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 |
|---|---|---|---|---|---|---|
StrictOrder_PreOrder`(Equivalence A eqA, StrictOrder A R, Proper _ (eqA==>eqA==>iffT) R) :
PreOrder (relation_disjunction R eqA).
Proof.
split.
- intros x. right. apply reflexivity.
- intros x y z [Hxy|Hxy] [Hyz|Hyz].
+ left. apply transitivity with y; auto.
+ left. eapply H1; try eassumption.
* apply reflexiv... | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | StrictOrder_PreOrder | |
StrictOrder_PartialOrder`(Equivalence A eqA, StrictOrder A R, Proper _ (eqA==>eqA==>iffT) R) :
PartialOrder eqA (relation_disjunction R eqA).
Proof.
intros. intros x y. compute. intuition auto.
- right; now apply symmetry.
- elim (StrictOrder_Irreflexive x).
eapply transitivity with y; eauto.
- now apply symmetry.
... | Lemma | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Export Corelib.Classes.CRelationClasses",
"Require Import Relation_Definitions"
] | Corelib/Classes/CMorphisms.v | StrictOrder_PartialOrder | |
crelation(A : Type) := A -> A -> Type. | Definition | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics"
] | Corelib/Classes/CRelationClasses.v | crelation | |
arrow(A B : Type) := A -> B. | Definition | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics"
] | Corelib/Classes/CRelationClasses.v | arrow | |
flip{A B C : Type} (f : A -> B -> C) := fun x y => f y x. | Definition | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics"
] | Corelib/Classes/CRelationClasses.v | flip | |
iffT(A B : Type) := ((A -> B) * (B -> A))%type.
Global Typeclasses Opaque flip arrow iffT. | Definition | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics"
] | Corelib/Classes/CRelationClasses.v | iffT | |
Reflexive(R : crelation 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"
] | Corelib/Classes/CRelationClasses.v | Reflexive | |
complement(R : crelation A) : crelation 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"
] | Corelib/Classes/CRelationClasses.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"
] | Corelib/Classes/CRelationClasses.v | complement_inverse | |
Irreflexive(R : crelation A) :=
irreflexivity : Reflexive (complement R). | Class | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics"
] | Corelib/Classes/CRelationClasses.v | Irreflexive | |
Symmetric(R : crelation 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"
] | Corelib/Classes/CRelationClasses.v | Symmetric | |
Asymmetric(R : crelation A) :=
asymmetry : forall {x y}, R x y -> (complement R y x : Type). | Class | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics"
] | Corelib/Classes/CRelationClasses.v | Asymmetric | |
Transitive(R : crelation 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"
] | Corelib/Classes/CRelationClasses.v | Transitive | |
PreOrder(R : crelation A) := {
#[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"
] | Corelib/Classes/CRelationClasses.v | PreOrder | |
StrictOrder(R : crelation A) := {
#[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"
] | Corelib/Classes/CRelationClasses.v | StrictOrder | |
PER(R : crelation A) := {
#[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"
] | Corelib/Classes/CRelationClasses.v | PER | |
Equivalence(R : crelation A) := {
#[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"
] | Corelib/Classes/CRelationClasses.v | Equivalence | |
AntisymmetriceqA `{equ : Equivalence eqA} (R : crelation 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"
] | Corelib/Classes/CRelationClasses.v | Antisymmetric | |
subrelation(R R' : crelation A) :=
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"
] | Corelib/Classes/CRelationClasses.v | subrelation | |
subrelation_symmetricR `(Symmetric R) : subrelation (flip R) R.
Proof. hnf. intros x y H'. red in H'. apply symmetry. assumption. Qed. | Lemma | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics"
] | Corelib/Classes/CRelationClasses.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"
] | Corelib/Classes/CRelationClasses.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"
] | Corelib/Classes/CRelationClasses.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"
] | Corelib/Classes/CRelationClasses.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"
] | Corelib/Classes/CRelationClasses.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"
] | Corelib/Classes/CRelationClasses.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"
] | Corelib/Classes/CRelationClasses.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"
] | Corelib/Classes/CRelationClasses.v | complement_Symmetric | |
RewriteRelation(RA : crelation A). | Class | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics"
] | Corelib/Classes/CRelationClasses.v | RewriteRelation | |
solve_crelation:=
match goal with
| [ |- ?R ?x ?x ] => reflexivity
| [ H : ?R ?x ?y |- ?R ?y ?x ] => symmetry ; exact H
end.
#[global]
Create HintDb crelations.
#[global]
Hint Extern 4 => solve_crelation : crelations. | Ltac | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics"
] | Corelib/Classes/CRelationClasses.v | solve_crelation | |
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"
] | Corelib/Classes/CRelationClasses.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"
] | Corelib/Classes/CRelationClasses.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"
] | Corelib/Classes/CRelationClasses.v | reduce | |
simpl_crelation:=
unfold flip, impl, arrow ; try reduce ; program_simpl ;
try ( solve [ dintuition auto with crelations ]).
Local Obligation Tactic := simpl_crelation. | Ltac | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics"
] | Corelib/Classes/CRelationClasses.v | simpl_crelation | |
iff_Reflexive: Reflexive iff := iff_refl.
#[global] | Instance | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics"
] | Corelib/Classes/CRelationClasses.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"
] | Corelib/Classes/CRelationClasses.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"
] | Corelib/Classes/CRelationClasses.v | iff_Transitive | |
iffT_Reflexive: Reflexive iffT.
Proof. firstorder. Defined.
#[global] | Instance | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics"
] | Corelib/Classes/CRelationClasses.v | iffT_Reflexive | |
iffT_Symmetric: Symmetric iffT.
Proof. firstorder. Defined.
#[global] | Instance | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics"
] | Corelib/Classes/CRelationClasses.v | iffT_Symmetric | |
iffT_Transitive: Transitive iffT.
Proof. firstorder. Defined. | Instance | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics"
] | Corelib/Classes/CRelationClasses.v | iffT_Transitive | |
relation_equivalence: crelation (crelation A) :=
fun R R' => forall x y, iffT (R x y) (R' x y).
Global Instance: RewriteRelation relation_equivalence.
Defined. | Definition | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics"
] | Corelib/Classes/CRelationClasses.v | relation_equivalence | |
relation_conjunction(R : crelation A) (R' : crelation A) : crelation A :=
fun x y => prod (R x y) (R' x y). | Definition | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics"
] | Corelib/Classes/CRelationClasses.v | relation_conjunction | |
relation_disjunction(R : crelation A) (R' : crelation A) : crelation A :=
fun x y => sum (R x y) (R' x y). | Definition | Corelib | [
"Require Export Corelib.Classes.Init",
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics"
] | Corelib/Classes/CRelationClasses.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"
] | Corelib/Classes/CRelationClasses.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"
] | Corelib/Classes/CRelationClasses.v | PartialOrder_inverse | |
equiv`{Equivalence A R} : relation A := R. | Definition | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Classes.Init",
"Require Import Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses",
"Require Import Corelib.Classes.Morphisms"
] | Corelib/Classes/Equivalence.v | equiv | |
pequiv`{PER A R} : relation A := R. | Definition | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Classes.Init",
"Require Import Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses",
"Require Import Corelib.Classes.Morphisms"
] | Corelib/Classes/Equivalence.v | pequiv | |
setoid_substH :=
match type of H with
?x === ?y => substitute H ; clear H x
end. | Ltac | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Classes.Init",
"Require Import Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses",
"Require Import Corelib.Classes.Morphisms"
] | Corelib/Classes/Equivalence.v | setoid_subst | |
setoid_subst_nofail:=
match goal with
| [ H : ?x === ?y |- _ ] => setoid_subst H ; setoid_subst_nofail
| _ => idtac
end. | Ltac | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Classes.Init",
"Require Import Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses",
"Require Import Corelib.Classes.Morphisms"
] | Corelib/Classes/Equivalence.v | setoid_subst_nofail | |
equiv_simplify_one:=
match goal with
| [ H : ?x === ?x |- _ ] => clear H
| [ H : ?x === ?y |- _ ] => setoid_subst H
| [ |- ?x =/= ?y ] => let name:=fresh "Hneq" in intro name
| [ |- ~ ?x === ?y ] => let name:=fresh "Hneq" in intro name
end. | Ltac | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Classes.Init",
"Require Import Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses",
"Require Import Corelib.Classes.Morphisms"
] | Corelib/Classes/Equivalence.v | equiv_simplify_one | |
equiv_simplify:= repeat equiv_simplify_one. | Ltac | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Classes.Init",
"Require Import Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses",
"Require Import Corelib.Classes.Morphisms"
] | Corelib/Classes/Equivalence.v | equiv_simplify | |
equivify_tac:=
match goal with
| [ s : Equivalence ?A ?R, H : ?R ?x ?y |- _ ] => change R with (@equiv A R s) in H
| [ s : Equivalence ?A ?R |- context C [ ?R ?x ?y ] ] => change (R x y) with (@equiv A R s x y)
end. | Ltac | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Classes.Init",
"Require Import Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses",
"Require Import Corelib.Classes.Morphisms"
] | Corelib/Classes/Equivalence.v | equivify_tac | |
equivify:= repeat equivify_tac. | Ltac | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Classes.Init",
"Require Import Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses",
"Require Import Corelib.Classes.Morphisms"
] | Corelib/Classes/Equivalence.v | equivify | |
respecting`(eqa : Equivalence A (R : relation A),
eqb : Equivalence B (R' : relation B)) : Type :=
{ morph : A -> B | respectful R R' morph morph }.
Program Instance respecting_equiv `(eqa : Equivalence A R, eqb : Equivalence B R') :
Equivalence (fun (f g : respecting eqa eqb) =>
... | Definition | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Classes.Init",
"Require Import Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses",
"Require Import Corelib.Classes.Morphisms"
] | Corelib/Classes/Equivalence.v | respecting | |
pointwise_reflexive{A} `(reflb : Reflexive B eqB) :
Reflexive (pointwise_relation A eqB) | 9.
Proof. firstorder. Qed.
#[global] | Instance | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Classes.Init",
"Require Import Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses",
"Require Import Corelib.Classes.Morphisms"
] | Corelib/Classes/Equivalence.v | pointwise_reflexive | |
pointwise_symmetric{A} `(symb : Symmetric B eqB) :
Symmetric (pointwise_relation A eqB) | 9.
Proof. firstorder. Qed.
#[global] | Instance | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Classes.Init",
"Require Import Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses",
"Require Import Corelib.Classes.Morphisms"
] | Corelib/Classes/Equivalence.v | pointwise_symmetric | |
pointwise_transitive{A} `(transb : Transitive B eqB) :
Transitive (pointwise_relation A eqB) | 9.
Proof. firstorder. Qed.
#[global] | Instance | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Classes.Init",
"Require Import Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses",
"Require Import Corelib.Classes.Morphisms"
] | Corelib/Classes/Equivalence.v | pointwise_transitive | |
pointwise_equivalence{A} `(eqb : Equivalence B eqB) :
Equivalence (pointwise_relation A eqB) | 9.
Proof. split; apply _. Qed. | Instance | Corelib | [
"Require Import Corelib.Program.Basics",
"Require Import Corelib.Program.Tactics",
"Require Import Corelib.Classes.Init",
"Require Import Relation_Definitions",
"Require Export Corelib.Classes.RelationClasses",
"Require Import Corelib.Classes.Morphisms"
] | Corelib/Classes/Equivalence.v | pointwise_equivalence | |
class_applyc := autoapply c with typeclass_instances. | Ltac | Corelib | [
"Require Import Corelib.Program.Basics"
] | Corelib/Classes/Init.v | class_apply | |
Unconvertible(A : Type) (a b : A) := unconvertible : unit. | Class | Corelib | [
"Require Import Corelib.Program.Basics"
] | Corelib/Classes/Init.v | Unconvertible | |
unconvertible:=
match goal with
| |- @Unconvertible _ ?x ?y => unify x y with typeclass_instances ; fail 1 "Convertible"
| |- _ => exact tt
end.
#[global]
Hint Extern 0 (@Unconvertible _ _ _) => unconvertible : typeclass_instances. | Ltac | Corelib | [
"Require Import Corelib.Program.Basics"
] | Corelib/Classes/Init.v | unconvertible | |
Proper(R : relation A) (m : A) : Prop :=
proper_prf : R 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 | Proper | |
ProperProxy(R : relation A) (m : A) : Prop :=
proper_proxy : R 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 | ProperProxy | |
ReflexiveProxy(R : relation A) : Prop :=
reflexive_proxy : forall x, R x x. | 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 | ReflexiveProxy | |
eq_proper_proxy(x : A) : ProperProxy (@eq A) x.
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 | eq_proper_proxy | |
reflexive_proper`{ReflexiveProxy R} (x : A) : Proper R x.
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 | reflexive_proper | |
reflexive_proper_proxy`(ReflexiveProxy R) (x : A) : ProperProxy R x.
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 | reflexive_proper_proxy | |
proper_proper_proxyx `(Proper R x) : ProperProxy R x.
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 | proper_proper_proxy | |
reflexive_reflexive_proxy`(Reflexive A R) : ReflexiveProxy 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 | reflexive_reflexive_proxy | |
respectful_hetero(A B : Type)
(C : A -> Type) (D : B -> Type)
(R : A -> B -> Prop)
(R' : forall (x : A) (y : B), C x -> D y -> Prop) :
(forall x : A, C x) -> (forall x : B, D x) -> Prop :=
fun f g => forall x y, R x y -> R' x y (f x) (g y). | 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 | respectful_hetero | |
respectful(R : relation A) (R' : relation B) : relation (A -> B) :=
Eval compute in @respectful_hetero A A (fun _ => B) (fun _ => B) R (fun _ _ => R'). | 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 | respectful | |
pointwise_relationA {B} (R : relation B) : relation (A -> B) :=
fun f g => forall a, R (f a) (g a). | 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 | pointwise_relation | |
rewrite_relation_pointwise{A B R} `{RewriteRelation B R}:
RewriteRelation (@pointwise_relation A B R).
Proof. split. 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 | rewrite_relation_pointwise | |
rewrite_relation_eq_dom{A B R} `{RewriteRelation B R}:
RewriteRelation (respectful (@Logic.eq A) R).
Proof. split. 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 | rewrite_relation_eq_dom | |
rewrite_relation_fun:= | 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 | rewrite_relation_fun | |
eq_rewrite_relation{A} : RewriteRelation (@eq A).
Proof. split. 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_rewrite_relation | |
eq_rewrite_relationA :=
solve [unshelve class_apply @eq_rewrite_relation].
Global Hint Extern 100 (@RewriteRelation ?A _) => eq_rewrite_relation A : typeclass_instances. | 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 | eq_rewrite_relation | |
find_rewrite_relationA R kont :=
assert (@RewriteRelation A R); [solve [unshelve typeclasses eauto]|]; kont R. | 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 | find_rewrite_relation | |
reflexive_proxy_tacA R :=
tryif has_evar R then | 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 | reflexive_proxy_tac | |
solve_respectfult :=
match goal with
| |- respectful _ _ _ _ =>
let H := fresh "H" in
intros ? ? H; solve_respectful ltac:(setoid_rewrite H; t)
| _ => t; reflexivity
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 | solve_respectful | |
solve_proper:= unfold Proper; solve_respectful ltac:(idtac). | 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 | solve_proper | |
f_equiv:=
match goal with
| |- ?R (?f ?x) (?f' _) =>
let T := type of x in
let Rx := fresh "R" in
evar (Rx : relation T);
let H := fresh in
assert (H : (Rx==>R)%signature f f');
unfold Rx in *; clear Rx; [ f_equiv | apply H; clear H; try reflexivity ]
| |- ?R ?f ?f' =>
solve [change (Pr... | 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 | f_equiv | |
forall_def: Type := forall x : A, P x. | 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 | forall_def | |
forall_relation(sig : forall a, relation (P a)) : relation (forall x, P x) :=
fun f g => forall a, sig a (f a) (g a). | 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 | forall_relation | |
pointwise_pointwise(R : relation B) :
relation_equivalence (pointwise_relation A R) (@eq A ==> R).
Proof. intros. split; reduce; subst; 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 | pointwise_pointwise | |
subrelation_respectful`(subl : subrelation A RA' RA, subr : subrelation B RB RB') :
subrelation (RA ==> RB) (RA' ==> RB').
Proof. unfold subrelation in *; 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 | subrelation_respectful | |
subrelation_reflR : @subrelation A R R.
Proof. unfold subrelation; 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 | subrelation_refl | |
subrelation_proper`(mor : Proper A R' m)
`(unc : Unconvertible (relation A) R R')
`(sub : subrelation A R' R) : Proper R m.
Proof.
intros. apply sub. apply mor.
Qed.
Global Instance proper_subrelation_proper :
Proper (subrelation ++> eq ==> impl) (@Proper A).
Proof. reduce. subst. first... | 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 | subrelation_proper | |
forall_subrelation(R S : forall x : A, relation (P x)) :
(forall a, subrelation (R a) (S a)) -> subrelation (forall_relation R) (forall_relation S).
Proof. intros H x y H0 a. apply H. apply H0. 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 | forall_subrelation | |
subrelation_tacT U :=
(is_ground T ; is_ground U ; class_apply @subrelation_refl) ||
class_apply @subrelation_respectful || class_apply @subrelation_refl.
#[global]
Hint Extern 3 (@subrelation _ ?T ?U) => subrelation_tac T U : typeclass_instances. | 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 | subrelation_tac | |
apply_subrelation: Prop := do_subrelation. | 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 | apply_subrelation | |
proper_subrelation:=
match goal with
[ H : apply_subrelation |- _ ] => clear H ; class_apply @subrelation_proper
end.
#[global]
Hint Extern 5 (@Proper _ ?H _) => proper_subrelation : typeclass_instances. | 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_subrelation | |
iff_impl_subrelation: subrelation iff impl | 2.
Proof. firstorder. Qed.
#[global] | 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 | iff_impl_subrelation | |
iff_flip_impl_subrelation: subrelation iff (flip impl) | 2.
Proof. firstorder. 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 | iff_flip_impl_subrelation | |
trans_contra_co_morphism`(Transitive A R) : Proper (R --> R ++> impl) R.
Next Obligation.
Proof.
intros R H x y H0 x0 y0 H1 H2.
transitivity x; auto.
transitivity x0; auto.
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 | trans_contra_co_morphism | |
trans_contra_inv_impl_morphism`(Transitive A R) {x} : Proper (R --> flip impl) (R x) | 3.
Next Obligation.
Proof.
intros R H x x0 y H0 H1.
transitivity y; auto.
Qed.
Global Program | 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 | trans_contra_inv_impl_morphism | |
trans_co_impl_morphism`(Transitive A R) {x} : Proper (R ++> impl) (R x) | 3.
Next Obligation.
Proof.
intros R H x x0 y H0 H1.
transitivity x0; auto.
Qed.
Global Program | 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 | trans_co_impl_morphism | |
trans_sym_co_inv_impl_morphism`(PER A R) {x} : Proper (R ++> flip impl) (R x) | 3.
Next Obligation.
Proof.
intros R H x x0 y H0 H1.
transitivity y; auto. symmetry; auto.
Qed.
Global Program Instance trans_sym_contra_impl_morphism
`(PER A R) {x} : Proper (R --> impl) (R x) | 3.
Next Obligation.
... | 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 | trans_sym_co_inv_impl_morphism | |
trans_co_eq_inv_impl_morphism`(Transitive A R) : Proper (R ==> (@eq A) ==> flip impl) R | 2.
Next Obligation.
Proof.
intros R H x y H0 y0 y1 e H2; destruct e.
transitivity y; auto.
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 | trans_co_eq_inv_impl_morphism | |
PER_morphism`(PER A R) : Proper (R ==> R ==> iff) R | 1.
Next Obligation.
Proof.
intros R H x y H0 x0 y0 H1.
split ; intros.
- transitivity x0; auto. transitivity x; auto. symmetry; auto.
- transitivity y; auto. transitivity y0; auto. symmetry; auto.
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 | PER_morphism | |
symmetric_equiv_flip`(Symmetric A R) : relation_equivalence R (flip R).
Proof. firstorder. Qed.
Global Program Instance compose_proper RA RB RC :
Proper ((RB ==> RC) ==> (RA ==> RB) ==> (RA ==> RC)) (@compose A B C).
Next Obligation.
Proof.
intros RA RB RC x y H x0 y0 H0 x1 y1 H1.
unfold compose. ... | 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 | symmetric_equiv_flip |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.