-naxiom Ph : ∀x.h x = O → x ◃ ∅.
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-ninductive eq2 (A : Type[1]) (a : A) : A → CProp[0] ≝
-| refl2 : eq2 A a a.
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-interpretation "eq2" 'eq T a b = (eq2 T a b).
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-ntheorem th_ch3: ¬∃a:caxs.∀x.ϕ a x = h x.
-*; #a; #H;
-ncut ((𝐂 a one) ⊆ { x | x ◃ ∅ }); (* bug *)
-nchange in xx with { x | h x
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-ntheorem cantor: ∀a:axs. ¬ (Z ⊆ R a ∧ R a ⊆ Z).
-#a; *; #ZRa; #RaZ;
-ncut (a ◃ R a); ##[ @2; ##[ napply one; ##] #x; #H; @; napply H; ##] #H1;
-ncut (a ◃ ∅); ##[
- napply (cover_trans … H1);
- #x; #H; nlapply (RaZ … H); #ABS; napply ABS; ##] #H2;
-ncut (a ∈ R a); ##[ napply ZRa; napply H2; ##] #H3;
-nelim H2 in H3;
-##[ nauto.
-##| nnormalize; nauto. ##] (* se lo lancio su entrambi fallisce di width *)
-nqed.
-
-ninductive deduct (A : nAx) (U : Ω^A) : A → CProp[0] ≝
-| drefl : ∀a.a ∈ U → deduct A U a
-| dcut : ∀a.∀i:𝐈 a. (∀y:𝐃 a i.deduct A U (𝐝 a i y)) → deduct A U a.
-
-notation " a ⊢ b " non associative with precedence 45 for @{ 'deduct $a $b }.
-interpretation "deduct" 'deduct a b = (deduct ? b a).
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-ntheorem th2_3_1 : ∀A:nAx.∀a:A.∀i:𝐈 a. a ⊢ 𝐈𝐦[𝐝 a i].
-#A; #a; #i;
-ncut (∀y:𝐃 a i.𝐝 a i y ⊢ 𝐈𝐦[𝐝 a i]); ##[ #y; @; @y; @; ##] #H1;
-napply (dcut … i); nassumption;
-nqed.
-
-ntheorem th2_3_2 :
- ∀A:nAx.∀a:A.∀i:𝐈 a.∀U,V. a ⊢ U → (∀u.u ∈ U → u ⊢ V) → a ⊢ V.
-#A; #a; #i; #U; #V; #aU; #xUxV;
-nelim aU;
-##[ nassumption;
-##| #b; #i; #dU; #dV; @2 i; nassumption;
-##]
-nqed.
-
-ntheorem th2_3 :
- ∀A:nAx.
- (∀a:A.∀i_star.(∃y:𝐃 a i_star.𝐝 a i_star y = a) ∨ ¬(∃y:𝐃 a i_star.𝐝 a i_star y = a)) →
- ∀a:A. a ◃ ∅ → ∃i:𝐈 a. ¬ a \in Z
-#A; #EM; #a; #H; nelim H;
-##[ #n; *;
-##| #b; #i_star; #IH1; #IH2;
- ncases (EM b i_star);
- ##[##2: #W; @i_star; napply W;
- ##| *; #y_star; #E; nlapply (IH2 y_star); nrewrite > E; #OK; napply OK;
- ##]
-##]
+alias symbol "eq" = "leibnitz's equality".
+alias symbol "eq" = "setoid1 eq".
+alias symbol "covers" = "covers".
+naxiom Ph : ∀x.h x = O \liff x ◃ ∅.
+
+nlemma replace_char:
+ ∀A:Ax.∀U,V.U ⊆ V → V ⊆ U → ∀a:A.a ◃ U → a ◃ V.
+#A; #U; #V; #a; #H1; #H2; #E; nelim E;
+##[ #b; #Hb; @; nauto;
+##| #b; #i; #H3; #H4; @2 i; #c; #Hc; nauto; ##]