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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "topology/igft.ma".
+
+alias symbol "covers" = "covers set".
+alias symbol "coverage" = "coverage cover".
+alias symbol "I" = "I".
+nlemma cover_ind':
+ ∀A:Ax.∀U,P:Ω^A.
+   (U ⊆ P) → (∀a:A.∀j:𝐈 a. 𝐂 a j ◃ U → 𝐂 a j ⊆ P → a ∈ P) →
+    ◃ U ⊆ P.
+ #A; #U; #P; #refl; #infty; #a; #H; nelim H
+  [ nauto | #b; #j; #K1; #K2; napply (infty … j) [ nassumption | napply K2]##]
+nqed.
+
+alias symbol "covers" = "covers".
+alias symbol "covers" = "covers set".
+alias symbol "covers" = "covers".
+alias symbol "covers" = "covers set".
+alias symbol "covers" = "covers".
+nlemma cover_ind'':
+ ∀A:Ax.∀U:Ω^A.∀P:A → CProp[0].
+  (∀a. a ∈ U → P a) → (∀a:A.∀j:𝐈 a. 𝐂 a j ◃ U → (∀b. b ∈ 𝐂 a j → P b) → P a) →
+   ∀b. b ◃ U → P b.
+   
+ #A; #U; #P; nletin V ≝ {x | P x}; napply (cover_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.
+
+nrecord uuAx : Type[1] ≝ {
+  uuS : Type[0];
+  uuC : uuS → option uuS
+}.
+
+ndefinition uuax : uuAx → Ax.
+#A; @ (uuS A)
+  [ #a; ncases (uuC … a) [ napply False | #_; napply unit]
+##| #a; ncases (uuC … a)
+     [ nnormalize; #H; napply (False_rect_Type1 … H)
+     | nnormalize; #b; #_; napply {x | x=b }]##]
+nqed.
+
+ncoercion uuax : ∀u:uuAx. Ax ≝ uuax on _u : uuAx to Ax.
+
+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; #P; #H; nassumption.
+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); nassumption.
+nqed.
+
+ninductive bool: Type[0] ≝
+   true: bool
+ | false: bool.
+
+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 ********)
+
+alias symbol "covers" = "covers".
+alias symbol "covers" = "covers".
+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. 𝐂 a i ◃ U → (∀b. b ∈ 𝐂 a i → P b) → 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; nauto
+  | #H; #_; ncut (uuC … b=uuC … b) [nauto] ncases (uuC … b) in ⊢ (???% → ?)
+    [ #E; napply False_rect_Type0; ncut (b=b) [nauto] ncases p in ⊢ (???% → ?)
+      [ #a; #K; #E2; napply H [ nauto | nrewrite > E2; nauto ]
+    ##| #a; #i; #K; #E2; nrewrite < E2 in i; nnormalize; nrewrite > E; nnormalize;
+        nauto]
+  ##| #a; #E;
+      ncut (a ◃ U)
+       [ nlapply E; nlapply (H ?) [nauto] ncases p
+          [ #x; #Hx; #K1; #_; ncases (K1 Hx)
+        ##| #x; #i; #Hx; #K1; #E2; napply Hx; ngeneralize in match i; nnormalize;
+            nrewrite > E2; nnormalize; #_; nauto]##]
+      #Hcut; 
+      nlapply (infty b); nnormalize; nrewrite > E; nnormalize; #H2;
+      napply (H2 one); #y; #E2; nrewrite > E2
+       [ napply Hcut
+     ##| napply (cover_rect A U memdec P refl infty a); napply Hcut]##]
+nqed.
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