The Coq Proof Assistant

# Library Coq.Init.Specif

Basic specifications : sets that may contain logical information
Set Implicit Arguments.
Require Import Notations.
Require Import Ltac.
Require Import Datatypes.
Require Import Logic.
Subsets and Sigma-types
(sig A P), or more suggestively {x:A | P x}, denotes the subset of elements of the type A which satisfy the predicate P. Similarly (sig2 A P Q), or {x:A | P x & Q x}, denotes the subset of elements of the type A which satisfy both P and Q.
#[universes(template)]
Inductive sig (A:Type) (P:A -> Prop) : Type :=
exist : forall x:A, P x -> sig P.
Register sig as core.sig.type.
Register exist as core.sig.intro.
Register sig_rect as core.sig.rect.
#[universes(template)]
Inductive sig2 (A:Type) (P Q:A -> Prop) : Type :=
exist2 : forall x:A, P x -> Q x -> sig2 P Q.
(sigT A P), or more suggestively {x:A & (P x)} is a Sigma-type. Similarly for (sigT2 A P Q), also written {x:A & (P x) & (Q x)}.
#[universes(template)]
Inductive sigT (A:Type) (P:A -> Type) : Type :=
existT : forall x:A, P x -> sigT P.
Register sigT as core.sigT.type.
Register existT as core.sigT.intro.
Register sigT_rect as core.sigT.rect.
#[universes(template)]
Inductive sigT2 (A:Type) (P Q:A -> Type) : Type :=
existT2 : forall x:A, P x -> Q x -> sigT2 P Q.
Arguments sig (A P)%_type.
Arguments sig2 (A P Q)%_type.
Arguments sigT (A P)%_type.
Arguments sigT2 (A P Q)%_type.
Notation "{ x | P }" := (sig (fun x => P)) : type_scope.
Notation "{ x | P & Q }" := (sig2 (fun x => P) (fun x => Q)) : type_scope.
Notation "{ x : A | P }" := (sig (A:=A) (fun x => P)) : type_scope.
Notation "{ x : A | P & Q }" := (sig2 (A:=A) (fun x => P) (fun x => Q)) :
type_scope.
Notation "{ x & P }" := (sigT (fun x => P)) : type_scope.
Notation "{ x & P & Q }" := (sigT2 (fun x => P) (fun x => Q)) : type_scope.
Notation "{ x : A & P }" := (sigT (A:=A) (fun x => P)) : type_scope.
Notation "{ x : A & P & Q }" := (sigT2 (A:=A) (fun x => P) (fun x => Q)) :
type_scope.
Notation "{ ' pat | P }" := (sig (fun pat => P)) : type_scope.
Notation "{ ' pat | P & Q }" := (sig2 (fun pat => P) (fun pat => Q)) : type_scope.
Notation "{ ' pat : A | P }" := (sig (A:=A) (fun pat => P)) : type_scope.
Notation "{ ' pat : A | P & Q }" := (sig2 (A:=A) (fun pat => P) (fun pat => Q)) :
type_scope.
Notation "{ ' pat & P }" := (sigT (fun pat => P)) : type_scope.
Notation "{ ' pat & P & Q }" := (sigT2 (fun pat => P) (fun pat => Q)) : type_scope.
Notation "{ ' pat : A & P }" := (sigT (A:=A) (fun pat => P)) : type_scope.
Notation "{ ' pat : A & P & Q }" := (sigT2 (A:=A) (fun pat => P) (fun pat => Q)) :
type_scope.
Add Printing Let sig.
Add Printing Let sig2.
Add Printing Let sigT.
Add Printing Let sigT2.
Projections of sig
An element y of a subset {x:A | (P x)} is the pair of an a of type A and of a proof h that a satisfies P. Then (proj1_sig y) is the witness a and (proj2_sig y) is the proof of (P a)
Section Subset_projections.
Variable A : Type.
Variable P : A -> Prop.
Definition proj1_sig (e:sig P) := match e with
| exist _ a b => a
end.
Definition proj2_sig (e:sig P) :=
match e return P (proj1_sig e) with
| exist _ a b => b
end.
Register proj1_sig as core.sig.proj1.
Register proj2_sig as core.sig.proj2.
End Subset_projections.
sig2 of a predicate can be projected to a sig.
This allows proj1_sig and proj2_sig to be usable with sig2.
The let statements occur in the body of the exist so that proj1_sig of a coerced X : sig2 P Q will unify with let (a, _, _) := X in a
Definition sig_of_sig2 (A : Type) (P Q : A -> Prop) (X : sig2 P Q) : sig P
:= exist P
(let (a, _, _) := X in a)
(let (x, p, _) as s return (P (let (a, _, _) := s in a)) := X in p).
Projections of sig2
An element y of a subset {x:A | (P x) & (Q x)} is the triple of an a of type A, a of a proof h that a satisfies P, and a proof h' that a satisfies Q. Then (proj1_sig (sig_of_sig2 y)) is the witness a, (proj2_sig (sig_of_sig2 y)) is the proof of (P a), and (proj3_sig y) is the proof of (Q a).
Section Subset_projections2.
Variable A : Type.
Variables P Q : A -> Prop.
Definition proj3_sig (e : sig2 P Q) :=
let (a, b, c) return Q (proj1_sig (sig_of_sig2 e)) := e in c.
End Subset_projections2.
Projections of sigT
An element x of a sigma-type {y:A & P y} is a dependent pair made of an a of type A and an h of type P a. Then, (projT1 x) is the first projection and (projT2 x) is the second projection, the type of which depends on the projT1.
Section Projections.
Variable A : Type.
Variable P : A -> Type.
Definition projT1 (x:sigT P) : A := match x with
| existT _ a _ => a
end.
Definition projT2 (x:sigT P) : P (projT1 x) :=
match x return P (projT1 x) with
| existT _ _ h => h
end.
Register projT1 as core.sigT.proj1.
Register projT2 as core.sigT.proj2.
End Projections.
Module SigTNotations.
Notation "( x ; y )" := (existT _ x y) (at level 0, format "( x ; '/ ' y )").
Notation "x .1" := (projT1 x) (at level 1, left associativity, format "x .1").
Notation "x .2" := (projT2 x) (at level 1, left associativity, format "x .2").
End SigTNotations.
Import SigTNotations.
sigT2 of a predicate can be projected to a sigT.
This allows projT1 and projT2 to be usable with sigT2.
The let statements occur in the body of the existT so that projT1 of a coerced X : sigT2 P Q will unify with let (a, _, _) := X in a
Definition sigT_of_sigT2 (A : Type) (P Q : A -> Type) (X : sigT2 P Q) : sigT P
:= existT P
(let (a, _, _) := X in a)
(let (x, p, _) as s return (P (let (a, _, _) := s in a)) := X in p).
Projections of sigT2
An element x of a sigma-type {y:A & P y & Q y} is a dependent pair made of an a of type A, an h of type P a, and an h' of type Q a. Then, (projT1 (sigT_of_sigT2 x)) is the first projection, (projT2 (sigT_of_sigT2 x)) is the second projection, and (projT3 x) is the third projection, the types of which depends on the projT1.
Section Projections2.
Variable A : Type.
Variables P Q : A -> Type.
Definition projT3 (e : sigT2 P Q) :=
let (a, b, c) return Q (projT1 (sigT_of_sigT2 e)) := e in c.
End Projections2.
Local Notation "x .3" := (projT3 x) (at level 1, left associativity, format "x .3").
sigT of a predicate is equivalent to sig
Definition sig_of_sigT (A : Type) (P : A -> Prop) (X : sigT P) : sig P
:= exist P (projT1 X) (projT2 X).
Definition sigT_of_sig (A : Type) (P : A -> Prop) (X : sig P) : sigT P
:= existT P (proj1_sig X) (proj2_sig X).
sigT2 of a predicate is equivalent to sig2
Definition sig2_of_sigT2 (A : Type) (P Q : A -> Prop) (X : sigT2 P Q) : sig2 P Q
:= exist2 P Q (projT1 (sigT_of_sigT2 X)) (projT2 (sigT_of_sigT2 X)) (projT3 X).
Definition sigT2_of_sig2 (A : Type) (P Q : A -> Prop) (X : sig2 P Q) : sigT2 P Q
:= existT2 P Q (proj1_sig (sig_of_sig2 X)) (proj2_sig (sig_of_sig2 X)) (proj3_sig X).
sig of a predicate on Props can be turned into ex
Definition ex_of_sig (A : Type) (P : A -> Prop) (X : sig P) : ex P
:= ex_intro P (proj1_sig X) (proj2_sig X).
sigT of a predicate on Props can be turned into ex
Definition ex_of_sigT (A : Type) (P : A -> Prop) (X : sigT P) : ex P
:= ex_of_sig (sig_of_sigT X).
sig2 of a predicate on Props can be turned into ex2
Definition ex2_of_sig2 (A : Type) (P Q : A -> Prop) (X : sig2 P Q) : ex2 P Q
:= ex_intro2 P Q (proj1_sig (sig_of_sig2 X)) (proj2_sig (sig_of_sig2 X)) (proj3_sig X).
sigT2 of a predicate on Props can be turned into ex2
Definition ex2_of_sigT2 (A : Type) (P Q : A -> Prop) (X : sigT2 P Q) : ex2 P Q
:= ex2_of_sig2 (sig2_of_sigT2 X).
η Principles
Definition sigT_eta {A P} (p : { a : A & P a })
: p = existT _ (projT1 p) (projT2 p).
Definition sig_eta {A P} (p : { a : A | P a })
: p = exist _ (proj1_sig p) (proj2_sig p).
Definition sigT2_eta {A P Q} (p : { a : A & P a & Q a })
: p = existT2 _ _ (projT1 (sigT_of_sigT2 p)) (projT2 (sigT_of_sigT2 p)) (projT3 p).
Definition sig2_eta {A P Q} (p : { a : A | P a & Q a })
: p = exist2 _ _ (proj1_sig (sig_of_sig2 p)) (proj2_sig (sig_of_sig2 p)) (proj3_sig p).
exists x : A, B is equivalent to inhabited {x : A | B}
Lemma exists_to_inhabited_sig {A P} : (exists x : A, P x) -> inhabited {x : A | P x}.
Lemma inhabited_sig_to_exists {A P} : inhabited {x : A | P x} -> exists x : A, P x.
Subtyping for prod
Section ProdSigT.
Variable A B : Type.
Definition sigT_of_prod (p : A * B) := (fst p; snd p).
Definition prod_of_sigT (s : { _ : A & B }) := (s.1, s.2).
Lemma sigT_prod_sigT p : sigT_of_prod (prod_of_sigT p) = p.
Lemma prod_sigT_prod s : prod_of_sigT (sigT_of_prod s) = s.
End ProdSigT.
Equality of sigma types
Import EqNotations.
Equality for sigT
Section sigT.
Local Unset Implicit Arguments.
Projecting an equality of a pair to equality of the first components
Definition 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.
Projecting an equality of a pair to equality of the second components
Definition 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.
Equality of sigT is itself a sigT (forwards-reasoning version)
Definition 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).
Equality of sigT is itself a sigT (backwards-reasoning version)
Definition 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.
Lemma 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).
Local Notation "(= u ; v )" := (eq_existT_curried u v) (at level 0, format "(= u ; '/ ' v )").
Lemma 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).
Lemma 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').
Theorem 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').
Curried version of proving equality of sigma types
Definition 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).
In order to have a performant inversion_sigma, we define specialized versions for when we have constructors on one or both sides of the equality
Definition 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 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.
Equality of sigT when the property is an hProp
Definition 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 _ _ _).
Equivalence of equality of sigT with a sigT of equality We could actually prove an isomorphism here, and not just <->, but for simplicity, we don't.
Definition 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 }.
Induction principle for @eq (sigT _)
Definition 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.
Definition eq_sigT_rec {A P u v} (Q : u = v :> { a : A & P a } -> Set) := eq_sigT_rect Q.
Definition eq_sigT_ind {A P u v} (Q : u = v :> { a : A & P a } -> Prop) := eq_sigT_rec Q.
In order to have a performant inversion_sigma, we define specialized versions for when we have constructors on one or both sides of the equality
Definition 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 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 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.
We want uncurried versions so inversion_sigma can accept intropatterns, but we use ex types for the induction hypothesis to avoid extraction errors about informative inductive types having Prop instances
Definition 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 eq_sigT_rec_uncurried {A P u v} (Q : u = v :> { a : A & P a } -> Set) := eq_sigT_rect_uncurried Q.
Definition eq_sigT_ind_uncurried {A P u v} (Q : u = v :> { a : A & P a } -> Prop) := eq_sigT_rec_uncurried Q.
Equivalence of equality of sigT involving hProps with equality of the first components
Definition 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).
Non-dependent classification of equality of sigT
Definition 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).
Classification of transporting across an equality of sigTs
Lemma 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)).
End sigT.
Global Arguments eq_existT_curried A P _ _ _ _ !p !q / .
Equality for sig
Section sig.
We define this as a Let rather than a Definition to avoid extraction errors about informative inductive types having Prop instances
Local Unset Implicit Arguments.
Projecting an equality of a pair to equality of the first components
Definition 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.
Projecting an equality of a pair to equality of the second components
Definition 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.
Equality of sig is itself a sig (forwards-reasoning version)
Definition 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.
Equality of sig is itself a sig (backwards-reasoning version)
Definition 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.
Lemma 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.
Curried version of proving equality of sigma types
Definition 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).
In order to have a performant inversion_sigma, we define specialized versions for when we have constructors on one or both sides of the equality
Definition 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 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.
Induction principle for @eq (sig _)
Definition 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.
Definition eq_sig_rec {A P u v} (Q : u = v :> { a : A | P a } -> Set) := eq_sig_rect Q.
Definition eq_sig_ind {A P u v} (Q : u = v :> { a : A | P a } -> Prop) := eq_sig_rec Q.
In order to have a performant inversion_sigma, we define specialized versions for when we have constructors on one or both sides of the equality
Definition 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 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 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.
We want uncurried versions so inversion_sigma can accept intropatterns, but we use ex types for the induction hypothesis to avoid extraction errors about informative inductive types having Prop instances
Definition 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 eq_sig_rec_uncurried {A P u v} (Q : u = v :> { a : A | P a } -> Set) := eq_sig_rect_uncurried Q.
Definition eq_sig_ind_uncurried {A P u v} (Q : u = v :> { a : A | P a } -> Prop) := eq_sig_rec_uncurried Q.
Equality of sig when the property is an hProp
Definition 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 _ _ _).
Equivalence of equality of sig with a sig of equality We could actually prove an isomorphism here, and not just <->, but for simplicity, we don't.
Definition 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 }.
Equivalence of equality of sig involving hProps with equality of the first components
Definition 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).
Lemma 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).
End sig.
Global Arguments eq_exist_curried A P _ _ _ _ !p !q / .
Equality for sigT2
Section sigT2.
Local Coercion sigT_of_sigT2 : sigT2 >-> sigT.
Local Coercion ex_of_ex2 : ex2 >-> ex.
Local Unset Implicit Arguments.
Projecting an equality of a pair to equality of the first components
Definition 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 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).
Projecting an equality of a pair to equality of the second components
Definition 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.
Projecting an equality of a pair to equality of the third components
Definition 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.
Equality of sigT2 is itself a sigT2 (forwards-reasoning version)
Definition 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.
Equality of sigT2 is itself a sigT2 (backwards-reasoning version)
Definition 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.
Lemma 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.
Curried version of proving equality of sigma types
Definition 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).
In order to have a performant inversion_sigma, we define specialized versions for when we have constructors on one or both sides of the equality
Definition 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 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.
Equality of sigT2 when the second property is an hProp
Definition 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 _ _ _).
Equivalence of equality of sigT2 with a sigT2 of equality We could actually prove an isomorphism here, and not just <->, but for simplicity, we don't.
Definition 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 }.
Induction principle for @eq (sigT2 _ _)
Definition 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.
Definition eq_sigT2_rec {A P Q u v} (R : u = v :> { a : A & P a & Q a } -> Set) := eq_sigT2_rect R.
Definition eq_sigT2_ind {A P Q u v} (R : u = v :> { a : A & P a & Q a } -> Prop) := eq_sigT2_rec R.
In order to have a performant inversion_sigma, we define specialized versions for when we have constructors on one or both sides of the equality
Definition 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 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 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.
We want uncurried versions so inversion_sigma can accept intropatterns, but we use ex2 types for the induction hypothesis to avoid extraction errors about informative inductive types having Prop instances
Definition 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 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 eq_sigT2_ind_uncurried {A P Q u v} (R : u = v :> { a : A & P a & Q a } -> Prop) := eq_sigT2_rec_uncurried R.
Equivalence of equality of sigT2 involving hProps with equality of the first components
Definition 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).
Non-dependent classification of equality of sigT
Definition 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).
Classification of transporting across an equality of sigT2s
Lemma 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 u.3).
End sigT2.
Global Arguments eq_existT2_curried A P Q _ _ _ _ _ _ !p !q !r / .
Equality for sig2
Section sig2.
Local Coercion sig_of_sig2 : sig2 >-> sig.
Local Coercion ex_of_ex2 : ex2 >-> ex.
Local Unset Implicit Arguments.
Projecting an equality of a pair to equality of the first components
Definition 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 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).
Projecting an equality of a pair to equality of the second components
Definition 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.
Projecting an equality of a pair to equality of the third components
Definition 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.
Equality of sig2 is itself a sig2 (fowards-reasoning version)
Definition 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.
Equality of sig2 is itself a sig2 (backwards-reasoning version)
Definition 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.
Lemma 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.
Curried version of proving equality of sigma types
Definition 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).
In order to have a performant inversion_sigma, we define specialized versions for when we have constructors on one or both sides of the equality
Definition 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 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.
Equality of sig2 when the second property is an hProp
Definition 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 _ _ _).
Equivalence of equality of sig2 with a sig2 of equality We could actually prove an isomorphism here, and not just <->, but for simplicity, we don't.
Definition 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 }.
Induction principle for @eq (sig2 _ _)
Definition 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.
Definition eq_sig2_rec {A P Q u v} (R : u = v :> { a : A | P a & Q a } -> Set) := eq_sig2_rect R.
Definition eq_sig2_ind {A P Q u v} (R : u = v :> { a : A | P a & Q a } -> Prop) := eq_sig2_rec R.
In order to have a performant inversion_sigma, we define specialized versions for when we have constructors on one or both sides of the equality
Definition 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 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 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.
We want uncurried versions so inversion_sigma can accept intropatterns, but we use ex2 types for the induction hypothesis to avoid extraction errors about informative inductive types having Prop instances
Definition 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 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 eq_sig2_ind_uncurried {A P Q u v} (R : u = v :> { a : A | P a & Q a } -> Prop) := eq_sig2_rec_uncurried R.
Equivalence of equality of sig2 involving hProps with equality of the first components
Definition 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).
Non-dependent classification of equality of sig
Definition 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).
Classification of transporting across an equality of sig2s
Lemma 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 dependent H in proj3_sig u).
End sig2.
Global Arguments eq_exist2_curried A P Q _ _ _ _ _ _ !p !q !r / .
sumbool is a boolean type equipped with the justification of their value
Inductive 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.
sumor is an option type equipped with the justification of why it may not be a regular value
#[universes(template)]
Inductive sumor (A:Type) (B:Prop) : Type :=
| inleft : A -> A + {B}
| inright : B -> A + {B}
where "A + { B }" := (sumor A B) : type_scope.
Add Printing If sumor.
Arguments inleft {A B} _ , [A] B _.
Arguments inright {A B} _ , A [B] _.
Various forms of the axiom of choice for specifications
Section Choice_lemmas.
Variables S S' : Set.
Variable R : S -> S' -> Prop.
Variable R' : S -> S' -> Set.
Variables R1 R2 : S -> Prop.
Lemma Choice :
(forall x:S, {y:S' | R x y}) -> {f:S -> S' | forall z:S, R z (f z)}.
Lemma Choice2 :
(forall x:S, {y:S' & R' x y}) -> {f:S -> S' & forall z:S, R' z (f z)}.
Lemma bool_choice :
(forall x:S, {R1 x} + {R2 x}) ->
{f:S -> bool | forall x:S, f x = true /\ R1 x \/ f x = false /\ R2 x}.
End Choice_lemmas.
Section Dependent_choice_lemmas.
Variable X : Type.
Variable R : X -> X -> Prop.
Lemma dependent_choice :
(forall x:X, {y | R x y}) ->
forall x0, {f : nat -> X | f O = x0 /\ forall n, R (f n) (f (S n))}.
End Dependent_choice_lemmas.
A result of type (Exc A) is either a normal value of type A or an error :
Inductive Exc [A:Type] : Type := value : A->(Exc A) | error : (Exc A).
It is implemented using the option type.
Section Exc.
Variable A : Type.
Definition Exc := option A.
Definition value := @Some A.
Definition error := @None A.
End Exc.
Arguments error {A}.
Definition except := False_rec.
Arguments except [P] _.
Theorem absurd_set : forall (A:Prop) (C:Set), A -> ~ A -> C.
#[global]
Hint Resolve left right inleft inright: core.
#[global]
Hint Resolve exist exist2 existT existT2: core.