| Basic specifications : Sets containing logical information |
Require Datatypes.
Require Logic.
Require LogicSyntax.
Section Subsets.
(sig A P), or more suggestively {x:A | (P x)}, denotes the subset of elements of the Set 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 Set A which satisfy both P and Q.
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Inductive sig [A:Set;P:A->Prop] : Set
:= exist : (x:A)(P x) -> (sig A P).
Inductive sig2 [A:Set;P,Q:A->Prop] : Set
:= exist2 : (x:A)(P x) -> (Q x) -> (sig2 A P Q).
(sigS A P), or more suggestively {x:A & (P x)}, is a subtle variant of subset where P is now of type Set. Similarly for (sigS2 A P Q), also written {x:A & (P x) & (Q x)}.
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Inductive sigS [A:Set;P:A->Set] : Set
:= existS : (x:A)(P x) -> (sigS A P).
Inductive sigS2 [A:Set;P,Q:A->Set] : Set
:= existS2 : (x:A)(P x) -> (Q x) -> (sigS2 A P Q).
End Subsets.
Add Printing Let sig.
Add Printing Let sig2.
Add Printing Let sigS.
Add Printing Let sigS2.
| Projections of sig |
Section Subset_projections.
Variable A:Set.
Variable P:A->Prop.
Definition proj1_sig :=
[e:(sig A P)]Cases e of (exist a b) => a end.
Definition proj2_sig :=
[e:(sig A P)]
<[e:(sig A P)](P (proj1_sig e))>Cases e of (exist a b) => b end.
End Subset_projections.
| Projections of sigS |
Section Projections.
Variable A:Set.
Variable P:A->Set.
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 (projS1 y) is the witness a and (projS2 y) is the proof of (P a)
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Definition projS1
:= [x:(sigS A P)]Cases x of (existS a _) => a end.
Definition projS2
:= [x:(sigS A P)]<[x:(sigS A P)](P (projS1 x))>
Cases x of (existS _ h) => h end.
End Projections.
Syntactic Definition ProjS1 := (projS1 ? ?).
Syntactic Definition ProjS2 := (projS2 ? ?).
Section Extended_booleans.
Syntax sumbool "{_}+{_}".
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Inductive sumbool [A,B:Prop] : Set
:= left : A -> (sumbool A B)
| right : B -> (sumbool A B).
Syntax sumor "_+{_}".
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Inductive sumor [A:Set;B:Prop] : Set
:= inleft : A -> (sumor A B)
| inright : B -> (sumor A B).
End Extended_booleans.
| Choice |
Section Choice_lemmas.
| The following lemmas state various forms of the axiom of choice |
Variables S,S':Set.
Variable R:S->S'->Prop.
Variable R':S->S'->Set.
Variables R1,R2 :S->Prop.
Lemma Choice : ((x:S)(sig ? [y:S'](R x y))) ->
(sig ? [f:S->S'](z:S)(R z (f z))).
Proof.
Intro H.
Exists [z:S]Cases (H z) of (exist y _) => y end.
Intro z; Elim (H z); Trivial.
Qed.
Lemma Choice2 : ((x:S)(sigS ? [y:S'](R' x y))) ->
(sigS ? [f:S->S'](z:S)(R' z (f z))).
Proof.
Intro H.
Exists [z:S]Cases (H z) of (existS y _) => y end.
Intro z; Elim (H z); Trivial.
Qed.
Lemma bool_choice :
((x:S)(sumbool (R1 x) (R2 x))) ->
(sig ? [f:S->bool] (x:S)( ((f x)=true /\ (R1 x))
\/ ((f x)=false /\ (R2 x)))).
Proof.
Intro H.
Exists [z:S]Cases (H z) of (left _) => true | (right _) => false end.
Intro z; Elim (H z); Auto.
Qed.
End Choice_lemmas.
A result of type (Exc A) is either a normal value of type A or an error : it is implemented using the option type.
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Definition Exc := option.
Definition value := Some.
Definition error := None.
Syntactic Definition Error := (error ?).
Syntactic Definition Value := (value ?).
Definition except := False_rec.
Syntactic Definition Except := (except ?).
Theorem absurd_set : (A:Prop)(C:Set)A->(~A)->C.
Proof.
Intros A C h1 h2.
Apply False_rec.
Apply (h2 h1).
Qed.
Hints Resolve left right inleft inright : core v62.
Sigma Type at Type level sigT
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Inductive sigT [A:Type;P:A->Type] : Type
:= existT : (x:A)(P x) -> (sigT A P).
Section projections_sigT.
Variable A:Type.
Variable P:A->Type.
Definition projT1
:= [H:(sigT A P)]Cases H of (existT x _) => x end.
Definition projT2
:= [H:(sigT A P)]<[H:(sigT A P)](P (projT1 H))>
Cases H of (existT x h) => h end.
End projections_sigT.