include "datatypes/bool.ma".
include "sets/sets.ma".
+(*
ninductive Admit : CProp[0] ≝ .
naxiom admit : Admit.
+*)
(* single = is for the abstract equality of setoids, == is for concrete
equalities (that may be lifted to the setoid level when needed *)
interpretation "bool eq" 'eq_low a b = (eq bool a b).
ndefinition BOOL : setoid.
-@bool; @(eq bool); ncases admit.nqed.
+@bool; @(eq bool); nnormalize; //; #x y; ##[ #E; ncases E; ##| #y H; ncases H; ##] //; nqed.
alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
alias id "refl" = "cic:/matita/ng/properties/relations/refl.fix(0,1,3)".
(*-----------------------------------------------------------------*) ⊢
list S ≡ carr1 TT.
+(* not as morphism *)
+nlemma Not_morphism : CProp[0] ⇒_1 CProp[0].
+@(λx:CProp[0].¬ x); #a b; *; #; @; /3/; nqed.
+
+unification hint 0 ≔ P : CProp[0];
+ A ≟ CPROP,
+ B ≟ CPROP,
+ M ≟ mk_unary_morphism1 ?? (λP.¬ P) (prop11 ?? Not_morphism)
+(*------------------------*)⊢
+ fun11 A B M P ≡ ¬ P.
+
(* XXX Ex setoid support *)
-nlemma Sig: ∀S,T:setoid.∀P: S → (T → CPROP).
- ∀y,z:T.y = z → (∀x.y=z → P x y = P x z) → (Ex S (λx.P x y)) =_1 (Ex S (λx.P x z)).
-#S T P y z Q E; @; *; #x Px; @x; nlapply (E x Q); *; /2/; nqed.
+nlemma Ex_morphism : ∀S:setoid.(S ⇒_1 CProp[0]) ⇒_1 CProp[0].
+#S; @(λP: S ⇒_1 CProp[0].Ex S P); #P Q E; @; *; #x Px; @x; ncases (E x x #); /2/; nqed.
+
+unification hint 0 ≔ S : setoid, P : S ⇒_1 CProp[0];
+ A ≟ unary_morphism1_setoid1 (setoid1_of_setoid S) CPROP,
+ B ≟ CPROP,
+ M ≟ mk_unary_morphism1 ?? (λP: S ⇒_1 CProp[0].Ex S P)
+ (prop11 ?? (Ex_morphism S))
+(*------------------------*)⊢
+ fun11 A B M P ≡ Ex S (fun11 S CPROP P).
+
+nlemma Ex_morphism_eta : ∀S:setoid.(S ⇒_1 CProp[0]) ⇒_1 CProp[0].
+#S; @(λP: S ⇒_1 CProp[0].Ex S (λx.P x)); #P Q E; @; *; #x Px; @x; ncases (E x x #); /2/; nqed.
+
+unification hint 0 ≔ S : setoid, P : S ⇒_1 CProp[0];
+ A ≟ unary_morphism1_setoid1 (setoid1_of_setoid S) CPROP,
+ B ≟ CPROP,
+ M ≟ mk_unary_morphism1 ?? (λP: S ⇒_1 CProp[0].Ex S (λx.P x))
+ (prop11 ?? (Ex_morphism_eta S))
+(*------------------------*)⊢
+ fun11 A B M P ≡ Ex S (λx.fun11 S CPROP P x).
+
+nlemma Ex_setoid : ∀S:setoid.(S ⇒_1 CPROP) → setoid.
+#T P; @ (Ex T (λx:T.P x)); @; ##[ #H1 H2; napply True |##*: //; ##] nqed.
+
+unification hint 0 ≔ T,P ;
+ S ≟ (Ex_setoid T P)
+(*---------------------------*) ⊢
+ Ex T (λx:T.P x) ≡ carr S.
+
+(* couts how many Ex we are traversing *)
+ninductive counter : Type[0] ≝
+ | End : counter
+ | Next : (bool → bool) → (* dummy arg please the notation mechanism *)
+ counter → counter.
+
+(* to rewrite terms (live in setoid) *)
+nlet rec mk_P (S, T : setoid) (n : counter) on n ≝
+ match n with [ End ⇒ T → CProp[0] | Next _ m ⇒ S → (mk_P S T m) ].
+
+nlet rec mk_F (S, T : setoid) (n : counter) on n ≝
+ match n with [ End ⇒ T | Next _ m ⇒ S → (mk_F S T m) ].
+
+nlet rec mk_E (S, T : setoid) (n : counter) on n : ∀f,g : mk_F S T n. CProp[0] ≝
+ match n with
+ [ End ⇒ λf,g:T. f = g
+ | Next q m ⇒ λf,g: mk_F S T (Next q m). ∀x:S.mk_E S T m (f x) (g x) ].
+
+nlet rec mk_H (S, T : setoid) (n : counter) on n :
+∀P1,P2: mk_P S T n.∀f,g : mk_F S T n. CProp[1] ≝
+ match n with
+ [ End ⇒ λP1,P2:mk_P S T End.λf,g:T. f = g → P1 f =_1 P2 g
+ | Next q m ⇒ λP1,P2: mk_P S T (Next q m).λf,g: mk_F S T (Next q m).
+ ∀x:S.mk_H S T m (P1 x) (P2 x) (f x) (g x) ].
+
+nlet rec mk_Ex (S, T : setoid) (n : counter) on n :
+∀P: mk_P S T n.∀f : mk_F S T n. CProp[0] ≝
+ match n with
+ [ End ⇒ λP:mk_P S T End.λf:T. P f
+ | Next q m ⇒ λP: mk_P S T (Next q m).λf: mk_F S T (Next q m).
+ ∃x:S.mk_Ex S T m (P x) (f x) ].
+
+nlemma Sig_generic : ∀S,T.∀n:counter.∀P,f,g.
+ mk_E S T n f g → mk_H S T n P P f g → mk_Ex S T n P f =_1 mk_Ex S T n P g.
+#S T n; nelim n; nnormalize;
+##[ #P f g E H; /2/;
+##| #q m IH P f g E H; @; *; #x Px; @x; ncases (IH … (E x) (H x)); /3/; ##]
+nqed.
-notation "∑" non associative with precedence 90 for @{Sig ?????}.
+(* to rewrite propositions (live in setoid1) *)
+nlet rec mk_P1 (S : setoid) (T : setoid1) (n : counter) on n ≝
+ match n with [ End ⇒ T → CProp[0] | Next _ m ⇒ S → (mk_P1 S T m) ].
-nlemma test : ∀S:setoid. ∀ee: S ⇒_1 S ⇒_1 CPROP.
- ∀x,y:setoid1_of_setoid S.x =_1 y → (Ex S (λw.ee x w ∧ True)) =_1 (Ex S (λw.ee y w ∧ True)).
-#S m x y E;
-napply (.=_1 (∑ E (λw,H.(H ╪_1 #)╪_1 #))).
-napply #.
+nlet rec mk_F1 (S : setoid) (T : setoid1) (n : counter) on n ≝
+ match n with [ End ⇒ T | Next _ m ⇒ S → (mk_F1 S T m) ].
+
+nlet rec mk_E1 (S : setoid) (T : setoid1) (n : counter) on n : ∀f,g : mk_F1 S T n. CProp[1] ≝
+ match n with
+ [ End ⇒ λf,g:T. f =_1 g
+ | Next q m ⇒ λf,g: mk_F1 S T (Next q m). ∀x:S.mk_E1 S T m (f x) (g x) ].
+
+nlet rec mk_H1 (S : setoid) (T : setoid1) (n : counter) on n :
+∀P1,P2: mk_P1 S T n.∀f,g : mk_F1 S T n. CProp[1] ≝
+ match n with
+ [ End ⇒ λP1,P2:mk_P1 S T End.λf,g:T. f = g → P1 f =_1 P2 g
+ | Next q m ⇒ λP1,P2: mk_P1 S T (Next q m).λf,g: mk_F1 S T (Next q m).
+ ∀x:S.mk_H1 S T m (P1 x) (P2 x) (f x) (g x) ].
+
+nlet rec mk_Ex1 (S : setoid) (T : setoid1) (n : counter) on n :
+∀P: mk_P1 S T n.∀f : mk_F1 S T n. CProp[0] ≝
+ match n with
+ [ End ⇒ λP:mk_P1 S T End.λf:T. P f
+ | Next q m ⇒ λP: mk_P1 S T (Next q m).λf: mk_F1 S T (Next q m).
+ ∃x:S.mk_Ex1 S T m (P x) (f x) ].
+
+nlemma Sig_generic1 : ∀S,T.∀n:counter.∀P,f,g.
+ mk_E1 S T n f g → mk_H1 S T n P P f g → mk_Ex1 S T n P f =_1 mk_Ex1 S T n P g.
+#S T n; nelim n; nnormalize;
+##[ #P f g E H; /2/;
+##| #q m IH P f g E H; @; *; #x Px; @x; ncases (IH … (E x) (H x)); /3/; ##]
nqed.
+(* notation "∑x1,...,xn. E / H ; P" were:
+ - x1...xn are bound in E and P, H is bound in P
+ - H is an identifier that will have the type of E in P
+ - P is the proof that the two existentially quantified predicates are equal*)
+notation > "∑ list1 ident x sep , . term 56 E / ident nE ; term 19 H" with precedence 20
+for @{ 'Sig_gen
+ ${ fold right @{ 'End } rec acc @{ ('Next (λ${ident x}.${ident x}) $acc) } }
+ ${ fold right @{ $E } rec acc @{ λ${ident x}.$acc } }
+ ${ fold right @{ λ${ident nE}.$H } rec acc @{ λ${ident x}.$acc } }
+}.
+
+interpretation "next" 'Next x y = (Next x y).
+interpretation "end" 'End = End.
+(*interpretation "sig_gen" 'Sig_gen n E H = (Sig_generic ?? n ??? E H).*)
+interpretation "sig_gen1" 'Sig_gen n E H = (Sig_generic1 ?? n ??? E H).
+
+nlemma test0 : ∀S:setoid. ∀P: S ⇒_1 CPROP.∀f,g:S → S.
+ (∀x:S.f x = g x) → (Ex S (λw.P (f w))) =_1 (Ex S (λw.P (g w))).
+#S P f g E; napply (∑w. E w / H ; ┼_1H); nqed.
+
+nlemma test : ∀S:setoid. ∀P: S ⇒_1 CPROP.∀f,g:S → S.
+ (∀x:S.f x = g x) → (Ex S (λw.P (f w)∧ True)) =_1 (Ex S (λw.P (g w)∧ True)).
+#S P f g E; napply (∑w. E w / H ; (┼_1H)╪_1#); nqed.
+
+nlemma test_bound : ∀S:setoid. ∀e,f: S ⇒_1 CPROP. e = f →
+ (Ex S (λw.e w ∧ True)) =_1 (Ex S (λw.f w ∧ True)).
+#S f g E; napply (.=_1 ∑x. E x x # / H ; (H ╪_1 #)); //; nqed.
+
nlemma test2 : ∀S:setoid. ∀ee: S ⇒_1 S ⇒_1 CPROP.
∀x,y:setoid1_of_setoid S.x =_1 y →
(True ∧ (Ex S (λw.ee x w ∧ True))) =_1 (True ∧ (Ex S (λw.ee y w ∧ True))).
-#S m x y E;
-napply (.=_1 #╪_1(∑ E (λw,H.(H ╪_1 #) ╪_1 #))).
-napply #.
-nqed.
-
-nlemma ex_setoid : ∀S:setoid.(S ⇒_1 CPROP) → setoid.
-#T P; @ (Ex T (λx:T.P x)); @;
-##[ #H1 H2; napply True |##*: //; ##]
-nqed.
-
-unification hint 0 ≔ T,P ; S ≟ (ex_setoid T P) ⊢
- Ex T (λx:T.P x) ≡ carr S.
+#S m x y E; napply (.=_1 #╪_1(∑w. E / E ; ((E ╪_1 #) ╪_1 #))). //; nqed.
nlemma test3 : ∀S:setoid. ∀ee: S ⇒_1 S ⇒_1 CPROP.
∀x,y:setoid1_of_setoid S.x =_1 y →
((Ex S (λw.ee x w ∧ True) ∨ True)) =_1 ((Ex S (λw.ee y w ∧ True) ∨ True)).
-#S m x y E;
-napply (.=_1 (∑ E (λw,H.(H ╪_1 #) ╪_1 #)) ╪_1 #).
-napply #.
-nqed.
+#S m x y E; napply (.=_1 (∑w. E / E ; ((E ╪_1 #) ╪_1 #)) ╪_1 #). //; nqed.
+
(* Ex setoid support end *)
ndefinition L_pi_ext : ∀S:Alpha.∀r:pitem S.Elang S.
##| #x; @; *;
##| #x; @; #H; nchange in H with ([?] =_0 ?); ##[ napply ((.=_0 H) E); ##]
napply ((.=_0 H) E^-1);
-##| #e1 e2 H1 H2;
+##| #e1 e2 H1 H2; (*
nchange in match (w1 ∈ 𝐋\p (?·?)) with ((∃_.?)∨?);
- nchange in match (w2 ∈ 𝐋\p (?·?)) with ((∃_.?)∨?);
+ nchange in match (w2 ∈ 𝐋\p (?·?)) with ((∃_.?)∨?); good! *)
napply (.= (#‡H2));
- napply (.=_1 (∑ E (λx1,H1.∑ E (λx2,H2.?)))╪_1 #); ##[
- ncut ((w1 = (x1@x2)) = (w2 = (x1@x2)));##[
- @; #X; ##[ napply ((.= H1^-1) X) | napply ((.= H2) X) ] ##] #X;
- napply ( (X‡#)‡#); ##]
- napply #;
-##| #e1 e2 H1 H2;
- nnormalize in ⊢ (???%%);
- napply (H1‡H2);
-##| #e H; nnormalize in ⊢ (???%%);
- napply (.=_1 (∑ E (λx1,H1.∑ E (λx2,H2.?)))); ##[
- ncut ((w1 = (x1@x2)) = (w2 = (x1@x2)));##[
- @; #X; ##[ napply ((.= H1^-1) X) | napply ((.= H2) X) ] ##] #X;
- napply ((X‡#)‡#); ##]
- napply #;##]
+ ncut (∀x1,x2. (w1 = (x1@x2)) = (w2 = (x1@x2)));##[
+ #x1 x2; @; #X; ##[ napply ((.= E^-1) X) | napply ((.= E) X) ] ##] #X;
+ napply ((∑w1,w2. X w1 w2 / H ; (H╪_1#)╪_1#) ╪_1 #);
+##| #e1 e2 H1 H2; napply (H1‡H2); (* good! *)
+##| #e H;
+ ncut (∀x1,x2.(w1 = (x1@x2)) = (w2 = (x1@x2)));##[
+ #x1 x2; @; #X; ##[ napply ((.= E^-1) X) | napply ((.= E) X) ] ##] #X;
+ (* nnormalize in ⊢ (???%%); good! (but a bit too hard) *)
+ napply (∑w1,w2. X w1 w2 / H ; (H╪_1#)╪_1#);
+##]
nqed.
unification hint 0 ≔ S : Alpha,e : pitem S;
(* theorem 16: 2 *)
nlemma oplus_cup : ∀S:Alpha.∀e1,e2:pre S.𝐋\p (e1 ⊕ e2) = 𝐋\p e1 ∪ 𝐋\p e2.
-#S r1; ncases r1; #e1 b1 r2; ncases r2; #e2 b2;
+#S r1; ncases r1; #e1 b1 r2; ncases r2; #e2 b2; (* oh my!
nwhd in ⊢ (???(??%)?);
nchange in ⊢(???%?) with (𝐋\p (e1 + e2) ∪ ϵ (b1 || b2));
-nchange in ⊢(???(??%?)?) with (𝐋\p (e1) ∪ 𝐋\p (e2));
+nchange in ⊢(???(??%?)?) with (𝐋\p (e1) ∪ 𝐋\p (e2)); *)
napply (.=_1 #╪_1 (epsilon_or ???));
napply (.=_1 (cupA…)^-1);
napply (.=_1 (cupA…)╪_1#);
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