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(* Copyright (c) 2009, Adam Chlipala
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions are met:
*
* - Redistributions of source code must retain the above copyright notice,
* this list of conditions and the following disclaimer.
* - Redistributions in binary form must reproduce the above copyright notice,
* this list of conditions and the following disclaimer in the documentation
* and/or other materials provided with the distribution.
* - The names of contributors may not be used to endorse or promote products
* derived from this software without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
* AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
* LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
* CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
* SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
* INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
* CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
* ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
* POSSIBILITY OF SUCH DAMAGE.
*)
Require Import Arith List TheoryList.
Require Import Axioms.
Require Import Syntax.
Set Implicit Arguments.
Definition row (T : Type) := list (name * T).
Fixpoint record (r : row Set) : Set :=
match r with
| nil => unit
| (_, T) :: r' => T * record r'
end%type.
Fixpoint kDen (k : kind) : Type :=
match k with
| KType => Set
| KName => name
| KArrow k1 k2 => kDen k1 -> kDen k2
| KRecord k1 => row (kDen k1)
end.
Fixpoint cfold T T' (f : name -> T -> T' -> T') (i : T') (r : row T) {struct r} : T' :=
match r with
| nil => i
| (n, v) :: r' => f n v (cfold f i r')
end.
Fixpoint cDen k (c : con kDen k) {struct c} : kDen k :=
match c in con _ k return kDen k with
| CVar _ x => x
| Arrow c1 c2 => cDen c1 -> cDen c2
| Poly _ c1 => forall x, cDen (c1 x)
| CAbs _ _ c1 => fun x => cDen (c1 x)
| CApp _ _ c1 c2 => (cDen c1) (cDen c2)
| Name n => n
| TRecord c1 => record (cDen c1)
| CEmpty _ => nil
| CSingle _ c1 c2 => (cDen c1, cDen c2) :: nil
| CConcat _ c1 c2 => cDen c1 ++ cDen c2
| CFold k1 k2 => @cfold _ _
| CGuarded _ _ _ _ c => cDen c
end.
Theorem subs_correct : forall k1 (c1 : con kDen k1) k2 (c2 : _ -> con kDen k2) c2',
subs c1 c2 c2'
-> cDen (c2 (cDen c1)) = cDen c2'.
induction 1; simpl; intuition; try (apply ext_eq_forallS || apply ext_eq);
repeat match goal with
| [ H : _ |- _ ] => rewrite H
end; intuition.
Qed.
Definition disjoint T (r1 r2 : row T) :=
AllS (fun p1 => AllS (fun p2 => fst p1 <> fst p2) r2) r1.
Definition dvar k (c1 c2 : con kDen (KRecord k)) :=
disjoint (cDen c1) (cDen c2).
Lemma AllS_app : forall T P (ls2 : list T),
AllS P ls2
-> forall ls1, AllS P ls1
-> AllS P (ls1 ++ ls2).
induction 2; simpl; intuition.
Qed.
Lemma AllS_weaken : forall T (P P' : T -> Prop),
(forall x, P x -> P' x)
-> forall ls,
AllS P ls
-> AllS P' ls.
induction 2; simpl; intuition.
Qed.
Theorem disjoint_symm : forall T (r1 r2 : row T),
disjoint r1 r2
-> disjoint r2 r1.
Hint Constructors AllS.
Hint Resolve AllS_weaken.
unfold disjoint; induction r2; simpl; intuition.
constructor.
eapply AllS_weaken; eauto.
intuition.
inversion H0; auto.
apply IHr2.
eapply AllS_weaken; eauto.
intuition.
inversion H0; auto.
Qed.
Lemma map_id : forall k (r : row k),
cfold (fun x x0 (x1 : row _) => (x, x0) :: x1) nil r = r.
induction r; simpl; intuition;
match goal with
| [ H : _ |- _ ] => rewrite H
end; intuition.
Qed.
Lemma map_dist : forall T1 T2 (f : T1 -> T2) (r2 r1 : row T1),
cfold (fun x x0 (x1 : row _) => (x, f x0) :: x1) nil (r1 ++ r2)
= cfold (fun x x0 (x1 : row _) => (x, f x0) :: x1) nil r1
++ cfold (fun x x0 (x1 : row _) => (x, f x0) :: x1) nil r2.
induction r1; simpl; intuition;
match goal with
| [ H : _ |- _ ] => rewrite H
end; intuition.
Qed.
Lemma fold_fuse : forall T1 T2 T3 (f : name -> T1 -> T2 -> T2) (i : T2) (f' : T3 -> T1) (c : row T3),
cfold f i (cfold (fun x x0 (x1 : row _) => (x, f' x0) :: x1) nil c)
= cfold (fun x x0 => f x (f' x0)) i c.
induction c; simpl; intuition;
match goal with
| [ H : _ |- _ ] => rewrite <- H
end; intuition.
Qed.
Scheme deq_mut := Minimality for deq Sort Prop
with disj_mut := Minimality for disj Sort Prop.
Theorem deq_correct : forall k (c1 c2 : con kDen k),
deq dvar c1 c2
-> cDen c1 = cDen c2.
Hint Resolve map_id map_dist fold_fuse AllS_app disjoint_symm.
Hint Extern 1 (_ = _) => unfold row; symmetry; apply app_ass.
apply (deq_mut (dvar := dvar)
(fun k (c1 c2 : con kDen k) =>
cDen c1 = cDen c2)
(fun k (c1 c2 : con kDen (KRecord k)) =>
disjoint (cDen c1) (cDen c2)));
simpl; intuition;
repeat (match goal with
| [ H : _ |- _ ] => rewrite H
| [ H : subs _ _ _ |- _ ] => rewrite <- (subs_correct H)
end; simpl; intuition); try congruence; unfold disjoint in *; intuition;
fold kDen in *; repeat match goal with
| [ H : AllS _ (_ :: _) |- _ ] => inversion H; clear H; subst; simpl in *
end; auto.
Qed.
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