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-rw-r--r--theories/Program/Basics.v2
-rw-r--r--theories/Program/Combinators.v2
-rw-r--r--theories/Program/Equality.v4
-rw-r--r--theories/Program/Program.v2
-rw-r--r--theories/Program/Subset.v14
-rw-r--r--theories/Program/Syntax.v2
-rw-r--r--theories/Program/Tactics.v2
-rw-r--r--theories/Program/Utils.v2
-rw-r--r--theories/Program/Wf.v23
9 files changed, 26 insertions, 27 deletions
diff --git a/theories/Program/Basics.v b/theories/Program/Basics.v
index 2c0f62ad..e5be0ca9 100644
--- a/theories/Program/Basics.v
+++ b/theories/Program/Basics.v
@@ -1,7 +1,7 @@
(* -*- coding: utf-8 -*- *)
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Program/Combinators.v b/theories/Program/Combinators.v
index 4d631e78..e246041b 100644
--- a/theories/Program/Combinators.v
+++ b/theories/Program/Combinators.v
@@ -1,7 +1,7 @@
(* -*- coding: utf-8 -*- *)
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Program/Equality.v b/theories/Program/Equality.v
index 04701ff5..a9aa30df 100644
--- a/theories/Program/Equality.v
+++ b/theories/Program/Equality.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
@@ -263,7 +263,7 @@ Class DependentEliminationPackage (A : Type) :=
Ltac elim_tac tac p :=
let ty := type of p in
- let eliminator := eval simpl in (elim (A:=ty)) in
+ let eliminator := eval simpl in (@elim (_ : DependentEliminationPackage ty)) in
tac p eliminator.
(** Specialization to do case analysis or induction.
diff --git a/theories/Program/Program.v b/theories/Program/Program.v
index 38f11231..5af6f4d7 100644
--- a/theories/Program/Program.v
+++ b/theories/Program/Program.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Program/Subset.v b/theories/Program/Subset.v
index 269556c2..50b89b5c 100644
--- a/theories/Program/Subset.v
+++ b/theories/Program/Subset.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
@@ -61,12 +61,12 @@ Ltac pi_subset_proofs := repeat pi_subset_proof.
Ltac clear_subset_proofs :=
abstract_subset_proofs ; simpl in * |- ; pi_subset_proofs ; clear_dups.
-Ltac pi := repeat progress f_equal ; apply proof_irrelevance.
+Ltac pi := repeat f_equal ; apply proof_irrelevance.
Lemma subset_eq : forall A (P : A -> Prop) (n m : sig P), n = m <-> `n = `m.
Proof.
- induction n.
- induction m.
+ destruct n as (x,p).
+ destruct m as (x',p').
simpl.
split ; intros ; subst.
@@ -79,14 +79,14 @@ Qed.
(* Somewhat trivial definition, but not unfolded automatically hence we can match on [match_eq ?A ?B ?x ?f]
in tactics. *)
-Definition match_eq (A B : Type) (x : A) (fn : forall (y : A | y = x), B) : B :=
+Definition match_eq (A B : Type) (x : A) (fn : {y : A | y = x} -> B) : B :=
fn (exist _ x eq_refl).
(* This is what we want to be able to do: replace the originaly matched object by a new,
propositionally equal one. If [fn] works on [x] it should work on any [y | y = x]. *)
-Lemma match_eq_rewrite : forall (A B : Type) (x : A) (fn : forall (y : A | y = x), B)
- (y : A | y = x),
+Lemma match_eq_rewrite : forall (A B : Type) (x : A) (fn : {y : A | y = x} -> B)
+ (y : {y:A | y = x}),
match_eq A B x fn = fn y.
Proof.
intros.
diff --git a/theories/Program/Syntax.v b/theories/Program/Syntax.v
index 269748b5..67e9a20c 100644
--- a/theories/Program/Syntax.v
+++ b/theories/Program/Syntax.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Program/Tactics.v b/theories/Program/Tactics.v
index fd55a553..0cf8d733 100644
--- a/theories/Program/Tactics.v
+++ b/theories/Program/Tactics.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Program/Utils.v b/theories/Program/Utils.v
index de2a76ab..e39128cb 100644
--- a/theories/Program/Utils.v
+++ b/theories/Program/Utils.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
diff --git a/theories/Program/Wf.v b/theories/Program/Wf.v
index 2a7a5e17..d89919b0 100644
--- a/theories/Program/Wf.v
+++ b/theories/Program/Wf.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
@@ -11,6 +11,7 @@
Require Import Coq.Init.Wf.
Require Import Coq.Program.Utils.
Require Import ProofIrrelevance.
+Require Import FunctionalExtensionality.
Local Open Scope program_scope.
@@ -32,14 +33,13 @@ Section Well_founded.
(* Notation Fix_F := (Fix_F_sub P F_sub) (only parsing). (* alias *) *)
(* Definition Fix (x:A) := Fix_F_sub P F_sub x (Rwf x). *)
- Hypothesis
- F_ext :
+ Hypothesis F_ext :
forall (x:A) (f g:forall y:{y:A | R y x}, P (`y)),
- (forall (y : A | R y x), f y = g y) -> F_sub x f = F_sub x g.
+ (forall y:{y : A | R y x}, f y = g y) -> F_sub x f = F_sub x g.
Lemma Fix_F_eq :
forall (x:A) (r:Acc R x),
- F_sub x (fun (y:A|R y x) => Fix_F_sub (`y) (Acc_inv r (proj2_sig y))) = Fix_F_sub x r.
+ F_sub x (fun y:{y:A | R y x} => Fix_F_sub (`y) (Acc_inv r (proj2_sig y))) = Fix_F_sub x r.
Proof.
destruct r using Acc_inv_dep; auto.
Qed.
@@ -50,7 +50,7 @@ Section Well_founded.
rewrite (proof_irrelevance (Acc R x) r s) ; auto.
Qed.
- Lemma Fix_eq : forall x:A, Fix_sub x = F_sub x (fun (y:A|R y x) => Fix_sub (proj1_sig y)).
+ Lemma Fix_eq : forall x:A, Fix_sub x = F_sub x (fun y:{ y:A | R y x} => Fix_sub (proj1_sig y)).
Proof.
intro x; unfold Fix_sub.
rewrite <- (Fix_F_eq ).
@@ -62,7 +62,8 @@ Section Well_founded.
forall x : A,
Fix_sub x =
let f_sub := F_sub in
- f_sub x (fun (y : A | R y x) => Fix_sub (`y)).
+ f_sub x (fun y: {y : A | R y x} => Fix_sub (`y)).
+ Proof.
exact Fix_eq.
Qed.
@@ -153,7 +154,7 @@ Section Fix_rects.
Hypothesis equiv_lowers:
forall x0 (g h: forall x: {y: A | R y x0}, P (proj1_sig x)),
- (forall x p p', g (exist (fun y: A => R y x0) x p) = h (exist _ x p')) ->
+ (forall x p p', g (exist (fun y: A => R y x0) x p) = h (exist (*FIXME shouldn't be needed *) (fun y => R y x0) x p')) ->
f g = f h.
(* From equiv_lowers, it follows that
@@ -221,8 +222,6 @@ Ltac fold_sub f :=
Module WfExtensionality.
- Require Import FunctionalExtensionality.
-
(** The two following lemmas allow to unfold a well-founded fixpoint definition without
restriction using the functional extensionality axiom. *)
@@ -231,10 +230,10 @@ Module WfExtensionality.
Program Lemma fix_sub_eq_ext :
forall (A : Type) (R : A -> A -> Prop) (Rwf : well_founded R)
(P : A -> Type)
- (F_sub : forall x : A, (forall (y : A | R y x), P y) -> P x),
+ (F_sub : forall x : A, (forall y:{y : A | R y x}, P (` y)) -> P x),
forall x : A,
Fix_sub A R Rwf P F_sub x =
- F_sub x (fun (y : A | R y x) => Fix_sub A R Rwf P F_sub y).
+ F_sub x (fun y:{y : A | R y x} => Fix_sub A R Rwf P F_sub (` y)).
Proof.
intros ; apply Fix_eq ; auto.
intros.
generated by cgit v1.2.3 (git 2.39.1) at 2024-06-02 08:54:48 +0000