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(***************************************************************************)
(* This is part of aac_tactics, it is distributed under the terms of the *)
(* GNU Lesser General Public License version 3 *)
(* (see file LICENSE for more details) *)
(* *)
(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *)
(***************************************************************************)
Require Import Arith NArith.
Require Import List.
Require Import FMapPositive FMapFacts.
Require Import RelationClasses Equality.
Set Implicit Arguments.
Set Asymmetric Patterns.
(** * Utilities for positive numbers
which we use as:
- indices for morphisms and symbols
- multiplicity of terms in sums *)
Notation idx := positive.
Fixpoint eq_idx_bool i j :=
match i,j with
| xH, xH => true
| xO i, xO j => eq_idx_bool i j
| xI i, xI j => eq_idx_bool i j
| _, _ => false
end.
Fixpoint idx_compare i j :=
match i,j with
| xH, xH => Eq
| xH, _ => Lt
| _, xH => Gt
| xO i, xO j => idx_compare i j
| xI i, xI j => idx_compare i j
| xI _, xO _ => Gt
| xO _, xI _ => Lt
end.
Notation pos_compare := idx_compare (only parsing).
(** Specification predicate for boolean binary functions *)
Inductive decide_spec {A} {B} (R : A -> B -> Prop) (x : A) (y : B) : bool -> Prop :=
| decide_true : R x y -> decide_spec R x y true
| decide_false : ~(R x y) -> decide_spec R x y false.
Lemma eq_idx_spec : forall i j, decide_spec (@eq _) i j (eq_idx_bool i j).
Proof.
induction i; destruct j; simpl; try (constructor; congruence).
case (IHi j); constructor; congruence.
case (IHi j); constructor; congruence.
Qed.
(** weak specification predicate for comparison functions: only the 'Eq' case is specified *)
Inductive compare_weak_spec A: A -> A -> comparison -> Prop :=
| pcws_eq: forall i, compare_weak_spec i i Eq
| pcws_lt: forall i j, compare_weak_spec i j Lt
| pcws_gt: forall i j, compare_weak_spec i j Gt.
Lemma pos_compare_weak_spec: forall i j, compare_weak_spec i j (pos_compare i j).
Proof. induction i; destruct j; simpl; try constructor; case (IHi j); intros; constructor. Qed.
Lemma idx_compare_reflect_eq: forall i j, idx_compare i j = Eq -> i=j.
Proof. intros i j. case (pos_compare_weak_spec i j); intros; congruence. Qed.
(** * Dependent types utilities *)
Notation cast T H u := (eq_rect _ T u _ H).
Section dep.
Variable U: Type.
Variable T: U -> Type.
Lemma cast_eq: (forall u v: U, {u=v}+{u<>v}) ->
forall A (H: A=A) (u: T A), cast T H u = u.
Proof. intros. rewrite <- Eqdep_dec.eq_rect_eq_dec; trivial. Qed.
Variable f: forall A B, T A -> T B -> comparison.
Definition reflect_eqdep := forall A u B v (H: A=B), @f A B u v = Eq -> cast T H u = v.
(* these lemmas have to remain transparent to get structural recursion
in the lemma [tcompare_weak_spec] below *)
Lemma reflect_eqdep_eq: reflect_eqdep ->
forall A u v, @f A A u v = Eq -> u = v.
Proof. intros H A u v He. apply (H _ _ _ _ eq_refl He). Defined.
Lemma reflect_eqdep_weak_spec: reflect_eqdep ->
forall A u v, compare_weak_spec u v (@f A A u v).
Proof.
intros. case_eq (f u v); try constructor.
intro H'. apply reflect_eqdep_eq in H'. subst. constructor. assumption.
Defined.
End dep.
(** * Utilities about (non-empty) lists and multisets *)
Inductive nelist (A : Type) : Type :=
| nil : A -> nelist A
| cons : A -> nelist A -> nelist A.
Notation "x :: y" := (cons x y).
Fixpoint nelist_map (A B: Type) (f: A -> B) l :=
match l with
| nil x => nil ( f x)
| cons x l => cons ( f x) (nelist_map f l)
end.
Fixpoint appne A l l' : nelist A :=
match l with
nil x => cons x l'
| cons t q => cons t (appne A q l')
end.
Notation "x ++ y" := (appne x y).
(** finite multisets are represented with ordered lists with multiplicities *)
Definition mset A := nelist (A*positive).
(** lexicographic composition of comparisons (this is a notation to keep it lazy) *)
Notation lex e f := (match e with Eq => f | _ => e end).
Section lists.
(** comparison functions *)
Section c.
Variables A B: Type.
Variable compare: A -> B -> comparison.
Fixpoint list_compare h k :=
match h,k with
| nil x, nil y => compare x y
| nil x, _ => Lt
| _, nil x => Gt
| u::h, v::k => lex (compare u v) (list_compare h k)
end.
End c.
Definition mset_compare A B compare: mset A -> mset B -> comparison :=
list_compare (fun un vm =>
let '(u,n) := un in
let '(v,m) := vm in
lex (compare u v) (pos_compare n m)).
Section list_compare_weak_spec.
Variable A: Type.
Variable compare: A -> A -> comparison.
Hypothesis Hcompare: forall u v, compare_weak_spec u v (compare u v).
(* this lemma has to remain transparent to get structural recursion
in the lemma [tcompare_weak_spec] below *)
Lemma list_compare_weak_spec: forall h k,
compare_weak_spec h k (list_compare compare h k).
Proof.
induction h as [|u h IHh]; destruct k as [|v k]; simpl; try constructor.
case (Hcompare a a0 ); try constructor.
case (Hcompare u v ); try constructor.
case (IHh k); intros; constructor.
Defined.
End list_compare_weak_spec.
Section mset_compare_weak_spec.
Variable A: Type.
Variable compare: A -> A -> comparison.
Hypothesis Hcompare: forall u v, compare_weak_spec u v (compare u v).
(* this lemma has to remain transparent to get structural recursion
in the lemma [tcompare_weak_spec] below *)
Lemma mset_compare_weak_spec: forall h k,
compare_weak_spec h k (mset_compare compare h k).
Proof.
apply list_compare_weak_spec.
intros [u n] [v m].
case (Hcompare u v); try constructor.
case (pos_compare_weak_spec n m); try constructor.
Defined.
End mset_compare_weak_spec.
(** (sorted) merging functions *)
Section m.
Variable A: Type.
Variable compare: A -> A -> comparison.
Definition insert n1 h1 :=
let fix insert_aux l2 :=
match l2 with
| nil (h2,n2) =>
match compare h1 h2 with
| Eq => nil (h1,Pplus n1 n2)
| Lt => (h1,n1):: nil (h2,n2)
| Gt => (h2,n2):: nil (h1,n1)
end
| (h2,n2)::t2 =>
match compare h1 h2 with
| Eq => (h1,Pplus n1 n2):: t2
| Lt => (h1,n1)::l2
| Gt => (h2,n2)::insert_aux t2
end
end
in insert_aux.
Fixpoint merge_msets l1 :=
match l1 with
| nil (h1,n1) => fun l2 => insert n1 h1 l2
| (h1,n1)::t1 =>
let fix merge_aux l2 :=
match l2 with
| nil (h2,n2) =>
match compare h1 h2 with
| Eq => (h1,Pplus n1 n2) :: t1
| Lt => (h1,n1):: merge_msets t1 l2
| Gt => (h2,n2):: l1
end
| (h2,n2)::t2 =>
match compare h1 h2 with
| Eq => (h1,Pplus n1 n2)::merge_msets t1 t2
| Lt => (h1,n1)::merge_msets t1 l2
| Gt => (h2,n2)::merge_aux t2
end
end
in merge_aux
end.
(** interpretation of a list with a constant and a binary operation *)
Variable B: Type.
Variable map: A -> B.
Variable b2: B -> B -> B.
Fixpoint fold_map l :=
match l with
| nil x => map x
| u::l => b2 (map u) (fold_map l)
end.
(** mapping and merging *)
Variable merge: A -> nelist B -> nelist B.
Fixpoint merge_map (l: nelist A): nelist B :=
match l with
| nil x => nil (map x)
| u::l => merge u (merge_map l)
end.
Variable ret : A -> B.
Variable bind : A -> B -> B.
Fixpoint fold_map' (l : nelist A) : B :=
match l with
| nil x => ret x
| u::l => bind u (fold_map' l)
end.
End m.
End lists.
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