The Coq Standard Library

# Library Coq.Init.Specif

Basic specifications : sets that may contain logical information
Set Implicit Arguments.
Require Import Notations.
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.
Inductive sig (A:Type) (P:A -> Prop) : Type :=
exist : forall x:A, P x -> sig P.
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)}.
Inductive sigT (A:Type) (P:A -> Type) : Type :=
existT : forall x:A, P x -> sigT P.
Inductive sigT2 (A:Type) (P Q:A -> Type) : Type :=
existT2 : forall x:A, P x -> Q x -> sigT2 P Q.
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 : 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.
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.
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.
End Projections.
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.
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).
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.
sumor is an option type equipped with the justification of why it may not be a regular value
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.
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.
Variables X : Set.
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.
Definition except := False_rec.
Theorem absurd_set : forall (A:Prop) (C:Set), A -> ~ A -> C.
Hint Resolve left right inleft inright: core.
Hint Resolve exist exist2 existT existT2: core.
Notation sigS := sigT (compat "8.2").
Notation existS := existT (compat "8.2").
Notation sigS_rect := sigT_rect (compat "8.2").
Notation sigS_rec := sigT_rec (compat "8.2").
Notation sigS_ind := sigT_ind (compat "8.2").
Notation projS1 := projT1 (compat "8.2").
Notation projS2 := projT2 (compat "8.2").
Notation sigS2 := sigT2 (compat "8.2").
Notation existS2 := existT2 (compat "8.2").
Notation sigS2_rect := sigT2_rect (compat "8.2").
Notation sigS2_rec := sigT2_rec (compat "8.2").
Notation sigS2_ind := sigT2_ind (compat "8.2").