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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 Eqdep.
Require Import Axioms.
Require Import Syntax.
Set Implicit Arguments.
Definition row (A : Type) : Type := name -> option A.
Definition record (r : row Set) := forall n, match r n with
| None => unit
| Some T => T
end.
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.
Definition disjoint T (r1 r2 : row T) :=
forall n, match r1 n, r2 n with
| Some _, Some _ => False
| _, _ => True
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 _ => fun _ => None
| CSingle _ c1 c2 => fun n => if name_eq_dec n (cDen c1) then Some (cDen c2) else None
| CConcat _ c1 c2 => fun n => match (cDen c1) n with
| None => (cDen c2) n
| v => v
end
| CMap k1 k2 => fun f r n => match r n with
| None => None
| Some T => Some (f T)
end
| TGuarded _ c1 c2 t => disjoint (cDen c1) (cDen c2) -> cDen t
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 dvar k (c1 c2 : con kDen (KRecord k)) :=
disjoint (cDen c1) (cDen c2).
Scheme deq_mut := Minimality for deq Sort Prop
with disj_mut := Minimality for disj Sort Prop.
Ltac deq_disj_correct scm :=
let t := repeat progress (simpl; intuition; subst) in
let rec use_disjoint' notDone E :=
match goal with
| [ H : disjoint _ _ |- _ ] =>
notDone H; generalize (H E); use_disjoint'
ltac:(fun H' =>
match H' with
| H => fail 1
| _ => notDone H'
end) E
| _ => idtac
end in
let use_disjoint := use_disjoint' ltac:(fun _ => idtac) in
apply (scm _ 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))); t;
repeat ((unfold row; apply ext_eq)
|| (match goal with
| [ H : _ |- _ ] => rewrite H; []
| [ H : subs _ _ _ |- _ ] => rewrite <- (subs_correct H)
end); t);
unfold disjoint; t;
repeat (match goal with
| [ |- context[match cDen ?C ?E with Some _ => _ | None => _ end] ] =>
use_disjoint E; destruct (cDen C E)
| [ |- context[if name_eq_dec ?N1 ?N2 then _ else _] ] =>
use_disjoint N1; use_disjoint N2; destruct (name_eq_dec N1 N2)
| [ _ : context[match cDen ?C ?E with Some _ => _ | None => _ end] |- _ ] =>
use_disjoint E; destruct (cDen C E)
| [ |- context[if ?E then _ else _] ] => destruct E
end; t).
Hint Unfold dvar.
Theorem deq_correct : forall k (c1 c2 : con kDen k),
deq dvar c1 c2
-> cDen c1 = cDen c2.
deq_disj_correct deq_mut.
Qed.
Theorem disj_correct : forall k (c1 c2 : con kDen (KRecord k)),
disj dvar c1 c2
-> disjoint (cDen c1) (cDen c2).
deq_disj_correct disj_mut.
Qed.
Definition tDen (t : con kDen KType) : Set := cDen t.
Theorem name_eq_dec_refl : forall n, name_eq_dec n n = left _ (refl_equal n).
intros; destruct (name_eq_dec n n); intuition; [
match goal with
| [ e : _ = _ |- _ ] => rewrite (UIP_refl _ _ e); reflexivity
end
| elimtype False; tauto
].
Qed.
Theorem cut_disjoint : forall n1 v r,
disjoint (fun n => if name_eq_dec n n1 then Some v else None) r
-> unit = match r n1 with
| Some T => T
| None => unit
end.
intros;
match goal with
| [ H : disjoint _ _ |- _ ] => generalize (H n1)
end; rewrite name_eq_dec_refl;
destruct (r n1); intuition.
Qed.
Implicit Arguments cut_disjoint [v r].
Fixpoint eDen t (e : exp dvar tDen t) {struct e} : tDen t :=
match e in exp _ _ t return tDen t with
| Var _ x => x
| App _ _ e1 e2 => (eDen e1) (eDen e2)
| Abs _ _ e1 => fun x => eDen (e1 x)
| ECApp _ c _ _ e1 Hsub => match subs_correct Hsub in _ = T return T with
| refl_equal => (eDen e1) (cDen c)
end
| ECAbs _ _ e1 => fun X => eDen (e1 X)
| Cast _ _ Heq e1 => match deq_correct Heq in _ = T return T with
| refl_equal => eDen e1
end
| Empty => fun _ => tt
| Single c _ e1 => fun n => if name_eq_dec n (cDen c) as B
return (match (match (if B then _ else _) with Some _ => if B then _ else _ | None => _ end)
with Some _ => _ | None => unit end)
then eDen e1 else tt
| Proj c _ _ e1 =>
match name_eq_dec_refl (cDen c) in _ = B
return (match (match (if B then _ else _) with
| Some _ => if B then _ else _
| None => _ end)
return Set
with Some _ => _ | None => _ end) with
| refl_equal => (eDen e1) (cDen c)
end
| Cut c _ c' Hdisj e1 => fun n =>
match name_eq_dec n (cDen c) as B return (match (match (if B then Some _ else None) with Some _ => if B then _ else _ | None => (cDen c') n end)
with Some T => T | None => unit end
-> match (cDen c') n with
| None => unit
| Some T => T
end) with
| left Heq => fun _ =>
match sym_eq Heq in _ = n' return match cDen c' n' return Set with Some _ => _ | None => _ end with
| refl_equal =>
match cut_disjoint _ (disj_correct Hdisj) in _ = T return T with
| refl_equal => tt
end
end
| right _ => fun x => x
end ((eDen e1) n)
| Concat c1 c2 e1 e2 => fun n =>
match (cDen c1) n as D return match D with
| None => unit
| Some T => T
end
-> match (match D with
| None => (cDen c2) n
| v => v
end) with
| None => unit
| Some T => T
end with
| None => fun _ => (eDen e2) n
| _ => fun x => x
end ((eDen e1) n)
| Guarded _ _ _ _ e1 => fun pf => eDen (e1 pf)
| GuardedApp _ _ _ _ e1 Hdisj => (eDen e1) (disj_correct Hdisj)
end.
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