--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "arithmetics/nat.ma".
+include "datatypes/bool.ma".
+
+ndefinition two ≝ S (S O).
+ndefinition natone ≝ S O.
+ndefinition four ≝ two * two.
+ndefinition eight ≝ two * four.
+ndefinition natS ≝ S.
+
+include "topology/igft.ma".
+
+nlemma hint_auto2 : ∀T.∀U,V:Ω^T.(∀x.x ∈ U → x ∈ V) → U ⊆ V./2/.nqed.
+
+ninductive Sigma (A: Type[0]) (P: A → CProp[0]) : Type[0] ≝
+ mk_Sigma: ∀a:A. P a → Sigma A P.
+
+(*<< To be moved in igft.ma *)
+ninductive ncover (A : nAx) (U : Ω^A) : A → CProp[0] ≝
+| ncreflexivity : ∀a. a ∈ U → ncover A U a
+| ncinfinity : ∀a. ∀i. (∀y.Sigma ? (λj.y = 𝐝 a i j) → ncover A U y) → ncover A U a.
+
+interpretation "ncovers" 'covers a U = (ncover ? U a).
+
+ntheorem ncover_cover_ok: ∀A:nAx.∀U.∀a:A. a ◃ U → cover (Ax_of_nAx A) U a.
+ #A; #U; #a; #H; nelim H
+ [ #n; #H1; @1; nassumption
+ | #a; #i; #IH; #H; @2 [ napply i ]
+ nnormalize; #y; *; #j; #E; nrewrite > E;
+ napply H;
+ /2/ ]
+nqed.
+
+ntheorem cover_ncover_ok: ∀A:Ax.∀U.∀a:A. a ◃ U → ncover (nAx_of_Ax A) U a.
+ #A; #U; #a; #H; nelim H
+ [ #n; #H1; @1; nassumption
+ | #a; #i; #IH; #H; @2 [ napply i ] #y; *; #j; #E; nrewrite > E; ncases j; #x; #K;
+ napply H; nnormalize; //.
+nqed.
+
+ndefinition ncoverage : ∀A:nAx.∀U:Ω^A.Ω^A ≝ λA,U.{ a | a ◃ U }.
+
+interpretation "ncoverage cover" 'coverage U = (ncoverage ? U).
+
+(*>> To be moved in igft.ma *)
+
+(*XXX
+nlemma ncover_ind':
+ ∀A:nAx.∀U,P:Ω^A.
+ (U ⊆ P) → (∀a:A.∀i:𝐈 a.(∀j. 𝐝 a i j ◃ U) → (∀j. 𝐝 a i j ∈ P) → a ∈ P) →
+ ◃ U ⊆ P.
+ #A; #U; #P; #refl; #infty; #a; #H; nelim H
+ [ // | #b; #j; #K1; #K2; napply infty; //; ##]
+nqed.
+
+alias symbol "covers" (instance 3) = "ncovers".
+nlemma cover_ind'':
+ ∀A:nAx.∀U:Ω^A.∀P:A → CProp[0].
+ (∀a. a ∈ U → P a) → (∀a:A.∀i:𝐈 a.(∀j. 𝐝 a i j ◃ U) → (∀j. P (𝐝 a i j)) → P a) →
+ ∀b. b ◃ U → P b.
+ #A; #U; #P; nletin V ≝ {x | P x}; napply (ncover_ind' … V).
+nqed.
+*)
+
+(*********** from Cantor **********)
+ninductive eq1 (A : Type[0]) : Type[0] → CProp[0] ≝
+| refl1 : eq1 A A.
+
+notation "hvbox( a break ∼ b)" non associative with precedence 40
+for @{ 'eqT $a $b }.
+
+interpretation "eq between types" 'eqT a b = (eq1 a b).
+
+ninductive unit : Type[0] ≝ one : unit.
+
+ninductive option (A: Type[0]) : Type[0] ≝
+ None: option A
+ | Some: A → option A
+ | Twice: A → A → option A.
+
+nrecord uuAx : Type[1] ≝ {
+ uuS : Type[0];
+ uuC : uuS → option uuS
+}.
+
+ndefinition uuax : uuAx → nAx.
+#A; @ (uuS A)
+ [ #a; napply unit
+##| #a; ncases (uuC … a); nnormalize
+ [ #_; napply False
+ | #_; #_; napply unit
+ | #_; #_; #_; napply bool ]
+##| #a; ncases (uuC … a); nnormalize
+ [ #_; #H; napply (False_rect_Type1 … H)
+ | #b; #_; #_; napply b
+ | #b1; #b2; #_; * [ napply b1 | napply b2]##]##]
+nqed.
+
+ncoercion uuax : ∀u:uuAx. nAx ≝ uuax on _u : uuAx to nAx.
+
+nlemma eq_rect_Type0_r':
+ ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → Type[0]. P a (refl A a) → P x p.
+ #A; #a; #x; #p; ncases p; //;
+nqed.
+
+nlemma eq_rect_Type0_r:
+ ∀A.∀a.∀P: ∀x:A. eq ? x a → Type[0]. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
+ #A; #a; #P; #p; #x0; #p0; napply (eq_rect_Type0_r' ??? p0); //.
+nqed.
+
+nrecord memdec (A: Type[0]) (U:Ω^A) : Type[0] ≝
+ { decide_mem:> A → bool;
+ decide_mem_ok: ∀x. decide_mem x = true → x ∈ U;
+ decide_mem_ko: ∀x. decide_mem x = false → ¬ (x ∈ U)
+ }.
+
+(*********** end from Cantor ********)
+
+nlemma csc_sym_eq: ∀A,x,y. eq A x y → eq A y x.
+ #A; #x; #y; #H; ncases H; @1.
+nqed.
+
+nlemma csc_eq_rect_CProp0_r':
+ ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. CProp[0]. P a → P x.
+ #A; #a; #x; #p; #P; #H;
+ napply (match csc_sym_eq ??? p return λa.λ_.P a with [ refl ⇒ H ]).
+nqed.
+
+nlet rec cover_rect
+ (A:uuAx) (U:Ω^(uuax A)) (memdec: memdec … U) (P:uuax A → Type[0])
+ (refl: ∀a:uuax A. a ∈ U → P a)
+ (infty: ∀a:uuax A.∀i: 𝐈 a.(∀j. 𝐝 a i j ◃ U) → (∀j.P (𝐝 a i j)) → P a)
+ (b:uuax A) (p: b ◃ U) on p : P b
+≝ ?.
+ nlapply (decide_mem_ok … memdec b); nlapply (decide_mem_ko … memdec b);
+ ncases (decide_mem … memdec b)
+ [ #_; #H; napply refl; /2/
+ | #H; #_; ncut (uuC … b=uuC … b) [//] ncases (uuC … b) in ⊢ (???% → ?)
+ [ #E;
+ nlapply (infty b); nnormalize; nrewrite > E; nnormalize; #H2;
+ napply (H2 one); #y; nelim y
+ ##| #a; #E;
+ ncut (a ◃ U)
+ [ nlapply E; nlapply (H ?); //; ncases p
+ [ #x; #Hx; #K1; #_; ncases (K1 Hx)
+ ##| #x; #i; #Hx; #K1; #E2; napply Hx; ngeneralize in match i; nnormalize;
+ nrewrite > E2; nnormalize; /2/ ]##]
+ #Hcut;
+ nlapply (infty b); nnormalize; nrewrite > E; nnormalize; #H2;
+ napply (H2 one); #y
+ [ napply Hcut
+ ##| napply (cover_rect A U memdec P refl infty a); // ]
+ ##| #a; #a1; #E;
+ ncut (a ◃ U)
+ [ nlapply E; nlapply (H ?) [//] ncases p
+ [ #x; #Hx; #K1; #_; ncases (K1 Hx)
+ ##| #x; #i; #Hx; #K1; #E2; napply Hx; ngeneralize in match i; nnormalize;
+ nrewrite > E2; nnormalize; #_; @1 (true); /2/ ]##]
+ #Hcut;
+ ncut (a1 ◃ U)
+ [ nlapply E; nlapply (H ?) [//] ncases p
+ [ #x; #Hx; #K1; #_; ncases (K1 Hx)
+ ##| #x; #i; #Hx; #K1; #E2; napply Hx; ngeneralize in match i; nnormalize;
+ nrewrite > E2; nnormalize; #_; @1 (false); /2/ ]##]
+ #Hcut1;
+ nlapply (infty b); nnormalize; nrewrite > E; nnormalize; #H2;
+ napply (H2 one); #y; ncases y; nnormalize
+ [##1,2: nassumption
+ | napply (cover_rect A U memdec P refl infty a); //
+ | napply (cover_rect A U memdec P refl infty a1); //]
+nqed.
+
+(********* Esempio:
+ let rec skip n =
+ match n with
+ [ O ⇒ 1
+ | S m ⇒
+ match m with
+ [ O ⇒ skipfact O
+ | S _ ⇒ S m * skipfact (pred m) * skipfact (pred m) ]]
+**)
+
+ntheorem psym_plus: ∀n,m. n + m = m + n.//.
+nqed.
+
+nlemma easy1: ∀n:nat. two * (S n) = two + two * n.//.
+nqed.
+
+ndefinition skipfact_dom: uuAx.
+ @ nat; #n; ncases n [ napply None | #m; ncases m [ napply (Some … O) | #_; napply (Twice … (pred m) (pred m)) ]
+nqed.
+
+ntheorem skipfact_base_dec:
+ memdec (uuax skipfact_dom) (mk_powerclass ? (λx: uuax skipfact_dom. False)).
+ nnormalize; @ (λ_.false); //. #_; #H; ndestruct.
+nqed.
+
+ntheorem skipfact_partial:
+ ∀n: uuax skipfact_dom. two * n ◃ mk_powerclass ? (λx: uuax skipfact_dom.False).
+ #n; nelim n
+ [ @2; nnormalize; //; #y; *; #a; ncases a
+ |
+ #m; nelim m; nnormalize
+ [ #H; @2; nnormalize; //;
+ #y; *; #a; #E; nrewrite > E; ncases a; nnormalize; //
+ ##| #p; #H1; #H2; @2; nnormalize; //;
+ #y; *; #a; #E; nrewrite > E; ncases a; nnormalize;
+ nrewrite < (plus_n_Sm …); // ]
+nqed.
+
+ndefinition skipfact: ∀n:nat. n ◃ mk_powerclass ? (λx: uuax skipfact_dom.False) → nat.
+ #n; #D; napply (cover_rect … skipfact_base_dec … n D)
+ [ #a; #H; nelim H
+ | #a; ncases a
+ [ nnormalize; #i; #_; #_; napply natone
+ | #m; ncases m
+ [ nnormalize; #_; #_; #H; napply H; @1
+ | #p; #i; nnormalize in i; #K;
+ #H; nnormalize in H;
+ napply (S m * H true * H false) ]
+nqed.
+
+nlemma test: skipfact four ? = four * two * two. ##[##2: napply (skipfact_partial two)]//.
+nqed.
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