predefini +# (matching par exemple), prendre garde si ce probleme s'etend a la +# confusion module/nom de librairie diff --git a/matcher.ml b/matcher.ml new file mode 100644 index 0000000..176b660 --- /dev/null +++ b/matcher.ml @@ -0,0 +1,980 @@ +(***************************************************************************) +(* 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} +*) + + +let debug = false +let time = false + + +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 + + + +type symbol = int +type var = int + + +(****************) +(* Search Monad *) +(****************) + + +(** The {!Search} module contains a search monad that allows to + express, in a legible maner, programs that solves combinatorial + problems + + @see the + inspiration of this module +*) +module Search : sig + (** A data type that represent a collection of ['a] *) + type 'a m + (** 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 + (** 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 +end += struct + + 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))) +end + +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) + | One of symbol + | Plus of (symbol * t * t) + | Zero of symbol + | Sym of (symbol * t array) + | Var of var + + val equal_aac : 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 + | TVar of var + + + (** {2 Constructors: we ensure that the terms are always + normalised} *) + val mk_TAC : symbol -> nf_term mset -> nf_term + val mk_TA : symbol -> nf_term list -> nf_term + val mk_TSym : symbol -> nf_term list -> nf_term + val mk_TVar : var -> 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 : t -> nf_term + val t_of_term : nf_term -> t +end + = struct + + type t = + Dot of (symbol * t * t) + | One of symbol + | Plus of (symbol * t * t) + | Zero of symbol + | Sym of (symbol * t array) + | Var of var + + 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 + | TVar of var + + + + (** {2 Comparison} *) + + let nf_term_compare = Pervasives.compare + let nf_equal a b = a = b + + (** {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' s l = match l with + | [t,0] -> TAC (s,[]) + | [t,1] -> t + | _ -> TAC (s,l) + let mk_TA' s l = match l with [t] -> t + | _ -> TA (s,l) + + + + (** [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. *) + 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 l' = + merge_ac nf_term_compare l l' + + let extract_A s t = + match t with + | TA (s',l) when s' = s -> l + | _ -> [t] + + let extract_AC s (t,ar) : nf_term mset = + match t with + | TAC (s',l) when s' = s -> List.map (fun (x,y) -> (x,y*ar)) l + | _ -> [t,ar] + + + (** {3 Constructors of {!nf_term}}*) + + let mk_TAC s (l : (nf_term *int) list) = + mk_TAC' s + (merge_map (fun u l -> merge (extract_AC s u) l) l) + let mk_TA s l = + mk_TA' s + (merge_map (fun u l -> (extract_A s u) @ l) l) + let mk_TSym s l = TSym (s,l) + let mk_TVar v = TVar v + + + (** {2 Printing function} *) + let print_binary_list single unit binary l = + 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 sprint_ac single (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_list single s l = + Printf.sprintf "[%i:AC]{%s}" + s + (print_binary_list + single + "0" + (fun x y -> x ^ " , " ^ y) + l + ) + + + let print_a single s l = + Printf.sprintf "[%i:A]{%s}" + 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) -> + Printf.sprintf "[%i:AC]{%s}" s + (sprint_ac + sprint_nf_term + l) + | TA (s,l) -> print_a sprint_nf_term s l + | TVar v -> Printf.sprintf "x%i" v + + + + (** {2 Conversion functions} *) + + (* rebuilds a tree out of a list *) + let rec binary_of_list f comb null l = + let l = List.rev l in + let rec aux = function + | [] -> null + | [t] -> f t + | t::q -> comb (aux q) (f t) + in + aux l + + let rec term_of_t : t -> nf_term = function + | Dot (s,l,r) -> + let l = term_of_t l in + let r = term_of_t r in + mk_TA s [l;r] + | Plus (s,l,r) -> + let l = term_of_t l in + let r = term_of_t r in + mk_TAC ( s) [l,1;r,1] + | One x -> + mk_TA ( x) [] + | Zero x -> + mk_TAC ( 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) + + 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)) + (Zero ( s)) + (linear l) + ) + | TA (s,l) -> + (binary_of_list + t_of_term + (fun l r -> Dot ( s,l,r)) + (One ( s)) + 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 + + + let equal_aac x y = + nf_equal (term_of_t x) (term_of_t 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 + | One _ as x -> x + | Zero _ 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 -> Util.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 + +(** 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 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 t >> + (fun left, right -> + a_nondet_split 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 (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 + +(** Try to affect the variable [x] to each left factor of [t]*) +let var_a_nondet_split ?(strict=false) env current x t = + a_nondet_split t + >> + (fun (l,r) -> + if strict && l=[] then fail() else + return ((Subst.add env x (mk_TA current l)), r) + ) + +(** Try to affect variable [x] to _each_ subset of t. *) +let var_ac_nondet_split ?(strict=false) (s : symbol) env (x : var) (t : nf_term mset) : (Subst.t * (nf_term mset)) m = + ac_nondet_split t + >> + (fun (subset,compl) -> + if strict && subset=[] then fail() else + return ((Subst.add env x (mk_TAC s subset)), compl) + ) + +(** 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 + | _ -> [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 + | _ -> [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 + | _ -> + begin match t with + | [] -> fail () + | t::q -> + if nf_equal v t + then return q + else fail () + end + + +(**************) +(* AC Removal *) +(**************) + +(** [fold_acc] 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 +*) + +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*) +let is_unit_AC s t = + nf_equal t (mk_TAC s []) + +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 + +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 + + + +(************) +(* 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 ?strict = + 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 + + 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 -> + begin match Subst.find env x with + | None -> + (var_ac_nondet_split ?strict 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 + | h :: q ->(* PAC =/= curent or PA *) + (ac_nondet_split t) + >> + (fun (left,right) -> + matching env h (mk_TAC current left) + >> + ( + 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 -> + var_a_nondet_split ?strict env current x t + >> (fun (env,residual)-> matchingA env current l residual) + | Some v -> + (left_factor current v t) + >> (fun residual -> matchingA env current l residual) + end + | h :: l -> + a_nondet_split t + >> (fun (t,r) -> + matching env h (mk_TA current 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 + matching + + + +(***********) +(* Subterm *) +(***********) + +(** [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 + l,r,f acc l t r + ) ([], l,acc) l + in acc + + +(** [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 (hence the tests on the length of the + lists). 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 Search.is_empty m + then + fail () + else + return (raw_p ,m) + +let subterm ?strict (raw_p:t) (raw_t:t): (int* t * Subst.t m) m= + let p = term_of_t raw_p in + let t = term_of_t raw_t in + let rec subterm (t:nf_term) : (t * Subst.t m) m= + match t with + | TAC (s,l) -> + (ac_nondet_split l) >> + (fun (left,right) -> + (subterm_AC s left) >> + (fun (p,m) -> + let p = if right = [] then p else + Plus (s,p,t_of_term (mk_TAC s right)) + in + return (p,m) + ) + + ) + | TA (s,l) -> + (a_nondet_middle l) + >> + (fun (left, middle, right) -> + (subterm_A s middle) >> + (fun (p,m) -> + let p = + if right = [] then p else + Dot (s,p,t_of_term (mk_TA s right)) + in + let p = + if left = [] then p else + Dot (s,t_of_term (mk_TA s left),p) + in + return (p,m) + ) + ) + | TSym (s, l) -> + let init = return_non_empty raw_p (matching ?strict 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 + | _ -> assert false + and subterm_AC s tl = + match tl with + [x,1] -> subterm x + | _ -> + return_non_empty raw_p (matching ?strict Subst.empty p (mk_TAC s tl)) + and subterm_A s tl = + match tl with + [x] -> subterm x + | _ -> + return_non_empty raw_p (matching ?strict Subst.empty p (mk_TA s tl)) + in + (subterm t >> fun (p,m) -> return (Terms.size p,p,m)) + +(* The functions we export, handlers for the previous ones. Some debug + information also *) +let subterm ?strict raw t = + let sols = time (subterm ?strict raw) t "%fs spent in subterm (including matching)\n" in + if debug then Printf.printf "%i possible solution(s)\n" + (Search.fold (fun (_,_,envm) acc -> count envm + acc) sols 0); + sols + + +let matcher ?strict p t = + let sols = time + (fun (p,t) -> + let p = (Terms.term_of_t p) in + let t = (Terms.term_of_t t) in + matching ?strict Subst.empty p t) (p,t) + "%fs spent in the matcher\n" + in + if debug then Printf.printf "%i solutions\n" (count sols); + sols + +(* A very basic way to interact with the envs, to choose a possible + solution *) +open Pp +let pp_env pt : Subst.t -> Pp.std_ppcmds = fun env -> + List.fold_left (fun acc (v,t) -> str (Printf.sprintf "x%i: " v) ++ pt t ++ str "; " ++ acc) (str "") (Subst.to_list env) + +let pp_envm pt : Subst.t Search.m -> Pp.std_ppcmds = fun m -> + let _,s = Search.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 pp_all pt : (int * Terms.t * Subst.t Search.m) Search.m -> Pp.std_ppcmds = fun m -> + let _,s = Search.fold + (fun (size,context,envm) (n,acc) -> + let s = str (Printf.sprintf "subterm %i\t" n) in + let s = s ++ (str "(context ") ++ pt context ++ (str ")\n") in + let s = s ++ str (Printf.sprintf "\t%i possible(s) substitutions" (Search.count envm) ) ++ fnl () in + let s = s ++ pp_envm pt envm in + n+1, s::acc + ) m (0,[]) in + List.fold_left (fun acc s -> s ++ str "\n" ++ acc) (str "") (s) + diff --git a/matcher.mli b/matcher.mli new file mode 100644 index 0000000..3e20e73 --- /dev/null +++ b/matcher.mli @@ -0,0 +1,192 @@ +(***************************************************************************) +(* 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). + + This module could be reused ouside of the Coq plugin. + + Matching modulo AAC a pattern [p] against a term [t] boils down to + finding a substitution [env] such that the pattern [p] instantiated + with [env] is equal to [t] modulo AAC. + + We proceed by structural decomposition of the pattern, trying 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] does + not match [a+a+b] ). + + We use a search monad {!Search} 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 AAC. In particular, this function + gives a solution to the aforementioned case ([x+x] against + [a+b+a]). +*) + +(** {2 Utility functions} *) + +type symbol = int +type var = int + +(** The {!Search} module contains a search monad that allows to + express our non-deterministic and back-tracking algorithm in a + legible maner. + + @see the + inspiration of this module +*) +module Search : +sig + (** A data type that represent a collection of ['a] *) + type 'a m + (** 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 + (** 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 +end + +(** 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 AAC. 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, Dot(4,[||])); + Dot(4, Var x, Sym(5,[|Sym(3,[||])|])) |])] + represents the pattern [forall x, f (x^1) (x*g b)], where [1] is the + unit associated with [( * )]. + *) + + type t = + Dot of (symbol * t * t) + | One of symbol + | Plus of (symbol * t * t) + | Zero of symbol + | Sym of (symbol * t array) + | Var of var + + (** Test for equality of terms modulo AAC (relies on the following + canonical representation of terms) *) + val equal_aac : 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 AAC + 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 AAC + iif [nf_equal (term_of_t a) (term_of_t b) = true] *) + val term_of_t : 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 -> Terms.t -> Terms.t -> Subst.t Search.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 -> Terms.t -> Terms.t -> (int * Terms.t * Subst.t Search.m) Search.m + +(** pretty printing of the solutions *) +val pp_all : (Terms.t -> Pp.std_ppcmds) -> (int * Terms.t * Subst.t Search.m) Search.m -> Pp.std_ppcmds diff --git a/theory.ml b/theory.ml new file mode 100644 index 0000000..45a76f1 --- /dev/null +++ b/theory.ml @@ -0,0 +1,708 @@ +(***************************************************************************) +(* 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 + +let ac_theory_path = ["AAC"] + +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 = + let map = + List.fold_left + (fun acc (i,t) -> assert (i > 0); add ty (Coq.Pos.of_int i) t acc) + (empty ty) + l + in to_fun ty null map +end + + +module Sym = struct + type pack = {ar: Term.constr; value: Term.constr ; morph: Term.constr} + + let path = ac_theory_path @ ["Sym"] + let typ = lazy (Coq.init_constant path "pack") + let mkPack = lazy (Coq.init_constant path "mkPack") + let typeof = lazy (Coq.init_constant path "type_of") + let relof = lazy (Coq.init_constant path "rel_of") + let value = lazy (Coq.init_constant path "value") + let null = lazy (Coq.init_constant path "null") + let mk_typeof : Coq.reltype -> int -> constr = fun (x,r) n -> + mkApp (Lazy.force typeof, [| x ; Coq.Nat.of_int n |]) + let mk_relof : Coq.reltype -> int -> constr = fun (x,r) n -> + mkApp (Lazy.force relof, [| x;r ; Coq.Nat.of_int n |]) + let mk_pack ((x,r): Coq.reltype) s = + mkApp (Lazy.force mkPack, [|x;r; s.ar;s.value;s.morph|]) + let null eqt witness = + let (x,r),e = eqt in + mkApp (Lazy.force null, [| x;r;e;witness |]) + +end + +module Sum = struct + type pack = {plus: Term.constr; zero: Term.constr ; opac: Term.constr} + + let path = ac_theory_path @ ["Sum"] + let typ = lazy (Coq.init_constant path "pack") + let mkPack = lazy (Coq.init_constant path "mkPack") + let cstr_zero = lazy (Coq.init_constant path "zero") + let cstr_plus = lazy (Coq.init_constant path "plus") + + let mk_pack: Coq.reltype -> pack -> Term.constr = fun (x,r) s -> + mkApp (Lazy.force mkPack , [| x ; r; s.plus; s.zero; s.opac|]) + + let zero: Coq.reltype -> constr -> constr = fun (x,r) pack -> + mkApp (Lazy.force cstr_zero, [| x;r;pack|]) + let plus: Coq.reltype -> constr -> constr = fun (x,r) pack -> + mkApp (Lazy.force cstr_plus, [| x;r;pack|]) + + +end + +module Prd = struct + type pack = {dot: Term.constr; one: Term.constr ; opa: Term.constr} + + let path = ac_theory_path @ ["Prd"] + let typ = lazy (Coq.init_constant path "pack") + let mkPack = lazy (Coq.init_constant path "mkPack") + let cstr_one = lazy (Coq.init_constant path "one") + let cstr_dot = lazy (Coq.init_constant path "dot") + + let mk_pack: Coq.reltype -> pack -> Term.constr = fun (x,r) s -> + mkApp (Lazy.force mkPack , [| x ; r; s.dot; s.one; s.opa|]) + let one: Coq.reltype -> constr -> constr = fun (x,r) pack -> + mkApp (Lazy.force cstr_one, [| x;r;pack|]) + let dot: Coq.reltype -> constr -> constr = fun (x,r) pack -> + mkApp (Lazy.force cstr_dot, [| x;r;pack|]) + + +end + + +let op_ac = lazy (Coq.init_constant ac_theory_path "Op_AC") +let op_a = lazy (Coq.init_constant ac_theory_path "Op_A") + +(** The constants from the inductive type *) +let _Tty = lazy (Coq.init_constant ac_theory_path "T") +let vTty = lazy (Coq.init_constant ac_theory_path "vT") + +let rsum = lazy (Coq.init_constant ac_theory_path "sum") +let rprd = lazy (Coq.init_constant ac_theory_path "prd") +let rsym = lazy (Coq.init_constant ac_theory_path "sym") +let vnil = lazy (Coq.init_constant ac_theory_path "vnil") +let vcons = lazy (Coq.init_constant ac_theory_path "vcons") +let eval = lazy (Coq.init_constant ac_theory_path "eval") +let decide_thm = lazy (Coq.init_constant ac_theory_path "decide") + +(** Rebuild an equality *) +let mk_equal = fun (x,r) left right -> + mkApp (r, [| left; right|]) + + +let anomaly msg = + Util.anomaly ("aac_tactics: " ^ msg) + +let user_error msg = + Util.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 raw morphisms ([Sym]) are coarser than [A] + operators, which in turn are coarser than [AC] operators. We + need to treat the units first. Indeed, considering [S] as a + morphism is problematic because [S O] is a unit. + + During this reification, we gather some informations that will + be used to rebuild Coq terms from AST ( type {!envs}) *) + + type pack = + | AC of Sum.pack (* the opac pack *) + | A of Prd.pack (* the opa pack *) + | Sym of Sym.pack (* the sym pack *) + + type head = + | Fun + | Plus + | Dot + | One + | Zero + + + (* discr is a map from {!Term.constr} to + (head * 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. + + However, for the transitivity process, we do not need to remember + if the constr was the unit or the binary operation. We will use + the projections from the module's record. + + The field [bloom] allows to give a unique number to each class we + found. *) + type envs = + { + discr : (constr , (head * pack) option) Hashtbl.t ; + + bloom : (pack, int ) Hashtbl.t; + bloom_back : (int, pack ) Hashtbl.t; + bloom_next : int ref; + } + + let empty_envs () = + { + discr = Hashtbl.create 17; + + bloom = Hashtbl.create 17; + bloom_back = Hashtbl.create 17; + bloom_next = ref 1; + } + + let add_bloom envs pack = + if Hashtbl.mem envs.bloom pack + then () + else + let x = ! (envs.bloom_next) in + Hashtbl.add envs.bloom pack x; + Hashtbl.add envs.bloom_back x pack; + incr (envs.bloom_next) + + let find_bloom envs pack = + try Hashtbl.find envs.bloom pack + with Not_found -> assert false + + + (** try to infer the kind of operator we are dealing with *) + let is_morphism rlt papp ar goal = + let typeof = Sym.mk_typeof rlt ar in + let relof = Sym.mk_relof rlt ar in + let env = Tacmach.pf_env goal in + let evar_map = Tacmach.project goal in + let m = Coq.Classes.mk_morphism typeof relof papp in + try + let _ = Tacmach.pf_check_type goal papp typeof in + let evar_map,m = Typeclasses.resolve_one_typeclass env evar_map m in + let pack = {Sym.ar = (Coq.Nat.of_int ar); Sym.value= papp; Sym.morph= m} in + Some (Coq.goal_update goal evar_map, Fun , Sym pack) + with + | Not_found -> None + | _ -> None + + (** gives back the class *) + let is_op gl (rlt: Coq.reltype) pack op null = + let (x,r) = rlt in + let ty = mkApp (pack ,[|x;r; op; null |]) in + let env = Tacmach.pf_env gl in + let evar_map = Tacmach.project gl in + try + let em,t = Typeclasses.resolve_one_typeclass env evar_map ty in + let op = Evarutil.nf_evar em op in + let null = Evarutil.nf_evar em null in + Some (Coq.goal_update gl em,t,op,null) + with Not_found -> None + + + let is_plus (rlt: Coq.reltype) plus goal= + let (zero,goal) = Coq.evar_unit goal rlt in + match is_op goal rlt (Lazy.force op_ac) plus zero + with + | None -> None + | Some (gl,s,plus,zero) -> + let pack = {Sum.plus=plus;Sum.zero=zero;Sum.opac=s} in + Some (gl, Plus , AC pack) + + let is_dot rlt dot goal= + let (one,goal) = Coq.evar_unit goal rlt in + match is_op goal rlt (Lazy.force op_a) dot one + with + | None -> None + | Some (goal,s,dot,one) -> + let pack = {Prd.dot=dot;Prd.one=one;Prd.opa=s} in + Some (goal, Dot, A pack) + + let is_zero rlt zero goal = + let (plus,goal) = Coq.evar_binary goal rlt in + match is_op goal rlt (Lazy.force op_ac) plus zero + with + | None -> None + | Some (goal,s,plus,zero) -> + let pack = {Sum.plus=plus;Sum.zero=zero;Sum.opac=s} in + Some (goal, Zero , AC pack) + + let is_one rlt one goal = + let (dot,goal) = Coq.evar_binary goal rlt in + match is_op goal rlt (Lazy.force op_a) dot one + with + | None -> None + | Some (goal,s,dot,one)-> + let pack = {Prd.dot=dot;Prd.one=one;Prd.opa=s} in + Some (goal, One, A pack) + + + let (>>) x y a = + match x a with | None -> y a | x -> x + + let is_unit rlt papp = + (is_zero rlt papp )>> (is_one rlt papp) + + let what_is_it rlt papp ar goal : + (Coq.goal_sigma * head * pack ) option = + match ar with + | 2 -> + begin + ( + (is_plus rlt papp)>> + (is_dot rlt papp)>> + (is_morphism rlt papp ar) + ) goal + end + | _ -> + is_morphism rlt papp ar goal + + + + + (** [discriminates goal envs rlt t ca] infer : + + - the nature of the application [t ca] (is the head symbol a + Plus, a Dot, so on), + + - 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 *) + let discriminate goal envs (rlt : Coq.reltype) t ca : Coq.goal_sigma * head * pack * constr * constr array = + let nb_params = Array.length ca in + let f cont x p_app ar = + assert (0 <= ar && ar <= nb_params); + match x with + | Some (goal, discr, pack) -> + Hashtbl.add envs.discr p_app (Some (discr, pack)); + add_bloom envs pack; + (goal, discr, pack, p_app, Array.sub ca (nb_params-ar) ar) + | None -> + begin + (* to memoise the failures *) + Hashtbl.add envs.discr p_app None; + (* will terminate, since [const] is + capped, and it is easy to find an + instance of a constant *) + cont goal (ar-1) + end + in + let rec aux goal ar :Coq.goal_sigma * head * pack * constr * constr array = + assert (0 <= ar && ar <= nb_params); + let p_app = mkApp (t, Array.sub ca 0 (nb_params - ar)) in + begin + try + begin match Hashtbl.find envs.discr p_app with + | None -> aux goal (ar-1) + | Some (discr,sym) -> + goal, + discr, + sym, + p_app, + Array.sub ca (nb_params -ar ) ar + end + with + Not_found -> (* Could not find this constr *) + f aux (what_is_it rlt p_app ar goal) p_app ar + end + in + match is_unit rlt (mkApp (t,ca)) goal with + | None -> aux goal (nb_params) + | x -> f (fun _ -> assert false) x (mkApp (t,ca)) 0 + + let discriminate goal envs rlt x = + try + match Coq.decomp_term x with + | Var _ -> + discriminate goal envs rlt x [| |] + | App (t,ca) -> + discriminate goal envs rlt t ca + | Construct c -> + discriminate goal envs rlt x [| |] + | _ -> + assert false + with + (* TODO: is it the only source of invalid use that fall into + this catch_all ? *) + e -> user_error "cannot handle this kind of hypotheses (higher order function symbols, variables occuring under a non-morphism, etc)." + + + (** [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.reltype ) envs : constr -> Matcher.Terms.t *Coq.goal_sigma = + let r_evar_map = ref (goal) in + let rec aux x = + match Coq.decomp_term x with + | Rel i -> Matcher.Terms.Var i + | _ -> + let goal, sym, pack , p_app, ca = discriminate (!r_evar_map) envs rlt x in + r_evar_map := goal; + let k = find_bloom envs pack + in + match sym with + | Plus -> + assert (Array.length ca = 2); + Matcher.Terms.Plus ( k, aux ca.(0), aux ca.(1)) + | Zero -> + assert (Array.length ca = 0); + Matcher.Terms.Zero ( k) + | Dot -> + assert (Array.length ca = 2); + Matcher.Terms.Dot ( k, aux ca.(0), aux ca.(1)) + | One -> + assert (Array.length ca = 0); + Matcher.Terms.One ( k) + | Fun -> + Matcher.Terms.Sym ( k, Array.map aux ca) + in + ( + fun x -> let r = aux x in r, !r_evar_map + ) + + (** {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] rebuilds a term in the raw representation *) + let raw_constr_of_t envs (rlt : Coq.reltype) (t : Matcher.Terms.t) : Term.constr = + 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 + let rec aux t = + match t with + | Matcher.Terms.Plus (s,l,r) -> + begin match Hashtbl.find envs.bloom_back s with + | AC s -> + mkApp (s.Sum.plus , [|(aux l);(aux r)|]) + | _ -> assert false + end + | Matcher.Terms.Dot (s,l,r) -> + begin match Hashtbl.find envs.bloom_back s with + | A s -> + mkApp (s.Prd.dot, [|(aux l);(aux r)|]) + | _ -> assert false + end + | Matcher.Terms.Sym (s,t) -> + begin match Hashtbl.find envs.bloom_back s with + | Sym s -> + mkApp (s.Sym.value, Array.map aux t) + | _ -> assert false + end + | Matcher.Terms.One x -> + begin match Hashtbl.find envs.bloom_back x with + | A s -> + s.Prd.one + | _ -> assert false + end + | Matcher.Terms.Zero x -> + begin match Hashtbl.find envs.bloom_back x with + | AC s -> + s.Sum.zero + | _ -> assert false + end + | Matcher.Terms.Var i -> add_set i; + let i = (Names.id_of_string (Printf.sprintf "x%i" i)) in + mkVar (i) + in + let x = fst rlt in + let rec cap c = function [] -> c + | t::q -> let i = (Names.id_of_string (Printf.sprintf "x%i" t)) in + cap (mkNamedProd i x c) q + in + cap ((aux t)) (get ()) + (* aux t is possible for printing only, but I wonder what + would be the status of such a term*) + + + (** {2 Building reified terms} *) + + (* Some informations to be able to build the maps *) + type reif_params = + { + a_null : constr; + ac_null : constr; + sym_null : constr; + a_ty : constr; + ac_ty : constr; + sym_ty : constr; + } + + type sigmas = { + env_sym : Term.constr; + env_sum : Term.constr; + env_prd : Term.constr; + } + (** A record containing the environments that will be needed by the + decision procedure, as a Coq constr. Contains also the functions + from the symbols (as ints) to indexes (as constr) *) + type sigma_maps = { + sym_to_pos : int -> Term.constr; + sum_to_pos : int -> Term.constr; + prd_to_pos : int -> Term.constr; + } + + (* convert the [envs] into the [sigma_maps] *) + let envs_to_list rlt envs = + let add x y (n,l) = (n+1, (x, ( n, y))::l) in + let nil = 1,[]in + Hashtbl.fold + (fun int pack (sym,sum,prd) -> + match pack with + |AC s -> + let s = Sum.mk_pack rlt s in + sym, add (int) s sum, prd + |A s -> + let s = Prd.mk_pack rlt s in + sym, sum, add (int) s prd + | Sym s -> + let s = Sym.mk_pack rlt s in + add (int) s sym, sum, prd + ) + envs.bloom_back + (nil,nil,nil) + + + + (** 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 eqt = + let rlt = fst eqt in + let plus,goal = Coq.evar_binary goal rlt in + let dot ,goal = Coq.evar_binary goal rlt in + let one ,goal = Coq.evar_unit goal rlt in + let zero,goal = Coq.evar_unit goal rlt in + + let (x,r) = rlt in + let ty_ac = ( mkApp (Lazy.force op_ac, [| x;r;plus;zero|])) in + let ty_a = ( mkApp (Lazy.force op_a, [| x;r;dot;one|])) in + let a ,goal= + try Coq.resolve_one_typeclass goal ty_a + with Not_found -> user_error ("Unable to infer a default A operator.") + in + let ac ,goal= + try Coq.resolve_one_typeclass goal ty_ac + with Not_found -> user_error ("Unable to infer a default AC operator.") + in + let plus = Coq.nf_evar goal plus in + let dot = Coq.nf_evar goal dot in + let one = Coq.nf_evar goal one in + let zero = Coq.nf_evar goal zero in + + let record = + {a_null = Prd.mk_pack rlt {Prd.dot=dot;Prd.one=one;Prd.opa=a}; + ac_null= Sum.mk_pack rlt {Sum.plus=plus;Sum.zero=zero;Sum.opac=ac}; + sym_null = Sym.null eqt zero; + a_ty = (mkApp (Lazy.force Prd.typ, [| x;r; |])); + ac_ty = ( mkApp (Lazy.force Sum.typ, [| x;r|])); + sym_ty = mkApp (Lazy.force Sym.typ, [|x;r|]) + } + in + goal, record + + (** [build_sigma_maps] given a envs and some reif_params, we are + able to build the sigmas *) + let build_sigma_maps goal eqt envs : sigmas * sigma_maps * Proof_type.goal Evd.sigma = + let rlt = fst eqt in + let goal,rp = build_reif_params goal eqt in + let ((_,sym), (_,sum),(_,prd)) = envs_to_list rlt envs in + let f = List.map (fun (x,(y,z)) -> (x,Coq.Pos.of_int y)) in + let g = List.map (fun (x,(y,z)) -> (y,z)) in + let sigmas = + {env_sym = Sigma.of_list rp.sym_ty rp.sym_null (g sym); + env_sum = Sigma.of_list rp.ac_ty rp.ac_null (g sum); + env_prd = Sigma.of_list rp.a_ty rp.a_null (g prd); + } in + let sigma_maps = { + sym_to_pos = (let sym = f sym in fun x -> (List.assoc x sym)); + sum_to_pos = (let sum = f sum in fun x -> (List.assoc x sum)); + prd_to_pos = (let prd = f prd in fun x -> (List.assoc x prd)); + } + in + sigmas, sigma_maps , goal + + + + (** builders for the reification *) + type reif_builders = + { + rsum: constr -> constr -> constr -> constr; + rprd: constr -> constr -> constr -> constr; + rzero: constr -> constr; + rone: constr -> constr; + rsym: constr -> constr array -> 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.reltype) (env_sym:constr) :reif_builders*Proof_type.tactic = + let x,r = rlt in + let tty = mkApp (Lazy.force _Tty, [| x ; r ; env_sym|]) in + (Coq.mk_letin' "mk_rsum" (mkApp (Lazy.force rsum, [| x; r; env_sym|]))) >> + (fun rsum -> (Coq.mk_letin' "mk_rprd" (mkApp (Lazy.force rprd, [| x; r; env_sym|]))) + >> fun rprd -> ( Coq.mk_letin' "mk_rsym" (mkApp (Lazy.force rsym, [| x;r; env_sym|]))) + >> fun rsym -> (Coq.mk_letin' "mk_vnil" (mkApp (Lazy.force vnil, [| x;r; env_sym|]))) + >> fun vnil -> Coq.mk_letin' "mk_vcons" (mkApp (Lazy.force vcons, [| x;r; env_sym|])) + >> fun vcons -> + return { + rsum = + begin fun idx l r -> + mkApp (rsum, [| idx ; Coq.mk_mset tty [l,1 ; r,1]|]) + end; + rzero = + begin fun idx -> + mkApp (rsum, [| idx ; Coq.mk_mset tty []|]) + end; + rprd = + begin fun idx l r -> + let lst = Coq.List.of_list tty [l;r] in + mkApp (rprd, [| idx; lst|]) + end; + rone = + begin fun idx -> + let lst = Coq.List.of_list tty [] in + mkApp (rprd, [| idx; lst|]) end; + rsym = + begin fun idx v -> + let vect = mk_vect vnil vcons v in + mkApp (rsym, [| idx; vect|]) + end; + } + ) + + + + type reifier = sigma_maps * reif_builders + + let mk_reifier eqt envs goal : sigmas * reifier * Proof_type.tactic = + let s, sm, goal = build_sigma_maps goal eqt envs in + let env_sym, env_sym_letin = Coq.mk_letin "env_sym_name" (s.env_sym) in + let env_prd, env_prd_letin = Coq.mk_letin "env_prd_name" (s.env_prd) in + let env_sum, env_sum_letin = Coq.mk_letin "env_sum_name" (s.env_sum) in + let s = {env_sym = env_sym; env_prd = env_prd; env_sum = env_sum} in + let rb , tac = mk_reif_builders (fst eqt) (s.env_sym) in + let evar_map = Tacmach.project goal in + let tac = Tacticals.tclTHENLIST ( + [ Refiner.tclEVARS evar_map; + env_prd_letin ; + env_sum_letin ; + env_sym_letin ; + tac + ]) in + s, (sm, rb), tac + + + (** [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.sum_to_pos s in + rb.rsum idx (aux l) (aux r) + | Matcher.Terms.Dot (s,l,r) -> + let idx = sm.prd_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.One s -> + let idx = sm.prd_to_pos s in + rb.rone idx + | Matcher.Terms.Zero s -> + let idx = sm.sum_to_pos s in + rb.rzero 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 new file mode 100644 index 0000000..85ad24b --- /dev/null +++ b/theory.mli @@ -0,0 +1,219 @@ +(***************************************************************************) +(* 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}). +*) + + + + +(** 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 + +(** The constr associated with our typeclasses for AC or A operators *) +val op_ac:Term.constr lazy_t +val op_a: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 + +(** {2 Packaging modules} *) + +(** 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_typeof rlt int] *) + val mk_typeof: Coq.reltype ->int ->Term.constr + + (** [mk_typeof rlt int] *) + val mk_relof: Coq.reltype ->int ->Term.constr + + (** [mk_pack rlt (ar,value,morph)] *) + val mk_pack: Coq.reltype -> pack -> Term.constr + + (** [null] builds a dummy symbol, given an {!Coq.eqtype} and a + constant *) + val null: Coq.eqtype -> Term.constr -> Term.constr + +end + +(** Dynamically typed AC operations *) +module Sum: +sig + + (** mimics the Coq record [Sum.pack] *) + type pack = {plus : Term.constr; zero: Term.constr ; opac: Term.constr} + + val typ: Term.constr lazy_t + + (** [mk_pack rlt (plus,zero,opac)] *) + val mk_pack: Coq.reltype -> pack -> Term.constr + + (** [zero rlt pack]*) + val zero: Coq.reltype -> Term.constr -> Term.constr + + (** [plus rlt pack]*) + val plus: Coq.reltype -> Term.constr -> Term.constr + +end + + +(** Dynamically typed A operations *) +module Prd: +sig + + (** mimics the Coq record [Prd.pack] *) + type pack = {dot: Term.constr; one: Term.constr ; opa: Term.constr} + + val typ: Term.constr lazy_t + + (** [mk_pack rlt (dot,one,opa)] *) + val mk_pack: Coq.reltype -> pack -> Term.constr + + (** [one rlt pack]*) + val one: Coq.reltype -> Term.constr -> Term.constr + + (** [dot rlt pack]*) + val dot: Coq.reltype -> Term.constr -> Term.constr + +end + + +(** [mk_equal rlt l r] builds the term [l == r] *) +val mk_equal : Coq.reltype -> Term.constr -> Term.constr -> Term.constr + + +(** {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; the + behaviour in this case is almost unspecified: we simply give + priority to AC 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 + + (** {1 Reification: from Coq terms to AST {!Matcher.Terms.t} } *) + + (** [t_of_constr goal rlt envs cstr] builds the abstract syntax tree + of the term [cstr]. 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 creates fresh evars; this is + why we give back the [goal], accordingly updated. *) + val t_of_constr : Coq.goal_sigma -> Coq.reltype -> envs -> Term.constr -> Matcher.Terms.t * Coq.goal_sigma + + (** {1 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. *) + + (** {2 Building raw, natural, terms} *) + + (** [raw_constr_of_t] rebuilds a term in the raw representation *) + val raw_constr_of_t : envs -> Coq.reltype -> Matcher.Terms.t -> Term.constr + + (** {2 Building reified terms} *) + + (** The reification environments, as Coq constrs *) + type sigmas = { + env_sym : Term.constr; + env_sum : Term.constr; + env_prd : Term.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.eqtype -> envs -> Coq.goal_sigma -> sigmas * reifier * 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