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-rw-r--r--theories/Arith/Between.v4
-rw-r--r--theories/Compat/Coq87.v3
-rw-r--r--theories/FSets/FSetCompat.v44
-rw-r--r--theories/Init/Tactics.v7
-rw-r--r--theories/Logic/FunctionalExtensionality.v3
-rw-r--r--theories/Program/Combinators.v12
-rw-r--r--theories/Sets/Powerset_facts.v91
7 files changed, 131 insertions, 33 deletions
diff --git a/theories/Arith/Between.v b/theories/Arith/Between.v
index 9b4071085..ead08b3eb 100644
--- a/theories/Arith/Between.v
+++ b/theories/Arith/Between.v
@@ -16,6 +16,8 @@ Implicit Types k l p q r : nat.
Section Between.
Variables P Q : nat -> Prop.
+ (** The [between] type expresses the concept
+ [forall i: nat, k <= i < l -> P i.]. *)
Inductive between k : nat -> Prop :=
| bet_emp : between k k
| bet_S : forall l, between k l -> P l -> between k (S l).
@@ -47,6 +49,8 @@ Section Between.
induction 1; auto with arith.
Qed.
+ (** The [exists_between] type expresses the concept
+ [exists i: nat, k <= i < l /\ Q i]. *)
Inductive exists_between k : nat -> Prop :=
| exists_S : forall l, exists_between k l -> exists_between k (S l)
| exists_le : forall l, k <= l -> Q l -> exists_between k (S l).
diff --git a/theories/Compat/Coq87.v b/theories/Compat/Coq87.v
index ef1737bf8..aeef9595d 100644
--- a/theories/Compat/Coq87.v
+++ b/theories/Compat/Coq87.v
@@ -15,3 +15,6 @@
and breaks at least fiat-crypto. *)
Declare ML Module "omega_plugin".
Unset Omega UseLocalDefs.
+
+
+Set Typeclasses Axioms Are Instances.
diff --git a/theories/FSets/FSetCompat.v b/theories/FSets/FSetCompat.v
index b1769da3d..31bc1cc31 100644
--- a/theories/FSets/FSetCompat.v
+++ b/theories/FSets/FSetCompat.v
@@ -165,13 +165,13 @@ End Backport_WSets.
(** * From new Sets to new ones *)
Module Backport_Sets
- (E:OrderedType.OrderedType)
- (M:MSetInterface.Sets with Definition E.t := E.t
- with Definition E.eq := E.eq
- with Definition E.lt := E.lt)
- <: FSetInterface.S with Module E:=E.
+ (O:OrderedType.OrderedType)
+ (M:MSetInterface.Sets with Definition E.t := O.t
+ with Definition E.eq := O.eq
+ with Definition E.lt := O.lt)
+ <: FSetInterface.S with Module E:=O.
- Include Backport_WSets E M.
+ Include Backport_WSets O M.
Implicit Type s : t.
Implicit Type x y : elt.
@@ -182,21 +182,21 @@ Module Backport_Sets
Definition min_elt_1 : forall s x, min_elt s = Some x -> In x s
:= M.min_elt_spec1.
Definition min_elt_2 : forall s x y,
- min_elt s = Some x -> In y s -> ~ E.lt y x
+ min_elt s = Some x -> In y s -> ~ O.lt y x
:= M.min_elt_spec2.
Definition min_elt_3 : forall s, min_elt s = None -> Empty s
:= M.min_elt_spec3.
Definition max_elt_1 : forall s x, max_elt s = Some x -> In x s
:= M.max_elt_spec1.
Definition max_elt_2 : forall s x y,
- max_elt s = Some x -> In y s -> ~ E.lt x y
+ max_elt s = Some x -> In y s -> ~ O.lt x y
:= M.max_elt_spec2.
Definition max_elt_3 : forall s, max_elt s = None -> Empty s
:= M.max_elt_spec3.
- Definition elements_3 : forall s, sort E.lt (elements s)
+ Definition elements_3 : forall s, sort O.lt (elements s)
:= M.elements_spec2.
Definition choose_3 : forall s s' x y,
- choose s = Some x -> choose s' = Some y -> Equal s s' -> E.eq x y
+ choose s = Some x -> choose s' = Some y -> Equal s s' -> O.eq x y
:= M.choose_spec3.
Definition lt_trans : forall s s' s'', lt s s' -> lt s' s'' -> lt s s''
:= @StrictOrder_Transitive _ _ M.lt_strorder.
@@ -211,7 +211,7 @@ Module Backport_Sets
[ apply EQ | apply LT | apply GT ]; auto.
Defined.
- Module E := E.
+ Module E := O.
End Backport_Sets.
@@ -342,13 +342,13 @@ End Update_WSets.
(** * From old Sets to new ones. *)
Module Update_Sets
- (E:Orders.OrderedType)
- (M:FSetInterface.S with Definition E.t := E.t
- with Definition E.eq := E.eq
- with Definition E.lt := E.lt)
- <: MSetInterface.Sets with Module E:=E.
+ (O:Orders.OrderedType)
+ (M:FSetInterface.S with Definition E.t := O.t
+ with Definition E.eq := O.eq
+ with Definition E.lt := O.lt)
+ <: MSetInterface.Sets with Module E:=O.
- Include Update_WSets E M.
+ Include Update_WSets O M.
Implicit Type s : t.
Implicit Type x y : elt.
@@ -359,21 +359,21 @@ Module Update_Sets
Definition min_elt_spec1 : forall s x, min_elt s = Some x -> In x s
:= M.min_elt_1.
Definition min_elt_spec2 : forall s x y,
- min_elt s = Some x -> In y s -> ~ E.lt y x
+ min_elt s = Some x -> In y s -> ~ O.lt y x
:= M.min_elt_2.
Definition min_elt_spec3 : forall s, min_elt s = None -> Empty s
:= M.min_elt_3.
Definition max_elt_spec1 : forall s x, max_elt s = Some x -> In x s
:= M.max_elt_1.
Definition max_elt_spec2 : forall s x y,
- max_elt s = Some x -> In y s -> ~ E.lt x y
+ max_elt s = Some x -> In y s -> ~ O.lt x y
:= M.max_elt_2.
Definition max_elt_spec3 : forall s, max_elt s = None -> Empty s
:= M.max_elt_3.
- Definition elements_spec2 : forall s, sort E.lt (elements s)
+ Definition elements_spec2 : forall s, sort O.lt (elements s)
:= M.elements_3.
Definition choose_spec3 : forall s s' x y,
- choose s = Some x -> choose s' = Some y -> Equal s s' -> E.eq x y
+ choose s = Some x -> choose s' = Some y -> Equal s s' -> O.eq x y
:= M.choose_3.
Instance lt_strorder : StrictOrder lt.
@@ -407,6 +407,6 @@ Module Update_Sets
Lemma compare_spec : forall s s', CompSpec eq lt s s' (compare s s').
Proof. intros; unfold compare; destruct M.compare; auto. Qed.
- Module E := E.
+ Module E := O.
End Update_Sets.
diff --git a/theories/Init/Tactics.v b/theories/Init/Tactics.v
index 5d0e7602a..47a971ef0 100644
--- a/theories/Init/Tactics.v
+++ b/theories/Init/Tactics.v
@@ -306,3 +306,10 @@ Ltac inversion_sigma_step :=
=> induction_sigma_in_using H @eq_sigT2_rect
end.
Ltac inversion_sigma := repeat inversion_sigma_step.
+
+(** A version of [time] that works for constrs *)
+Ltac time_constr tac :=
+ let eval_early := match goal with _ => restart_timer end in
+ let ret := tac () in
+ let eval_early := match goal with _ => finish_timing ( "Tactic evaluation" ) end in
+ ret.
diff --git a/theories/Logic/FunctionalExtensionality.v b/theories/Logic/FunctionalExtensionality.v
index ac95ddd0c..82b04d132 100644
--- a/theories/Logic/FunctionalExtensionality.v
+++ b/theories/Logic/FunctionalExtensionality.v
@@ -221,13 +221,12 @@ Tactic Notation "extensionality" "in" hyp(H) :=
(* If we [subst H], things break if we already have another equation of the form [_ = H] *)
destruct Heq; rename H_out into H.
-(** Eta expansion follows from extensionality. *)
+(** Eta expansion is built into Coq. *)
Lemma eta_expansion_dep {A} {B : A -> Type} (f : forall x : A, B x) :
f = fun x => f x.
Proof.
intros.
- extensionality x.
reflexivity.
Qed.
diff --git a/theories/Program/Combinators.v b/theories/Program/Combinators.v
index 90db10ef1..237d878bf 100644
--- a/theories/Program/Combinators.v
+++ b/theories/Program/Combinators.v
@@ -22,15 +22,13 @@ Open Scope program_scope.
Lemma compose_id_left : forall A B (f : A -> B), id ∘ f = f.
Proof.
intros.
- unfold id, compose.
- symmetry. apply eta_expansion.
+ reflexivity.
Qed.
Lemma compose_id_right : forall A B (f : A -> B), f ∘ id = f.
Proof.
intros.
- unfold id, compose.
- symmetry ; apply eta_expansion.
+ reflexivity.
Qed.
Lemma compose_assoc : forall A B C D (f : A -> B) (g : B -> C) (h : C -> D),
@@ -47,9 +45,7 @@ Hint Rewrite <- @compose_assoc : core.
Lemma flip_flip : forall A B C, @flip A B C ∘ flip = id.
Proof.
- unfold flip, compose.
intros.
- extensionality x ; extensionality y ; extensionality z.
reflexivity.
Qed.
@@ -57,9 +53,7 @@ Qed.
Lemma prod_uncurry_curry : forall A B C, @prod_uncurry A B C ∘ prod_curry = id.
Proof.
- simpl ; intros.
- unfold prod_uncurry, prod_curry, compose.
- extensionality x ; extensionality y ; extensionality z.
+ intros.
reflexivity.
Qed.
diff --git a/theories/Sets/Powerset_facts.v b/theories/Sets/Powerset_facts.v
index 2dd559a95..209c22f71 100644
--- a/theories/Sets/Powerset_facts.v
+++ b/theories/Sets/Powerset_facts.v
@@ -40,6 +40,11 @@ Section Sets_as_an_algebra.
auto 6 with sets.
Qed.
+ Theorem Empty_set_zero_right : forall X:Ensemble U, Union U X (Empty_set U) = X.
+ Proof.
+ auto 6 with sets.
+ Qed.
+
Theorem Empty_set_zero' : forall x:U, Add U (Empty_set U) x = Singleton U x.
Proof.
unfold Add at 1; auto using Empty_set_zero with sets.
@@ -131,6 +136,17 @@ Section Sets_as_an_algebra.
elim H'; intros x0 H'0; elim H'0; auto with sets.
Qed.
+ Lemma Distributivity_l
+ : forall (A B C : Ensemble U),
+ Intersection U (Union U A B) C =
+ Union U (Intersection U A C) (Intersection U B C).
+ Proof.
+ intros A B C.
+ rewrite Intersection_commutative.
+ rewrite Distributivity.
+ f_equal; apply Intersection_commutative.
+ Qed.
+
Theorem Distributivity' :
forall A B C:Ensemble U,
Union U A (Intersection U B C) =
@@ -251,6 +267,81 @@ Section Sets_as_an_algebra.
intros; apply Definition_of_covers; auto with sets.
Qed.
+ Lemma Disjoint_Intersection:
+ forall A s1 s2, Disjoint A s1 s2 -> Intersection A s1 s2 = Empty_set A.
+ Proof.
+ intros. apply Extensionality_Ensembles. split.
+ * destruct H.
+ intros x H1. unfold In in *. exfalso. intuition. apply (H _ H1).
+ * intuition.
+ Qed.
+
+ Lemma Intersection_Empty_set_l:
+ forall A s, Intersection A (Empty_set A) s = Empty_set A.
+ Proof.
+ intros. auto with sets.
+ Qed.
+
+ Lemma Intersection_Empty_set_r:
+ forall A s, Intersection A s (Empty_set A) = Empty_set A.
+ Proof.
+ intros. auto with sets.
+ Qed.
+
+ Lemma Seminus_Empty_set_l:
+ forall A s, Setminus A (Empty_set A) s = Empty_set A.
+ Proof.
+ intros. apply Extensionality_Ensembles. split.
+ * intros x H1. destruct H1. unfold In in *. assumption.
+ * intuition.
+ Qed.
+
+ Lemma Seminus_Empty_set_r:
+ forall A s, Setminus A s (Empty_set A) = s.
+ Proof.
+ intros. apply Extensionality_Ensembles. split.
+ * intros x H1. destruct H1. unfold In in *. assumption.
+ * intuition.
+ Qed.
+
+ Lemma Setminus_Union_l:
+ forall A s1 s2 s3,
+ Setminus A (Union A s1 s2) s3 = Union A (Setminus A s1 s3) (Setminus A s2 s3).
+ Proof.
+ intros. apply Extensionality_Ensembles. split.
+ * intros x H. inversion H. inversion H0; intuition.
+ * intros x H. constructor; inversion H; inversion H0; intuition.
+ Qed.
+
+ Lemma Setminus_Union_r:
+ forall A s1 s2 s3,
+ Setminus A s1 (Union A s2 s3) = Setminus A (Setminus A s1 s2) s3.
+ Proof.
+ intros. apply Extensionality_Ensembles. split.
+ * intros x H. inversion H. constructor. intuition. contradict H1. intuition.
+ * intros x H. inversion H. inversion H0. constructor; intuition. inversion H4; intuition.
+ Qed.
+
+ Lemma Setminus_Disjoint_noop:
+ forall A s1 s2,
+ Intersection A s1 s2 = Empty_set A -> Setminus A s1 s2 = s1.
+ Proof.
+ intros. apply Extensionality_Ensembles. split.
+ * intros x H1. inversion_clear H1. intuition.
+ * intros x H1. constructor; intuition. contradict H.
+ apply Inhabited_not_empty.
+ exists x. intuition.
+ Qed.
+
+ Lemma Setminus_Included_empty:
+ forall A s1 s2,
+ Included A s1 s2 -> Setminus A s1 s2 = Empty_set A.
+ Proof.
+ intros. apply Extensionality_Ensembles. split.
+ * intros x H1. inversion_clear H1. contradiction H2. intuition.
+ * intuition.
+ Qed.
+
End Sets_as_an_algebra.
Hint Resolve Empty_set_zero Empty_set_zero' Union_associative Union_add
generated by cgit v1.2.3 (git 2.39.1) at 2024-05-15 06:42:03 +0000