### changeset 635:75c7a69354d6

Coq formalization uses TDisjoint
author Adam Chlipala Tue, 24 Feb 2009 16:08:14 -0500 6302b10dbe0e de8333ef1a0c src/coq/Name.v src/coq/Semantics.v src/coq/Syntax.v 3 files changed, 17 insertions(+), 129 deletions(-) [+]
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```--- a/src/coq/Name.v	Tue Feb 24 15:54:05 2009 -0500
+++ b/src/coq/Name.v	Tue Feb 24 16:08:14 2009 -0500
@@ -25,47 +25,7 @@
* POSSIBILITY OF SUCH DAMAGE.
*)

-Set Implicit Arguments.
+Require Import String.

-
-Fixpoint name' (n : nat) : Type :=
-  match n with
-    | O => Empty_set
-    | S n' => option (name' n')
-  end.
-
-Definition name'_eq_dec : forall n (x y : name' n), {x = y} + {x <> y}.
-  Hint Extern 1 (_ = _ -> False) => congruence.
-
-  induction n; simpl; intuition;
-    repeat match goal with
-             | [ x : Empty_set |- _ ] => destruct x
-             | [ x : option _ |- _ ] => destruct x
-           end; intuition;
-    match goal with
-      | [ IH : _, n1 : name' _, n2 : name' _ |- _ ] =>
-        destruct (IHn n1 n0); subst; intuition
-    end.
-Qed.
-
-Definition badName' n (P : name' n -> bool) :
-  {nm : _ | P nm = false} + {forall nm, P nm = true}.
-  Hint Constructors sig.
-  Hint Extern 1 (_ = true) =>
-    match goal with
-      | [ nm : option _ |- _ ] => destruct nm
-    end; auto.
-
-  induction n; simpl; intuition;
-    match goal with
-      | [ IH : forall P : _ -> _,_ |- _ ] =>
-        case_eq (P None);
-        destruct (IH (fun nm => P (Some nm))); firstorder eauto
-    end.
-Qed.
-
-Parameter numNames : nat.
-Definition name := name' (S numNames).
-Definition name_eq_dec : forall (x y : name), {x = y} + {x <> y} := @name'_eq_dec _.
-Definition badName : forall P : name -> bool, {nm : _ | P nm = false} + {forall nm, P nm = true} := @badName' _.
-Definition defaultName : name := None.
+Definition name := string.
+Definition name_eq_dec : forall x y : name, {x = y} + {x <> y} := string_dec.```
```--- a/src/coq/Semantics.v	Tue Feb 24 15:54:05 2009 -0500
+++ b/src/coq/Semantics.v	Tue Feb 24 16:08:14 2009 -0500
@@ -48,13 +48,11 @@
| KRecord k1 => row (kDen k1)
end.

-Fixpoint kDefault (k : kind) : kDen k :=
-  match k return kDen k with
-    | KType => unit
-    | KName => defaultName
-    | KArrow _ k2 => fun _ => kDefault k2
-    | KRecord _ => fun _ => None
-  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
@@ -75,13 +73,7 @@
| None => None
| Some T => Some (f T)
end
-    | CGuarded _ _ c1 c2 c =>
-      if badName (fun n => match (cDen c1) n, (cDen c2) n with
-                            | Some _, Some _ => false
-                            | _, _ => true
-                          end)
-        then kDefault _
-        else cDen c
+    | 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',
@@ -93,41 +85,14 @@
end; intuition.
Qed.

-Definition disjoint T (r1 r2 : row T) :=
-  forall n, match r1 n, r2 n with
-              | Some _, Some _ => False
-              | _, _ => True
-            end.
Definition dvar k (c1 c2 : con kDen (KRecord k)) :=
disjoint (cDen c1) (cDen c2).

-Theorem known_badName : forall T (r1 r2 : row T) T' (v1 v2 : T'),
-  disjoint r1 r2
-  -> (if badName (fun n => match r1 n, r2 n with
-                            | Some _, Some _ => false
-                            | _, _ => true
-                          end)
-    then v1
-    else v2) = v2.
-  intros; match goal with
-            | [ |- context[if ?E then _ else _] ] => destruct E
-          end; firstorder;
-  match goal with
-    | [ H : disjoint _ _, x : name |- _ ] =>
-      generalize (H x);
-        repeat match goal with
-                 | [ |- context[match ?E with None => _ | Some _ => _ end] ] => destruct E
-               end; tauto || congruence
-  end.
-Qed.
-
-Hint Rewrite known_badName using solve [ auto ] : Semantics.
-
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; autorewrite with Semantics) in
+  let t := repeat progress (simpl; intuition; subst) in

let rec use_disjoint' notDone E :=
match goal with
@@ -163,28 +128,6 @@
| [ |- context[if ?E then _ else _] ] => destruct E
end; t).

-Lemma bool_disjoint : forall T (r1 r2 : row T),
-  (forall nm : name,
-    match r1 nm with
-      | Some _ => match r2 nm with
-                    | Some _ => false
-                    | None => true
-                  end
-      | None => true
-    end = true)
-  -> disjoint r1 r2.
-  intros; intro;
-    match goal with
-      | [ H : _, n : name |- _ ] => generalize (H n)
-    end;
-    repeat match goal with
-             | [ |- context[match ?E with Some _ => _ | None => _ end] ] => destruct E
-           end; tauto || discriminate.
-Qed.
-
-Implicit Arguments bool_disjoint [T r1 r2].
-
-Hint Resolve bool_disjoint.
Hint Unfold dvar.

Theorem deq_correct : forall k (c1 c2 : con kDen k),
@@ -225,8 +168,6 @@

Implicit Arguments cut_disjoint [v r].

-Set Printing All.
-
Fixpoint eDen t (e : exp dvar tDen t) {struct e} : tDen t :=
match e in exp _ _ t return tDen t with
| Var _ x => x
@@ -286,13 +227,6 @@
| _ => fun x => x
end ((eDen e1) n)

-    | Guarded _ c1 c2 _ e1 =>
-      match badName (fun n => match (cDen c1) n, (cDen c2) n with
-                                | Some _, Some _ => false
-                                | _, _ => true
-                              end)
-      as BN return (if BN return Set then _ else _) with
-        | inleft _ => tt
-        | inright pf => eDen (e1 (bool_disjoint pf))
-      end
+    | Guarded _ _ _ _ e1 => fun pf => eDen (e1 pf)
+    | GuardedApp _ _ _ _ e1 Hdisj => (eDen e1) (disj_correct Hdisj)
end.```
```--- a/src/coq/Syntax.v	Tue Feb 24 15:54:05 2009 -0500
+++ b/src/coq/Syntax.v	Tue Feb 24 16:08:14 2009 -0500
@@ -54,7 +54,7 @@
| CSingle : forall k, con KName -> con k -> con (KRecord k)
| CConcat : forall k, con (KRecord k) -> con (KRecord k) -> con (KRecord k)
| CMap : forall k1 k2, con (KArrow (KArrow k1 k2) (KArrow (KRecord k1) (KRecord k2)))
-  | CGuarded : forall k1 k2, con (KRecord k1) -> con (KRecord k1) -> con k2 -> con k2.
+  | TGuarded : forall k, con (KRecord k) -> con (KRecord k) -> con KType -> con KType.

Variable dvar : forall k, con (KRecord k) -> con (KRecord k) -> Type.

@@ -91,11 +91,11 @@
subs c2 c2'
-> subs c3 c3'
-> subs (fun x => CConcat (c2 x) (c3 x)) (CConcat c2' c3')
-    | S_CGuarded : forall k2 k3 (c2 c3 : _ -> con (KRecord k2)) (c4 : _ -> con k3) c2' c3' c4',
+    | S_TGuarded : forall k2 (c2 c3 : _ -> con (KRecord k2)) c4 c2' c3' c4',
subs c2 c2'
-> subs c3 c3'
-> subs c4 c4'
-      -> subs (fun x => CGuarded (c2 x) (c3 x) (c4 x)) (CGuarded c2' c3' c4').
+      -> subs (fun x => TGuarded (c2 x) (c3 x) (c4 x)) (TGuarded c2' c3' c4').
End subs.

Inductive disj : forall k, con (KRecord k) -> con (KRecord k) -> Prop :=
@@ -157,13 +157,6 @@
-> deq (CApp (CApp (CMap k1 k2) f) (CConcat (CSingle c1 c2) c3))
(CConcat (CSingle c1 (CApp f c2)) (CApp (CApp (CMap k1 k2) f) c3))

-  | Eq_Guarded_Remove : forall k1 k2 (c1 c2 : con (KRecord k1)) (c : con k2),
-    disj c1 c2
-    -> deq (CGuarded c1 c2 c) c
-  | Eq_Guarded_Cong : forall k1 k2 (c1 c2 : con (KRecord k1)) (c c' : con k2),
-    (dvar c1 c2 -> deq c c')
-    -> deq (CGuarded c1 c2 c) (CGuarded c1 c2 c')
-
| Eq_Map_Ident : forall k c,
deq (CApp (CApp (CMap k k) (CAbs (fun x => CVar x))) c) c
| Eq_Map_Dist : forall k1 k2 f c1 c2,
@@ -188,5 +181,6 @@
| Proj : forall c t c', exp (TRecord (CConcat (CSingle c t) c')) -> exp t
| Cut : forall c t c', disj (CSingle c t) c' -> exp (TRecord (CConcat (CSingle c t) c')) -> exp (TRecord c')
| Concat : forall c1 c2, exp (TRecord c1) -> exp (TRecord c2) -> exp (TRecord (CConcat c1 c2))
-  | Guarded : forall k (c1 c2 : con (KRecord k)) c, (dvar c1 c2 -> exp c) -> exp (CGuarded c1 c2 c).
+  | Guarded : forall k (c1 c2 : con (KRecord k)) c, (dvar c1 c2 -> exp c) -> exp (TGuarded c1 c2 c)
+  | GuardedApp : forall k (c1 c2 : con (KRecord k)) t, exp (TGuarded c1 c2 t) -> disj c1 c2 -> exp t.
End vars.```