fact stringlengths 6 2.88k | type stringclasses 17
values | library stringclasses 2
values | imports listlengths 0 16 | filename stringclasses 89
values | symbolic_name stringlengths 1 36 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
Reflexive_partial_app_morphism`(Proper (A -> B) (R ==> R') m, ProperProxy A R x) :
Proper R' (m x).
Proof. simpl_relation. Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | Reflexive_partial_app_morphism | |
flip_respectful(R : relation A) (R' : relation B) :
relation_equivalence (flip (R ==> R')) (flip R ==> flip R').
Proof.
intros.
unfold flip, respectful.
split ; intros ; intuition.
Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | flip_respectful | |
flip1`(subrelation A R' R) : subrelation (flip (flip R')) R.
Proof. firstorder. Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | flip1 | |
flip2`(subrelation A R R') : subrelation R (flip (flip R')).
Proof. firstorder. Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | flip2 | |
eq_subrelation`(Reflexive A R) : subrelation (@eq A) R.
Proof. simpl_relation. Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | eq_subrelation | |
proper_flip_proper`(mor : Proper A R m) : Proper (flip R) m := mor. | Definition | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | proper_flip_proper | |
proper_eq(x : A) : Proper (@eq A) x.
Proof. intros. reflexivity. Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | proper_eq | |
PartialApplication. | Class | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | PartialApplication | |
normalization_done: Prop := did_normalization. | CoInductive | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | normalization_done | |
Params{A : Type} (of : A) (arity : nat).
#[global] Instance eq_pars : Params (@eq) 1 := {}.
#[global] Instance iff_pars : Params (@iff) 0 := {}.
#[global] Instance impl_pars : Params (@impl) 0 := {}.
#[global] Instance flip_pars : Params (@flip) 4 := {}. | Class | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | Params | |
partial_application_tactic:=
let rec do_partial_apps H m cont :=
match m with
| ?m' ?x => class_apply @Reflexive_partial_app_morphism ;
[(do_partial_apps H m' ltac:(idtac))|clear H]
| _ => cont
end
in
let rec do_partial H ar m :=
lazymatch ar with
| 0%nat => do_partial_apps H... | Ltac | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | partial_application_tactic | |
proper_proper{A} : Proper (relation_equivalence ==> eq ==> iff) (@Proper A).
Proof.
intros x y H y0 y1 e; destruct e.
reduce in H.
split ; red ; intros H0.
- apply H, H0.
- apply H, H0.
Qed. | Instance | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | proper_proper | |
proper_reflexive:=
match goal with
| [ _ : normalization_done |- _ ] => fail 1
| _ => class_apply proper_eq || class_apply @reflexive_proper
end.
#[global]
Hint Extern 1 (subrelation (flip _) _) => class_apply @flip1 : typeclass_instances.
#[global]
Hint Extern 1 (subrelation _ (flip _)) => class_apply @fl... | Ltac | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | proper_reflexive | |
Normalizes(m : relation A) (m' : relation A) : Prop :=
normalizes : relation_equivalence m m'. | Class | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | Normalizes | |
proper_normalizes_proper`(Normalizes R0 R1, Proper A R1 m) : Proper R0 m.
Proof.
eapply proper_proper; eauto.
Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | proper_normalizes_proper | |
flip_atomR : Normalizes R (flip (flip R)).
Proof.
firstorder.
Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | flip_atom | |
flip_arrow{A : Type} {B : Type}
`(NA : Normalizes A R (flip R'''), NB : Normalizes B R' (flip R'')) :
Normalizes (A -> B) (R ==> R') (flip (R''' ==> R'')%signature).
Proof.
unfold Normalizes in *.
unfold relation_equivalence in *.
unfold predicate_equivalence in *. simpl in *.
unfold respectful. unfold ... | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | flip_arrow | |
normalizes:=
match goal with
| [ |- Normalizes _ (respectful _ _) _ ] => class_apply @flip_arrow
| _ => class_apply @flip_atom
end. | Ltac | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | normalizes | |
proper_normalization:=
match goal with
| [ _ : normalization_done |- _ ] => fail 1
| [ _ : apply_subrelation |- @Proper _ ?R _ ] =>
let H := fresh "H" in
set(H:=did_normalization) ; class_apply @proper_normalizes_proper
end.
#[global]
Hint Extern 1 (Normalizes _ _ _) => normalizes : typeclass_i... | Ltac | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | proper_normalization | |
proper_sym_flip:
forall `(Symmetric A R1)`(Proper (A->B) (R1==>R2) f),
Proper (R1==>flip R2) f.
Proof.
intros A R1 Sym B R2 f Hf.
intros x x' Hxx'. apply Hf, Sym, Hxx'.
Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | proper_sym_flip | |
proper_sym_flip_2:
forall `(Symmetric A R1)`(Symmetric B R2)`(Proper (A->B->C) (R1==>R2==>R3) f),
Proper (R1==>R2==>flip R3) f.
Proof.
intros A R1 Sym1 B R2 Sym2 C R3 f Hf.
intros x x' Hxx' y y' Hyy'. apply Hf; auto.
Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | proper_sym_flip_2 | |
proper_sym_impl_iff: forall `(Symmetric A R)`(Proper _ (R==>impl) f),
Proper (R==>iff) f.
Proof.
intros A R Sym f Hf x x' Hxx'. repeat red in Hf. split; eauto.
Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | proper_sym_impl_iff | |
proper_sym_impl_iff_2:
forall `(Symmetric A R)`(Symmetric B R')`(Proper _ (R==>R'==>impl) f),
Proper (R==>R'==>iff) f.
Proof.
intros A R Sym B R' Sym' f Hf x x' Hxx' y y' Hyy'.
repeat red in Hf. split; eauto.
Qed. | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | proper_sym_impl_iff_2 | |
PartialOrder_proper`(PartialOrder A eqA R) :
Proper (eqA==>eqA==>iff) R.
Proof.
intros.
apply proper_sym_impl_iff_2. 1-2: auto with relations.
intros x x' Hx y y' Hy Hr.
transitivity x.
- generalize (partial_order_equivalence x x'); compute; intuition.
- transitivity y; auto.
generalize (partial_order_equivalence y... | Instance | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | PartialOrder_proper | |
PartialOrder_StrictOrder`(PartialOrder A eqA R) :
StrictOrder (relation_conjunction R (complement eqA)).
Proof.
split; compute.
- intros x (_,Hx). apply Hx, Equivalence_Reflexive.
- intros x y z (Hxy,Hxy') (Hyz,Hyz'). split.
+ apply PreOrder_Transitive with y; assumption.
+ intro Hxz.
apply Hxy'.
apply pa... | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | PartialOrder_StrictOrder | |
StrictOrder_PreOrder`(Equivalence A eqA, StrictOrder A R, Proper _ (eqA==>eqA==>iff) R) :
PreOrder (relation_disjunction R eqA).
Proof.
split.
- intros x. right. reflexivity.
- intros x y z [Hxy|Hxy] [Hyz|Hyz].
+ left. transitivity y; auto.
+ left. eapply H1; eauto.
* apply reflexivity.
* now apply symmetr... | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | StrictOrder_PreOrder | |
StrictOrder_PartialOrder`(Equivalence A eqA, StrictOrder A R, Proper _ (eqA==>eqA==>iff) R) :
PartialOrder eqA (relation_disjunction R eqA).
Proof.
intros. intros x y. compute. intuition auto with relations.
elim (StrictOrder_Irreflexive x).
transitivity y; auto.
Qed.
#[global]
Hint Extern 4 (StrictOrder (relation_c... | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses"
] | Corelib/Classes/Morphisms.v | StrictOrder_PartialOrder | |
Acc_pt_morphism{A:Type}(E R : A->A->Prop)
`(Equivalence _ E) `(Proper _ (E==>E==>iff) R) :
Proper (E==>iff) (Acc R).
Proof.
apply proper_sym_impl_iff.
- auto with relations.
- intros x y EQ WF. apply Acc_intro; intros z Hz.
rewrite <- EQ in Hz. now apply Acc_inv with x.
Qed. | Instance | Corelib | [
"Require Import Corelib.Classes.Morphisms",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics"
] | Corelib/Classes/Morphisms_Prop.v | Acc_pt_morphism | |
Acc_rel_morphism{A:Type} :
Proper (relation_equivalence ==> Logic.eq ==> iff) (@Acc A).
Proof.
apply proper_sym_impl_iff_2.
- red; now symmetry.
- red; now symmetry.
- intros R R' EQ a a' Ha WF. subst a'.
induction WF as [x _ WF']. constructor.
intros y Ryx. now apply WF', EQ.
Qed. | Instance | Corelib | [
"Require Import Corelib.Classes.Morphisms",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics"
] | Corelib/Classes/Morphisms_Prop.v | Acc_rel_morphism | |
well_founded_morphism{A : Type} :
Proper (relation_equivalence ==> iff) (@well_founded A).
Proof.
unfold well_founded. solve_proper.
Qed. | Instance | Corelib | [
"Require Import Corelib.Classes.Morphisms",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics"
] | Corelib/Classes/Morphisms_Prop.v | well_founded_morphism | |
Reflexive(R : relation A) :=
reflexivity : forall x : A, R x x. | Class | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | Reflexive | |
complement(R : relation A) : relation A := fun x y => R x y -> False. | Definition | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | complement | |
complement_inverseR : complement (flip R) = flip (complement R).
Proof. reflexivity. Qed. | Lemma | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | complement_inverse | |
Irreflexive(R : relation A) :=
irreflexivity : Reflexive (complement R). | Class | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | Irreflexive | |
Symmetric(R : relation A) :=
symmetry : forall {x y}, R x y -> R y x. | Class | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | Symmetric | |
Asymmetric(R : relation A) :=
asymmetry : forall {x y}, R x y -> R y x -> False. | Class | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | Asymmetric | |
Transitive(R : relation A) :=
transitivity : forall {x y z}, R x y -> R y z -> R x z. | Class | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | Transitive | |
PreOrder(R : relation A) : Prop := {
#[global] PreOrder_Reflexive :: Reflexive R | 2 ;
#[global] PreOrder_Transitive :: Transitive R | 2 }. | Class | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | PreOrder | |
StrictOrder(R : relation A) : Prop := {
#[global] StrictOrder_Irreflexive :: Irreflexive R ;
#[global] StrictOrder_Transitive :: Transitive R }. | Class | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | StrictOrder | |
PER(R : relation A) : Prop := {
#[global] PER_Symmetric :: Symmetric R | 3 ;
#[global] PER_Transitive :: Transitive R | 3 }. | Class | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | PER | |
Equivalence(R : relation A) : Prop := {
#[global] Equivalence_Reflexive :: Reflexive R ;
#[global] Equivalence_Symmetric :: Symmetric R ;
#[global] Equivalence_Transitive :: Transitive R }. | Class | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | Equivalence | |
AntisymmetriceqA `{equ : Equivalence eqA} (R : relation A) :=
antisymmetry : forall {x y}, R x y -> R y x -> eqA x y. | Class | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | Antisymmetric | |
subrelation(R R' : relation A) : Prop :=
is_subrelation : forall {x y}, R x y -> R' x y. | Class | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | subrelation | |
subrelation_symmetricR `(Symmetric R) : subrelation (flip R) R.
Proof. hnf. intros x y H0. red in H0. apply symmetry. assumption. Qed. | Lemma | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | subrelation_symmetric | |
flip_Reflexive`{Reflexive R} : Reflexive (flip R).
Proof. tauto. Qed.
Program Definition flip_Irreflexive `(Irreflexive R) : Irreflexive (flip R) :=
irreflexivity (R:=R).
Program Definition flip_Symmetric `(Symmetric R) : Symmetric (flip R) :=
fun x y H => symmetry (R:=R) H.
Program Defin... | Lemma | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | flip_Reflexive | |
flip_PreOrder`(PreOrder R) : PreOrder (flip R).
Proof. firstorder. Qed. | Lemma | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | flip_PreOrder | |
flip_StrictOrder`(StrictOrder R) : StrictOrder (flip R).
Proof. firstorder. Qed. | Lemma | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | flip_StrictOrder | |
flip_PER`(PER R) : PER (flip R).
Proof. firstorder. Qed. | Lemma | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | flip_PER | |
flip_Equivalence`(Equivalence R) : Equivalence (flip R).
Proof. firstorder. Qed. | Lemma | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | flip_Equivalence | |
complement_Irreflexive`(Reflexive R)
: Irreflexive (complement R).
Proof. firstorder. Qed. | Lemma | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | complement_Irreflexive | |
complement_Symmetric`(Symmetric R) : Symmetric (complement R).
Proof. firstorder. Qed. | Lemma | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | complement_Symmetric | |
RewriteRelation(RA : relation A). | Class | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | RewriteRelation | |
inverse_impl_rewrite_relation: RewriteRelation (flip impl) | 3 := {}.
#[global] | Instance | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | inverse_impl_rewrite_relation | |
impl_rewrite_relation: RewriteRelation impl | 3 := {}.
#[global] | Instance | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | impl_rewrite_relation | |
iff_rewrite_relation: RewriteRelation iff | 2 := {}. | Instance | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | iff_rewrite_relation | |
equivalence_rewrite_relation`(eqa : Equivalence A eqA) : RewriteRelation eqA :=
Build_RewriteRelation _. | Definition | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | equivalence_rewrite_relation | |
equiv_rewrite_relationR :=
tryif is_evar R then fail
else class_apply equivalence_rewrite_relation.
#[global]
Hint Extern 10 (@RewriteRelation ?A ?R) => equiv_rewrite_relation R : typeclass_instances. | Ltac | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | equiv_rewrite_relation | |
solve_relation:=
match goal with
| [ |- ?R ?x ?x ] => reflexivity
| [ H : ?R ?x ?y |- ?R ?y ?x ] => symmetry ; exact H
end.
#[global]
Create HintDb relations.
#[global]
Hint Extern 4 => solve_relation : relations. | Ltac | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | solve_relation | |
reduce_hypH :=
match type of H with
| context [ _ <-> _ ] => fail 1
| _ => red in H ; try reduce_hyp H
end. | Ltac | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | reduce_hyp | |
reduce_goal:=
match goal with
| [ |- _ <-> _ ] => fail 1
| _ => red ; intros ; try reduce_goal
end.
Tactic Notation "reduce" "in" hyp(Hid) := reduce_hyp Hid. | Ltac | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | reduce_goal | |
reduce:= reduce_goal.
Tactic Notation "apply" "*" constr(t) :=
first [ refine t | refine (t _) | refine (t _ _) | refine (t _ _ _) | refine (t _ _ _ _) |
refine (t _ _ _ _ _) | refine (t _ _ _ _ _ _) | refine (t _ _ _ _ _ _ _) ]. | Ltac | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | reduce | |
simpl_relation:=
unfold flip, impl, arrow ; try reduce ; program_simpl ;
try ( solve [ dintuition auto with relations ]).
Local Obligation Tactic := try solve [ simpl_relation ]. | Ltac | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | simpl_relation | |
iff_Reflexive: Reflexive iff := iff_refl.
#[global] | Instance | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | iff_Reflexive | |
iff_Symmetric: Symmetric iff := iff_sym.
#[global] | Instance | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | iff_Symmetric | |
iff_Transitive: Transitive iff := iff_trans. | Instance | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | iff_Transitive | |
Tlist: Type := Tnil : Tlist | Tcons : Type -> Tlist -> Tlist.
Local Infix "::" := Tcons. | Inductive | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | Tlist | |
arrows(l : Tlist) (r : Type) : Type :=
match l with
| Tnil => r
| A :: l' => A -> arrows l' r
end. | Fixpoint | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | arrows | |
unary_operationA := arrows (A::Tnil) A. | Definition | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | unary_operation | |
binary_operationA := arrows (A::A::Tnil) A. | Definition | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | binary_operation | |
ternary_operationA := arrows (A::A::A::Tnil) A. | Definition | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | ternary_operation | |
unary_predicateA := predicate (A::Tnil). | Definition | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | unary_predicate | |
binary_relationA := predicate (A::A::Tnil). | Definition | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | binary_relation | |
predicate_all(l : Tlist) : predicate l -> Prop :=
match l with
| Tnil => fun f => f
| A :: tl => fun f => forall x : A, predicate_all tl (f x)
end. | Fixpoint | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | predicate_all | |
predicate_exists(l : Tlist) : predicate l -> Prop :=
match l with
| Tnil => fun f => f
| A :: tl => fun f => exists x : A, predicate_exists tl (f x)
end. | Fixpoint | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | predicate_exists | |
pointwise_extension{T : Type} (op : binary_operation T)
(l : Tlist) : binary_operation (arrows l T) :=
match l with
| Tnil => fun R R' => op R R'
| A :: tl => fun R R' =>
fun x => pointwise_extension op tl (R x) (R' x)
end. | Fixpoint | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | pointwise_extension | |
pointwise_lifting(op : binary_relation Prop) (l : Tlist) : binary_relation (predicate l) :=
match l with
| Tnil => fun R R' => op R R'
| A :: tl => fun R R' =>
forall x, pointwise_lifting op tl (R x) (R' x)
end. | Fixpoint | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | pointwise_lifting | |
predicate_equivalence{l : Tlist} : binary_relation (predicate l) :=
pointwise_lifting iff l. | Definition | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | predicate_equivalence | |
predicate_implication{l : Tlist} :=
pointwise_lifting impl l. | Definition | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | predicate_implication | |
predicate_intersection:= pointwise_extension and. | Definition | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | predicate_intersection | |
predicate_union:= pointwise_extension or.
Infix "/∙\" := predicate_intersection (at level 80, right associativity) : predicate_scope.
Infix "\∙/" := predicate_union (at level 85, right associativity) : predicate_scope. | Definition | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | predicate_union | |
true_predicate{l : Tlist} : predicate l :=
match l with
| Tnil => True
| A :: tl => fun _ => @true_predicate tl
end. | Fixpoint | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | true_predicate | |
false_predicate{l : Tlist} : predicate l :=
match l with
| Tnil => False
| A :: tl => fun _ => @false_predicate tl
end. | Fixpoint | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | false_predicate | |
relation_equivalence: relation (relation A) :=
@predicate_equivalence (_::_::Tnil).
Global Instance relation_equivalence_rewrite_relation: RewriteRelation relation_equivalence := {}. | Definition | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | relation_equivalence | |
relation_conjunction(R : relation A) (R' : relation A) : relation A :=
@predicate_intersection (A::A::Tnil) R R'. | Definition | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | relation_conjunction | |
relation_disjunction(R : relation A) (R' : relation A) : relation A :=
@predicate_union (A::A::Tnil) R R'. | Definition | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | relation_disjunction | |
PartialOrdereqA `{equ : Equivalence A eqA} R `{preo : PreOrder A R} :=
partial_order_equivalence : relation_equivalence eqA (relation_conjunction R (flip R)). | Class | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | PartialOrder | |
PartialOrder_inverse`(PartialOrder eqA R) : PartialOrder eqA (flip R).
Proof. firstorder. Qed. | Lemma | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Relations.Relation_Definitions"
] | Corelib/Classes/RelationClasses.v | PartialOrder_inverse | |
DefaultRelationA (R : relation A).
Register DefaultRelation as rewrite.DefaultRelation. | Class | Corelib | [
"Require Corelib.Classes.CRelationClasses Corelib.Classes.CMorphisms",
"Require Import Corelib.Classes.Morphisms Corelib.Classes.Morphisms_Prop",
"Require Export Corelib.Classes.RelationClasses Corelib.Relations.Relation_Definitions",
"Require Import Corelib.Classes.Equivalence Corelib.Program.Basics",
"Req... | Corelib/Classes/SetoidTactics.v | DefaultRelation | |
default_relation`{DefaultRelation A R} := R. | Definition | Corelib | [
"Require Corelib.Classes.CRelationClasses Corelib.Classes.CMorphisms",
"Require Import Corelib.Classes.Morphisms Corelib.Classes.Morphisms_Prop",
"Require Export Corelib.Classes.RelationClasses Corelib.Relations.Relation_Definitions",
"Require Import Corelib.Classes.Equivalence Corelib.Program.Basics",
"Req... | Corelib/Classes/SetoidTactics.v | default_relation | |
equivalence_default`(Equivalence A R) : DefaultRelation R | 4.
Defined. | Instance | Corelib | [
"Require Corelib.Classes.CRelationClasses Corelib.Classes.CMorphisms",
"Require Import Corelib.Classes.Morphisms Corelib.Classes.Morphisms_Prop",
"Require Export Corelib.Classes.RelationClasses Corelib.Relations.Relation_Definitions",
"Require Import Corelib.Classes.Equivalence Corelib.Program.Basics",
"Req... | Corelib/Classes/SetoidTactics.v | equivalence_default | |
setoidreplaceH t :=
let Heq := fresh "Heq" in
cut(H) ; unfold default_relation ; [ intro Heq ; setoid_rewrite Heq ; clear Heq | t ]. | Ltac | Corelib | [
"Require Corelib.Classes.CRelationClasses Corelib.Classes.CMorphisms",
"Require Import Corelib.Classes.Morphisms Corelib.Classes.Morphisms_Prop",
"Require Export Corelib.Classes.RelationClasses Corelib.Relations.Relation_Definitions",
"Require Import Corelib.Classes.Equivalence Corelib.Program.Basics",
"Req... | Corelib/Classes/SetoidTactics.v | setoidreplace | |
setoidreplaceinH H' t :=
let Heq := fresh "Heq" in
cut(H) ; unfold default_relation ; [ intro Heq ; setoid_rewrite Heq in H' ; clear Heq | t ]. | Ltac | Corelib | [
"Require Corelib.Classes.CRelationClasses Corelib.Classes.CMorphisms",
"Require Import Corelib.Classes.Morphisms Corelib.Classes.Morphisms_Prop",
"Require Export Corelib.Classes.RelationClasses Corelib.Relations.Relation_Definitions",
"Require Import Corelib.Classes.Equivalence Corelib.Program.Basics",
"Req... | Corelib/Classes/SetoidTactics.v | setoidreplacein | |
setoidreplaceinatH H' t occs :=
let Heq := fresh "Heq" in
cut(H) ; unfold default_relation ; [ intro Heq ; setoid_rewrite Heq in H' at occs ; clear Heq | t ]. | Ltac | Corelib | [
"Require Corelib.Classes.CRelationClasses Corelib.Classes.CMorphisms",
"Require Import Corelib.Classes.Morphisms Corelib.Classes.Morphisms_Prop",
"Require Export Corelib.Classes.RelationClasses Corelib.Relations.Relation_Definitions",
"Require Import Corelib.Classes.Equivalence Corelib.Program.Basics",
"Req... | Corelib/Classes/SetoidTactics.v | setoidreplaceinat | |
setoidreplaceatH t occs :=
let Heq := fresh "Heq" in
cut(H) ; unfold default_relation ; [ intro Heq ; setoid_rewrite Heq at occs ; clear Heq | t ].
Tactic Notation "setoid_replace" constr(x) "with" constr(y) :=
setoidreplace (default_relation x y) idtac.
Tactic Notation "setoid_replace" constr(x) "with" const... | Ltac | Corelib | [
"Require Corelib.Classes.CRelationClasses Corelib.Classes.CMorphisms",
"Require Import Corelib.Classes.Morphisms Corelib.Classes.Morphisms_Prop",
"Require Export Corelib.Classes.RelationClasses Corelib.Relations.Relation_Definitions",
"Require Import Corelib.Classes.Equivalence Corelib.Program.Basics",
"Req... | Corelib/Classes/SetoidTactics.v | setoidreplaceat | |
red_subst_eq_morphismconcl :=
match concl with
| @Logic.eq ?A ==> ?R' => red ; intros ; subst ; red_subst_eq_morphism R'
| ?R ==> ?R' => red ; intros ; red_subst_eq_morphism R'
| _ => idtac
end. | Ltac | Corelib | [
"Require Corelib.Classes.CRelationClasses Corelib.Classes.CMorphisms",
"Require Import Corelib.Classes.Morphisms Corelib.Classes.Morphisms_Prop",
"Require Export Corelib.Classes.RelationClasses Corelib.Relations.Relation_Definitions",
"Require Import Corelib.Classes.Equivalence Corelib.Program.Basics",
"Req... | Corelib/Classes/SetoidTactics.v | red_subst_eq_morphism | |
destruct_proper:=
match goal with
| [ |- @Proper ?A ?R ?m ] => red
end. | Ltac | Corelib | [
"Require Corelib.Classes.CRelationClasses Corelib.Classes.CMorphisms",
"Require Import Corelib.Classes.Morphisms Corelib.Classes.Morphisms_Prop",
"Require Export Corelib.Classes.RelationClasses Corelib.Relations.Relation_Definitions",
"Require Import Corelib.Classes.Equivalence Corelib.Program.Basics",
"Req... | Corelib/Classes/SetoidTactics.v | destruct_proper | |
reverse_arrowsx :=
match x with
| @Logic.eq ?A ==> ?R' => revert_last ; reverse_arrows R'
| ?R ==> ?R' => do 3 revert_last ; reverse_arrows R'
| _ => idtac
end. | Ltac | Corelib | [
"Require Corelib.Classes.CRelationClasses Corelib.Classes.CMorphisms",
"Require Import Corelib.Classes.Morphisms Corelib.Classes.Morphisms_Prop",
"Require Export Corelib.Classes.RelationClasses Corelib.Relations.Relation_Definitions",
"Require Import Corelib.Classes.Equivalence Corelib.Program.Basics",
"Req... | Corelib/Classes/SetoidTactics.v | reverse_arrows | |
default_add_morphism_tactic:=
unfold flip ; intros ;
(try destruct_proper) ;
match goal with
| [ |- (?x ==> ?y) _ _ ] => red_subst_eq_morphism (x ==> y) ; reverse_arrows (x ==> y)
end. | Ltac | Corelib | [
"Require Corelib.Classes.CRelationClasses Corelib.Classes.CMorphisms",
"Require Import Corelib.Classes.Morphisms Corelib.Classes.Morphisms_Prop",
"Require Export Corelib.Classes.RelationClasses Corelib.Relations.Relation_Definitions",
"Require Import Corelib.Classes.Equivalence Corelib.Program.Basics",
"Req... | Corelib/Classes/SetoidTactics.v | default_add_morphism_tactic | |
add_morphism_tactic:= default_add_morphism_tactic.
#[global] Obligation Tactic := program_simpl.
#[export] Obligation Tactic := program_simpl. | Ltac | Corelib | [
"Require Corelib.Classes.CRelationClasses Corelib.Classes.CMorphisms",
"Require Import Corelib.Classes.Morphisms Corelib.Classes.Morphisms_Prop",
"Require Export Corelib.Classes.RelationClasses Corelib.Relations.Relation_Definitions",
"Require Import Corelib.Classes.Equivalence Corelib.Program.Basics",
"Req... | Corelib/Classes/SetoidTactics.v | add_morphism_tactic | |
SF64classify:= SFclassify prec. | Definition | Corelib | [
"From Corelib Require Import BinNums IntDef Uint63Axioms",
"From Corelib Require Import SpecFloat PrimFloat FloatOps"
] | Corelib/Floats/FloatAxioms.v | SF64classify |
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