Proof Assistant Projects
Collection
Digesting proof assistant libraries for AI ingestion. • 103 items • Updated • 3
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 |
|---|---|---|---|---|---|---|---|---|---|---|
value : Type | :=
| Int : nat -> value
| Bool : bool -> value. | Inductive | value | examples | examples/EvalWithExc.v | [
"Coq.Strings.String",
"ExtLib.Structures.Monads",
"ExtLib.Data.Monads.EitherMonad",
"ExtLib.Data.String",
"MonadNotation"
] | [] | Syntax and values of a simple language * | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 |
exp : Type | :=
| ConstI : nat -> exp
| ConstB : bool -> exp
| Plus : exp -> exp -> exp
| If : exp -> exp -> exp -> exp. | Inductive | exp | examples | examples/EvalWithExc.v | [
"Coq.Strings.String",
"ExtLib.Structures.Monads",
"ExtLib.Data.Monads.EitherMonad",
"ExtLib.Data.String",
"MonadNotation"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
asInt (v : value) : m nat | :=
match v with
| Int n => ret n
| _ =>
(** if we don't have an integer, signal an error using
** [raise] from the MoandExc instance
**)
raise ("expected integer got bool")%string
end. | Definition | asInt | examples | examples/EvalWithExc.v | [
"Coq.Strings.String",
"ExtLib.Structures.Monads",
"ExtLib.Data.Monads.EitherMonad",
"ExtLib.Data.String",
"MonadNotation"
] | [
"value"
] | Functions that get [nat] or [bool] values from a [value] * | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 |
asBool (v : value) : m bool | :=
match v with
| Bool b => ret b
| _ => raise ("expected bool got integer")%string
end. | Definition | asBool | examples | examples/EvalWithExc.v | [
"Coq.Strings.String",
"ExtLib.Structures.Monads",
"ExtLib.Data.Monads.EitherMonad",
"ExtLib.Data.String",
"MonadNotation"
] | [
"value"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
eval' (e : exp) : m value | :=
match e with
(** when there is no error, we can just return (i.e. [ret])
** the answer
**)
| ConstI i => ret (Int i)
| ConstB b => ret (Bool b)
| Plus l r =>
(** evaluate the sub-terms to numbers **)
l <- eval' l ;;
l <- asInt l ;;
r <- ... | Fixpoint | eval' | examples | examples/EvalWithExc.v | [
"Coq.Strings.String",
"ExtLib.Structures.Monads",
"ExtLib.Data.Monads.EitherMonad",
"ExtLib.Data.String",
"MonadNotation"
] | [
"asBool",
"asInt",
"exp",
"value"
] | The main evaluator routine returns a [value], but since we are
** working in the [m] monad, we return [m value]
* | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 |
eval : exp -> string + value | :=
eval' (m := sum string). | Definition | eval | examples | examples/EvalWithExc.v | [
"Coq.Strings.String",
"ExtLib.Structures.Monads",
"ExtLib.Data.Monads.EitherMonad",
"ExtLib.Data.String",
"MonadNotation"
] | [
"eval'",
"exp",
"value"
] | Wrap the [eval] function up with the monad instance that we
** want to use
* | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 |
update1 : istate A B unit | :=
modify_ function1. | Definition | update1 | examples | examples/indexedstate.v | [
"ExtLib.Data.Monads.IStateMonad",
"ExtLib.Structures.IXMonad",
"IxMonadNotation"
] | [
"istate",
"modify_"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
update2 : istate B C unit | :=
modify_ function2. | Definition | update2 | examples | examples/indexedstate.v | [
"ExtLib.Data.Monads.IStateMonad",
"ExtLib.Structures.IXMonad",
"IxMonadNotation"
] | [
"istate",
"modify_"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
compose : istate A C unit | :=
update1 ;;
update2. | Definition | compose | examples | examples/indexedstate.v | [
"ExtLib.Data.Monads.IStateMonad",
"ExtLib.Structures.IXMonad",
"IxMonadNotation"
] | [
"istate",
"update1",
"update2"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
repeatM `{Monad M} (n : nat) `(x : A) (p : A -> M A) : M unit | :=
match n with
| O => ret tt
| S n => y <- p x;;
repeatM n y p
end. | Fixpoint | repeatM | examples | examples/Notations.v | [
"ExtLib.Structures.Monad",
"MonadNotation",
"MonadLetNotation"
] | [
"Monad"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
repeatM `{Monad M} (n : nat) `(x : A) (p : A -> M A) : M unit | :=
match n with
| O => ret tt
| S n => let* y := p x in
repeatM n y p
end. | Fixpoint | repeatM | examples | examples/Notations.v | [
"ExtLib.Structures.Monad",
"MonadNotation",
"MonadLetNotation"
] | [
"Monad"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
PrinterMonad : Type -> Type | :=
writerT (@show_mon _ ShowScheme_string_compose) ident. | Definition | PrinterMonad | examples | examples/Printing.v | [
"Coq.Strings.String",
"ExtLib.Structures.MonadWriter",
"ExtLib.Data.PPair",
"ExtLib.Data.Monads.WriterMonad",
"ExtLib.Data.Monads.IdentityMonad",
"ExtLib.Programming.Show"
] | [
"ShowScheme_string_compose",
"ident",
"writerT"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
print {T : Type} {ST : Show T} (val : T) : PrinterMonad unit | :=
@MonadWriter.tell _ (@show_mon _ ShowScheme_string_compose) _ _
(@show _ ST val _ show_inj (@show_mon _ ShowScheme_string_compose)). | Definition | print | examples | examples/Printing.v | [
"Coq.Strings.String",
"ExtLib.Structures.MonadWriter",
"ExtLib.Data.PPair",
"ExtLib.Data.Monads.WriterMonad",
"ExtLib.Data.Monads.IdentityMonad",
"ExtLib.Programming.Show"
] | [
"MonadWriter",
"PrinterMonad",
"Show",
"ShowScheme_string_compose"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
printString (str : string) : PrinterMonad unit | :=
@MonadWriter.tell _ (@show_mon _ ShowScheme_string_compose) _ _
(@show_exact str _ show_inj (@show_mon _ ShowScheme_string_compose)). | Definition | printString | examples | examples/Printing.v | [
"Coq.Strings.String",
"ExtLib.Structures.MonadWriter",
"ExtLib.Data.PPair",
"ExtLib.Data.Monads.WriterMonad",
"ExtLib.Data.Monads.IdentityMonad",
"ExtLib.Programming.Show"
] | [
"MonadWriter",
"PrinterMonad",
"ShowScheme_string_compose",
"show_exact"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
runPrinter {T : Type} (c : PrinterMonad T) : T * string | :=
let '(ppair val str) := unIdent (runWriterT c) in
(val, str ""%string). | Definition | runPrinter | examples | examples/Printing.v | [
"Coq.Strings.String",
"ExtLib.Structures.MonadWriter",
"ExtLib.Data.PPair",
"ExtLib.Data.Monads.WriterMonad",
"ExtLib.Data.Monads.IdentityMonad",
"ExtLib.Programming.Show"
] | [
"PrinterMonad"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
GameValue : Type | := Z. | Definition | GameValue | examples | examples/StateGame.v | [
"Coq.ZArith.ZArith_base",
"Coq.Strings.String",
"Coq.Strings.Ascii",
"ExtLib.Data.Monads.StateMonad",
"ExtLib.Structures.Monads",
"MonadNotation"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
GameState : Type | := (prod bool Z). | Definition | GameState | examples | examples/StateGame.v | [
"Coq.ZArith.ZArith_base",
"Coq.Strings.String",
"Coq.Strings.Ascii",
"ExtLib.Data.Monads.StateMonad",
"ExtLib.Structures.Monads",
"MonadNotation"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
playGame (s: string) {struct s}: m GameValue | :=
match s with
| EmptyString =>
v <- get ;;
let '(on, score) := v in ret score
| String x xs =>
v <- get ;;
let '(on, score) := v in
match x, on with
| "a", true => put (on, score + 1)
| "b", true => put (on, score - 1)
| "c", _ => pu... | Fixpoint | playGame | examples | examples/StateGame.v | [
"Coq.ZArith.ZArith_base",
"Coq.Strings.String",
"Coq.Strings.Ascii",
"ExtLib.Data.Monads.StateMonad",
"ExtLib.Structures.Monads",
"MonadNotation"
] | [
"GameValue",
"get",
"put"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
startState: GameState | := (false, 0). | Definition | startState | examples | examples/StateGame.v | [
"Coq.ZArith.ZArith_base",
"Coq.Strings.String",
"Coq.Strings.Ascii",
"ExtLib.Data.Monads.StateMonad",
"ExtLib.Structures.Monads",
"MonadNotation"
] | [
"GameState"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
main : GameValue | :=
(@evalState GameState GameValue (playGame (state GameState) "abcaaacbbcabbab") startState). | Definition | main | examples | examples/StateGame.v | [
"Coq.ZArith.ZArith_base",
"Coq.Strings.String",
"Coq.Strings.Ascii",
"ExtLib.Data.Monads.StateMonad",
"ExtLib.Structures.Monads",
"MonadNotation"
] | [
"GameState",
"GameValue",
"evalState",
"playGame",
"startState",
"state"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
foo : stateT unit option unit | :=
ret tt. | Definition | foo | examples | examples/StateTMonad.v | [
"ExtLib",
"Monad",
"OptionMonad",
"StateMonad"
] | [
"stateT"
] | Now the definition succeeds | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 |
contains_both (v1 v2 : V) (s : set) : bool | :=
contains v1 s && contains v2 s. | Definition | contains_both | examples | examples/UsingSets.v | [
"Coq.Bool.Bool",
"ExtLib.Structures.Sets",
"ExtLib.Structures.Reducible",
"ExtLib.Structures.Functor",
"ExtLib.Data.Set.ListSet",
"ExtLib.ExtLib"
] | [
"contains"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
toList (s : set) : list V | :=
fold (@cons _) nil s. | Definition | toList | examples | examples/UsingSets.v | [
"Coq.Bool.Bool",
"ExtLib.Structures.Sets",
"ExtLib.Structures.Reducible",
"ExtLib.Structures.Functor",
"ExtLib.Data.Set.ListSet",
"ExtLib.ExtLib"
] | [
"fold"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
RTest : Set | := mkRTest {
a : bool ; b : nat ; c : bool
}. | Record | RTest | examples | examples/WithDemo.v | [
"List",
"ExtLib.Programming.With"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Struct_RTest : Struct RTest | := {
fields := ((@existT _ _ _ a) :: (@existT _ _ _ b) :: (@existT _ _ _ c):: nil) ;
ctor := mkRTest
}. | Instance | Struct_RTest | examples | examples/WithDemo.v | [
"List",
"ExtLib.Programming.With"
] | [
"RTest",
"Struct"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Acc_RTest_a : Accessor a | := { acc := Here }. | Instance | Acc_RTest_a | examples | examples/WithDemo.v | [
"List",
"ExtLib.Programming.With"
] | [
"Accessor"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Acc_RTest_b : Accessor b | := { acc := Next Here }. | Instance | Acc_RTest_b | examples | examples/WithDemo.v | [
"List",
"ExtLib.Programming.With"
] | [
"Accessor"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Acc_RTest_c : Accessor c | := { acc := Next (Next Here) }. | Instance | Acc_RTest_c | examples | examples/WithDemo.v | [
"List",
"ExtLib.Programming.With"
] | [
"Accessor"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
compose (A B C : Type) (f : A -> B) (g : B -> C) : A -> C | :=
fun x => g (f x). | Definition | compose | scratch | scratch/FunctorFromMonad.v | [
"Relations",
"ExtLib.Data.Fun",
"ExtLib.Structures.Proper",
"ExtLib.Structures.Monad",
"ExtLib.Structures.FunctorRelations",
"ExtLib.Structures.MonadLaws"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
pure (T : Type) : T -> m T | := @ret _ _ _. | Definition | pure | scratch | scratch/FunctorFromMonad.v | [
"Relations",
"ExtLib.Data.Fun",
"ExtLib.Structures.Proper",
"ExtLib.Structures.Monad",
"ExtLib.Structures.FunctorRelations",
"ExtLib.Structures.MonadLaws"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
fapply (T U : Type) (f : m (T -> U)) (x : m T) : m U | :=
bind f (fun f => bind x (fun x => ret (f x))). | Definition | fapply | scratch | scratch/FunctorFromMonad.v | [
"Relations",
"ExtLib.Data.Fun",
"ExtLib.Structures.Proper",
"ExtLib.Structures.Monad",
"ExtLib.Structures.FunctorRelations",
"ExtLib.Structures.MonadLaws"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
fun_app_proper (A B : Type) (rA : relation A) (rB : relation B) (pA : Proper rA) (pB : Proper rB) (f : A -> B) x :
proper f -> proper x ->
proper (f x). | Proof.
intros. apply H. auto.
Qed. | Instance | fun_app_proper | scratch | scratch/FunctorFromMonad.v | [
"Relations",
"ExtLib.Data.Fun",
"ExtLib.Structures.Proper",
"ExtLib.Structures.Monad",
"ExtLib.Structures.FunctorRelations",
"ExtLib.Structures.MonadLaws"
] | [
"apply"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
fun_abs (A B : Type) (rA : relation A) (rB : relation B) (pA : Proper rA) (pB : Proper rB) (f : A -> B) :
(forall x, proper x -> proper (f x)) ->
(forall x y, proper x -> proper y -> rA x y -> rB (f x) (f y)) ->
proper (fun x => f x). | Proof.
intros. split; auto; eapply H.
Qed. | Instance | fun_abs | scratch | scratch/FunctorFromMonad.v | [
"Relations",
"ExtLib.Data.Fun",
"ExtLib.Structures.Proper",
"ExtLib.Structures.Monad",
"ExtLib.Structures.FunctorRelations",
"ExtLib.Structures.MonadLaws"
] | [
"split"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
prove_proper x k | :=
match x with
| _ => match goal with
| [ H : proper x |- _ ] => k H
end
| bind ?A ?B =>
prove_proper A ltac:(fun a => prove_proper B ltac:(fun b =>
let H := fresh in
assert (H : proper x); [ eapply bind_proper; eauto with typeclass_instances |... | Ltac | prove_proper | scratch | scratch/FunctorFromMonad.v | [
"Relations",
"ExtLib.Data.Fun",
"ExtLib.Structures.Proper",
"ExtLib.Structures.Monad",
"ExtLib.Structures.FunctorRelations",
"ExtLib.Structures.MonadLaws"
] | [
"apply",
"assert",
"fun_abs"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
PReflexive_stuff : PReflexive
(pfun_ext (pfun_ext (pfun_ext rC pA) (Proper_pfun pB pC))
(Proper_pfun pA pB)). | Proof. intuition. Qed. | Instance | PReflexive_stuff | scratch | scratch/FunctorFromMonad.v | [
"Relations",
"ExtLib.Data.Fun",
"ExtLib.Structures.Proper",
"ExtLib.Structures.Monad",
"ExtLib.Structures.FunctorRelations",
"ExtLib.Structures.MonadLaws"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
bind_law : forall (f : A -> B) (g : B -> C),
proper f -> proper g ->
mleq (pfun_ext rC pA)
(fapply (fapply (pure (@compose A B C)) (pure f)) (pure g))
(pure (compose f g)). | Proof.
unfold fapply, pure, compose; simpl; intros.
propers.
(eapply ptransitive; [ | | | | eapply (@bind_associativity _ _ _ _ MonadLaws_mleq) | ]); eauto with typeclass_instances; propers.
(eapply ptransitive; [ | | | | eapply (@bind_of_return _ _ _ _ MonadLaws_mleq) | ]); eauto with typeclass_instan... | Theorem | bind_law | scratch | scratch/FunctorFromMonad.v | [
"Relations",
"ExtLib.Data.Fun",
"ExtLib.Structures.Proper",
"ExtLib.Structures.Monad",
"ExtLib.Structures.FunctorRelations",
"ExtLib.Structures.MonadLaws"
] | [
"compose",
"fapply",
"pure"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Any (T : Type) : Prop. | Class | Any | Core | theories/Core/Any.v | [] | [] | This class should be used when no requirements are needed * | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Any_a (T : Type) : Any T | := {}. | Instance | Any_a | Core | theories/Core/Any.v | [] | [
"Any"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
RESOLVE (T : Type) : Type | := T. | Definition | RESOLVE | Core | theories/Core/Any.v | [] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
CmpDec (T : Type) (equ : T -> T -> Prop) (ltu : T -> T -> Prop) : Type | :=
{ cmp_dec : T -> T -> comparison }. | Class | CmpDec | Core | theories/Core/CmpDec.v | [
"Coq.Bool.Bool",
"Coq.Classes.RelationClasses",
"ExtLib.Tactics.Consider"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
CmpDec_Correct T (equ ltu : T -> T -> Prop) (ED : CmpDec equ ltu) : Prop | :=
{ cmp_dec_correct : forall x y : T,
match cmp_dec x y with
| Eq => equ x y
| Lt => ltu x y
| Gt => ltu y x
end }. | Class | CmpDec_Correct | Core | theories/Core/CmpDec.v | [
"Coq.Bool.Bool",
"Coq.Classes.RelationClasses",
"ExtLib.Tactics.Consider"
] | [
"CmpDec"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
cmp_case (P Q R : Prop) : comparison -> Prop | :=
| CaseEq : P -> cmp_case P Q R Eq
| CaseLt : Q -> cmp_case P Q R Lt
| CaseGt : R -> cmp_case P Q R Gt. | Inductive | cmp_case | Core | theories/Core/CmpDec.v | [
"Coq.Bool.Bool",
"Coq.Classes.RelationClasses",
"ExtLib.Tactics.Consider"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
eq_pair (a b : T * U) : Prop | :=
eqt (fst a) (fst b) /\ equ (snd a) (snd b). | Definition | eq_pair | Core | theories/Core/CmpDec.v | [
"Coq.Bool.Bool",
"Coq.Classes.RelationClasses",
"ExtLib.Tactics.Consider"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
lt_pair (a b : T * U) : Prop | :=
ltt (fst a) (fst b) \/ (eqt (fst a) (fst b) /\ ltu (snd a) (snd b)). | Definition | lt_pair | Core | theories/Core/CmpDec.v | [
"Coq.Bool.Bool",
"Coq.Classes.RelationClasses",
"ExtLib.Tactics.Consider"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
CmpDec_pair : CmpDec eq_pair lt_pair | :=
{ cmp_dec := fun a b =>
let '(al,ar) := a in
let '(bl,br) := b in
match cmp_dec al bl with
| Eq => cmp_dec ar br
| x => x
end }. | Instance | CmpDec_pair | Core | theories/Core/CmpDec.v | [
"Coq.Bool.Bool",
"Coq.Classes.RelationClasses",
"ExtLib.Tactics.Consider"
] | [
"CmpDec",
"eq_pair",
"lt_pair"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
CmpDec_Correct_pair : CmpDec_Correct CmpDec_pair. | Proof.
constructor. destruct x; destruct y; unfold eq_pair, lt_pair; simpl in *.
generalize (cmp_dec_correct t t0); destruct (cmp_dec t t0); simpl; intros; auto.
generalize (cmp_dec_correct u u0); destruct (cmp_dec u u0); simpl; intros; auto.
Qed. | Instance | CmpDec_Correct_pair | Core | theories/Core/CmpDec.v | [
"Coq.Bool.Bool",
"Coq.Classes.RelationClasses",
"ExtLib.Tactics.Consider"
] | [
"CmpDec_Correct",
"CmpDec_pair",
"eq_pair",
"lt_pair"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
decideP (P : Prop) {D : Decidable P} : {P} + {~P} | :=
match @Decidable_witness P D as X return (X = true -> P) -> (X = false -> ~P) -> {P} + {~P} with
| true => fun pf _ => left (pf eq_refl)
| false => fun _ pf => right (pf eq_refl)
end (@Decidable_sound _ D) (@Decidable_complete_alt _ D). | Definition | decideP | Core | theories/Core/Decision.v | [
"Coq.Classes",
"DecidableClass"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
cases_ifd Hn | :=
match goal with
|- context[if ?d then ?tt else ?ff] =>
let Hnt := fresh Hn "t" in
let Hnf := fresh Hn "f" in
destruct d as [Hnt | Hnf] end. | Ltac | cases_ifd | Core | theories/Core/Decision.v | [
"Coq.Classes",
"DecidableClass"
] | [
"context"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
decide_decideP {P:Prop }`{Decidable P} {R:Type} (a b : R) :
(if (decide P) then a else b) = (if (decideP P) then a else b). | Proof.
symmetry. unfold decide.
destruct (decideP P).
- rewrite Decidable_complete; auto.
- rewrite Decidable_sound_alt; auto.
Qed. | Lemma | decide_decideP | Core | theories/Core/Decision.v | [
"Coq.Classes",
"DecidableClass"
] | [
"decideP"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
EquivDec_refl_left {T : Type} {c : EqDec T (@eq T)} :
forall (n : T), equiv_dec n n = left (refl_equal _). | Proof.
intros. destruct (equiv_dec n n); try congruence.
Require Eqdep_dec.
rewrite (Eqdep_dec.UIP_dec (A := T) (@equiv_dec _ _ _ c) e (refl_equal _)).
reflexivity.
Qed. | Theorem | EquivDec_refl_left | Core | theories/Core/EquivDec.v | [
"Coq.Classes",
"EquivDec",
"Eqdep_dec"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
RelDec (T : Type) (equ : T -> T -> Prop) : Type | :=
{ rel_dec : T -> T -> bool }. | Class | RelDec | Core | theories/Core/RelDec.v | [
"Coq.Bool.Bool",
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
RelDec_Correct T (equ : T -> T -> Prop) (ED : RelDec equ) : Prop | :=
{ rel_dec_correct : forall x y : T, rel_dec x y = true <-> equ x y }. | Class | RelDec_Correct | Core | theories/Core/RelDec.v | [
"Coq.Bool.Bool",
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [
"RelDec"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
"a ?[ r ] b" | := (@rel_dec _ r _ a b) (at level 30, b at next level). | Notation | a ?[ r ] b | Core | theories/Core/RelDec.v | [
"Coq.Bool.Bool",
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
eq_dec {T : Type} {ED : RelDec (@eq T)} | := rel_dec. | Definition | eq_dec | Core | theories/Core/RelDec.v | [
"Coq.Bool.Bool",
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [
"RelDec"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
neg_rel_dec_correct : forall {x y}, ~R x y <-> rel_dec x y = false. | Proof. intros x y. destruct (bool_dec (rel_dec x y) true) ; constructor ; intros ;
repeat
match goal with
| [ |- ~ _ ] => unfold not ; intros
| [ H1 : ?P, H2 : ~?P |- _ ] => specialize (H2 H1) ; contradiction
| [ H1 : ?P = true, H2 : ?P = false |- _ ] => rewrite H1 in H2 ; discriminate
... | Definition | neg_rel_dec_correct | Core | theories/Core/RelDec.v | [
"Coq.Bool.Bool",
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [
"apply"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
rel_dec_p (x:T) (y:T) : {R x y} + {~R x y}. | Proof. destruct (bool_dec (rel_dec x y) true) as [H | H].
apply rel_dec_correct in H ; eauto.
apply not_true_is_false in H ; apply neg_rel_dec_correct in H ; eauto.
Qed. | Definition | rel_dec_p | Core | theories/Core/RelDec.v | [
"Coq.Bool.Bool",
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [
"apply",
"neg_rel_dec_correct"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
neg_rel_dec_p (x:T) (y:T) : {~R x y} + {R x y}. | Proof. destruct (rel_dec_p x y) ; [ right | left ] ; auto. Qed. | Definition | neg_rel_dec_p | Core | theories/Core/RelDec.v | [
"Coq.Bool.Bool",
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [
"rel_dec_p"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
rel_dec_eq_true : forall x y,
eqt x y -> rel_dec x y = true. | Proof.
intros. eapply rel_dec_correct in H. assumption.
Qed. | Theorem | rel_dec_eq_true | Core | theories/Core/RelDec.v | [
"Coq.Bool.Bool",
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
rel_dec_neq_false : forall x y,
~eqt x y -> rel_dec x y = false. | Proof.
intros. remember (x ?[ eqt ] y).
symmetry in Heqb.
destruct b; try reflexivity.
exfalso. eapply (@rel_dec_correct _ _ _ rc) in Heqb. auto.
Qed. | Theorem | rel_dec_neq_false | Core | theories/Core/RelDec.v | [
"Coq.Bool.Bool",
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
rel_dec_sym : Symmetric eqt -> forall x y,
x ?[ eqt ] y = y ?[ eqt ] x. | Proof.
intros.
remember (x ?[ eqt ] y); remember (y ?[ eqt ] x); intuition.
destruct b; destruct b0; auto.
{ symmetry in Heqb; symmetry in Heqb0.
eapply (@rel_dec_correct _ _ _ rc) in Heqb.
symmetry in Heqb.
eapply (@rel_dec_correct _ _ _ rc) in Heqb.
congruence. }
{ symmetry... | Theorem | rel_dec_sym | Core | theories/Core/RelDec.v | [
"Coq.Bool.Bool",
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
RelDec_from_dec
: RelDec R | :=
{| rel_dec := fun a b =>
match f a b with
| left _ => true
| right _ => false
end |}. | Definition | RelDec_from_dec | Core | theories/Core/RelDec.v | [
"Coq.Bool.Bool",
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [
"RelDec"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
RelDec_Correct_eq_typ : RelDec_Correct RelDec_from_dec. | Proof.
constructor.
intros.
unfold rel_dec; simpl.
destruct (f x y).
- tauto.
- split.
+ inversion 1.
+ intro. apply n in H. tauto.
Qed. | Instance | RelDec_Correct_eq_typ | Core | theories/Core/RelDec.v | [
"Coq.Bool.Bool",
"Coq.Classes.RelationClasses",
"Coq.Setoids.Setoid"
] | [
"RelDec_Correct",
"RelDec_from_dec",
"apply",
"split"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
RelDec_eq : RelDec (@eq bool) | :=
{ rel_dec := fun x y => match x , y with
| true , true
| false , false => true
| _ , _=> false
end }. | Instance | RelDec_eq | Data | theories/Data/Bool.v | [
"ExtLib.Core.RelDec"
] | [
"RelDec"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
RelDec_Correct_eq_bool : RelDec_Correct RelDec_eq. | constructor. destruct x; destruct y; auto; simpl; intuition.
Qed. | Instance | RelDec_Correct_eq_bool | Data | theories/Data/Bool.v | [
"ExtLib.Core.RelDec"
] | [
"RelDec_Correct",
"RelDec_eq"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
RelDec_ascii : RelDec (@eq Ascii.ascii) | :=
{ rel_dec := Ascii.eqb }. | Instance | RelDec_ascii | Data | theories/Data/Char.v | [
"Coq.Strings.Ascii",
"ExtLib.Data.Bool",
"ExtLib.Tactics.Consider",
"ExtLib.Core.RelDec",
"Ascii"
] | [
"RelDec"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
RelDec_Correct_ascii : RelDec_Correct RelDec_ascii. | Proof.
constructor; auto using Ascii.eqb_eq.
Qed. | Instance | RelDec_Correct_ascii | Data | theories/Data/Char.v | [
"Coq.Strings.Ascii",
"ExtLib.Data.Bool",
"ExtLib.Tactics.Consider",
"ExtLib.Core.RelDec",
"Ascii"
] | [
"RelDec_Correct",
"RelDec_ascii"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Reflect_ascii_dec a b : Reflect (Ascii.eqb a b) (a = b) (a <> b). | Proof.
apply iff_to_reflect; auto using Ascii.eqb_eq.
Qed. | Instance | Reflect_ascii_dec | Data | theories/Data/Char.v | [
"Coq.Strings.Ascii",
"ExtLib.Data.Bool",
"ExtLib.Tactics.Consider",
"ExtLib.Core.RelDec",
"Ascii"
] | [
"Reflect",
"apply",
"iff_to_reflect"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
digit2ascii (n:nat) : Ascii.ascii | :=
match n with
| 0 => "0"
| 1 => "1"
| 2 => "2"
| 3 => "3"
| 4 => "4"
| 5 => "5"
| 6 => "6"
| 7 => "7"
| 8 => "8"
| 9 => "9"
| n => ascii_of_nat (n - 10 + nat_of_ascii "A")
end%char. | Definition | digit2ascii | Data | theories/Data/Char.v | [
"Coq.Strings.Ascii",
"ExtLib.Data.Bool",
"ExtLib.Tactics.Consider",
"ExtLib.Core.RelDec",
"Ascii"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
chr_newline : ascii | :=
Eval compute in ascii_of_nat 10. | Definition | chr_newline | Data | theories/Data/Char.v | [
"Coq.Strings.Ascii",
"ExtLib.Data.Bool",
"ExtLib.Tactics.Consider",
"ExtLib.Core.RelDec",
"Ascii"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Checked : option T -> Type | :=
| Success : forall {v}, F v -> Checked (Some v)
| Failure : Checked None. | Inductive | Checked | Data | theories/Data/Checked.v | [] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
succeeded (o : option T) (d : Checked o) : bool | :=
match d with
| Success _ _ => true
| Failure => false
end. | Definition | succeeded | Data | theories/Data/Checked.v | [] | [
"Checked"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
failed (o : option T) (d : Checked o) : bool | :=
match d with
| Success _ _ => false
| Failure => true
end. | Definition | failed | Data | theories/Data/Checked.v | [] | [
"Checked"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
asOption (o : option T) (d : Checked o) : option (match o with
| None => False
| Some x => F x
end) | :=
match d in Checked o return option match o with
| None => False
| Some x => F x
end
with
| Success _ x => Some x
| Failure => None
end. | Definition | asOption | Data | theories/Data/Checked.v | [] | [
"Checked"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
eq_sym_eq
: forall T (a b : T) (pf : a = b) (F : T -> Type) val,
match eq_sym pf in _ = x return F x with
| eq_refl => val
end =
match pf in _ = x return F x -> F a with
| eq_refl => fun x => x
end val. | Proof.
destruct pf. reflexivity.
Defined. | Lemma | eq_sym_eq | Data | theories/Data/Eq.v | [
"ExtLib.Data.Eq.UIP_trans"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
match_eq_sym_eq
: forall T (a b : T) (pf : a = b) F X,
match pf in _ = t return F t with
| eq_refl => match eq_sym pf in _ = t return F t with
| eq_refl => X
end
end = X. | Proof.
destruct pf. reflexivity.
Defined. | Lemma | match_eq_sym_eq | Data | theories/Data/Eq.v | [
"ExtLib.Data.Eq.UIP_trans"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
match_eq_sym_eq'
: forall T (a b : T) (pf : a = b) F X,
match eq_sym pf in _ = t return F t with
| eq_refl => match pf in _ = t return F t with
| eq_refl => X
end
end = X. | Proof.
destruct pf. reflexivity.
Defined. | Lemma | match_eq_sym_eq' | Data | theories/Data/Eq.v | [
"ExtLib.Data.Eq.UIP_trans"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
match_eq_match_eq
: forall T F (a b : T) (pf : a = b) X Y,
X = Y ->
match pf in _ = T return F T with
| eq_refl => X
end =
match pf in _ = T return F T with
| eq_refl => Y
end. | Proof.
intros. subst. auto.
Defined. | Lemma | match_eq_match_eq | Data | theories/Data/Eq.v | [
"ExtLib.Data.Eq.UIP_trans"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
eq_sym_eq_trans
: forall T (a b c : T) (pf : a = b) (pf' : b = c),
eq_sym (eq_trans pf pf') =
eq_trans (eq_sym pf') (eq_sym pf). | Proof.
clear. destruct pf. destruct pf'. reflexivity.
Defined. | Lemma | eq_sym_eq_trans | Data | theories/Data/Eq.v | [
"ExtLib.Data.Eq.UIP_trans"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
eq_Const_eq
: forall T (a b : T) (pf : a = b) (R : Type) val,
match pf in _ = x return R with
| eq_refl => val
end = val. | Proof.
destruct pf. reflexivity.
Defined. | Lemma | eq_Const_eq | Data | theories/Data/Eq.v | [
"ExtLib.Data.Eq.UIP_trans"
] | [] | Particular Instances * | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 |
eq_Arr_eq
: forall T (a b : T) (pf : a = b) (F G : T -> Type) val x,
match pf in _ = x return F x -> G x with
| eq_refl => val
end x =
match pf in _ = x return G x with
| eq_refl => val match eq_sym pf in _ = x return F x with
| eq_refl => x
end
... | Proof.
destruct pf. reflexivity.
Defined. | Lemma | eq_Arr_eq | Data | theories/Data/Eq.v | [
"ExtLib.Data.Eq.UIP_trans"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
eq_sym_eq_sym : forall (T : Type) (a b : T) (pf : a = b),
eq_sym (eq_sym pf) = pf. | Proof. destruct pf. reflexivity. Defined. | Lemma | eq_sym_eq_sym | Data | theories/Data/Eq.v | [
"ExtLib.Data.Eq.UIP_trans"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
autorewrite_eq_rw | :=
repeat progress (autorewrite with eq_rw;
repeat match goal with
| |- context [ match ?X in @eq _ _ _ return _ -> _ with
| eq_refl => _
end ] => rewrite (eq_Arr_eq X)
... | Ltac | autorewrite_eq_rw | Data | theories/Data/Eq.v | [
"ExtLib.Data.Eq.UIP_trans"
] | [
"context",
"eq_Arr_eq"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
fin : nat -> Type | :=
| F0 : forall {n}, fin (S n)
| FS : forall {n}, fin n -> fin (S n). | Inductive | fin | Data | theories/Data/Fin.v | [
"Coq.Lists.List",
"ExtLib.Core.RelDec",
"ExtLib.Tactics.EqDep",
"ExtLib.Tactics.Injection"
] | [] | `fin n` corresponds to "naturals less than `n`",
i.e. a finite set of size n
* | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 |
fin_all (n : nat) : list (fin n) | :=
match n as n return list (fin n) with
| 0 => nil
| S n => @F0 n :: List.map (@FS _) (fin_all n)
end%list. | Fixpoint | fin_all | Data | theories/Data/Fin.v | [
"Coq.Lists.List",
"ExtLib.Core.RelDec",
"ExtLib.Tactics.EqDep",
"ExtLib.Tactics.Injection"
] | [
"fin"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
fin_all_In : forall {n} (f : fin n),
List.In f (fin_all n). | Proof.
induction n; intros.
inversion f.
remember (S n). destruct f.
simpl; firstorder.
inversion Heqn0. subst.
simpl. right. apply List.in_map. auto.
Qed. | Theorem | fin_all_In | Data | theories/Data/Fin.v | [
"Coq.Lists.List",
"ExtLib.Core.RelDec",
"ExtLib.Tactics.EqDep",
"ExtLib.Tactics.Injection"
] | [
"apply",
"fin",
"fin_all"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
fin_case : forall n (f : fin (S n)),
f = F0 \/ exists f', f = FS f'. | Proof.
intros. generalize (fin_all_In f). intros.
destruct H; auto.
eapply List.in_map_iff in H. right. destruct H.
exists x. intuition.
Qed. | Theorem | fin_case | Data | theories/Data/Fin.v | [
"Coq.Lists.List",
"ExtLib.Core.RelDec",
"ExtLib.Tactics.EqDep",
"ExtLib.Tactics.Injection"
] | [
"fin",
"fin_all_In"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
fin0_elim (f : fin 0) : forall T, T | :=
match f in fin n return match n with
| 0 => forall T, T
| _ => unit
end with
| F0 _ => tt
| FS _ _ => tt
end. | Definition | fin0_elim | Data | theories/Data/Fin.v | [
"Coq.Lists.List",
"ExtLib.Core.RelDec",
"ExtLib.Tactics.EqDep",
"ExtLib.Tactics.Injection"
] | [
"fin"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
pf_lt (n m : nat) : Prop | :=
match n , m with
| 0 , S _ => True
| S n , S m => pf_lt n m
| _ , _ => False
end. | Fixpoint | pf_lt | Data | theories/Data/Fin.v | [
"Coq.Lists.List",
"ExtLib.Core.RelDec",
"ExtLib.Tactics.EqDep",
"ExtLib.Tactics.Injection"
] | [] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
make (m n : nat) {struct m} : pf_lt n m -> fin m | :=
match n as n , m as m return pf_lt n m -> fin m with
| 0 , 0 => @False_rect _
| 0 , S n => fun _ => F0
| S n , 0 => @False_rect _
| S n , S m => fun pf => FS (make m n pf)
end. | Fixpoint | make | Data | theories/Data/Fin.v | [
"Coq.Lists.List",
"ExtLib.Core.RelDec",
"ExtLib.Tactics.EqDep",
"ExtLib.Tactics.Injection"
] | [
"fin",
"pf_lt"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
"'##' n" | := (@make _ n I) (at level 0). | Notation | '##' n | Data | theories/Data/Fin.v | [
"Coq.Lists.List",
"ExtLib.Core.RelDec",
"ExtLib.Tactics.EqDep",
"ExtLib.Tactics.Injection"
] | [
"make"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Injective_FS {n : nat} (a b : fin n)
: Injective (FS a = FS b). | refine {| result := a = b |}.
abstract (intro ; inversion H ; eapply inj_pair2 in H1 ; assumption).
Defined. | Instance | Injective_FS | Data | theories/Data/Fin.v | [
"Coq.Lists.List",
"ExtLib.Core.RelDec",
"ExtLib.Tactics.EqDep",
"ExtLib.Tactics.Injection"
] | [
"Injective",
"fin",
"inj_pair2"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
fin_eq_dec {n} (x : fin n) {struct x} : fin n -> bool | :=
match x in fin n' return fin n' -> bool with
| F0 _ => fun y => match y with
| F0 _ => true
| _ => false
end
| FS n' x' => fun y : fin (S n') =>
match y in fin n'' return (match n'' with
| 0 =>... | Fixpoint | fin_eq_dec | Data | theories/Data/Fin.v | [
"Coq.Lists.List",
"ExtLib.Core.RelDec",
"ExtLib.Tactics.EqDep",
"ExtLib.Tactics.Injection"
] | [
"fin"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
RelDec_fin_eq (n : nat) : RelDec (@eq (fin n)) | :=
{ rel_dec := fin_eq_dec }. | Instance | RelDec_fin_eq | Data | theories/Data/Fin.v | [
"Coq.Lists.List",
"ExtLib.Core.RelDec",
"ExtLib.Tactics.EqDep",
"ExtLib.Tactics.Injection"
] | [
"RelDec",
"fin",
"fin_eq_dec"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
RelDec_Correct_fin_eq (n : nat)
: RelDec_Correct (RelDec_fin_eq n). | Proof.
constructor.
induction x. simpl.
intro. destruct (fin_case y) ; subst.
intuition.
destruct H ; subst.
intuition; auto with *; try congruence.
(* inversion H.*)
intro ; destruct (fin_case y) ; subst ; simpl.
intuition ; try congruence.
inversion H.
destruct H ; subst.
split ; intro.
f_equ... | Instance | RelDec_Correct_fin_eq | Data | theories/Data/Fin.v | [
"Coq.Lists.List",
"ExtLib.Core.RelDec",
"ExtLib.Tactics.EqDep",
"ExtLib.Tactics.Injection"
] | [
"RelDec_Correct",
"RelDec_fin_eq",
"apply",
"fin_case",
"inv_all",
"split"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Functor_Fun : Functor (Fun A) | :=
{ fmap _A _B g f x := g (f x) }. | Instance | Functor_Fun | Data | theories/Data/Fun.v | [
"ExtLib.Data.PreFun",
"ExtLib.Structures.Functor",
"ExtLib.Structures.Applicative",
"ExtLib.Structures.CoFunctor",
"ExtLib.Structures.Monoid",
"PreFun"
] | [
"Fun",
"Functor"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
CoFunctor_Fun T : CoFunctor (fun x => x -> T) | :=
{| cofmap := fun _ _ g f => fun x => f (g x) |}. | Instance | CoFunctor_Fun | Data | theories/Data/Fun.v | [
"ExtLib.Data.PreFun",
"ExtLib.Structures.Functor",
"ExtLib.Structures.Applicative",
"ExtLib.Structures.CoFunctor",
"ExtLib.Structures.Monoid",
"PreFun"
] | [
"CoFunctor"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Functor_functor F G (fF : Functor F) (fG : Functor G) : Functor (fun x => F (G x)) | :=
{| fmap := fun _ _ g => @fmap F _ _ _ (@fmap G _ _ _ g) |}. | Instance | Functor_functor | Data | theories/Data/Fun.v | [
"ExtLib.Data.PreFun",
"ExtLib.Structures.Functor",
"ExtLib.Structures.Applicative",
"ExtLib.Structures.CoFunctor",
"ExtLib.Structures.Monoid",
"PreFun"
] | [
"Functor"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
CoFunctor_functor F G (fF : Functor F) (fG : CoFunctor G) : CoFunctor (fun x => F (G x)) | :=
{| cofmap := fun _ _ g => @fmap F _ _ _ (@cofmap G _ _ _ g) |}. | Instance | CoFunctor_functor | Data | theories/Data/Fun.v | [
"ExtLib.Data.PreFun",
"ExtLib.Structures.Functor",
"ExtLib.Structures.Applicative",
"ExtLib.Structures.CoFunctor",
"ExtLib.Structures.Monoid",
"PreFun"
] | [
"CoFunctor",
"Functor"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
Functor_cofunctor F G (fF : CoFunctor F) (fG : Functor G) : CoFunctor (fun x => F (G x)) | :=
{| cofmap := fun _ _ g => @cofmap F _ _ _ (@fmap G _ _ _ g) |}. | Instance | Functor_cofunctor | Data | theories/Data/Fun.v | [
"ExtLib.Data.PreFun",
"ExtLib.Structures.Functor",
"ExtLib.Structures.Applicative",
"ExtLib.Structures.CoFunctor",
"ExtLib.Structures.Monoid",
"PreFun"
] | [
"CoFunctor",
"Functor"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 | |
CoFunctor_cofunctor F G (fF : CoFunctor F) (fG : CoFunctor G) : Functor (fun x => F (G x)) | :=
{| fmap := fun _ _ g => @cofmap F _ _ _ (@cofmap G _ _ _ g) |}. | Instance | CoFunctor_cofunctor | Data | theories/Data/Fun.v | [
"ExtLib.Data.PreFun",
"ExtLib.Structures.Functor",
"ExtLib.Structures.Applicative",
"ExtLib.Structures.CoFunctor",
"ExtLib.Structures.Monoid",
"PreFun"
] | [
"CoFunctor",
"Functor"
] | https://github.com/coq-community/coq-ext-lib | ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 |
End of preview. Expand in Data Studio
Coq-ExtLib
Structured dataset from coq-ext-lib — Extended standard library with monads and data structures.
Source
- Repository: https://github.com/coq-community/coq-ext-lib
- Commit:
ddd03d257f6b85a93bfaa0ed4d03658e0ddf5075 - Files: 129
- License: bsd-2-clause
Schema
| Column | Type | Description |
|---|---|---|
| statement | string | Declaration signature/claim with the leading keyword removed (verbatim slice); the full declaration minus its proof |
| proof | string | Verbatim proof/body, empty if the declaration has none |
| type | string | Declaration keyword |
| symbolic_name | string | Declaration identifier |
| library | string | Sub-library |
| filename | string | Repository-relative source path |
| imports | list[string] | File-level Require/Import modules |
| deps | list[string] | Intra-corpus identifiers referenced |
| docstring | string | Preceding documentation comment, empty if absent |
| source_url | string | Upstream repository |
| commit | string | Upstream commit extracted |
Statistics
- Entries: 1,019
- With proof: 1,005 (98.6%)
- With docstring: 54 (5.3%)
- Libraries: 15
By type
| Type | Count |
|---|---|
| Instance | 251 |
| Definition | 238 |
| Lemma | 129 |
| Fixpoint | 114 |
| Theorem | 62 |
| Class | 60 |
| Notation | 57 |
| Inductive | 46 |
| Ltac | 37 |
| Record | 11 |
| Hypothesis | 11 |
| Coercion | 2 |
| CoInductive | 1 |
Example
exp : Type
:=
| ConstI : nat -> exp
| ConstB : bool -> exp
| Plus : exp -> exp -> exp
| If : exp -> exp -> exp -> exp.
- type: Inductive | symbolic_name:
exp| examples/EvalWithExc.v
Use
Each declaration is split into a statement (signature/claim) and a proof (body) that are disjoint
and together form the complete declaration, for proof modeling, autoformalization, retrieval, and
dependency analysis via deps.
Citation
@misc{coq_extlib_dataset,
title = {Coq-ExtLib},
author = {Norton, Charles},
year = {2026},
note = {Extracted from https://github.com/coq-community/coq-ext-lib, commit ddd03d257f6b},
url = {https://huggingface.co/datasets/phanerozoic/Coq-ExtLib}
}
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