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 |
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inhabited_sig_to_exists{A P} : inhabited {x : A | P x} -> exists x : A, P x.
Proof.
intros [[x y]];exists x;exact y.
Qed. | Lemma | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | inhabited_sig_to_exists | |
sigT_of_prod(p : A * B) := (fst p; snd p). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | sigT_of_prod | |
prod_of_sigT(s : { _ : A & B }) := (s.1, s.2). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | prod_of_sigT | |
sigT_prod_sigTp : sigT_of_prod (prod_of_sigT p) = p.
Proof. destruct p; reflexivity. Qed. | Lemma | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | sigT_prod_sigT | |
prod_sigT_prods : prod_of_sigT (sigT_of_prod s) = s.
Proof. destruct s; reflexivity. Qed. | Lemma | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | prod_sigT_prod | |
projT1_eq{A} {P : A -> Type} {u v : { a : A & P a }} (p : u = v)
: u.1 = v.1
:= f_equal (fun x => x.1) p. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | projT1_eq | |
projT2_eq{A} {P : A -> Type} {u v : { a : A & P a }} (p : u = v)
: rew projT1_eq p in u.2 = v.2
:= rew dependent p in eq_refl. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | projT2_eq | |
eq_existT_uncurried{A : Type} {P : A -> Type} {u1 v1 : A} {u2 : P u1} {v2 : P v1}
(pq : { p : u1 = v1 & rew p in u2 = v2 })
: (u1; u2) = (v1; v2).
Proof.
destruct pq as [p q].
destruct q; simpl in *.
destruct p; reflexivity.
Defined. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_existT_uncurried | |
eq_sigT_uncurried{A : Type} {P : A -> Type} (u v : { a : A & P a })
(pq : { p : u.1 = v.1 & rew p in u.2 = v.2 })
: u = v.
Proof.
destruct u as [u1 u2], v as [v1 v2]; simpl in *.
apply eq_existT_uncurried; exact pq.
Defined. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT_uncurried | |
eq_existT_curried{A : Type} {P : A -> Type} {u1 v1 : A} {u2 : P u1} {v2 : P v1}
(p : u1 = v1) (q : rew p in u2 = v2) : (u1; u2) = (v1; v2).
Proof.
apply eq_sigT_uncurried; exists p; exact q.
Defined.
Local Notation "(= u ; v )" := (eq_existT_curried u v) (at level 0, format "(= u ; '/ ' v )"). | Lemma | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_existT_curried | |
eq_existT_curried_map{A A' P P'} (f:A -> A') (g:forall u:A, P u -> P' (f u))
{u1 v1 : A} {u2 : P u1} {v2 : P v1} (p : u1 = v1) (q : rew p in u2 = v2) :
f_equal (fun x => (f x.1; g x.1 x.2)) (= p; q) =
(= f_equal f p; f_equal_dep2 f g p q).
Proof.
destruct p, q. reflexivity.
Defined. | Lemma | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_existT_curried_map | |
eq_existT_curried_trans{A P} {u1 v1 w1 : A} {u2 : P u1} {v2 : P v1} {w2 : P w1}
(p : u1 = v1) (q : rew p in u2 = v2)
(p' : v1 = w1) (q': rew p' in v2 = w2) :
eq_trans (= p; q) (= p'; q') =
(= eq_trans p p'; eq_trans_map p p' q q').
Proof.
destruct p', q'. reflexivity.
Defined. | Lemma | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_existT_curried_trans | |
eq_existT_curried_congr{A P} {u1 v1 : A} {u2 : P u1} {v2 : P v1}
{p p' : u1 = v1} {q : rew p in u2 = v2} {q': rew p' in u2 = v2}
(r : p = p') : rew [fun H => rew H in u2 = v2] r in q = q' -> (= p; q) = (= p'; q').
Proof.
destruct r, 1. reflexivity.
Qed. | Theorem | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_existT_curried_congr | |
eq_sigT{A : Type} {P : A -> Type} (u v : { a : A & P a })
(p : u.1 = v.1) (q : rew p in u.2 = v.2)
: u = v
:= eq_sigT_uncurried u v (existT _ p q). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT | |
eq_existT_l{A : Type} {P : A -> Type} {u1 : A} {u2 : P u1} {v : { a : A & P a }}
(p : u1 = v.1) (q : rew p in u2 = v.2) : (u1; u2) = v
:= eq_sigT (u1; u2) v p q. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_existT_l | |
eq_existT_r{A : Type} {P : A -> Type} {u : { a : A & P a }} {v1 : A} {v2 : P v1}
(p : u.1 = v1) (q : rew p in u.2 = v2) : u = (v1; v2)
:= eq_sigT u (v1; v2) p q. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_existT_r | |
eq_sigT_hprop{A P} (P_hprop : forall (x : A) (p q : P x), p = q)
(u v : { a : A & P a })
(p : u.1 = v.1)
: u = v
:= eq_sigT u v p (P_hprop _ _ _). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT_hprop | |
eq_sigT_uncurried_iff{A P}
(u v : { a : A & P a })
: u = v <-> { p : u.1 = v.1 & rew p in u.2 = v.2 }.
Proof.
split; [ intro; subst; exists eq_refl; reflexivity | apply eq_sigT_uncurried ].
Defined. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT_uncurried_iff | |
eq_sigT_rect{A P} {u v : { a : A & P a }} (Q : u = v -> Type)
(f : forall p q, Q (eq_sigT u v p q))
: forall p, Q p.
Proof. intro p; specialize (f (projT1_eq p) (projT2_eq p)); destruct u, p; exact f. Defined. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT_rect | |
eq_sigT_rec{A P u v} (Q : u = v :> { a : A & P a } -> Set) := eq_sigT_rect Q. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT_rec | |
eq_sigT_ind{A P u v} (Q : u = v :> { a : A & P a } -> Prop) := eq_sigT_rec Q. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT_ind | |
eq_sigT_rect_existT_l{A P} {u1 u2 v} (Q : _ -> Type)
(f : forall p q, Q (@eq_existT_l A P u1 u2 v p q))
: forall p, Q p
:= eq_sigT_rect Q f. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT_rect_existT_l | |
eq_sigT_rect_existT_r{A P} {u v1 v2} (Q : _ -> Type)
(f : forall p q, Q (@eq_existT_r A P u v1 v2 p q))
: forall p, Q p
:= eq_sigT_rect Q f. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT_rect_existT_r | |
eq_sigT_rect_existT{A P} {u1 u2 v1 v2} (Q : _ -> Type)
(f : forall p q, Q (@eq_existT_curried A P u1 v1 u2 v2 p q))
: forall p, Q p
:= eq_sigT_rect Q f. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT_rect_existT | |
eq_sigT_rect_uncurried{A P} {u v : { a : A & P a }} (Q : u = v -> Type)
(f : forall pq : exists p : u.1 = v.1, _, Q (eq_sigT u v (ex_proj1 pq) (ex_proj2 pq)))
: forall p, Q p
:= eq_sigT_rect Q (fun p q => f (ex_intro _ p q)). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT_rect_uncurried | |
eq_sigT_rec_uncurried{A P u v} (Q : u = v :> { a : A & P a } -> Set) := eq_sigT_rect_uncurried Q. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT_rec_uncurried | |
eq_sigT_ind_uncurried{A P u v} (Q : u = v :> { a : A & P a } -> Prop) := eq_sigT_rec_uncurried Q. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT_ind_uncurried | |
eq_sigT_hprop_iff{A P} (P_hprop : forall (x : A) (p q : P x), p = q)
(u v : { a : A & P a })
: u = v <-> (u.1 = v.1)
:= conj (fun p => f_equal (@projT1 _ _) p) (eq_sigT_hprop P_hprop u v). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT_hprop_iff | |
eq_sigT_nondep{A B : Type} (u v : { a : A & B })
(p : u.1 = v.1) (q : u.2 = v.2)
: u = v
:= @eq_sigT _ _ u v p (eq_trans (rew_const _ _) q). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT_nondep | |
rew_sigT{A x} {P : A -> Type} (Q : forall a, P a -> Prop) (u : { p : P x & Q x p }) {y} (H : x = y)
: rew [fun a => { p : P a & Q a p }] H in u
= existT
(Q y)
(rew H in u.1)
(rew dependent H in (u.2)).
Proof.
destruct H, u; reflexivity.
Defined. | Lemma | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | rew_sigT | |
proj1_sig_eq{A} {P : A -> Prop} {u v : { a : A | P a }} (p : u = v)
: proj1_sig u = proj1_sig v
:= f_equal (@proj1_sig _ _) p. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | proj1_sig_eq | |
proj2_sig_eq{A} {P : A -> Prop} {u v : { a : A | P a }} (p : u = v)
: rew proj1_sig_eq p in proj2_sig u = proj2_sig v
:= rew dependent p in eq_refl. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | proj2_sig_eq | |
eq_exist_uncurried{A : Type} {P : A -> Prop} {u1 v1 : A} {u2 : P u1} {v2 : P v1}
(pq : { p : u1 = v1 | rew p in u2 = v2 })
: exist _ u1 u2 = exist _ v1 v2.
Proof.
destruct pq as [p q].
destruct q; simpl in *.
destruct p; reflexivity.
Defined. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_exist_uncurried | |
eq_sig_uncurried{A : Type} {P : A -> Prop} (u v : { a : A | P a })
(pq : { p : proj1_sig u = proj1_sig v | rew p in proj2_sig u = proj2_sig v })
: u = v.
Proof.
destruct u as [u1 u2], v as [v1 v2]; simpl in *.
apply eq_exist_uncurried; exact pq.
Defined. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig_uncurried | |
eq_exist_curried{A : Type} {P : A -> Prop} {u1 v1 : A} {u2 : P u1} {v2 : P v1}
(p : u1 = v1) (q : rew p in u2 = v2) : exist P u1 u2 = exist P v1 v2.
Proof.
apply eq_sig_uncurried; exists p; exact q.
Defined. | Lemma | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_exist_curried | |
eq_sig{A : Type} {P : A -> Prop} (u v : { a : A | P a })
(p : proj1_sig u = proj1_sig v) (q : rew p in proj2_sig u = proj2_sig v)
: u = v
:= eq_sig_uncurried u v (exist _ p q). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig | |
eq_exist_l{A : Type} {P : A -> Prop} {u1 : A} {u2 : P u1} {v : { a : A | P a }}
(p : u1 = proj1_sig v) (q : rew p in u2 = proj2_sig v) : exist _ u1 u2 = v
:= eq_sig (exist _ u1 u2) v p q. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_exist_l | |
eq_exist_r{A : Type} {P : A -> Prop} {u : { a : A | P a }} {v1 : A} {v2 : P v1}
(p : proj1_sig u = v1) (q : rew p in proj2_sig u = v2) : u = exist _ v1 v2
:= eq_sig u (exist _ v1 v2) p q. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_exist_r | |
eq_sig_rect{A P} {u v : { a : A | P a }} (Q : u = v -> Type)
(f : forall p q, Q (eq_sig u v p q))
: forall p, Q p.
Proof. intro p; specialize (f (proj1_sig_eq p) (proj2_sig_eq p)); destruct u, p; exact f. Defined. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig_rect | |
eq_sig_rec{A P u v} (Q : u = v :> { a : A | P a } -> Set) := eq_sig_rect Q. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig_rec | |
eq_sig_ind{A P u v} (Q : u = v :> { a : A | P a } -> Prop) := eq_sig_rec Q. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig_ind | |
eq_sig_rect_exist_l{A P} {u1 u2 v} (Q : _ -> Type)
(f : forall p q, Q (@eq_exist_l A P u1 u2 v p q))
: forall p, Q p
:= eq_sig_rect Q f. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig_rect_exist_l | |
eq_sig_rect_exist_r{A P} {u v1 v2} (Q : _ -> Type)
(f : forall p q, Q (@eq_exist_r A P u v1 v2 p q))
: forall p, Q p
:= eq_sig_rect Q f. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig_rect_exist_r | |
eq_sig_rect_exist{A P} {u1 u2 v1 v2} (Q : _ -> Type)
(f : forall p q, Q (@eq_exist_curried A P u1 v1 u2 v2 p q))
: forall p, Q p
:= eq_sig_rect Q f. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig_rect_exist | |
eq_sig_rect_uncurried{A P} {u v : { a : A | P a }} (Q : u = v -> Type)
(f : forall pq : exists p : proj1_sig u = proj1_sig v, _, Q (eq_sig u v (ex_proj1 pq) (ex_proj2 pq)))
: forall p, Q p
:= eq_sig_rect Q (fun p q => f (ex_intro _ p q)). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig_rect_uncurried | |
eq_sig_rec_uncurried{A P u v} (Q : u = v :> { a : A | P a } -> Set) := eq_sig_rect_uncurried Q. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig_rec_uncurried | |
eq_sig_ind_uncurried{A P u v} (Q : u = v :> { a : A | P a } -> Prop) := eq_sig_rec_uncurried Q. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig_ind_uncurried | |
eq_sig_hprop{A} {P : A -> Prop} (P_hprop : forall (x : A) (p q : P x), p = q)
(u v : { a : A | P a })
(p : proj1_sig u = proj1_sig v)
: u = v
:= eq_sig u v p (P_hprop _ _ _). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig_hprop | |
eq_sig_uncurried_iff{A} {P : A -> Prop}
(u v : { a : A | P a })
: u = v <-> { p : proj1_sig u = proj1_sig v | rew p in proj2_sig u = proj2_sig v }.
Proof.
split; [ intro; subst; exists eq_refl; reflexivity | apply eq_sig_uncurried ].
Defined. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig_uncurried_iff | |
eq_sig_hprop_iff{A} {P : A -> Prop} (P_hprop : forall (x : A) (p q : P x), p = q)
(u v : { a : A | P a })
: u = v <-> (proj1_sig u = proj1_sig v)
:= conj (fun p => f_equal (@proj1_sig _ _) p) (eq_sig_hprop P_hprop u v). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig_hprop_iff | |
rew_sig{A x} {P : A -> Type} (Q : forall a, P a -> Prop) (u : { p : P x | Q x p }) {y} (H : x = y)
: rew [fun a => { p : P a | Q a p }] H in u
= exist
(Q y)
(rew H in proj1_sig u)
(rew dependent H in proj2_sig u).
Proof.
destruct H, u; reflexivity.
Defined. | Lemma | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | rew_sig | |
sigT_of_sigT2_eq{A} {P Q : A -> Type} {u v : { a : A & P a & Q a }} (p : u = v)
: u = v :> { a : A & P a }
:= f_equal _ p. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | sigT_of_sigT2_eq | |
projT1_of_sigT2_eq{A} {P Q : A -> Type} {u v : { a : A & P a & Q a }} (p : u = v)
: u.1 = v.1
:= projT1_eq (sigT_of_sigT2_eq p). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | projT1_of_sigT2_eq | |
projT2_of_sigT2_eq{A} {P Q : A -> Type} {u v : { a : A & P a & Q a }} (p : u = v)
: rew projT1_of_sigT2_eq p in u.2 = v.2
:= rew dependent p in eq_refl. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | projT2_of_sigT2_eq | |
projT3_eq{A} {P Q : A -> Type} {u v : { a : A & P a & Q a }} (p : u = v)
: rew projT1_of_sigT2_eq p in u.3 = v.3
:= rew dependent p in eq_refl. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | projT3_eq | |
eq_existT2_uncurried{A : Type} {P Q : A -> Type}
{u1 v1 : A} {u2 : P u1} {v2 : P v1} {u3 : Q u1} {v3 : Q v1}
(pqr : { p : u1 = v1
& rew p in u2 = v2 & rew p in u3 = v3 })
: existT2 _ _ u1 u2 u3 = existT2 _ _ v1 v2 v3.
Proof.
destruct pqr as [p q r].
destruct r... | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_existT2_uncurried | |
eq_sigT2_uncurried{A : Type} {P Q : A -> Type} (u v : { a : A & P a & Q a })
(pqr : { p : u.1 = v.1
& rew p in u.2 = v.2 & rew p in u.3 = v.3 })
: u = v.
Proof.
destruct u as [u1 u2 u3], v as [v1 v2 v3]; simpl in *.
apply eq_existT2_uncurried; exact pqr.
Defined. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT2_uncurried | |
eq_existT2_curried{A : Type} {P Q : A -> Type} {u1 v1 : A} {u2 : P u1} {v2 : P v1} {u3 : Q u1} {v3 : Q v1}
(p : u1 = v1) (q : rew p in u2 = v2) (r : rew p in u3 = v3) : existT2 P Q u1 u2 u3 = existT2 P Q v1 v2 v3.
Proof.
apply eq_sigT2_uncurried; exists p; exact q + exact r.
Defined. | Lemma | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_existT2_curried | |
eq_sigT2{A : Type} {P Q : A -> Type} (u v : { a : A & P a & Q a })
(p : u.1 = v.1)
(q : rew p in u.2 = v.2)
(r : rew p in u.3 = v.3)
: u = v
:= eq_sigT2_uncurried u v (existT2 _ _ p q r). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT2 | |
eq_existT2_l{A : Type} {P Q : A -> Type} {u1 : A} {u2 : P u1} {u3 : Q u1} {v : { a : A & P a & Q a }}
(p : u1 = v.1) (q : rew p in u2 = v.2) (r : rew p in u3 = v.3) : existT2 P Q u1 u2 u3 = v
:= eq_sigT2 (existT2 P Q u1 u2 u3) v p q r. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_existT2_l | |
eq_existT2_r{A : Type} {P Q : A -> Type} {u : { a : A & P a & Q a }} {v1 : A} {v2 : P v1} {v3 : Q v1}
(p : u.1 = v1) (q : rew p in u.2 = v2) (r : rew p in u.3 = v3) : u = existT2 P Q v1 v2 v3
:= eq_sigT2 u (existT2 P Q v1 v2 v3) p q r. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_existT2_r | |
eq_sigT2_hprop{A P Q} (Q_hprop : forall (x : A) (p q : Q x), p = q)
(u v : { a : A & P a & Q a })
(p : u = v :> { a : A & P a })
: u = v
:= eq_sigT2 u v (projT1_eq p) (projT2_eq p) (Q_hprop _ _ _). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT2_hprop | |
eq_sigT2_uncurried_iff{A P Q}
(u v : { a : A & P a & Q a })
: u = v
<-> { p : u.1 = v.1
& rew p in u.2 = v.2 & rew p in u.3 = v.3 }.
Proof.
split; [ intro; subst; exists eq_refl; reflexivity | apply eq_sigT2_uncurried ].
Defined. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT2_uncurried_iff | |
eq_sigT2_rect{A P Q} {u v : { a : A & P a & Q a }} (R : u = v -> Type)
(f : forall p q r, R (eq_sigT2 u v p q r))
: forall p, R p.
Proof.
intro p.
specialize (f (projT1_of_sigT2_eq p) (projT2_of_sigT2_eq p) (projT3_eq p)).
destruct u, p; exact f.
Defined. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT2_rect | |
eq_sigT2_rec{A P Q u v} (R : u = v :> { a : A & P a & Q a } -> Set) := eq_sigT2_rect R. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT2_rec | |
eq_sigT2_ind{A P Q u v} (R : u = v :> { a : A & P a & Q a } -> Prop) := eq_sigT2_rec R. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT2_ind | |
eq_sigT2_rect_existT2_l{A P Q} {u1 u2 u3 v} (R : _ -> Type)
(f : forall p q r, R (@eq_existT2_l A P Q u1 u2 u3 v p q r))
: forall p, R p
:= eq_sigT2_rect R f. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT2_rect_existT2_l | |
eq_sigT2_rect_existT2_r{A P Q} {u v1 v2 v3} (R : _ -> Type)
(f : forall p q r, R (@eq_existT2_r A P Q u v1 v2 v3 p q r))
: forall p, R p
:= eq_sigT2_rect R f. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT2_rect_existT2_r | |
eq_sigT2_rect_existT2{A P Q} {u1 u2 u3 v1 v2 v3} (R : _ -> Type)
(f : forall p q r, R (@eq_existT2_curried A P Q u1 v1 u2 v2 u3 v3 p q r))
: forall p, R p
:= eq_sigT2_rect R f. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT2_rect_existT2 | |
eq_sigT2_rect_uncurried{A P Q} {u v : { a : A & P a & Q a }} (R : u = v -> Type)
(f : forall pqr : exists2 p : u.1 = v.1, _ & _, R (eq_sigT2 u v (ex_proj1 pqr) (ex_proj2 pqr) (ex_proj3 pqr)))
: forall p, R p
:= eq_sigT2_rect R (fun p q r => f (ex_intro2 _ _ p q r)). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT2_rect_uncurried | |
eq_sigT2_rec_uncurried{A P Q u v} (R : u = v :> { a : A & P a & Q a } -> Set) := eq_sigT2_rect_uncurried R. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT2_rec_uncurried | |
eq_sigT2_ind_uncurried{A P Q u v} (R : u = v :> { a : A & P a & Q a } -> Prop) := eq_sigT2_rec_uncurried R. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT2_ind_uncurried | |
eq_sigT2_hprop_iff{A P Q} (Q_hprop : forall (x : A) (p q : Q x), p = q)
(u v : { a : A & P a & Q a })
: u = v <-> (u = v :> { a : A & P a })
:= conj (fun p => f_equal (@sigT_of_sigT2 _ _ _) p) (eq_sigT2_hprop Q_hprop u v). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT2_hprop_iff | |
eq_sigT2_nondep{A B C : Type} (u v : { a : A & B & C })
(p : u.1 = v.1) (q : u.2 = v.2) (r : u.3 = v.3)
: u = v
:= @eq_sigT2 _ _ _ u v p (eq_trans (rew_const _ _) q) (eq_trans (rew_const _ _) r). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sigT2_nondep | |
rew_sigT2{A x} {P : A -> Type} (Q R : forall a, P a -> Prop)
(u : { p : P x & Q x p & R x p })
{y} (H : x = y)
: rew [fun a => { p : P a & Q a p & R a p }] H in u
= existT2
(Q y)
(R y)
(rew H in u.1)
(rew dependent H in u.2)
(rew dependent H in... | Lemma | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | rew_sigT2 | |
sig_of_sig2_eq{A} {P Q : A -> Prop} {u v : { a : A | P a & Q a }} (p : u = v)
: u = v :> { a : A | P a }
:= f_equal _ p. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | sig_of_sig2_eq | |
proj1_sig_of_sig2_eq{A} {P Q : A -> Prop} {u v : { a : A | P a & Q a }} (p : u = v)
: proj1_sig u = proj1_sig v
:= proj1_sig_eq (sig_of_sig2_eq p). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | proj1_sig_of_sig2_eq | |
proj2_sig_of_sig2_eq{A} {P Q : A -> Prop} {u v : { a : A | P a & Q a }} (p : u = v)
: rew proj1_sig_of_sig2_eq p in proj2_sig u = proj2_sig v
:= rew dependent p in eq_refl. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | proj2_sig_of_sig2_eq | |
proj3_sig_eq{A} {P Q : A -> Prop} {u v : { a : A | P a & Q a }} (p : u = v)
: rew proj1_sig_of_sig2_eq p in proj3_sig u = proj3_sig v
:= rew dependent p in eq_refl. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | proj3_sig_eq | |
eq_exist2_uncurried{A} {P Q : A -> Prop}
{u1 v1 : A} {u2 : P u1} {v2 : P v1} {u3 : Q u1} {v3 : Q v1}
(pqr : { p : u1 = v1
| rew p in u2 = v2 & rew p in u3 = v3 })
: exist2 _ _ u1 u2 u3 = exist2 _ _ v1 v2 v3.
Proof.
destruct pqr as [p q r].
destruct r, q, p; si... | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_exist2_uncurried | |
eq_sig2_uncurried{A} {P Q : A -> Prop} (u v : { a : A | P a & Q a })
(pqr : { p : proj1_sig u = proj1_sig v
| rew p in proj2_sig u = proj2_sig v & rew p in proj3_sig u = proj3_sig v })
: u = v.
Proof.
destruct u as [u1 u2 u3], v as [v1 v2 v3]; simpl in *.
apply eq_exist2_u... | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig2_uncurried | |
eq_exist2_curried{A : Type} {P Q : A -> Prop} {u1 v1 : A} {u2 : P u1} {v2 : P v1} {u3 : Q u1} {v3 : Q v1}
(p : u1 = v1) (q : rew p in u2 = v2) (r : rew p in u3 = v3) : exist2 P Q u1 u2 u3 = exist2 P Q v1 v2 v3.
Proof.
apply eq_sig2_uncurried; exists p; exact q + exact r.
Defined. | Lemma | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_exist2_curried | |
eq_sig2{A} {P Q : A -> Prop} (u v : { a : A | P a & Q a })
(p : proj1_sig u = proj1_sig v)
(q : rew p in proj2_sig u = proj2_sig v)
(r : rew p in proj3_sig u = proj3_sig v)
: u = v
:= eq_sig2_uncurried u v (exist2 _ _ p q r). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig2 | |
eq_exist2_l{A : Type} {P Q : A -> Prop} {u1 : A} {u2 : P u1} {u3 : Q u1} {v : { a : A | P a & Q a }}
(p : u1 = proj1_sig v) (q : rew p in u2 = proj2_sig v) (r : rew p in u3 = proj3_sig v) : exist2 P Q u1 u2 u3 = v
:= eq_sig2 (exist2 P Q u1 u2 u3) v p q r. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_exist2_l | |
eq_exist2_r{A : Type} {P Q : A -> Prop} {u : { a : A | P a & Q a }} {v1 : A} {v2 : P v1} {v3 : Q v1}
(p : proj1_sig u = v1) (q : rew p in proj2_sig u = v2) (r : rew p in proj3_sig u = v3) : u = exist2 P Q v1 v2 v3
:= eq_sig2 u (exist2 P Q v1 v2 v3) p q r. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_exist2_r | |
eq_sig2_hprop{A} {P Q : A -> Prop} (Q_hprop : forall (x : A) (p q : Q x), p = q)
(u v : { a : A | P a & Q a })
(p : u = v :> { a : A | P a })
: u = v
:= eq_sig2 u v (proj1_sig_eq p) (proj2_sig_eq p) (Q_hprop _ _ _). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig2_hprop | |
eq_sig2_uncurried_iff{A P Q}
(u v : { a : A | P a & Q a })
: u = v
<-> { p : proj1_sig u = proj1_sig v
| rew p in proj2_sig u = proj2_sig v & rew p in proj3_sig u = proj3_sig v }.
Proof.
split; [ intro; subst; exists eq_refl; reflexivity | apply eq_sig2_uncurried ].
Defined. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig2_uncurried_iff | |
eq_sig2_rect{A P Q} {u v : { a : A | P a & Q a }} (R : u = v -> Type)
(f : forall p q r, R (eq_sig2 u v p q r))
: forall p, R p.
Proof.
intro p.
specialize (f (proj1_sig_of_sig2_eq p) (proj2_sig_of_sig2_eq p) (proj3_sig_eq p)).
destruct u, p; exact f.
Defined. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig2_rect | |
eq_sig2_rec{A P Q u v} (R : u = v :> { a : A | P a & Q a } -> Set) := eq_sig2_rect R. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig2_rec | |
eq_sig2_ind{A P Q u v} (R : u = v :> { a : A | P a & Q a } -> Prop) := eq_sig2_rec R. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig2_ind | |
eq_sig2_rect_exist2_l{A P Q} {u1 u2 u3 v} (R : _ -> Type)
(f : forall p q r, R (@eq_exist2_l A P Q u1 u2 u3 v p q r))
: forall p, R p
:= eq_sig2_rect R f. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig2_rect_exist2_l | |
eq_sig2_rect_exist2_r{A P Q} {u v1 v2 v3} (R : _ -> Type)
(f : forall p q r, R (@eq_exist2_r A P Q u v1 v2 v3 p q r))
: forall p, R p
:= eq_sig2_rect R f. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig2_rect_exist2_r | |
eq_sig2_rect_exist2{A P Q} {u1 u2 u3 v1 v2 v3} (R : _ -> Type)
(f : forall p q r, R (@eq_exist2_curried A P Q u1 v1 u2 v2 u3 v3 p q r))
: forall p, R p
:= eq_sig2_rect R f. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig2_rect_exist2 | |
eq_sig2_rect_uncurried{A P Q} {u v : { a : A | P a & Q a }} (R : u = v -> Type)
(f : forall pqr : exists2 p : proj1_sig u = proj1_sig v, _ & _, R (eq_sig2 u v (ex_proj1 pqr) (ex_proj2 pqr) (ex_proj3 pqr)))
: forall p, R p
:= eq_sig2_rect R (fun p q r => f (ex_intro2 _ _ p q r)). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig2_rect_uncurried | |
eq_sig2_rec_uncurried{A P Q u v} (R : u = v :> { a : A | P a & Q a } -> Set) := eq_sig2_rect_uncurried R. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig2_rec_uncurried | |
eq_sig2_ind_uncurried{A P Q u v} (R : u = v :> { a : A | P a & Q a } -> Prop) := eq_sig2_rec_uncurried R. | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig2_ind_uncurried | |
eq_sig2_hprop_iff{A} {P Q : A -> Prop} (Q_hprop : forall (x : A) (p q : Q x), p = q)
(u v : { a : A | P a & Q a })
: u = v <-> (u = v :> { a : A | P a })
:= conj (fun p => f_equal (@sig_of_sig2 _ _ _) p) (eq_sig2_hprop Q_hprop u v). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig2_hprop_iff | |
eq_sig2_nondep{A} {B C : Prop} (u v : @sig2 A (fun _ => B) (fun _ => C))
(p : proj1_sig u = proj1_sig v) (q : proj2_sig u = proj2_sig v) (r : proj3_sig u = proj3_sig v)
: u = v
:= @eq_sig2 _ _ _ u v p (eq_trans (rew_const _ _) q) (eq_trans (rew_const _ _) r). | Definition | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | eq_sig2_nondep | |
rew_sig2{A x} {P : A -> Type} (Q R : forall a, P a -> Prop)
(u : { p : P x | Q x p & R x p })
{y} (H : x = y)
: rew [fun a => { p : P a | Q a p & R a p }] H in u
= exist2
(Q y)
(R y)
(rew H in proj1_sig u)
(rew dependent H in proj2_sig u)
(rew ... | Lemma | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | rew_sig2 | |
sumbool(A B:Prop) : Set :=
| left : A -> {A} + {B}
| right : B -> {A} + {B}
where "{ A } + { B }" := (sumbool A B) : type_scope.
Add Printing If sumbool.
Arguments left {A B} _, [A] B _.
Arguments right {A B} _ , A [B] _.
Register sumbool as core.sumbool.type. | Inductive | Corelib | [
"Require Import Notations",
"Require Import Ltac",
"Require Import Datatypes",
"Require Import Logic"
] | Corelib/Init/Specif.v | sumbool |
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