fact stringlengths 6 2.88k | type stringclasses 17
values | library stringclasses 2
values | imports listlengths 0 16 | filename stringclasses 89
values | symbolic_name stringlengths 1 36 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
_Has_J_Has_J_forward@{sa se sp;la le lp} eq Has_refl
{has_J : Has_J@{sa se sp;la le lp} eq Has_refl} : Has_J_forward@{sa se sp;la le lp} eq Has_refl
:= Has_J_Has_J_forward _ _.
Hint Resolve _Has_J_Has_J_forward : rewrite_instances. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac"
] | Corelib/Init/Equality.v | _Has_J_Has_J_forward | |
Has_J_r_Has_J_r_forward@{sa se sp;la le lp} eq Has_refl
{has_J : Has_J_r@{sa se sp;la le lp} eq Has_refl} :
forall (A : Type@{sa ; la}) (x : A) (P : forall y : A, eq A y x -> Type@{sp ; lp}) y e,
P y e -> P x (refl A x) :=
fun A x P y e => J_r_eliminator _ _ (fun y e => P y e -> P _ _) (fun x => x) _ _. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac"
] | Corelib/Init/Equality.v | Has_J_r_Has_J_r_forward | |
_Has_J_r_Has_J_r_forward@{sa se sp;la le lp} eq Has_refl
{has_J : Has_J_r@{sa se sp;la le lp} eq Has_refl} : Has_J_r_forward@{sa se sp;la le lp} eq Has_refl
:= Has_J_r_Has_J_r_forward _ _.
Hint Resolve _Has_J_r_Has_J_r_forward : rewrite_instances. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac"
] | Corelib/Init/Equality.v | _Has_J_r_Has_J_r_forward | |
uint:=
| Nil
| D0 (_:uint)
| D1 (_:uint)
| D2 (_:uint)
| D3 (_:uint)
| D4 (_:uint)
| D5 (_:uint)
| D6 (_:uint)
| D7 (_:uint)
| D8 (_:uint)
| D9 (_:uint)
| Da (_:uint)
| Db (_:uint)
| Dc (_:uint)
| Dd (_:uint)
| De (_:uint)
| Df (_:uint). | Inductive | Corelib | [
"Require Import Datatypes Specif Decimal"
] | Corelib/Init/Hexadecimal.v | uint | |
signed_int:= Pos (d:uint) | Neg (d:uint).
Abbreviation int := signed_int. | Variant | Corelib | [
"Require Import Datatypes Specif Decimal"
] | Corelib/Init/Hexadecimal.v | signed_int | |
hexadecimal:=
| Hexadecimal (i:int) (f:uint)
| HexadecimalExp (i:int) (f:uint) (e:Decimal.int). | Variant | Corelib | [
"Require Import Datatypes Specif Decimal"
] | Corelib/Init/Hexadecimal.v | hexadecimal | |
Equalityfor uint. | Scheme | Corelib | [
"Require Import Datatypes Specif Decimal"
] | Corelib/Init/Hexadecimal.v | Equality | |
Equalityfor int. | Scheme | Corelib | [
"Require Import Datatypes Specif Decimal"
] | Corelib/Init/Hexadecimal.v | Equality | |
Equalityfor hexadecimal.
Abbreviation int_eq_dec := signed_int_eq_dec.
Abbreviation int_beq := signed_int_beq.
Abbreviation internal_int_dec_lb := internal_signed_int_dec_lb.
Abbreviation internal_int_dec_bl := internal_signed_int_dec_bl.
Declare Scope hex_uint_scope.
Delimit Scope hex_uint_scope with huint.
Bind Scop... | Scheme | Corelib | [
"Require Import Datatypes Specif Decimal"
] | Corelib/Init/Hexadecimal.v | Equality | |
nb_digitsd :=
match d with
| Nil => O
| D0 d | D1 d | D2 d | D3 d | D4 d | D5 d | D6 d | D7 d | D8 d | D9 d
| Da d | Db d | Dc d | Dd d | De d | Df d =>
S (nb_digits d)
end. | Fixpoint | Corelib | [
"Require Import Datatypes Specif Decimal"
] | Corelib/Init/Hexadecimal.v | nb_digits | |
nzheadd :=
match d with
| D0 d => nzhead d
| _ => d
end. | Fixpoint | Corelib | [
"Require Import Datatypes Specif Decimal"
] | Corelib/Init/Hexadecimal.v | nzhead | |
unormd :=
match nzhead d with
| Nil => zero
| d => d
end. | Definition | Corelib | [
"Require Import Datatypes Specif Decimal"
] | Corelib/Init/Hexadecimal.v | unorm | |
normd :=
match d with
| Pos d => Pos (unorm d)
| Neg d =>
match nzhead d with
| Nil => Pos zero
| d => Neg d
end
end. | Definition | Corelib | [
"Require Import Datatypes Specif Decimal"
] | Corelib/Init/Hexadecimal.v | norm | |
opp(d:int) :=
match d with
| Pos d => Neg d
| Neg d => Pos d
end. | Definition | Corelib | [
"Require Import Datatypes Specif Decimal"
] | Corelib/Init/Hexadecimal.v | opp | |
abs(d:int) : uint :=
match d with
| Pos d => d
| Neg d => d
end. | Definition | Corelib | [
"Require Import Datatypes Specif Decimal"
] | Corelib/Init/Hexadecimal.v | abs | |
revapp(d d' : uint) :=
match d with
| Nil => d'
| D0 d => revapp d (D0 d')
| D1 d => revapp d (D1 d')
| D2 d => revapp d (D2 d')
| D3 d => revapp d (D3 d')
| D4 d => revapp d (D4 d')
| D5 d => revapp d (D5 d')
| D6 d => revapp d (D6 d')
| D7 d => revapp d (D7 d')
| D8 d => revapp d (D8 d')
| D9 ... | Fixpoint | Corelib | [
"Require Import Datatypes Specif Decimal"
] | Corelib/Init/Hexadecimal.v | revapp | |
revd := revapp d Nil. | Definition | Corelib | [
"Require Import Datatypes Specif Decimal"
] | Corelib/Init/Hexadecimal.v | rev | |
appd d' := revapp (rev d) d'. | Definition | Corelib | [
"Require Import Datatypes Specif Decimal"
] | Corelib/Init/Hexadecimal.v | app | |
app_intd1 d2 :=
match d1 with Pos d1 => Pos (app d1 d2) | Neg d1 => Neg (app d1 d2) end. | Definition | Corelib | [
"Require Import Datatypes Specif Decimal"
] | Corelib/Init/Hexadecimal.v | app_int | |
nztaild :=
let fix aux d_rev :=
match d_rev with
| D0 d_rev => let (r, n) := aux d_rev in pair r (S n)
| _ => pair d_rev O
end in
let (r, n) := aux (rev d) in pair (rev r) n. | Definition | Corelib | [
"Require Import Datatypes Specif Decimal"
] | Corelib/Init/Hexadecimal.v | nztail | |
nztail_intd :=
match d with
| Pos d => let (r, n) := nztail d in pair (Pos r) n
| Neg d => let (r, n) := nztail d in pair (Neg r) n
end. | Definition | Corelib | [
"Require Import Datatypes Specif Decimal"
] | Corelib/Init/Hexadecimal.v | nztail_int | |
del_headn d :=
match n with
| O => d
| S n =>
match d with
| Nil => zero
| D0 d | D1 d | D2 d | D3 d | D4 d | D5 d | D6 d | D7 d | D8 d | D9 d
| Da d | Db d | Dc d | Dd d | De d | Df d =>
del_head n d
end
end. | Fixpoint | Corelib | [
"Require Import Datatypes Specif Decimal"
] | Corelib/Init/Hexadecimal.v | del_head | |
del_head_intn d :=
match d with
| Pos d => del_head n d
| Neg d => del_head n d
end. | Definition | Corelib | [
"Require Import Datatypes Specif Decimal"
] | Corelib/Init/Hexadecimal.v | del_head_int | |
del_tailn d := rev (del_head n (rev d)). | Definition | Corelib | [
"Require Import Datatypes Specif Decimal"
] | Corelib/Init/Hexadecimal.v | del_tail | |
del_tail_intn d :=
match d with
| Pos d => Pos (del_tail n d)
| Neg d => Neg (del_tail n d)
end. | Definition | Corelib | [
"Require Import Datatypes Specif Decimal"
] | Corelib/Init/Hexadecimal.v | del_tail_int | |
succd :=
match d with
| Nil => D1 Nil
| D0 d => D1 d
| D1 d => D2 d
| D2 d => D3 d
| D3 d => D4 d
| D4 d => D5 d
| D5 d => D6 d
| D6 d => D7 d
| D7 d => D8 d
| D8 d => D9 d
| D9 d => Da d
| Da d => Db d
| Db d => Dc d
| Dc d => Dd d
| Dd d => De d
| De d => Df d
| Df d => D0 (succ d)... | Fixpoint | Corelib | [
"Require Import Datatypes Specif Decimal"
] | Corelib/Init/Hexadecimal.v | succ | |
doubled :=
match d with
| Nil => Nil
| D0 d => D0 (double d)
| D1 d => D2 (double d)
| D2 d => D4 (double d)
| D3 d => D6 (double d)
| D4 d => D8 (double d)
| D5 d => Da (double d)
| D6 d => Dc (double d)
| D7 d => De (double d)
| D8 d => D0 (succ_double d)
| D9 d => D2 (succ_double d)
| Da d ... | Fixpoint | Corelib | [
"Require Import Datatypes Specif Decimal"
] | Corelib/Init/Hexadecimal.v | double | |
True: Prop :=
I : True.
Register True as core.True.type.
Register I as core.True.I. | Inductive | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | True | |
False: Prop :=.
Register False as core.False.type. | Inductive | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | False | |
not(A:Prop) := A -> False. | Definition | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | not | |
notT(A:Type) := A -> False. | Definition | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | notT | |
and(A B:Prop) : Prop :=
conj : A -> B -> A /\ B
where "A /\ B" := (and A B) : type_scope.
Register and as core.and.type.
Register conj as core.and.conj. | Inductive | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | and | |
proj1: A /\ B -> A.
Proof.
destruct 1; trivial.
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | proj1 | |
proj2: A /\ B -> B.
Proof.
destruct 1; trivial.
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | proj2 | |
or(A B:Prop) : Prop :=
| or_introl : A -> A \/ B
| or_intror : B -> A \/ B
where "A \/ B" := (or A B) : type_scope.
Arguments or_introl [A B] _, [A] B _.
Arguments or_intror [A B] _, A [B] _.
Register or as core.or.type. | Inductive | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | or | |
iff(A B:Prop) := (A -> B) /\ (B -> A). | Definition | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | iff | |
iff_refl: forall A:Prop, A <-> A.
Proof.
split; auto.
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | iff_refl | |
iff_trans: forall A B C:Prop, (A <-> B) -> (B <-> C) -> (A <-> C).
Proof.
intros A B C [H1 H2] [H3 H4]; split; auto.
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | iff_trans | |
iff_sym: forall A B:Prop, (A <-> B) -> (B <-> A).
Proof.
intros A B [H1 H2]; split; auto.
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | iff_sym | |
and_iff_compat_l: forall A B C : Prop,
(B <-> C) -> (A /\ B <-> A /\ C).
Proof.
intros ? ? ? [Hl Hr]; split; intros [? ?]; (split; [ assumption | ]);
[apply Hl | apply Hr]; assumption.
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | and_iff_compat_l | |
and_iff_compat_r: forall A B C : Prop,
(B <-> C) -> (B /\ A <-> C /\ A).
Proof.
intros ? ? ? [Hl Hr]; split; intros [? ?]; (split; [ | assumption ]);
[apply Hl | apply Hr]; assumption.
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | and_iff_compat_r | |
or_iff_compat_l: forall A B C : Prop,
(B <-> C) -> (A \/ B <-> A \/ C).
Proof.
intros ? ? ? [Hl Hr]; split; (intros [?|?]; [left; assumption| right]);
[apply Hl | apply Hr]; assumption.
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | or_iff_compat_l | |
or_iff_compat_r: forall A B C : Prop,
(B <-> C) -> (B \/ A <-> C \/ A).
Proof.
intros ? ? ? [Hl Hr]; split; (intros [?|?]; [left| right; assumption]);
[apply Hl | apply Hr]; assumption.
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | or_iff_compat_r | |
imp_iff_compat_l: forall A B C : Prop,
(B <-> C) -> ((A -> B) <-> (A -> C)).
Proof.
intros ? ? ? [Hl Hr]; split; intros H ?; [apply Hl | apply Hr]; apply H; assumption.
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | imp_iff_compat_l | |
imp_iff_compat_r: forall A B C : Prop,
(B <-> C) -> ((B -> A) <-> (C -> A)).
Proof.
intros ? ? ? [Hl Hr]; split; intros H ?; [apply H, Hr | apply H, Hl]; assumption.
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | imp_iff_compat_r | |
not_iff_compat: forall A B : Prop,
(A <-> B) -> (~ A <-> ~B).
Proof.
intros; apply imp_iff_compat_r; assumption.
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | not_iff_compat | |
neg_false: forall A : Prop, ~ A <-> (A <-> False).
Proof.
intro A; unfold not; split.
- intro H; split; [exact H | intro H1; elim H1].
- intros [H _]; exact H.
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | neg_false | |
and_cancel_l: forall A B C : Prop,
(B -> A) -> (C -> A) -> ((A /\ B <-> A /\ C) <-> (B <-> C)).
Proof.
intros A B C Hl Hr.
split; [ | apply and_iff_compat_l]; intros [HypL HypR]; split; intros.
+ apply HypL; split; [apply Hl | ]; assumption.
+ apply HypR; split; [apply Hr | ]; assumption.
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | and_cancel_l | |
and_cancel_r: forall A B C : Prop,
(B -> A) -> (C -> A) -> ((B /\ A <-> C /\ A) <-> (B <-> C)).
Proof.
intros A B C Hl Hr.
split; [ | apply and_iff_compat_r]; intros [HypL HypR]; split; intros.
+ apply HypL; split; [ | apply Hl ]; assumption.
+ apply HypR; split; [ | apply Hr ]; assumption.
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | and_cancel_r | |
and_comm: forall A B : Prop, A /\ B <-> B /\ A.
Proof.
intros; split; intros [? ?]; split; assumption.
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | and_comm | |
and_assoc: forall A B C : Prop, (A /\ B) /\ C <-> A /\ B /\ C.
Proof.
intros; split; [ intros [[? ?] ?]| intros [? [? ?]]]; repeat split; assumption.
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | and_assoc | |
or_cancel_l: forall A B C : Prop,
(B -> ~ A) -> (C -> ~ A) -> ((A \/ B <-> A \/ C) <-> (B <-> C)).
Proof.
intros ? ? ? Fl Fr; split; [ | apply or_iff_compat_l]; intros [Hl Hr]; split; intros.
{ destruct Hl; [ right | destruct Fl | ]; assumption. }
{ destruct Hr; [ right | destruct Fr | ]; assumption. }
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | or_cancel_l | |
or_cancel_r: forall A B C : Prop,
(B -> ~ A) -> (C -> ~ A) -> ((B \/ A <-> C \/ A) <-> (B <-> C)).
Proof.
intros ? ? ? Fl Fr; split; [ | apply or_iff_compat_r]; intros [Hl Hr]; split; intros.
{ destruct Hl; [ left | | destruct Fl ]; assumption. }
{ destruct Hr; [ left | | destruct Fr ]; assumption. }
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | or_cancel_r | |
or_comm: forall A B : Prop, (A \/ B) <-> (B \/ A).
Proof.
intros; split; (intros [? | ?]; [ right | left ]; assumption).
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | or_comm | |
or_assoc: forall A B C : Prop, (A \/ B) \/ C <-> A \/ B \/ C.
Proof.
intros; split; [ intros [[?|?]|?]| intros [?|[?|?]]].
+ left; assumption.
+ right; left; assumption.
+ right; right; assumption.
+ left; left; assumption.
+ left; right; assumption.
+ right; assumption.
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | or_assoc | |
iff_and: forall A B : Prop, (A <-> B) -> (A -> B) /\ (B -> A).
Proof.
intros A B []; split; trivial.
Qed. | Lemma | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | iff_and | |
iff_to_and: forall A B : Prop, (A <-> B) <-> (A -> B) /\ (B -> A).
Proof.
intros; split; intros [Hl Hr]; (split; intros; [ apply Hl | apply Hr]); assumption.
Qed. | Lemma | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | iff_to_and | |
ex(A:Type) (P:A -> Prop) : Prop :=
ex_intro : forall x:A, P x -> ex (A:=A) P.
Register ex as core.ex.type.
Register ex_intro as core.ex.intro. | Inductive | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | ex | |
ex_proj1(x:ex P) : A :=
match x with ex_intro _ a _ => a end. | Definition | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | ex_proj1 | |
ex_proj2(x:ex P) : P (ex_proj1 x) :=
match x with ex_intro _ _ b => b end.
Register ex_proj1 as core.ex.proj1.
Register ex_proj2 as core.ex.proj2. | Definition | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | ex_proj2 | |
ex2(A:Type) (P Q:A -> Prop) : Prop :=
ex_intro2 : forall x:A, P x -> Q x -> ex2 (A:=A) P Q. | Inductive | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | ex2 | |
ex_of_ex2(A : Prop) (P Q : A -> Prop) (X : ex2 P Q) : ex P
:= ex_intro P
(let (a, _, _) := X in a)
(let (x, p, _) as s return (P (let (a, _, _) := s in a)) := X in p). | Definition | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | ex_of_ex2 | |
ex_proj3(x:ex2 P Q) : Q (ex_proj1 (ex_of_ex2 x)) :=
match x with ex_intro2 _ _ _ _ b => b end. | Definition | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | ex_proj3 | |
all(A:Type) (P:A -> Prop) := forall x:A, P x.
Register all as core.all. | Definition | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | all | |
inst: forall x:A, all (fun x => P x) -> P x.
Proof.
unfold all; auto.
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | inst | |
gen: forall (B:Prop) (f:forall y:A, B -> P y), B -> all P.
Proof.
red; auto.
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | gen | |
eq(A:Type) (x:A) : A -> Prop :=
eq_refl : x = x :>A
where "x = y :> A" := (@eq A x y) : type_scope.
Arguments eq {A} x _.
Arguments eq_refl {A x} , [A] x.
Arguments eq_ind [A] x P _ y _ : rename.
Arguments eq_rec [A] x P _ y _ : rename.
Arguments eq_rect [A] x P _ y _ : rename. | Inductive | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | eq | |
Rewritingfor eq.
Register eq_rew_dep as core.eq.rect_dep.
Register eq_rew_dep as core.eq.ind_dep.
Register eq_rew_r_dep as core.eq.rect_r_dep.
Register eq_rew_r_dep as core.eq.ind_r_dep.
Arguments eq_sym_involutive [A]%_type_scope [x y] e : rename. | Scheme | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | Rewriting | |
absurd: forall A C:Prop, A -> ~ A -> C.
Proof.
unfold not; intros A C h1 h2.
destruct (h2 h1).
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | absurd | |
eq_trans: x = y -> y = z -> x = z.
Proof.
destruct 2; trivial.
Defined.
Register eq_trans as core.eq.trans. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | eq_trans | |
eq_trans_r: x = y -> z = y -> x = z.
Proof.
destruct 2; trivial.
Defined. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | eq_trans_r | |
f_equal: x = y -> f x = f y.
Proof.
destruct 1; trivial.
Defined.
Register f_equal as core.eq.congr. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | f_equal | |
not_eq_sym: x <> y -> y <> x.
Proof.
red; intros h1 h2; apply h1; destruct h2; trivial.
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | not_eq_sym | |
eq_sind_r:
forall (A:Type) (x:A) (P:A -> SProp), P x -> forall y:A, y = x -> P y.
Proof.
intros A x P H y H0. elim eq_sym with (1 := H0); assumption.
Defined. | Definition | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | eq_sind_r | |
eq_ind_r:
forall (A:Type) (x:A) (P:A -> Prop), P x -> forall y:A, y = x -> P y.
intros A x P H y H0. elim eq_sym with (1 := H0); assumption.
Defined.
Register eq_ind_r as core.eq.ind_r.
Register eq_sind_r as core.eq.sind_r. | Definition | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | eq_ind_r | |
eq_rec_r:
forall (A:Type) (x:A) (P:A -> Set), P x -> forall y:A, y = x -> P y.
intros A x P H y H0; elim eq_sym with (1 := H0); assumption.
Defined. | Definition | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | eq_rec_r | |
eq_rect_r:
forall (A:Type) (x:A) (P:A -> Type), P x -> forall y:A, y = x -> P y.
intros A x P H y H0; elim eq_sym with (1 := H0); assumption.
Defined.
Register eq_rect_r as core.eq.rect_r.
Register eq_rec_r as core.eq.rec_r. | Definition | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | eq_rect_r | |
f_equal_dep(H: x = y) : rew H in f x = f y.
Proof.
destruct H; reflexivity.
Defined. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | f_equal_dep | |
f_equal_dep2{A A' B B'} (f : A -> A') (g : forall a:A, B a -> B' (f a))
{x1 x2 : A} {y1 : B x1} {y2 : B x2} (H : x1 = x2) :
rew H in y1 = y2 -> rew f_equal f H in g x1 y1 = g x2 y2.
Proof.
destruct H, 1. reflexivity.
Defined. | Lemma | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | f_equal_dep2 | |
rew_opp_rA (P:A->Type) (x y:A) (H:x=y) (a:P y) : rew H in rew <- H in a = a.
Proof.
destruct H.
reflexivity.
Defined. | Lemma | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | rew_opp_r | |
rew_opp_lA (P:A->Type) (x y:A) (H:x=y) (a:P x) : rew <- H in rew H in a = a.
Proof.
destruct H.
reflexivity.
Defined. | Lemma | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | rew_opp_l | |
f_equal2:
forall (A1 A2 B:Type) (f:A1 -> A2 -> B) (x1 y1:A1)
(x2 y2:A2), x1 = y1 -> x2 = y2 -> f x1 x2 = f y1 y2.
Proof.
destruct 1; destruct 1; reflexivity.
Qed.
Register f_equal2 as core.eq.congr2. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | f_equal2 | |
f_equal3:
forall (A1 A2 A3 B:Type) (f:A1 -> A2 -> A3 -> B) (x1 y1:A1)
(x2 y2:A2) (x3 y3:A3),
x1 = y1 -> x2 = y2 -> x3 = y3 -> f x1 x2 x3 = f y1 y2 y3.
Proof.
destruct 1; destruct 1; destruct 1; reflexivity.
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | f_equal3 | |
f_equal4:
forall (A1 A2 A3 A4 B:Type) (f:A1 -> A2 -> A3 -> A4 -> B)
(x1 y1:A1) (x2 y2:A2) (x3 y3:A3) (x4 y4:A4),
x1 = y1 -> x2 = y2 -> x3 = y3 -> x4 = y4 -> f x1 x2 x3 x4 = f y1 y2 y3 y4.
Proof.
destruct 1; destruct 1; destruct 1; destruct 1; reflexivity.
Qed. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | f_equal4 | |
f_equal5:
forall (A1 A2 A3 A4 A5 B:Type) (f:A1 -> A2 -> A3 -> A4 -> A5 -> B)
(x1 y1:A1) (x2 y2:A2) (x3 y3:A3) (x4 y4:A4) (x5 y5:A5),
x1 = y1 ->
x2 = y2 ->
x3 = y3 -> x4 = y4 -> x5 = y5 -> f x1 x2 x3 x4 x5 = f y1 y2 y3 y4 y5.
Proof.
destruct 1; destruct 1; destruct 1; destruct 1; destruct 1; reflexiv... | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | f_equal5 | |
f_equal_composeA B C (a b:A) (f:A->B) (g:B->C) (e:a=b) :
f_equal g (f_equal f e) = f_equal (fun a => g (f a)) e.
Proof.
destruct e. reflexivity.
Defined. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | f_equal_compose | |
eq_trans_refl_lA (x y:A) (e:x=y) : eq_trans eq_refl e = e.
Proof.
destruct e. reflexivity.
Defined. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | eq_trans_refl_l | |
eq_trans_refl_rA (x y:A) (e:x=y) : eq_trans e eq_refl = e.
Proof.
destruct e. reflexivity.
Defined. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | eq_trans_refl_r | |
eq_trans_sym_inv_lA (x y:A) (e:x=y) : eq_trans (eq_sym e) e = eq_refl.
Proof.
destruct e; reflexivity.
Defined. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | eq_trans_sym_inv_l | |
eq_trans_sym_inv_rA (x y:A) (e:x=y) : eq_trans e (eq_sym e) = eq_refl.
Proof.
destruct e; reflexivity.
Defined. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | eq_trans_sym_inv_r | |
eq_trans_assocA (x y z t:A) (e:x=y) (e':y=z) (e'':z=t) :
eq_trans e (eq_trans e' e'') = eq_trans (eq_trans e e') e''.
Proof.
destruct e''; reflexivity.
Defined. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | eq_trans_assoc | |
rew_mapA B (P:B->Type) (f:A->B) x1 x2 (H:x1=x2) (y:P (f x1)) :
rew [fun x => P (f x)] H in y = rew f_equal f H in y.
Proof.
destruct H; reflexivity.
Defined. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | rew_map | |
eq_trans_map{A B} {x1 x2 x3:A} {y1:B x1} {y2:B x2} {y3:B x3}
(H1:x1=x2) (H2:x2=x3) (H1': rew H1 in y1 = y2) (H2': rew H2 in y2 = y3) :
rew eq_trans H1 H2 in y1 = y3.
Proof.
destruct H2. exact (eq_trans H1' H2').
Defined. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | eq_trans_map | |
map_subst{A} {P Q:A->Type} (f : forall x, P x -> Q x) {x y} (H:x=y) (z:P x) :
rew H in f x z = f y (rew H in z).
Proof.
destruct H. reflexivity.
Defined. | Lemma | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | map_subst | |
map_subst_map{A B} {P:A->Type} {Q:B->Type} (f:A->B) (g : forall x, P x -> Q (f x))
{x y} (H:x=y) (z:P x) :
rew f_equal f H in g x z = g y (rew H in z).
Proof.
destruct H. reflexivity.
Defined. | Lemma | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | map_subst_map | |
rew_swapA (P:A->Type) x1 x2 (H:x1=x2) (y1:P x1) (y2:P x2) : rew H in y1 = y2 -> y1 = rew <- H in y2.
Proof.
destruct H. trivial.
Defined. | Lemma | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | rew_swap | |
rew_composeA (P:A->Type) x1 x2 x3 (H1:x1=x2) (H2:x2=x3) (y:P x1) :
rew H2 in rew H1 in y = rew (eq_trans H1 H2) in y.
Proof.
destruct H2. reflexivity.
Defined. | Lemma | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | rew_compose | |
eq_id_comm_lA (f:A->A) (Hf:forall a, a = f a) a : f_equal f (Hf a) = Hf (f a).
Proof.
unfold f_equal.
rewrite <- (eq_trans_sym_inv_l (Hf a)).
destruct (Hf a) at 1 2.
destruct (Hf a).
reflexivity.
Defined. | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | eq_id_comm_l | |
eq_id_comm_rA (f:A->A) (Hf:forall a, f a = a) a : f_equal f (Hf a) = Hf (f a).
Proof.
unfold f_equal.
rewrite <- (eq_trans_sym_inv_l (Hf (f (f a)))).
set (Hfsymf := fun a => eq_sym (Hf a)).
change (eq_sym (Hf (f (f a)))) with (Hfsymf (f (f a))).
pattern (Hfsymf (f (f a))).
destruct (eq_id_comm_l f Hfsymf (f... | Theorem | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | eq_id_comm_r | |
eq_refl_map_distrA B x (f:A->B) : f_equal f (eq_refl x) = eq_refl (f x).
Proof.
reflexivity.
Qed. | Lemma | Corelib | [
"Require Export Notations",
"Require Import Ltac"
] | Corelib/Init/Logic.v | eq_refl_map_distr |
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