fact stringlengths 17 6.18k | type stringclasses 17 values | library stringclasses 3 values | imports listlengths 0 12 | filename stringclasses 115 values | symbolic_name stringlengths 1 30 | docstring stringclasses 1 value |
|---|---|---|---|---|---|---|
Transitive_Roption : Transitive R -> Transitive Roption. Proof. clear. compute. intros. destruct H0; auto. inversion H1. constructor; auto. subst. eapply H; eassumption. Qed. Global Instance Injective_Roption_None : Injective (Roption None None). refine {| result := True |}. auto. Defined. Global Instance Injective_Roption_None_Some a : Injective (Roption None (Some a)). refine {| result := False |}. inversion 1. Defined. Global Instance Injective_Roption_Some_None a : Injective (Roption (Some a) None). refine {| result := False |}. inversion 1. Defined. Global Instance Injective_Roption_Some_Some a b : Injective (Roption (Some a) (Some b)). refine {| result := R a b |}. inversion 1. auto. Defined. Global Instance Injective_Proper_Roption_Some x : Injective (Proper Roption (Some x)). refine {| result := R x x |}. abstract (inversion 1; assumption). Defined. | Lemma | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Option.v | Transitive_Roption | |
eq_option_eq : forall T (a b : T) (pf : a = b) (F : _ -> Type) val, match pf in _ = x return option (F x) with | eq_refl => val end = match val with | None => None | Some val => Some match pf in _ = x return F x with | eq_refl => val end end. Proof. destruct pf. destruct val; reflexivity. Defined. #[global] Hint Rewrite eq_option_eq : eq_rw. | Lemma | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Option.v | eq_option_eq | |
Eqpair : relation (T * U) := | Eqpair_both : forall a b c d, rT a b -> rU c d -> Eqpair (a,c) (b,d). Global Instance Reflexive_Eqpair {RrT : Reflexive rT} {RrU : Reflexive rU} : Reflexive Eqpair. Proof. red. destruct x. constructor; reflexivity. Qed. Global Instance Symmetric_Eqpair {RrT : Symmetric rT} {RrU : Symmetric rU} : Symmetric Eqpair. Proof. red. inversion 1; constructor; symmetry; assumption. Qed. Global Instance Transitive_Eqpair {RrT : Transitive rT} {RrU : Transitive rU} : Transitive Eqpair. Proof. red. inversion 1; inversion 1; constructor; etransitivity; eauto. Qed. Global Instance Injective_Eqpair a b c d : Injective (Eqpair (a,b) (c,d)). refine {| result := rT a c /\ rU b d |}. abstract (inversion 1; auto). Defined. | Inductive | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Pair.v | Eqpair | |
R_pair : T * U -> T * U -> Prop := | L : forall l l' r r', RT l l' -> R_pair (l,r) (l',r') | R : forall l r r', RU r r' -> R_pair (l,r) (l,r'). Hypothesis wf_RT : well_founded RT. Hypothesis wf_RU : well_founded RU. | Inductive | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Pair.v | R_pair | |
wf_R_pair : well_founded R_pair. Proof. red. intro x. destruct x. generalize dependent u. apply (well_founded_ind wf_RT (fun t => forall u : U, Acc R_pair (t, u))) . do 2 intro. apply (well_founded_ind wf_RU (fun u => Acc R_pair (x,u))). intros. constructor. destruct y. remember (t0,u). remember (x,x0). inversion 1; subst; inversion H4; inversion H3; clear H4 H3; subst; eauto. Defined. | Theorem | theories | [
"Require Import Coq.",
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Pair.v | wf_R_pair | |
pIn_split_l {A B : Type} (lst : plist (pprod A B)) (p : pprod A B) (H : pIn p lst) : (pIn (pfst p) (fst (split lst))). Proof. destruct p; simpl. induction lst; simpl in *. + destruct H. + destruct t; simpl. destruct (split lst); simpl. destruct H as [H | H]. { inv_all. tauto. } { tauto. } Qed. | Lemma | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import Coq."
] | theories/Data/PList.v | pIn_split_l | |
pIn_split_r {A B : Type} (lst : plist (pprod A B)) (p : pprod A B) (H : pIn p lst) : (pIn (psnd p) (snd (split lst))). Proof. destruct p; simpl. induction lst; simpl in *. + destruct H. + destruct t; simpl. destruct (split lst); simpl. destruct H. { inv_all; tauto. } { tauto. } Qed. | Lemma | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import Coq."
] | theories/Data/PList.v | pIn_split_r | |
pIn_app_iff (A : Type) (l l' : plist A) (a : A) : pIn a (app l l') <-> pIn a l \/ pIn a l'. Proof. induction l; simpl; intuition congruence. Qed. | Lemma | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import Coq."
] | theories/Data/PList.v | pIn_app_iff | |
inb_sound (x : A) (lst : plist A) (H : inb x lst = true) : pIn x lst. Proof. induction lst; simpl in *; try congruence. consider (x ?[ eq ] t); intros; subst. + left; reflexivity. + right; apply IHlst; assumption. Qed. | Lemma | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import Coq."
] | theories/Data/PList.v | inb_sound | |
inb_complete (x : A) (lst : plist A) (H : pIn x lst) : inb x lst = true. Proof. induction lst; simpl in *; try intuition congruence. consider (x ?[ eq ] t); intros; destruct H as [H | H]; try congruence. apply IHlst; assumption. Qed. | Lemma | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import Coq."
] | theories/Data/PList.v | inb_complete | |
nodup_sound (lst : plist A) (H : nodup lst = true) : pNoDup lst. Proof. induction lst. + constructor. + simpl in *. rewrite andb_true_iff in H; destruct H as [H1 H2]. rewrite negb_true_iff in H1. constructor. * intro H. apply inb_complete in H. intuition congruence. * apply IHlst; assumption. Qed. | Lemma | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import Coq."
] | theories/Data/PList.v | nodup_sound | |
nodup_complete (lst : plist A) (H : pNoDup lst) : nodup lst = true. Proof. induction lst. + constructor. + simpl in *. rewrite andb_true_iff. inversion H; subst; split; clear H. * apply eq_true_not_negb. intros H; apply H2. apply inb_sound; assumption. * apply IHlst; assumption. Qed. | Lemma | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import Coq."
] | theories/Data/PList.v | nodup_complete | |
Instance Functor_plist. | Existing | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import Coq."
] | theories/Data/PList.v | Instance | |
poption : Type@{i} := | pSome : T -> poption | pNone. Global Instance Injective_pSome@{} a b : Injective (pSome a = pSome b) := { result := a = b ; injection := fun pf => match pf in _ = X return a = match X with | pSome y => y | _ => a end with | eq_refl => eq_refl end }. Global Instance Injective_pSome_pNone a : Injective (pSome a = pNone) := { result := False ; injection := fun pf => match pf in _ = X return match X return Prop with | pSome y => True | _ => False end with | eq_refl => I end }. Global Instance Injective_pNone_pSome@{} a : Injective (pNone = pSome a) := { result := False ; injection := fun pf => match pf in _ = X return match X return Prop with | pNone => True | _ => False end with | eq_refl => I end }. | Inductive | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/POption.v | poption | |
fmap_poption @{} (x : poption@{i} T) : poption@{j} U := match x with | pNone => pNone@{j} | pSome x => pSome@{j} (f x) end. | Definition | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/POption.v | fmap_poption | |
ap_poption @{} (f : poption@{i} (T -> U)) (x : poption@{i} T) : poption@{j} U := match f , x with | pSome f , pSome x => pSome (f x) | _ , _ => pNone end. | Definition | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/POption.v | ap_poption | |
Functor_poption @{i} : Functor@{i i} poption@{i} := {| fmap := @fmap_poption@{i i} |}. #[global] | Definition | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/POption.v | Functor_poption | |
Instance Functor_poption. | Existing | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/POption.v | Instance | |
Applicative_poption @{i} : Applicative@{i i} poption@{i} := {| pure := @pSome@{i} ; ap := @ap_poption |}. #[global] | Definition | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/POption.v | Applicative_poption | |
Instance Applicative_poption. | Existing | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/POption.v | Instance | |
Fun @{d c} (A : Type@{d}) (B : Type@{c}) := A -> B. | Definition | theories | [
"From Coq.Classes Require Import Morphisms.",
"From Coq.Relations Require Import Relations."
] | theories/Data/PreFun.v | Fun | |
compose @{uA uB uC} {A:Type@{uA}} {B:Type@{uB}} {C : Type@{uC}} (g : B -> C) (f : A -> B) : A -> C := fun x => g (f x). | Definition | theories | [
"From Coq.Classes Require Import Morphisms.",
"From Coq.Relations Require Import Relations."
] | theories/Data/PreFun.v | compose | |
and_True_iff : forall P, (P /\ True) <-> P. Proof. intuition. Qed. | Lemma | theories | [
"From Coq Require Import Setoid."
] | theories/Data/Prop.v | and_True_iff | |
and_and_iff : forall P, (P /\ P) <-> P. Proof. intuition. Qed. | Lemma | theories | [
"From Coq Require Import Setoid."
] | theories/Data/Prop.v | and_and_iff | |
and_assoc : forall P Q R, (P /\ Q /\ R) <-> ((P /\ Q) /\ R). Proof. intuition. Qed. | Lemma | theories | [
"From Coq Require Import Setoid."
] | theories/Data/Prop.v | and_assoc | |
and_comm : forall P Q, (P /\ Q) <-> (Q /\ P). Proof. intuition. Qed. | Lemma | theories | [
"From Coq Require Import Setoid."
] | theories/Data/Prop.v | and_comm | |
and_False_iff : forall P, (P /\ False) <-> False. Proof. intuition. Qed. | Lemma | theories | [
"From Coq Require Import Setoid."
] | theories/Data/Prop.v | and_False_iff | |
and_cancel : forall P Q R : Prop, (P -> (Q <-> R)) -> ((P /\ Q) <-> (P /\ R)). Proof. intuition. Qed. | Lemma | theories | [
"From Coq Require Import Setoid."
] | theories/Data/Prop.v | and_cancel | |
and_iff : forall P Q R S : Prop, (P <-> R) -> (P -> (Q <-> S)) -> ((P /\ Q) <-> (R /\ S)). Proof. clear; intuition. Qed. (** Or/Disjunction **) | Lemma | theories | [
"From Coq Require Import Setoid."
] | theories/Data/Prop.v | and_iff | |
or_False_iff : forall P, (P \/ False) <-> P. Proof. intuition. Qed. | Lemma | theories | [
"From Coq Require Import Setoid."
] | theories/Data/Prop.v | or_False_iff | |
or_or_iff : forall P, (P \/ P) <-> P. Proof. intuition. Qed. | Lemma | theories | [
"From Coq Require Import Setoid."
] | theories/Data/Prop.v | or_or_iff | |
or_assoc : forall P Q R, (P \/ Q \/ R) <-> ((P \/ Q) \/ R). Proof. intuition. Qed. | Lemma | theories | [
"From Coq Require Import Setoid."
] | theories/Data/Prop.v | or_assoc | |
or_comm : forall P Q, (P \/ Q) <-> (Q \/ P). Proof. intuition. Qed. | Lemma | theories | [
"From Coq Require Import Setoid."
] | theories/Data/Prop.v | or_comm | |
or_True_iff : forall P, (P \/ True) <-> True. Proof. intuition. Qed. (** Implication **) | Lemma | theories | [
"From Coq Require Import Setoid."
] | theories/Data/Prop.v | or_True_iff | |
impl_True_iff : forall (P : Prop), (True -> P) <-> P. Proof. clear; intros; tauto. Qed. | Lemma | theories | [
"From Coq Require Import Setoid."
] | theories/Data/Prop.v | impl_True_iff | |
impl_iff : forall P Q R S : Prop, (P <-> R) -> (P -> (Q <-> S)) -> ((P -> Q) <-> (R -> S)). Proof. clear. intuition. Qed. | Lemma | theories | [
"From Coq Require Import Setoid."
] | theories/Data/Prop.v | impl_iff | |
impl_eq : forall (P Q : Prop), P = Q -> (P -> Q). Proof. clear. intros; subst; auto. Qed. | Lemma | theories | [
"From Coq Require Import Setoid."
] | theories/Data/Prop.v | impl_eq | |
uncurry : forall (P Q R : Prop), (P /\ Q -> R) <-> (P -> Q -> R). Proof. clear. tauto. Qed. (** Forall **) | Lemma | theories | [
"From Coq Require Import Setoid."
] | theories/Data/Prop.v | uncurry | |
forall_iff : forall T P Q, (forall x, P x <-> Q x) -> ((forall x : T, P x) <-> (forall x : T, Q x)). Proof. intros. setoid_rewrite H. reflexivity. Qed. | Lemma | theories | [
"From Coq Require Import Setoid."
] | theories/Data/Prop.v | forall_iff | |
forall_impl : forall {T} (P Q : T -> Prop), (forall x, P x -> Q x) -> (forall x, P x) -> (forall x, Q x). Proof. clear. intuition. Qed. (** Exists **) | Lemma | theories | [
"From Coq Require Import Setoid."
] | theories/Data/Prop.v | forall_impl | |
exists_iff : forall T P Q, (forall x, P x <-> Q x) -> ((exists x : T, P x) <-> (exists x : T, Q x)). Proof. intros. setoid_rewrite H. reflexivity. Qed. | Lemma | theories | [
"From Coq Require Import Setoid."
] | theories/Data/Prop.v | exists_iff | |
exists_impl : forall {T} (P Q : T -> Prop), (forall x, P x -> Q x) -> (exists x, P x) -> (exists x, Q x). Proof. clear. intuition. destruct H0; eauto. Qed. | Lemma | theories | [
"From Coq Require Import Setoid."
] | theories/Data/Prop.v | exists_impl | |
iff_eq : forall (P Q : Prop), P = Q -> (P <-> Q). Proof. clear. intros; subst; reflexivity. Qed. | Lemma | theories | [
"From Coq Require Import Setoid."
] | theories/Data/Prop.v | iff_eq | |
eq_sigT_rw : forall T U F (a b : T) (pf : a = b) s, match pf in _ = x return @sigT U (F x) with | eq_refl => s end = @existT U (F b) (projT1 s) match pf in _ = x return F x (projT1 s) with | eq_refl => (projT2 s) end. Proof. destruct pf. destruct s; reflexivity. Qed. #[global] Hint Rewrite eq_sigT_rw : eq_rw. | Lemma | theories | [
"From Coq.Classes Require EquivDec.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/SigT.v | eq_sigT_rw | |
stream : Type := | snil : stream | scons : T -> stream -> stream. | CoInductive | theories | [] | theories/Data/Stream.v | stream | |
deprecated_bool_cmp (l r : bool) : comparison := match l , r with | true , false => Gt | false , true => Lt | true , true | false , false => Eq end. #[deprecated(since="8.12",note="Use Bool.compare instead.")] | Definition | theories | [
"From Coq Require Import String Program PeanoNat.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/String.v | deprecated_bool_cmp | |
bool_cmp := deprecated_bool_cmp. | Notation | theories | [
"From Coq Require Import String Program PeanoNat.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/String.v | bool_cmp | |
deprecated_ascii_cmp (l r : Ascii.ascii) : comparison := match l , r with | Ascii.Ascii l1 l2 l3 l4 l5 l6 l7 l8 , Ascii.Ascii r1 r2 r3 r4 r5 r6 r7 r8 => bool_cmp l8 r8 >> bool_cmp l7 r7 >> bool_cmp l6 r6 >> bool_cmp l5 r5 >> bool_cmp l4 r4 >> bool_cmp l3 r3 >> bool_cmp l2 r2 >> bool_cmp l1 r1 end. #[deprecated(since="8.15",note="Use Ascii.compare instead.")] | Definition | theories | [
"From Coq Require Import String Program PeanoNat.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/String.v | deprecated_ascii_cmp | |
ascii_cmp := deprecated_ascii_cmp. Global Instance RelDec_string : RelDec (@eq string) := {| rel_dec := String.eqb |}. Global Instance RelDec_Correct_string : RelDec_Correct RelDec_string. Proof. constructor; auto using String.eqb_eq. Qed. Global Instance Reflect_string_dec a b : Reflect (String.eqb a b) (a = b) (a <> b). Proof. apply iff_to_reflect; auto using String.eqb_eq. Qed. | Notation | theories | [
"From Coq Require Import String Program PeanoNat.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/String.v | ascii_cmp | |
deprecated_string_cmp (l r : string) : comparison := match l , r with | EmptyString , EmptyString => Eq | EmptyString , _ => Lt | _ , EmptyString => Gt | String l ls , String r rs => ascii_cmp l r >> deprecated_string_cmp ls rs end. #[deprecated(since="8.15",note="Use String.compare instead.")] | Fixpoint | theories | [
"From Coq Require Import String Program PeanoNat.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/String.v | deprecated_string_cmp | |
string_cmp := deprecated_string_cmp. | Notation | theories | [
"From Coq Require Import String Program PeanoNat.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/String.v | string_cmp | |
_xxx : forall m n, 1 < m -> ~ n < m -> 0 < n. Proof. destruct n; destruct m; intros. inversion H. exfalso. apply H0. etransitivity. 2: eassumption. repeat constructor. inversion H. now apply Nat.lt_0_succ. Qed. | Lemma | theories | [
"From Coq Require Import String Program PeanoNat.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/String.v | _xxx | |
Fixpoint nat2string (n:nat) {measure n}: string := match Nat.ltb n modulus as x return Nat.ltb n modulus = x -> string with | true => fun _ => String (digit2ascii n) EmptyString | false => fun pf => let m := Nat.div n modulus in append (nat2string m) (String (digit2ascii (n - modulus * m)) EmptyString) end (@Logic.eq_refl _ (Nat.ltb n modulus)). | Program | theories | [
"From Coq Require Import String Program PeanoNat.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/String.v | Fixpoint | |
Obligation . eapply Nat.div_lt; auto. consider (Nat.ltb n modulus); try congruence. intros. eapply _xxx; eassumption. Defined. | Next | theories | [
"From Coq Require Import String Program PeanoNat.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/String.v | Obligation | |
nat2string10 : nat -> string. refine (@nat2string 10 _). repeat constructor. Defined. | Definition | theories | [
"From Coq Require Import String Program PeanoNat.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/String.v | nat2string10 | |
nat2string2 : nat -> string. refine (@nat2string 2 _). repeat constructor. Defined. | Definition | theories | [
"From Coq Require Import String Program PeanoNat.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/String.v | nat2string2 | |
nat2string8 : nat -> string. refine (@nat2string 8 _). repeat constructor. Defined. | Definition | theories | [
"From Coq Require Import String Program PeanoNat.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/String.v | nat2string8 | |
nat2string16 : nat -> string. refine (@nat2string 16 _). repeat constructor. Defined. Global Instance Foldable_string : Foldable string ascii := fun _ f base => fix go ls := match ls with | EmptyString => base | String l ls => f l (go ls) end. | Definition | theories | [
"From Coq Require Import String Program PeanoNat.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/String.v | nat2string16 | |
R_string_len : string -> string -> Prop := | R_s_len : forall n m, length n < length m -> R_string_len n m. | Inductive | theories | [
"From Coq Require Import String Program PeanoNat.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/String.v | R_string_len | |
wf_R_string_len : well_founded R_string_len. Proof. constructor. intros. refine (@Fix _ _ wf_R_lt (fun n : nat => forall ls : string, n = length ls -> Acc R_string_len ls) (fun x rec ls pfls => Acc_intro _ _) _ _ refl_equal). refine ( match ls as ls return x = length ls -> forall z : string, R_string_len z ls -> Acc R_string_len z with | EmptyString => fun (pfls : x = 0) z pf => _ | String l ls => fun pfls z pf => rec _ (match pf in R_string_len xs ys return x = length ys -> R_nat_lt (length xs) x with | R_s_len n m pf' => fun pf_eq => match eq_sym pf_eq in _ = x return R_nat_lt (length n) x with | refl_equal => R_lt pf' end end pfls) _ eq_refl end pfls). clear - pf; abstract (inversion pf; subst; simpl in *; inversion H). Defined. | Theorem | theories | [
"From Coq Require Import String Program PeanoNat.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/String.v | wf_R_string_len | |
Monoid_string_append : Monoid string := {| monoid_plus := append ; monoid_unit := EmptyString |}. | Definition | theories | [
"From Coq Require Import String Program PeanoNat.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/String.v | Monoid_string_append | |
sum_eq : T + U -> T + U -> Prop := | Inl_eq : forall a b, eqT a b -> sum_eq (inl a) (inl b) | Inr_eq : forall a b, eqU a b -> sum_eq (inr a) (inr b). Variable EDT : RelDec eqT. Variable EDU : RelDec eqU. Global Instance RelDec_equ_sum : RelDec (sum_eq) := { rel_dec := fun x y => match x , y with | inl x , inl y => rel_dec x y | inr x , inr y => rel_dec x y | inl _ , inr _ => false | inr _ , inl _ => false end }. Variable EDCT : RelDec_Correct EDT. Variable EDCU : RelDec_Correct EDU. Global Instance RelDec_Correct_equ_sum : RelDec_Correct RelDec_equ_sum. Proof. constructor; destruct x; destruct y; split; simpl in *; intros; repeat match goal with | [ H : context [ rel_dec ?X ?Y ] |- _ ] => consider (rel_dec X Y); intros; subst | [ |- context [ rel_dec ?X ?Y ] ] => consider (rel_dec X Y); intros; subst end; intuition; try solve [ constructor; auto | congruence ]. + inversion H. intuition. + inversion H. + inversion H. + inversion H; intuition. Qed. | Inductive | theories | [
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Data/Sum.v | sum_eq | |
pmap_lookup' (ts : pmap Type) (p : positive) : option Type := match p with | xH => pmap_here ts | xI p => pmap_lookup' (pmap_right ts) p | xO p => pmap_lookup' (pmap_left ts) p end. | Fixpoint | theories | [
"From Coq Require Import PArith.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import PArith."
] | theories/Data/SumN.v | pmap_lookup' | |
OneOf (ts : pmap Type) : Type := mkOneOf { index : positive ; value : match pmap_lookup' ts index with | None => Empty_set | Some T => T end }. | Record | theories | [
"From Coq Require Import PArith.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import PArith."
] | theories/Data/SumN.v | OneOf | |
Into {ts} {T : Type} (n : positive) (pf : pmap_lookup' ts n = Some T) : T -> OneOf ts := match pf in _ = X return match X with | Some T => T | None => Empty_set end -> OneOf ts with | eq_refl => @mkOneOf ts n end. | Definition | theories | [
"From Coq Require Import PArith.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import PArith."
] | theories/Data/SumN.v | Into | |
asNth' {ts : pmap Type} (p p' : positive) : match pmap_lookup' ts p' with | None => Empty_set | Some T => T end -> option (match pmap_lookup' ts p with | None => Empty_set | Some T => T end) := match p as p , p' as p' return match pmap_lookup' ts p' with | None => Empty_set | Some T => T end -> option (match pmap_lookup' ts p with | None => Empty_set | Some T => T end) with | xH , xH => Some | xI p , xI p' => asNth' p p' | xO p , xO p' => asNth' p p' | _ , _ => fun _ => None end. | Fixpoint | theories | [
"From Coq Require Import PArith.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import PArith."
] | theories/Data/SumN.v | asNth' | |
asNth {ts : pmap Type} (p : positive) (oe : OneOf ts) : option (match pmap_lookup' ts p with | None => Empty_set | Some T => T end) := @asNth' ts p oe.(index ts) oe.(value ts). | Definition | theories | [
"From Coq Require Import PArith.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import PArith."
] | theories/Data/SumN.v | asNth | |
OutOf {ts} {T : Type} (n : positive) (pf : pmap_lookup' ts n = Some T) : OneOf ts -> option T := match pf in _ = X return OneOf ts -> option match X with | None => Empty_set:Type | Some T => T end with | eq_refl => @asNth ts n end. | Definition | theories | [
"From Coq Require Import PArith.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import PArith."
] | theories/Data/SumN.v | OutOf | |
asNth'_get_lookup : forall p ts v, asNth' (ts:=ts) p p v = Some v. Proof. induction p; simpl; intros; auto. Qed. | Lemma | theories | [
"From Coq Require Import PArith.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import PArith."
] | theories/Data/SumN.v | asNth'_get_lookup | |
Outof_Into : forall ts T p pf v, @OutOf ts T p pf (@Into ts T p pf v) = Some v. Proof using. unfold OutOf, Into. intros. repeat rewrite (eq_Arr_eq pf). repeat rewrite (eq_Const_eq pf). repeat rewrite (eq_Const_eq (eq_sym pf)). unfold asNth. simpl. rewrite asNth'_get_lookup. { generalize dependent (pmap_lookup' ts p). intros. subst. reflexivity. } Qed. | Theorem | theories | [
"From Coq Require Import PArith.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import PArith."
] | theories/Data/SumN.v | Outof_Into | |
asNth_eq : forall ts p oe v, @asNth ts p oe = Some v -> oe = {| index := p ; value := v |}. Proof. unfold asNth. destruct oe; simpl. revert value0. revert index0. revert ts. induction p; destruct index0; simpl; intros; solve [ congruence | eapply IHp in H; inversion H; clear H IHp; subst; auto ]. Qed. | Theorem | theories | [
"From Coq Require Import PArith.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import PArith."
] | theories/Data/SumN.v | asNth_eq | |
pmap_elim (R : Type) (ts : pmap T) : Type := match ts with | Empty => R | Branch None l r => pmap_elim (pmap_elim R r) l | Branch (Some x) l r => F x -> pmap_elim (pmap_elim R r) l end. | Fixpoint | theories | [
"From Coq Require Import PArith.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import PArith."
] | theories/Data/SumN.v | pmap_elim | |
pmap_lookup'_Empty (p : positive) : pmap_lookup' Empty p = None := match p with | xH => eq_refl | xO p => pmap_lookup'_Empty p | xI p => pmap_lookup'_Empty p end. | Fixpoint | theories | [
"From Coq Require Import PArith.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import PArith."
] | theories/Data/SumN.v | pmap_lookup'_Empty | |
OneOf_Empty (f : OneOf Empty) : False. Proof. destruct f. rewrite pmap_lookup'_Empty in *. intuition congruence. Defined. | Definition | theories | [
"From Coq Require Import PArith.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import PArith."
] | theories/Data/SumN.v | OneOf_Empty | |
pmap_lookup'_eq p m : pmap_lookup p m = pmap_lookup' m p. Proof. generalize dependent m. induction p; intuition. simpl. destruct m. simpl. rewrite pmap_lookup'_Empty. reflexivity. simpl in *. apply IHp. simpl in *. destruct m. simpl. rewrite pmap_lookup'_Empty. reflexivity. simpl. apply IHp. Defined. Global Instance Injective_OneOf m i1 i2 v1 v2 : Injective (@eq (OneOf m) {| index := i1 ; value := v1 |} {| index := i2 ; value := v2 |}) := { result := exists pf : i2 = i1, v1 = match pf in _ = T return match pmap_lookup' m T with | None => Empty_set | Some T => T end with | eq_refl => v2 end ; injection := fun H => match H in _ = h return exists pf : index _ h = i1 , v1 = match pf in (_ = T) return match pmap_lookup' m T with | Some T0 => T0 | None => Empty_set end with | eq_refl => value _ h end with | eq_refl => @ex_intro _ _ eq_refl eq_refl end }. | Lemma | theories | [
"From Coq Require Import PArith.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"From Coq Require Import PArith."
] | theories/Data/SumN.v | pmap_lookup'_eq | |
vector (T : Type) (n : nat) : Type := match n with | 0 => unit | S n => prod T (vector T n) end. | Fixpoint | theories | [
"Require Import ExtLib."
] | theories/Data/Tuple.v | vector | |
get {T} {n : nat} (f : fin n) : vector T n -> T := match f in fin n return vector T n -> T with | F0 n => fun v : T * vector T n => fst v | FS n f => fun v : T * vector T n => get f (snd v) end. | Fixpoint | theories | [
"Require Import ExtLib."
] | theories/Data/Tuple.v | get | |
put {T} {n : nat} (f : fin n) (t : T) : vector T n -> vector T n := match f in fin n return vector T n -> vector T n with | F0 _ => fun v => (t, snd v) | FS _ f => fun v => (fst v, put f t (snd v)) end. | Fixpoint | theories | [
"Require Import ExtLib."
] | theories/Data/Tuple.v | put | |
get_put_eq : forall {T n} (v : vector T n) (f : fin n) val, get f (put f val v) = val. Proof. induction n. { inversion f. } { remember (S n). destruct f. inversion Heqn0; subst; intros; reflexivity. inversion Heqn0; subst; simpl; auto. } Qed. | Theorem | theories | [
"Require Import ExtLib."
] | theories/Data/Tuple.v | get_put_eq | |
get_put_neq : forall {T n} (v : vector T n) (f f' : fin n) val, f <> f' -> get f (put f' val v) = get f v. Proof. induction n. { inversion f. } { remember (S n); destruct f. { inversion Heqn0; clear Heqn0; subst; intros. destruct (fin_case f'); try congruence. destruct H0; subst. auto. } { inversion Heqn0; clear Heqn0; subst; intros. destruct (fin_case f'). subst; auto. destruct H0; subst. simpl. eapply IHn. congruence. } } Qed. | Theorem | theories | [
"Require Import ExtLib."
] | theories/Data/Tuple.v | get_put_neq | |
vector_tl {T : Type} {n : nat} (v : vector T (S n)) : vector T n := snd v. | Definition | theories | [
"Require Import ExtLib."
] | theories/Data/Tuple.v | vector_tl | |
vector_hd {T : Type} {n : nat} (v : vector T (S n)) : T := fst v. | Definition | theories | [
"Require Import ExtLib."
] | theories/Data/Tuple.v | vector_hd | |
vector T : nat -> Type := | Vnil : vector T 0 | Vcons : forall {n}, T -> vector T n -> vector T (S n). | Inductive | theories | [
"Require Import ExtLib."
] | theories/Data/Vector.v | vector | |
vector_hd n (v : vector T (S n)) : T := match v in vector _ n' return match n' with | 0 => unit | S _ => T end with | Vnil => tt | Vcons _ x _ => x end. | Definition | theories | [
"Require Import ExtLib."
] | theories/Data/Vector.v | vector_hd | |
vector_tl n (v : vector T (S n)) : vector T n := match v in vector _ n' return match n' with | 0 => unit | S n => vector T n end with | Vnil => tt | Vcons _ _ x => x end. | Definition | theories | [
"Require Import ExtLib."
] | theories/Data/Vector.v | vector_tl | |
vector_eta : forall n (v : vector T n), v = match n as n return vector T n -> vector T n with | 0 => fun _ => Vnil _ | S n => fun v => Vcons (vector_hd v) (vector_tl v) end v. Proof. destruct v; auto. Qed. | Theorem | theories | [
"Require Import ExtLib."
] | theories/Data/Vector.v | vector_eta | |
get {n : nat} (f : fin n) : vector T n -> T := match f in fin n return vector T n -> T with | F0 n => @vector_hd _ | FS n f => fun v => get f (vector_tl v) end. | Fixpoint | theories | [
"Require Import ExtLib."
] | theories/Data/Vector.v | get | |
put {n : nat} (f : fin n) (t : T) : vector T n -> vector T n := match f in fin n return vector T n -> vector T n with | F0 _ => fun v => Vcons t (vector_tl v) | FS _ f => fun v => Vcons (vector_hd v) (put f t (vector_tl v)) end. | Fixpoint | theories | [
"Require Import ExtLib."
] | theories/Data/Vector.v | put | |
get_put_eq : forall {n} (v : vector T n) (f : fin n) val, get f (put f val v) = val. Proof. induction n. { inversion f. } { remember (S n). destruct f. inversion Heqn0; subst; intros; reflexivity. inversion Heqn0; subst; simpl; auto. } Qed. | Theorem | theories | [
"Require Import ExtLib."
] | theories/Data/Vector.v | get_put_eq | |
get_put_neq : forall {n} (v : vector T n) (f f' : fin n) val, f <> f' -> get f (put f' val v) = get f v. Proof. induction n. { inversion f. } { remember (S n); destruct f. { inversion Heqn0; clear Heqn0; subst; intros. destruct (fin_case f'); try congruence. destruct H0; subst. auto. } { inversion Heqn0; clear Heqn0; subst; intros. destruct (fin_case f'). subst; auto. destruct H0; subst. simpl. eapply IHn. congruence. } } Qed. | Theorem | theories | [
"Require Import ExtLib."
] | theories/Data/Vector.v | get_put_neq | |
ForallV : forall n, vector T n -> Prop := | ForallV_nil : ForallV (Vnil _) | ForallV_cons : forall n e es, P e -> @ForallV n es -> ForallV (Vcons e es). | Inductive | theories | [
"Require Import ExtLib."
] | theories/Data/Vector.v | ForallV | |
ForallV_vector_hd n (v : vector T (S n)) (f : ForallV v) : P (vector_hd v) := match f in @ForallV n v return match n as n return vector T n -> Prop with | 0 => fun _ => True | S _ => fun v => P (vector_hd v) end v with | ForallV_nil => I | ForallV_cons _ _ _ pf _ => pf end. | Definition | theories | [
"Require Import ExtLib."
] | theories/Data/Vector.v | ForallV_vector_hd | |
ForallV_vector_tl n (v : vector T (S n)) (f : ForallV v) : ForallV (vector_tl v) := match f in @ForallV n v return match n as n return vector T n -> Prop with | 0 => fun _ => True | S _ => fun v => ForallV (vector_tl v) end v with | ForallV_nil => I | ForallV_cons _ _ _ _ pf => pf end. | Definition | theories | [
"Require Import ExtLib."
] | theories/Data/Vector.v | ForallV_vector_tl | |
vector_dec {n} (a : vector T n) : forall b : vector T n, {a = b} + {a <> b} := match a in vector _ n return forall b : vector T n, {a = b} + {a <> b} with | Vnil => fun b => left match b in vector _ 0 with | Vnil => eq_refl end | Vcons _ a a' => fun b => match b as b in vector _ (S n) return forall a', (forall a : vector T n, {a' = a} + {a' <> a}) -> {Vcons a a' = b} + {Vcons a a' <> b} with | Vcons _ b b' => fun a' rec => match Tdec a b , rec b' with | left pf , left pf' => left match pf , pf' with | eq_refl , eq_refl => eq_refl end | right pf , _ => right (fun x : Vcons a a' = Vcons b b' => pf match x in _ = z return a = vector_hd z with | eq_refl => eq_refl end) | left _ , right pf => right (fun x : Vcons a a' = Vcons b b' => pf match x in _ = z return a' = vector_tl z with | eq_refl => eq_refl end) end end a' (@vector_dec _ a') end. | Fixpoint | theories | [
"Require Import ExtLib."
] | theories/Data/Vector.v | vector_dec | |
vector_In : forall {n}, vector T n -> Prop := | vHere : forall n rst, @vector_In (S n) (Vcons a rst) | vNext : forall n rst b, @vector_In n rst -> @vector_In (S n) (Vcons b rst). | Inductive | theories | [
"Require Import ExtLib."
] | theories/Data/Vector.v | vector_In | |
ForallV_vector_In : forall {n} t (vs : vector T n) P, ForallV P vs -> vector_In t vs -> P t. Proof. induction 2. - eapply (ForallV_vector_hd H). - eapply IHvector_In. eapply (ForallV_vector_tl H). Qed. | Lemma | theories | [
"Require Import ExtLib."
] | theories/Data/Vector.v | ForallV_vector_In | |
vector_map {n} (v : vector T n) : vector U n := match v with | Vnil => Vnil _ | Vcons _ v vs => Vcons (f v) (vector_map vs) end. | Fixpoint | theories | [
"Require Import ExtLib."
] | theories/Data/Vector.v | vector_map | |
hlist_to_tuple ps (h : hlist (fun x : Type => x) ps) : asTuple ps := match h in hlist _ ps return asTuple ps with | Hnil => tt | Hcons x h => (x,hlist_to_tuple h) end. | Fixpoint | theories | [
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Generic/Data.v | hlist_to_tuple | |
itype (ps : list Type) : Type := | Inj : Type -> itype ps | Rec : hlist (fun x => x) ps -> itype ps | Sum : itype ps -> itype ps -> itype ps | Prod : itype ps -> itype ps -> itype ps | Sig : forall T : Type, (T -> itype ps) -> itype ps | Pi : forall T : Type, (T -> itype ps) -> itype ps | Get : forall T : Type, member T ps -> (T -> itype ps) -> itype ps | Unf : forall T : Type, member T ps -> T -> itype ps -> itype ps. | Inductive | theories | [
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Generic/Data.v | itype | |
Unit {ps} := @Inj ps unit. | Definition | theories | [
"Require Import Coq.",
"Require Import ExtLib.",
"Require Import ExtLib.",
"Require Import ExtLib."
] | theories/Generic/Data.v | Unit |
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