alias symbol "covers" (instance 2) = "covers set".
alias symbol "covers" (instance 3) = "covers".
ntheorem transitivity: ∀A:Ax.∀a:A.∀U,V. a ◃ U → U ◃ V → a ◃ V.
-#A; #a; #U; #V; #aU; #UV;
-nelim aU;
-##[ #c; #H; nauto;
-##| #c; #i; #HCU; #H; @2 i; nauto;
-##]
+#A; #a; #U; #V; #aU; #UV; nelim aU; nauto depth=4;
nqed.
ndefinition emptyset: ∀A.Ω^A ≝ λA.{x | False}.
naxiom EM : ∀A:Ax.∀a:A.∀i_star.(a ∈ 𝐂 a i_star) ∨ ¬( a ∈ 𝐂 a i_star).
+alias symbol "covers" = "covers".
ntheorem th2_3 :
∀A:Ax.∀a:A. a ◃ ∅ → ∃i. ¬ a ∈ 𝐂 a i.
#A; #a; #H; nelim H;
##[ #n; *;
-##| #b; #i_star; #IH1; #IH2;
- ncases (EM … b i_star);
- ##[##2: (* nauto; *) #W; @i_star; napply W;
- ##| nauto;
- ##]
+##| #b; #i_star; #IH1; #IH2; ncases (EM … b i_star); nauto;
##]
nqed.
}.
ndefinition uax : uAx → Ax.
-#A; @ (uax_ A) (λx.unit); #a; #_; napply (𝐂 a ?); nlapply one; ncases (with_ A a); nauto;
+#A; @ (uax_ A) (λx.unit); #a; #_;
+napply (𝐂 a ?); nlapply one; ncases (with_ A a); nauto;
nqed.
ncoercion uax : ∀u:uAx. Ax ≝ uax on _u : uAx to Ax.
ntheorem col2_4 :
- ∀A:uAx.∀a:A. a ◃ ∅ → ¬ a ∈ 𝐂 a ?. ##[ (* bug *) ##2: nnormalize; napply one; ##]
+ ∀A:uAx.∀a:A. a ◃ ∅ → ¬ a ∈ 𝐂 a one.
#A; #a; #H; nelim H;
##[ #n; *;
-##| #b; #i_star; #IH1; #IH2; #H3; nlapply (IH2 … H3); #H4; nauto;
+##| #b; #i_star; #IH1; #IH2; #H3; nlapply (IH2 … H3); nauto;
##]
nqed.
ndefinition Z : Ω^axs ≝ { x | x ◃ ∅ }.
ntheorem cover_monotone: ∀A:Ax.∀a:A.∀U,V.U ⊆ V → a ◃ U → a ◃ V.
-#A; #a; #U; #V; #HUV; #H; nelim H;
-##[ nauto;
-##| #b; #i; #HCU; #W; @2 i; #x; nauto; ##]
+#A; #a; #U; #V; #HUV; #H; nelim H; nauto depth=4;
nqed.
ntheorem th3_1: ¬∃a:axs.Z ⊆ S a ∧ S a ⊆ Z.
*; #a; *; #ZSa; #SaZ;
ncut (a ◃ Z); ##[
nlapply (axiom_cond … a one); #AxCon; nchange in AxCon with (a ◃ S a);
- (* nauto; *) napply (cover_monotone … AxCon); nassumption; ##] #H;
-ncut (a ◃ ∅); ##[ napply (transitivity … H); #x; #E; napply E; ##] #H1;
+ napply (cover_monotone … AxCon); nassumption; ##] #H;
+ncut (a ◃ ∅); ##[ napply (transitivity … H); nwhd in match Z; nauto; ##] #H1;
ncut (¬ a ∈ S a); ##[ napply (col2_4 … H1); ##] #H2;
ncut (a ∈ S a); ##[ napply ZSa; napply H1; ##] #H3;
nauto;
naxiom h : nat → nat.
-naxiom Ph : ∀x.h x = O → x ◃ ∅.
+alias symbol "eq" = "leibnitz's equality".
+alias symbol "eq" = "setoid1 eq".
+alias symbol "covers" = "covers".
+alias symbol "eq" = "leibnitz's equality".
+naxiom Ph : ∀x.h x = O \liff x ◃ ∅.
-ninductive eq2 (A : Type[1]) (a : A) : A → CProp[0] ≝
-| refl2 : eq2 A a a.
-
-interpretation "eq2" 'eq T a b = (eq2 T a b).
-
-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
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-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).
-
-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;
- ##]
-##]
+nlemma replace_char:
+ ∀A:Ax.∀U,V.U ⊆ V → V ⊆ U → ∀a:A.a ◃ U → a ◃ V.
+#A; #U; #V; #UV; #VU; #a; #aU; nelim aU; nauto;
nqed.
-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).
-
-nrecord uAx : Type[1] ≝ {
- uax_ : Ax;
- with_ : ∀a:uax_.𝐈 a ∼ unit
-}.
-
-ndefinition uax : uAx → Ax.
-*; #A; #E; @ A (λx.unit); #a; ncases (E a);
-##[ #i; napply (𝐃 a i);
-##| #i; nnormalize; #j; napply (𝐝 a i j);
-##]
+ntheorem th_ch3: ¬∃a:caxs.∀x.ϕ a x = h x.
+*; #a; #H;
+ncut (a ◃ { x | x ◃ ∅}); ##[
+ napply (replace_char … { x | h x = O }); ##[ ##1,2: #x; ncases (Ph x); nauto; ##]
+ napply (replace_char … { x | ϕ a x = O }); ##[##1,2: #x; nrewrite > (H x); nauto; ##]
+ napply (axiom_cond … a one); ##] #H1;
+ncut (a ◃ ∅); ##[ napply (transitivity … H1); nauto; ##] #H2;
+nlapply (col2_4 …H2); #H3;
+ncut (a ∈ 𝐂 a one); ##[
+ nnormalize; ncases (Ph a); nrewrite > (H a); nauto; ##] #H4;
+nauto;
nqed.
-ncoercion uax : ∀u:unAx. nAx ≝ uax on _u : unAx to nAx.
-
-
-nlemma cor_2_5 : ∀A:unAx.∀a:A.∀i.a ⊢ ∅ → ¬(a ∈ 𝐈𝐦[𝐝 a i]).
-#A; #a; #i; #H; nelim H in i;
-##[ #w; *;
-##| #a; #i; #IH1; #IH2;
-
-
-
\ No newline at end of file
projects / helm.git / blobdiff