statement stringlengths 1 1.02k | proof stringlengths 0 5.19k | type stringclasses 13
values | symbolic_name stringlengths 1 30 | library stringclasses 15
values | filename stringclasses 117
values | imports listlengths 0 16 | deps listlengths 0 16 | docstring stringclasses 50
values | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
nat_Show : Show nat | := { show := nat_show }. | Instance | nat_Show | Programming | theories/Programming/Show.v | [
"Coq",
"Ascii",
"String",
"Coq.Program",
"Wf",
"BinPos",
"ZArith",
"ExtLib.Structures.Monoid",
"ExtLib.Structures.Reducible",
"ExtLib.Programming.Injection",
"ExtLib.Data.Char",
"ExtLib.Data.String",
"ExtLib.Data.Fun",
"ExtLib.Core.RelDec",
"ShowNotation"
] | [
"Show",
"nat_show"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Show_positive : Show positive | :=
fun x => nat_show (Pos.to_nat x). | Instance | Show_positive | Programming | theories/Programming/Show.v | [
"Coq",
"Ascii",
"String",
"Coq.Program",
"Wf",
"BinPos",
"ZArith",
"ExtLib.Structures.Monoid",
"ExtLib.Structures.Reducible",
"ExtLib.Programming.Injection",
"ExtLib.Data.Char",
"ExtLib.Data.String",
"ExtLib.Data.Fun",
"ExtLib.Core.RelDec",
"ShowNotation"
] | [
"Show",
"nat_show",
"to_nat"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Show_Z : Show Z | :=
fun x =>
match x with
| Z0 => "0"%char
| Zpos p => show p
| Zneg p => "-"%char << show p
end. | Instance | Show_Z | Programming | theories/Programming/Show.v | [
"Coq",
"Ascii",
"String",
"Coq.Program",
"Wf",
"BinPos",
"ZArith",
"ExtLib.Structures.Monoid",
"ExtLib.Structures.Reducible",
"ExtLib.Programming.Injection",
"ExtLib.Data.Char",
"ExtLib.Data.String",
"ExtLib.Data.Fun",
"ExtLib.Core.RelDec",
"ShowNotation"
] | [
"Show"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
pair_Show@{a m t}
{A : Type@{a}} {B : Type@{a}} {AS:Show A} {BS:Show B}
: Show@{_ t} (A*B) | :=
fun p =>
let (a,b) := p in
"("%char << show a << ","%char << show b << ")"%char. | Definition | pair_Show | Programming | theories/Programming/Show.v | [
"Coq",
"Ascii",
"String",
"Coq.Program",
"Wf",
"BinPos",
"ZArith",
"ExtLib.Structures.Monoid",
"ExtLib.Structures.Reducible",
"ExtLib.Programming.Injection",
"ExtLib.Data.Char",
"ExtLib.Data.String",
"ExtLib.Data.Fun",
"ExtLib.Core.RelDec",
"ShowNotation"
] | [
"Show"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Ctor {T : Type} (ls : list {x : Type & T -> x}) : Type | :=
match ls with
| nil => T
| a :: b => (projT1 a) -> Ctor b
end. | Fixpoint | Ctor | Programming | theories/Programming/With.v | [
"Coq.Lists.List"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Struct (T : Type) : Type | := {
fields : list {x : Type & T -> x} ;
ctor : Ctor fields
}. | Class | Struct | Programming | theories/Programming/With.v | [
"Coq.Lists.List"
] | [
"Ctor"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Mem : list {x : Type & T -> x} -> Type | :=
| Here : forall a b, Mem ((@existT _ _ U a) :: b)
| Next : forall a b, Mem b -> Mem (a :: b). | Inductive | Mem | Programming | theories/Programming/With.v | [
"Coq.Lists.List"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
applyRest (f : list {x : Type & T -> x}) : Ctor f -> T | :=
match f as f return Ctor f -> T with
| nil => fun x => x
| a :: b => fun acc => applyRest b (acc ((projT2 a) rec))
end. | Fixpoint | applyRest | Programming | theories/Programming/With.v | [
"Coq.Lists.List"
] | [
"Ctor"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
applyUntil (f : list {x : Type & T -> x}) (n : Mem U f) : Ctor f -> T | :=
match n in Mem _ f return Ctor f -> T with
| Here a b => fun ctor => applyRest b (ctor v)
| Next a b i => fun ctor => applyUntil b i (ctor ((projT2 a) rec))
end. | Fixpoint | applyUntil | Programming | theories/Programming/With.v | [
"Coq.Lists.List"
] | [
"Ctor",
"Mem",
"applyRest"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
structWith {U : Type} (v : U) (n : Mem U fields) : T | :=
applyUntil v fields n ctor. | Definition | structWith | Programming | theories/Programming/With.v | [
"Coq.Lists.List"
] | [
"Mem",
"applyUntil"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Accessor {T U : Type} {strt : Struct T} (f : T -> U) : Type | := {
acc : Mem T U fields
}. | Class | Accessor | Programming | theories/Programming/With.v | [
"Coq.Lists.List"
] | [
"Mem",
"Struct"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
wrapWith {T U : Type} (t : T) (f : T -> U) (v : U)
(_strt : Struct T) (_acc : Accessor f) | :=
@structWith _ _ t _ v acc. | Definition | wrapWith | Programming | theories/Programming/With.v | [
"Coq.Lists.List"
] | [
"Accessor",
"Struct",
"structWith"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
"{$ x 'with' y ':=' v $}" | := (@wrapWith _ _ x y v _ _) : struct_scope. | Notation | {$ x 'with' y ':=' v $} | Programming | theories/Programming/With.v | [
"Coq.Lists.List"
] | [
"wrapWith"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
wf_anti_sym T (R : T -> T -> Prop) (wf : well_founded R)
: Irreflexive R. | Proof.
refine (fun a =>
(@Fix _ _ wf
(fun x => x = a -> R x a ->False)
(fun x rec pf pfr =>
rec _ match eq_sym pf in _ = t return R x t with
| eq_refl => pfr
end pf pfr)) a eq_refl).
Qed. | Lemma | wf_anti_sym | Recur | theories/Recur/Facts.v | [
"Coq.Classes.RelationClasses"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
guard A (R : A -> A -> Prop) (n : nat) (wfR : well_founded R)
{struct n}: well_founded R | :=
match n with
| 0 => wfR
| S n => fun x => Acc_intro x (fun y _ => guard n (guard n wfR) y)
end. | Fixpoint | guard | Recur | theories/Recur/GenRec.v | [
"Coq.Classes.RelationClasses"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Hstep : forall (x : A) (f g : forall y : A, R y x -> P y),
(forall (y : A) (p : R y x), r (f y p) (g y p)) ->
r (@F x f) (@F x g). | Hypothesis | Hstep | Recur | theories/Recur/GenRec.v | [
"Coq.Classes.RelationClasses"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | ||
Fix_F_equiv_inv
: forall (x : A) (r' s' : Acc R x),
r (Fix_F _ F r') (Fix_F _ F s'). | Proof.
intro x; induction (Rwf x); intros.
rewrite <- (Fix_F_eq _ F r'); rewrite <- (Fix_F_eq _ F s'); intros.
eapply Hstep.
eauto.
Qed. | Lemma | Fix_F_equiv_inv | Recur | theories/Recur/GenRec.v | [
"Coq.Classes.RelationClasses"
] | [
"Hstep"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Fix_equiv
: forall x : A,
r (Fix Rwf P F x) (@F x (fun (y : A) (_ : R y x) => Fix Rwf P F y)). | Proof.
intro x; unfold Fix.
rewrite <- Fix_F_eq.
apply Hstep; intros.
apply Fix_F_equiv_inv.
Qed. | Theorem | Fix_equiv | Recur | theories/Recur/GenRec.v | [
"Coq.Classes.RelationClasses"
] | [
"Fix_F_equiv_inv",
"Hstep",
"apply"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
well_founded_R : well_founded R. | Hypothesis | well_founded_R | Recur | theories/Recur/Measure.v | [
"Coq.Classes",
"RelationClasses",
"Coq.Arith",
"Wf_nat"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | ||
compose (a b : T) : Prop | :=
R (f a) (f b). | Definition | compose | Recur | theories/Recur/Measure.v | [
"Coq.Classes",
"RelationClasses",
"Coq.Arith",
"Wf_nat"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
well_founded_compose : well_founded compose | :=
(fun t =>
(@Fix _ R well_founded_R (fun x => forall y, f y = x -> Acc compose y)
(fun x recur y pf =>
@Acc_intro _ compose y (fun y' (pf' : R (f y') (f y)) =>
recur _ match pf in _ = t return R (f y') t with
... | Definition | well_founded_compose | Recur | theories/Recur/Measure.v | [
"Coq.Classes",
"RelationClasses",
"Coq.Arith",
"Wf_nat"
] | [
"compose",
"well_founded_R"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
mlt : T -> T -> Prop | :=
compose m lt. | Definition | mlt | Recur | theories/Recur/Measure.v | [
"Coq.Classes",
"RelationClasses",
"Coq.Arith",
"Wf_nat"
] | [
"compose",
"lt"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
well_founded_mlt : well_founded mlt | :=
@well_founded_compose T nat m lt Wf_nat.lt_wf. | Definition | well_founded_mlt | Recur | theories/Recur/Measure.v | [
"Coq.Classes",
"RelationClasses",
"Coq.Arith",
"Wf_nat"
] | [
"lt",
"mlt",
"well_founded_compose"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
wf_rightTrans : well_founded (rightTrans R). | Proof.
red.
eapply Fix. eapply wf_R. clear.
intros. constructor. intros.
revert H.
induction H0.
{ intros. eauto. }
{ intros.
eapply IHrightTrans; clear IHrightTrans.
specialize (H1 _ H). inversion H1.
intros. eapply H2. eapply RTFin. eassumption. }
Defined. | Theorem | wf_rightTrans | Recur | theories/Recur/Relation.v | [
"ExtLib.Relations.TransitiveClosure"
] | [
"rightTrans"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
wf_leftTrans : well_founded (leftTrans R). | Proof.
red.
eapply Fix. eapply wf_R. clear.
intros. constructor. intros.
revert H.
induction H0.
{ intros. eauto. }
{ intros.
eapply IHleftTrans; clear IHleftTrans.
intros. eapply H1. auto.
eapply LTFin. auto. }
Defined. | Theorem | wf_leftTrans | Recur | theories/Recur/Relation.v | [
"ExtLib.Relations.TransitiveClosure"
] | [
"leftTrans"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
compose (x z : T) : Prop | :=
exists y, R1 x y /\ R2 y z. | Definition | compose | Relations | theories/Relations/Compose.v | [] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
makeRefl (x : T) : T -> Prop | :=
| RRefl : makeRefl x x
| RStep : forall y, R x y -> makeRefl x y. | Inductive | makeRefl | Relations | theories/Relations/TransitiveClosure.v | [
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [] | Reflexivity * | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 |
Refl_makeRefl : Reflexive makeRefl. | Proof.
constructor.
Qed. | Instance | Refl_makeRefl | Relations | theories/Relations/TransitiveClosure.v | [
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [
"makeRefl"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Refl_makeTrans : Transitive R -> Transitive makeRefl. | Proof.
intro. intro. intros. inversion H0; clear H0; subst; auto.
inversion H1; clear H1; subst; auto using RStep.
apply RStep. etransitivity; eauto.
Qed. | Instance | Refl_makeTrans | Relations | theories/Relations/TransitiveClosure.v | [
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [
"apply",
"makeRefl"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
makeTrans (x y : T) : Prop | :=
| TStep : R x y -> makeTrans x y
| TTrans : forall z, makeTrans x z -> makeTrans z y -> makeTrans x y. | Inductive | makeTrans | Relations | theories/Relations/TransitiveClosure.v | [
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [] | Transitivity * | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 |
Trans_makeTrans : Transitive makeTrans. | Proof.
intro. intros; eapply TTrans; eassumption.
Qed. | Instance | Trans_makeTrans | Relations | theories/Relations/TransitiveClosure.v | [
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [
"makeTrans"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Trans_makeRefl : Reflexive R -> Reflexive makeTrans. | Proof.
intro. intro. apply TStep. reflexivity.
Qed. | Instance | Trans_makeRefl | Relations | theories/Relations/TransitiveClosure.v | [
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [
"apply",
"makeTrans"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
leftTrans (x y : T) : Prop | :=
| LTFin : R x y -> leftTrans x y
| LTStep : forall z, R x z -> leftTrans z y -> leftTrans x y. | Inductive | leftTrans | Relations | theories/Relations/TransitiveClosure.v | [
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
rightTrans (x y : T) : Prop | :=
| RTFin : R x y -> rightTrans x y
| RTStep : forall z, rightTrans x z -> R z y -> rightTrans x y. | Inductive | rightTrans | Relations | theories/Relations/TransitiveClosure.v | [
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
leftTrans_rightTrans_acc x y (l : leftTrans y x) : forall z, rightTrans z y -> rightTrans z x | :=
match l with
| LTFin pf => fun z pfR => RTStep pfR pf
| LTStep _ pf pfL => fun z pfR =>
leftTrans_rightTrans_acc pfL (RTStep pfR pf)
end. | Fixpoint | leftTrans_rightTrans_acc | Relations | theories/Relations/TransitiveClosure.v | [
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [
"leftTrans",
"rightTrans"
] | Equivalence of definitions of transitivity * | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 |
rightTrans_leftTrans_acc x y (l : rightTrans x y) : forall z, leftTrans y z -> leftTrans x z | :=
match l with
| RTFin pf => fun z pfR => LTStep pf pfR
| RTStep _ pf pfL => fun z pfR =>
rightTrans_leftTrans_acc pf (LTStep pfL pfR)
end. | Fixpoint | rightTrans_leftTrans_acc | Relations | theories/Relations/TransitiveClosure.v | [
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [
"leftTrans",
"rightTrans"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
leftTrans_rightTrans : forall x y,
leftTrans x y <-> rightTrans x y. | Proof.
split.
{ destruct 1. apply RTFin; assumption.
eapply leftTrans_rightTrans_acc. eassumption. eapply RTFin. eassumption. }
{ destruct 1. apply LTFin. assumption.
eapply rightTrans_leftTrans_acc. eassumption. eapply LTFin. eassumption. }
Qed. | Theorem | leftTrans_rightTrans | Relations | theories/Relations/TransitiveClosure.v | [
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [
"apply",
"leftTrans",
"leftTrans_rightTrans_acc",
"rightTrans",
"rightTrans_leftTrans_acc",
"split"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
leftTrans_makeTrans_acc x y (l : leftTrans x y) : makeTrans x y | :=
match l with
| LTFin pf => TStep pf
| LTStep _ pf pfL =>
TTrans (TStep pf) (leftTrans_makeTrans_acc pfL)
end. | Fixpoint | leftTrans_makeTrans_acc | Relations | theories/Relations/TransitiveClosure.v | [
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [
"leftTrans",
"makeTrans"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
leftTrans_trans x y (l : leftTrans x y) : forall z (r : leftTrans y z), leftTrans x z | :=
match l with
| LTFin pf => fun _ pfL => LTStep pf pfL
| LTStep _ pf pfL => fun _ pfR => LTStep pf (leftTrans_trans pfL pfR)
end. | Fixpoint | leftTrans_trans | Relations | theories/Relations/TransitiveClosure.v | [
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [
"leftTrans"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
makeTrans_leftTrans : forall s s',
makeTrans s s' <-> leftTrans s s'. | Proof.
split; intros.
{ induction H. eapply LTFin. eassumption.
eapply leftTrans_trans; eassumption. }
{ apply leftTrans_makeTrans_acc. assumption. }
Qed. | Theorem | makeTrans_leftTrans | Relations | theories/Relations/TransitiveClosure.v | [
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [
"apply",
"leftTrans",
"leftTrans_makeTrans_acc",
"leftTrans_trans",
"makeTrans",
"split"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
makeTrans_rightTrans : forall s s',
makeTrans s s' <-> rightTrans s s'. | Proof.
intros. etransitivity. apply makeTrans_leftTrans. apply leftTrans_rightTrans.
Qed. | Theorem | makeTrans_rightTrans | Relations | theories/Relations/TransitiveClosure.v | [
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [
"apply",
"leftTrans_rightTrans",
"makeTrans",
"makeTrans_leftTrans",
"rightTrans"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
RTStep_left : forall x y z : T, R x y -> rightTrans y z -> rightTrans x z. | intros. revert H. revert x.
induction H0.
{ intros. eapply RTStep. eapply RTFin. eassumption. eassumption. }
{ intros. eapply RTStep. eapply IHrightTrans. eassumption. eassumption. }
Defined. | Definition | RTStep_left | Relations | theories/Relations/TransitiveClosure.v | [
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [
"rightTrans"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
makeTrans_idem : forall s s',
makeTrans (makeTrans R) s s' <-> makeTrans R s s'. | Proof.
split.
{ induction 1; eauto using TTrans. }
{ eapply TStep. }
Qed. | Theorem | makeTrans_idem | Relations | theories/Relations/TransitiveClosure.v | [
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [
"makeTrans",
"split"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
makeTrans_makeRefl_comm : forall s s',
makeTrans (makeRefl R) s s' <-> makeRefl (makeTrans R) s s'. | Proof.
split.
{ induction 1;
repeat match goal with
| [ H : makeRefl _ _ _ |- _ ] => inversion H; clear H; subst
end; eauto using RRefl, RStep, TStep, TTrans. }
{ intros. inversion H; clear H; subst; auto. apply TStep. apply RRefl.
induction H0; eauto using RStep, TSt... | Theorem | makeTrans_makeRefl_comm | Relations | theories/Relations/TransitiveClosure.v | [
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [
"apply",
"makeRefl",
"makeTrans",
"split"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
makeRefl_idem : forall s s',
makeRefl (makeRefl R) s s' <-> makeRefl R s s'. | Proof.
split; inversion 1; subst; eauto using RStep, RRefl.
Qed. | Theorem | makeRefl_idem | Relations | theories/Relations/TransitiveClosure.v | [
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [
"makeRefl",
"split"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Applicative@{d c} (T : Type@{d} -> Type@{c}) | :=
{ pure : forall {A : Type@{d}}, A -> T A
; ap : forall {A B : Type@{d}}, T (A -> B) -> T A -> T B
}. | Class | Applicative | Structures | theories/Structures/Applicative.v | [
"ExtLib",
"Functor",
"ApplicativeNotation"
] | [
"pure"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
"f <*> x" | := (ap f x) (at level 52, left associativity). | Notation | f <*> x | Structures | theories/Structures/Applicative.v | [
"ExtLib",
"Functor",
"ApplicativeNotation"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
liftA@{d c} {T : Type@{d} -> Type@{c}} {AT:Applicative@{d c} T} {A B : Type@{d}} (f:A -> B) (aT:T A) : T B | := pure f <*> aT. | Definition | liftA | Structures | theories/Structures/Applicative.v | [
"ExtLib",
"Functor",
"ApplicativeNotation"
] | [
"Applicative",
"pure"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
liftA2@{d c} {T : Type@{d} -> Type@{c}} {AT:Applicative@{d c} T} {A B C : Type@{d}} (f:A -> B -> C) (aT:T A) (bT:T B) : T C | := liftA f aT <*> bT. | Definition | liftA2 | Structures | theories/Structures/Applicative.v | [
"ExtLib",
"Functor",
"ApplicativeNotation"
] | [
"Applicative",
"liftA"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Functor_Applicative@{} : Functor T | :=
{ fmap := @liftA _ _ }. | Instance | Functor_Applicative | Structures | theories/Structures/Applicative.v | [
"ExtLib",
"Functor",
"ApplicativeNotation"
] | [
"Functor",
"liftA"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
LeftUnit : Type | :=
lunit : forall a, equ (op u a) a. | Class | LeftUnit | Structures | theories/Structures/BinOps.v | [] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
RightUnit : Type | :=
runit : forall a, equ (op a u) a. | Class | RightUnit | Structures | theories/Structures/BinOps.v | [] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Commutative : Type | :=
commut : forall a b, equ (op a b) (op b a). | Class | Commutative | Structures | theories/Structures/BinOps.v | [] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Associative : Type | :=
assoc : forall a b c, equ (op (op a b) c) (op a (op b c)). | Class | Associative | Structures | theories/Structures/BinOps.v | [] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
CoFunctor@{d c} (F : Type@{d} -> Type@{c}) : Type | :=
{ cofmap : forall {A B : Type@{d}}, (B -> A) -> F A -> F B }. | Class | CoFunctor | Structures | theories/Structures/CoFunctor.v | [
"ExtLib.Core.Any"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
CoPFunctor@{d c p} (F : Type@{d} -> Type@{c}) : Type | :=
{ CoFunP : Type@{d} -> Type@{p}
; copfmap : forall {A B : Type@{d}} {P : CoFunP B}, (B -> A) -> F A -> F B
}. | Class | CoPFunctor | Structures | theories/Structures/CoFunctor.v | [
"ExtLib.Core.Any"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
CoPFunctor_From_CoFunctor@{d c p} (F : Type@{d} -> Type@{c}) (F_ : CoFunctor@{d c} F) : CoPFunctor@{d c p} F | :=
{| CoFunP := Any@{p}
; copfmap := fun _ _ _ f x => cofmap f x
|}. | Instance | CoPFunctor_From_CoFunctor | Structures | theories/Structures/CoFunctor.v | [
"ExtLib.Core.Any"
] | [
"Any",
"CoFunctor",
"CoPFunctor"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
CoMonad (m : Type -> Type) : Type | :=
{ extract : forall {A}, m A -> A
; extend : forall {A B}, (m A -> B) -> m A -> m B
}. | Class | CoMonad | Structures | theories/Structures/CoMonad.v | [] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
coret {A: Type} | := extract (A:=A). | Definition | coret | Structures | theories/Structures/CoMonad.v | [] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
cobind {A B: Type} | := extend (A:=A) (B:=B). | Definition | cobind | Structures | theories/Structures/CoMonad.v | [] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
CoMonadLaws | :=
{
extend_extract: forall (A B:Type),
extend (B:=A) extract = id ;
extract_extend: forall (A B:Type) {f},
extract ∘ extend (A:=A) (B:=B) f = f;
extend_extend:forall (A B:Type) {f g},
extend (A:=B) (B:=A) f ∘ extend (A:=A) g = extend (f ∘ extend g)
}. | Class | CoMonadLaws | Structures | theories/Structures/CoMonadLaws.v | [
"Coq.Program.Basics",
"ExtLib.Structures.CoMonad"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
UIP_refl : forall {x : A} (p1 : x = x), p1 = refl_equal _. | intros.
eapply Eqdep_dec.UIP_dec. apply EquivDec.equiv_dec.
Qed. | Theorem | UIP_refl | Structures | theories/Structures/EqDep.v | [
"Coq.Logic",
"Eqdep_dec",
"EquivDec",
"ExtLib.Core.RelDec",
"ExtLib.Tactics.Consider"
] | [
"apply"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
UIP_equal : forall {x y : A} (p1 p2 : x = y), p1 = p2. | eapply Eqdep_dec.UIP_dec. apply EquivDec.equiv_dec.
Qed. | Theorem | UIP_equal | Structures | theories/Structures/EqDep.v | [
"Coq.Logic",
"Eqdep_dec",
"EquivDec",
"ExtLib.Core.RelDec",
"ExtLib.Tactics.Consider"
] | [
"apply"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
inj_pair2 :
forall (P:A -> Type) (p:A) (x y:P p),
existT P p x = existT P p y -> x = y. | Proof.
intros. eapply Eqdep_dec.inj_pair2_eq_dec; auto.
Qed. | Lemma | inj_pair2 | Structures | theories/Structures/EqDep.v | [
"Coq.Logic",
"Eqdep_dec",
"EquivDec",
"ExtLib.Core.RelDec",
"ExtLib.Tactics.Consider"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
equiv_dec_refl_left : forall a, @EquivDec.equiv_dec _ _ _ dec a a = left eq_refl. | Proof.
intros. destruct (EquivDec.equiv_dec a a); try congruence.
f_equal. apply UIP_equal.
Qed. | Theorem | equiv_dec_refl_left | Structures | theories/Structures/EqDep.v | [
"Coq.Logic",
"Eqdep_dec",
"EquivDec",
"ExtLib.Core.RelDec",
"ExtLib.Tactics.Consider"
] | [
"UIP_equal",
"apply"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
EqDec_RelDec : EquivDec.EqDec T (@eq T). | Proof.
red; intros.
consider (x ?[ eq ] y); intros; subst; auto.
left. reflexivity.
Qed. | Instance | EqDec_RelDec | Structures | theories/Structures/EqDep.v | [
"Coq.Logic",
"Eqdep_dec",
"EquivDec",
"ExtLib.Core.RelDec",
"ExtLib.Tactics.Consider"
] | [
"consider"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Foldable : Type | :=
{ fold_mon : forall m {M : Monoid m}, (A -> m) -> T -> m }. | Class | Foldable | Structures | theories/Structures/Foldable.v | [
"Coq.Lists.List",
"ExtLib.Structures.Monoid"
] | [
"Monoid"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
fold (R : Type) (f : A -> R -> R) (init : R) (s : T) : R | :=
@fold_mon Foldable_T (R -> R)
{| monoid_plus := fun f g x => f (g x)
; monoid_unit := fun x => x
|} f s init. | Definition | fold | Structures | theories/Structures/Foldable.v | [
"Coq.Lists.List",
"ExtLib.Structures.Monoid"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
toList : T -> list A | :=
fold_mon (M := {| monoid_plus := @List.app A
; monoid_unit := nil |}) (fun x => x :: nil). | Definition | toList | Structures | theories/Structures/Foldable.v | [
"Coq.Lists.List",
"ExtLib.Structures.Monoid"
] | [
"app"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
FoldableOk | :=
{ fold_ind : forall m (M : Monoid m) (ML : MonoidLaws M) (P : m -> Prop) f u,
P (monoid_unit M) ->
(forall x y z,
Add x y z ->
P (@fold_mon Foldable_T m M f y) ->
P (monoid_plus M (f x) (@fold_mon Foldable_T m M f z))) ->... | Class | FoldableOk | Structures | theories/Structures/Foldable.v | [
"Coq.Lists.List",
"ExtLib.Structures.Monoid"
] | [
"Monoid",
"MonoidLaws"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Functor@{d c} (F : Type@{d} -> Type@{c}) : Type | :=
{ fmap : forall {A B : Type@{d}}, (A -> B) -> F A -> F B }. | Class | Functor | Structures | theories/Structures/Functor.v | [
"ExtLib.Core.Any"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
ID@{d} {T : Type@{d}} (f : T -> T) : Prop | :=
forall x : T, f x = x. | Definition | ID | Structures | theories/Structures/Functor.v | [
"ExtLib.Core.Any"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
"f <$> x" | := (@fmap _ _ _ _ f x) (at level 52, left associativity). | Notation | f <$> x | Structures | theories/Structures/Functor.v | [
"ExtLib.Core.Any"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
FunctorLaws {F} (Functor_F : Functor F) | :=
{ fmap_id : forall {T} (x : F T), fmap id x = x
; fmap_compose : forall {T U V} (f : T -> U) (g : U -> V) (x : F T),
fmap (compose g f) x = fmap g (fmap f x)
}. | Class | FunctorLaws | Structures | theories/Structures/FunctorLaws.v | [
"Coq.Relations",
"Relations",
"ExtLib.Data.Fun",
"ExtLib.Structures.Functor"
] | [
"Functor",
"compose"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
IxMonad@{d c} (m : Type@{d} -> Type@{d} -> Type@{c} -> Type@{c}) : Type | :=
{ ret : forall {i : Type@{d}} {a : Type@{d}}, a -> m i i a
; bind : forall {i j k a b: Type@{d}}, m i j a -> (a -> m j k b) -> m i k b
}. | Class | IxMonad | Structures | theories/Structures/IXMonad.v | [
"ExtLib.Structures.Monad",
"Applicative"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
"c >>= f" | := (@bind _ _ _ _ _ _ _ c f) (at level 50, left associativity) : ixmonad_scope. | Notation | c >>= f | Structures | theories/Structures/IXMonad.v | [
"ExtLib.Structures.Monad",
"Applicative"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
"f =<< c" | := (@bind _ _ _ _ _ _ _ c f) (at level 51, right associativity) : ixmonad_scope. | Notation | f =<< c | Structures | theories/Structures/IXMonad.v | [
"ExtLib.Structures.Monad",
"Applicative"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
"x <- c1 ;; c2" | := (@bind _ _ _ _ _ _ _ c1 (fun x => c2))
(at level 100, c1 at next level, right associativity) : ixmonad_scope. | Notation | x <- c1 ;; c2 | Structures | theories/Structures/IXMonad.v | [
"ExtLib.Structures.Monad",
"Applicative"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
"e1 ;; e2" | := (_ <- e1%ixmonad ;; e2%ixmonad)%ixmonad
(at level 100, right associativity) : ixmonad_scope. | Notation | e1 ;; e2 | Structures | theories/Structures/IXMonad.v | [
"ExtLib.Structures.Monad",
"Applicative"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
"' pat <- c1 ;; c2" | :=
(@bind _ _ _ _ _ _ _ c1 (fun x => match x with pat => c2 end))
(at level 100, pat pattern, c1 at next level, right associativity) : monad_scope. | Notation | ' pat <- c1 ;; c2 | Structures | theories/Structures/IXMonad.v | [
"ExtLib.Structures.Monad",
"Applicative"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Applicative_Monad {m i} {M:IxMonad m}
: Applicative (m i i) | :=
{| Applicative.pure := fun (A : Type) (X : A) => ret X
; Applicative.ap := fun (A B : Type) (X : m i i (A -> B)) (X0 : m i i A) => bind X (fun X1 : A -> B => bind X0 (fun X2 : A => ret (X1 X2)))
|}. | Instance | Applicative_Monad | Structures | theories/Structures/IXMonad.v | [
"ExtLib.Structures.Monad",
"Applicative"
] | [
"Applicative",
"IxMonad",
"pure"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Map : Type | :=
{ empty : map
; add : K -> V -> map -> map
; remove : K -> map -> map
; lookup : K -> map -> option V
; union : map -> map -> map
}. | Class | Map | Structures | theories/Structures/Maps.v | [
"RelationClasses",
"ExtLib.Structures.Monad",
"ExtLib.Structures.Reducible"
] | [
"empty",
"remove"
] | General Maps * | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 |
MapOk (M : Map) : Type | :=
{ mapsto : K -> V -> map -> Prop
; mapsto_empty : forall k v, ~mapsto k v empty
; mapsto_lookup : forall k v m, lookup k m = Some v <-> mapsto k v m
; mapsto_add_eq : forall m k v, mapsto k v (add k v m)
; mapsto_add_neq : forall m k v k',
~R k k' ->
forall v',... | Class | MapOk | Structures | theories/Structures/Maps.v | [
"RelationClasses",
"ExtLib.Structures.Monad",
"ExtLib.Structures.Reducible"
] | [
"Map",
"empty",
"remove"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
contains (k : K) (m : map) : bool | :=
match lookup k m with
| None => false
| Some _ => true
end. | Definition | contains | Structures | theories/Structures/Maps.v | [
"RelationClasses",
"ExtLib.Structures.Monad",
"ExtLib.Structures.Reducible"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
singleton (k : K) (v : V) : map | :=
add k v empty. | Definition | singleton | Structures | theories/Structures/Maps.v | [
"RelationClasses",
"ExtLib.Structures.Monad",
"ExtLib.Structures.Reducible"
] | [
"empty"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
combine (f : K -> V -> V -> V) (m1 m2 : map) : map | :=
fold (fun k_v acc =>
let '(k,v) := k_v in
match lookup k acc with
| None => add k v acc
| Some v' => add k (f k v v') acc
end) m2 m1. | Definition | combine | Structures | theories/Structures/Maps.v | [
"RelationClasses",
"ExtLib.Structures.Monad",
"ExtLib.Structures.Reducible"
] | [
"fold"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
filter (f : K -> V -> bool) (m : map) : map | :=
fold (fun k_v acc =>
let '(k,v) := k_v in
if f k v
then add k v acc
else acc) empty m. | Definition | filter | Structures | theories/Structures/Maps.v | [
"RelationClasses",
"ExtLib.Structures.Monad",
"ExtLib.Structures.Reducible"
] | [
"empty",
"fold"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
submap_with (le : V -> V -> bool) (m1 m2 : map) : bool | :=
fold (fun k_v (acc : bool) =>
if acc then
let '(k,v) := k_v in
match lookup k m2 with
| None => false
| Some v' => le v v'
end
else false) true m1. | Definition | submap_with | Structures | theories/Structures/Maps.v | [
"RelationClasses",
"ExtLib.Structures.Monad",
"ExtLib.Structures.Reducible"
] | [
"fold"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Monad@{d c} (m : Type@{d} -> Type@{c}) : Type | :=
{ ret : forall {t : Type@{d}}, t -> m t
; bind : forall {t u : Type@{d}}, m t -> (t -> m u) -> m u
}. | Class | Monad | Structures | theories/Structures/Monad.v | [
"ExtLib.Structures.Functor",
"ExtLib.Structures.Applicative",
"MonadBaseNotation"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
liftM@{d c}
{m : Type@{d} -> Type@{c}}
{M : Monad m}
{T U : Type@{d}} (f : T -> U) : m T -> m U | :=
fun x => bind x (fun x => ret (f x)). | Definition | liftM | Structures | theories/Structures/Monad.v | [
"ExtLib.Structures.Functor",
"ExtLib.Structures.Applicative",
"MonadBaseNotation"
] | [
"Monad"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
liftM2@{d c}
{m : Type@{d} -> Type@{c}}
{M : Monad m}
{T U V : Type@{d}} (f : T -> U -> V) : m T -> m U -> m V | :=
Eval cbv beta iota zeta delta [ liftM ] in
fun x y => bind x (fun x => liftM (f x) y). | Definition | liftM2 | Structures | theories/Structures/Monad.v | [
"ExtLib.Structures.Functor",
"ExtLib.Structures.Applicative",
"MonadBaseNotation"
] | [
"Monad",
"liftM"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
liftM3@{d c}
{m : Type@{d} -> Type@{c}}
{M : Monad m}
{T U V W : Type@{d}} (f : T -> U -> V -> W) : m T -> m U -> m V -> m W | :=
Eval cbv beta iota zeta delta [ liftM2 ] in
fun x y z => bind x (fun x => liftM2 (f x) y z). | Definition | liftM3 | Structures | theories/Structures/Monad.v | [
"ExtLib.Structures.Functor",
"ExtLib.Structures.Applicative",
"MonadBaseNotation"
] | [
"Monad",
"liftM2"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
apM@{d c}
{m : Type@{d} -> Type@{c}}
{M : Monad m}
{A B : Type@{d}} (fM:m (A -> B)) (aM:m A) : m B | :=
bind fM (fun f => liftM f aM). | Definition | apM | Structures | theories/Structures/Monad.v | [
"ExtLib.Structures.Functor",
"ExtLib.Structures.Applicative",
"MonadBaseNotation"
] | [
"Monad",
"liftM"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
mcompose@{c d}
{m:Type@{d}->Type@{c}}
{M: Monad m}
{T U V:Type@{d}}
(f: T -> m U) (g: U -> m V): (T -> m V) | :=
fun x => bind (f x) g. | Definition | mcompose | Structures | theories/Structures/Monad.v | [
"ExtLib.Structures.Functor",
"ExtLib.Structures.Applicative",
"MonadBaseNotation"
] | [
"Monad"
] | Left-to-right composition of Kleisli arrows. | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 |
join@{d c} {m : Type@{d} -> Type@{c}} {a} `{Monad m} : m (m a) -> m a | :=
fun x => bind x (fun y => y). | Definition | join | Structures | theories/Structures/Monad.v | [
"ExtLib.Structures.Functor",
"ExtLib.Structures.Applicative",
"MonadBaseNotation"
] | [
"Monad"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
"c >>= f" | := (@bind _ _ _ _ c f) (at level 58, left associativity) : monad_scope. | Notation | c >>= f | Structures | theories/Structures/Monad.v | [
"ExtLib.Structures.Functor",
"ExtLib.Structures.Applicative",
"MonadBaseNotation"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
"f =<< c" | := (@bind _ _ _ _ c f) (at level 61, right associativity) : monad_scope. | Notation | f =<< c | Structures | theories/Structures/Monad.v | [
"ExtLib.Structures.Functor",
"ExtLib.Structures.Applicative",
"MonadBaseNotation"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
"f >=> g" | := (@mcompose _ _ _ _ _ f g) (at level 61, right associativity) : monad_scope. | Notation | f >=> g | Structures | theories/Structures/Monad.v | [
"ExtLib.Structures.Functor",
"ExtLib.Structures.Applicative",
"MonadBaseNotation"
] | [
"mcompose"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
"e1 ;; e2" | := (@bind _ _ _ _ e1%monad (fun _ => e2%monad))%monad
(at level 61, right associativity) : monad_scope. | Notation | e1 ;; e2 | Structures | theories/Structures/Monad.v | [
"ExtLib.Structures.Functor",
"ExtLib.Structures.Applicative",
"MonadBaseNotation"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
"x <- c1 ;; c2" | := (@bind _ _ _ _ c1 (fun x => c2))
(at level 61, c1 at next level, right associativity) : monad_scope. | Notation | x <- c1 ;; c2 | Structures | theories/Structures/Monad.v | [
"ExtLib.Structures.Functor",
"ExtLib.Structures.Applicative",
"MonadBaseNotation"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 |
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