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Streamless-ex-sig-dec.v
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Streamless-ex-sig-dec.v
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Require Import Coq.Arith.Compare_dec.
Require Import Coq.Logic.ConstructiveEpsilon.
Section Streamless.
(* We assume that we have some sort of pairing function, enabling us to encode
pairs of natural numbers as a single natural number (and decode them again). *)
Variable encoding : nat->nat->nat.
(* Decoding functions: *)
Variable dec1:nat->nat.
Variable dec2:nat->nat.
(* We need to know that decoding works*)
Axiom com1: forall n m:nat, (dec1 (encoding n m)) = n.
Axiom com2: forall n m:nat, (dec2 (encoding n m)) = m.
Hint Resolve com1 com2.
Definition DecidableEq (A: Set) := forall a b: A, {a=b} + {~ a=b}.
(* Each of the definitions come in a pair and single number version *)
Definition StreamlessSigOneNat (A:Set):= forall g:nat->A,
{ij : nat | dec1 ij < dec2 ij /\ g(dec1 ij)=g(dec2 ij)}.
Definition StreamlessSig (A:Set):= forall g:nat->A,
{ij : nat*nat | fst ij < snd ij /\ g(fst ij)=g(snd ij)}.
Definition StreamlessEx(A:Set):= forall g:nat->A,exists j i, i<j /\ g(i) = g(j).
Definition StreamlessExOneNat(A:Set):= forall g:nat->A,exists i, dec1 i<dec2 i /\ g(dec1 i) = g(dec2 i).
Definition functional_extensionality (A B:Set):=
forall (f g : A-> B), (forall x, f x = g x) -> f = g.
(* First we show that the paired/unpaired versions of streamlessEx are equvalent, and then the same for StreamlessSig *)
Theorem strExToStrExEnc(A:Set): StreamlessEx A -> StreamlessExOneNat A.
Proof.
unfold StreamlessEx.
unfold StreamlessExOneNat.
intros.
pose (H g).
destruct e.
destruct H0.
exists (encoding x0 x).
rewrite com1.
rewrite com2.
auto.
Qed.
Hint Resolve strExToStrExEnc.
Theorem strExEncToStrEx(A:Set): StreamlessExOneNat A -> StreamlessEx A.
Proof.
unfold StreamlessEx.
unfold StreamlessExOneNat.
intros.
pose (H g).
destruct e.
destruct H0.
exists (dec2 x).
exists (dec1 x).
auto.
Qed.
Hint Resolve strExEncToStrEx.
Theorem strSetToStrSetEnc(A:Set): StreamlessSig A -> StreamlessSigOneNat A.
Proof.
unfold StreamlessSig.
unfold StreamlessSigOneNat.
intros.
pose (H g).
destruct s.
destruct x.
exists (encoding n n0).
rewrite com1.
rewrite com2.
auto.
Qed.
Hint Resolve strSetToStrSetEnc.
Theorem strSetEncToStrSet(A:Set): StreamlessSigOneNat A -> StreamlessSig A.
Proof.
unfold StreamlessSig.
unfold StreamlessSigOneNat.
intros.
pose (H g).
destruct s.
exists (dec1 x, dec2 x).
auto.
Qed.
Hint Resolve strSetEncToStrSet.
Lemma streamlessExToStrSig(A:Set)(A_dec: DecidableEq A) : StreamlessEx A -> StreamlessSig A.
intros.
assert (StreamlessSigOneNat A).
pose (strExToStrExEnc A H).
intro.
apply constructive_indefinite_ground_description_nat;auto.
intros.
pose (lt_dec (dec1 n) (dec2 n)).
pose (A_dec (g (dec1 n)) (g (dec2 n))).
inversion s0;inversion s1;firstorder.
apply (strSetEncToStrSet A H0).
Qed.
Lemma LimitedAc (A:Set) (Ma : StreamlessSig A) (g f : nat -> A) (id : g=f): proj1_sig (Ma g) = proj1_sig (Ma f).
Proof.
rewrite id.
auto.
Qed.
Hint Resolve LimitedAc.
(* A pair of natural numbers are equal if their components are *)
Lemma pairNatEqNat:(forall a b:nat*nat,{a=b}+{~a=b}).
Proof.
intros.
destruct a.
destruct b.
destruct ((NPeano.Nat.eq_dec n n1)); destruct ((NPeano.Nat.eq_dec n0 n2));auto;right;intro;inversion H;firstorder.
Qed.
Hint Resolve pairNatEqNat.
Lemma strSigAndFuncExtImpliesDecA (A:Set) (Ma:StreamlessSig A) (fext: functional_extensionality nat A): forall a b :A, {a=b}+{not (a=b)} .
Proof.
intros.
set(fa := (fun x:nat=> a)).
pose (Ma fa).
pose (proj1_sig s).
set(fap := (fun x:nat=> if (NPeano.Nat.eq_dec x (fst p)) then b else a)).
assert (a=b -> fa=fap);auto.
intro.
apply (fext fa fap).
intro.
unfold fa.
unfold fap.
destruct ((NPeano.Nat.eq_dec x (fst p)));auto.
assert (a=b -> proj1_sig (Ma fa)=proj1_sig (Ma fap));auto.
assert ({ proj1_sig (Ma fa) = proj1_sig (Ma fap)}+{~ proj1_sig (Ma fa) = proj1_sig (Ma fap)});auto.
inversion H1;auto.
assert (fst (proj1_sig (Ma fa))=fst (proj1_sig (Ma fap)));auto.
elim H2;auto.
assert (snd (proj1_sig (Ma fa))=snd (proj1_sig (Ma fap)));auto.
elim H2;auto.
unfold p in fap.
unfold s in fap.
left.
pose (proj1_sig (Ma fap) ).
pose (proj2_sig (Ma fap) ).
pose (fap (fst (proj1_sig (Ma fap)))).
pose (fap (snd (proj1_sig (Ma fap)))).
assert (fap (fst (proj1_sig (Ma fap))) = b).
rewrite<- H3.
unfold fap.
elim (NPeano.Nat.eq_dec (fst (proj1_sig (Ma fa))) (fst (proj1_sig (Ma fa))));auto.
intros.
contradiction b0;firstorder.
assert (fap (snd (proj1_sig (Ma fap))) = a).
rewrite<- H4.
unfold fap.
elim (NPeano.Nat.eq_dec (fst (proj1_sig (Ma fa))) (snd (proj1_sig (Ma fa))));auto;intros.
contradict a3.
apply NPeano.Nat.lt_neq;auto.
apply (proj2_sig s).
elim (NPeano.Nat.eq_dec (snd (proj1_sig (Ma fa))) (fst (proj1_sig (Ma fa))));auto;intros.
contradict a3.
auto.
rewrite<- H5.
rewrite<- H6.
destruct a0.
auto.
Qed.
End Streamless.