commit by user utente2

diff --git a/weblib/basics/logic.ma b/weblib/basics/logic.ma
index 3f92d6aad834d81fdc0bc2bd803f2cb41f85f998..5e97f9b5c1bdd4b27da7613edad60fa8c644b917 100644 (file)
--- a/weblib/basics/logic.ma
+++ b/weblib/basics/logic.ma
@@ -20,42 +20,42 @@ inductive eq (A:Type[1]) (x:A) : A → Prop ≝
 interpretation "leibnitz's equality" 'eq t x y = (eq t x y).
 
 lemma eq_rect_r:
- ∀A.∀a,x.∀p:\ 5A href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/A\ 6 ? x a.∀P: 
- ∀x:A. x \ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6 a → Type[2]. P a (refl A a) → P x p.
+ ∀A.∀a,x.∀p:\ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 ? x a.∀P: 
+ ∀x:A. x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 a → Type[2]. P a (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 A a) → P x p.
  #A #a #x #p (cases p) // qed.
 
 lemma eq_ind_r :
- ∀A.∀a.∀P: ∀x:A. x \ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6 a → Prop. P a (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 A a) → 
-   ∀x.∀p:\ 5A href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/A\ 6 ? x a.P x p.
- #A #a #P #p #x0 #p0; @(\ 5A href="cic:/matita/basics/logic/eq_rect_r.def(1)"\ 6eq_rect_r\ 5/A\ 6 ? ? ? p0) //; qed.
+ ∀A.∀a.∀P: ∀x:A. x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 a → Prop. P a (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 A a) → 
+   ∀x.∀p:\ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 ? x a.P x p.
+ #A #a #P #p #x0 #p0; @(\ 5a href="cic:/matita/basics/logic/eq_rect_r.def(1)"\ 6eq_rect_r\ 5/a\ 6 ? ? ? p0) //; qed.
 
 lemma eq_rect_Type2_r:
-  ∀A.∀a.∀P: ∀x:A. \ 5A href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/A\ 6 ? x a → Type[2]. P a (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 A a) → 
-    ∀x.∀p:\ 5A href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/A\ 6 ? x a.P x p.
+  ∀A.∀a.∀P: ∀x:A. \ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 ? x a → Type[2]. P a (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 A a) → 
+    ∀x.∀p:\ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 ? x a.P x p.
   #A #a #P #H #x #p (generalize in match H) (generalize in match P)
   cases p; //; qed.
 
-theorem rewrite_l: ∀A:Type[1].∀x.∀P:A → Type[1]. P x → ∀y. x \ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6 y → P y.
+theorem rewrite_l: ∀A:Type[1].∀x.∀P:A → Type[1]. P x → ∀y. x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 y → P y.
 #A #x #P #Hx #y #Heq (cases Heq); //; qed.
 
-theorem sym_eq: ∀A.∀x,y:A. x \ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6 y → y \ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6 x.
-#A #x #y #Heq @(\ 5A href="cic:/matita/basics/logic/rewrite_l.def(1)"\ 6rewrite_l\ 5/A\ 6 A x (λz.z\ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6x)); //; qed.
+theorem sym_eq: ∀A.∀x,y:A. x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 y → y \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 x.
+#A #x #y #Heq @(\ 5a href="cic:/matita/basics/logic/rewrite_l.def(1)"\ 6rewrite_l\ 5/a\ 6 A x (λz.z\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6x)); //; qed.
 
-theorem rewrite_r: ∀A:Type[1].∀x.∀P:A → Type[1]. P x → ∀y. y \ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6 x → P y.
-#A #x #P #Hx #y #Heq (cases (\ 5A href="cic:/matita/basics/logic/sym_eq.def(2)"\ 6sym_eq\ 5/A\ 6 ? ? ? Heq)); //; qed.
+theorem rewrite_r: ∀A:Type[1].∀x.∀P:A → Type[1]. P x → ∀y. y \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 x → P y.
+#A #x #P #Hx #y #Heq (cases (\ 5a href="cic:/matita/basics/logic/sym_eq.def(2)"\ 6sym_eq\ 5/a\ 6 ? ? ? Heq)); //; qed.
 
-theorem eq_coerc: ∀A,B:Type[0].A→(A\ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6B)→B.
+theorem eq_coerc: ∀A,B:Type[0].A→(A\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6B)→B.
 #A #B #Ha #Heq (elim Heq); //; qed.
 
-theorem trans_eq : ∀A.∀x,y,z:A. x \ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6 y → y \ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6 z → x \ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6 z.
+theorem trans_eq : ∀A.∀x,y,z:A. x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 y → y \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 z → x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 z.
 #A #x #y #z #H1 #H2 >H1; //; qed.
 
-theorem eq_f: ∀A,B.∀f:A→B.∀x,y:A. x\ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6y → f x \ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6 f y.
+theorem eq_f: ∀A,B.∀f:A→B.∀x,y:A. x\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6y → f x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 f y.
 #A #B #f #x #y #H >H; //; qed.
 
 (* deleterio per auto? *)
 theorem eq_f2: ∀A,B,C.∀f:A→B→C.
-∀x1,x2:A.∀y1,y2:B. x1\ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6x2 → y1\ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6y2 → f x1 y1 \ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6 f x2 y2.
+∀x1,x2:A.∀y1,y2:B. x1\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6x2 → y1\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6y2 → f x1 y1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 f x2 y2.
 #A #B #C #f #x1 #x2 #y1 #y2 #E1 #E2 >E1; >E2; //; qed. 
 
 (* hint to genereric equality 
@@ -80,12 +80,12 @@ inductive False: Prop ≝ .
 λA. A → False. *)
 
 inductive Not (A:Prop): Prop ≝
-nmk: (A → \ 5A href="cic:/matita/basics/logic/False.ind(1,0,0)"\ 6False\ 5/A\ 6) → Not A.
+nmk: (A → \ 5a href="cic:/matita/basics/logic/False.ind(1,0,0)"\ 6False\ 5/a\ 6) → Not A.
 
 
 interpretation "logical not" 'not x = (Not x).
 
-theorem absurd : ∀A:Prop. A → \ 5A title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/A\ 6A → \ 5A href="cic:/matita/basics/logic/False.ind(1,0,0)"\ 6False\ 5/A\ 6.
+theorem absurd : ∀A:Prop. A → \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6A → \ 5a href="cic:/matita/basics/logic/False.ind(1,0,0)"\ 6False\ 5/a\ 6.
 #A #H #Hn (elim Hn); /2/; qed.
 
 (*
@@ -93,13 +93,13 @@ ntheorem absurd : ∀ A,C:Prop. A → ¬A → C.
 #A; #C; #H; #Hn; nelim (Hn H).
 nqed. *)
 
-theorem not_to_not : ∀A,B:Prop. (A → B) → \ 5A title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/A\ 6B →\ 5A title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/A\ 6A.
+theorem not_to_not : ∀A,B:Prop. (A → B) → \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6B →\ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6A.
 /4/; qed.
 
 (* inequality *)
 interpretation "leibnitz's non-equality" 'neq t x y = (Not (eq t x y)).
 
-theorem sym_not_eq: ∀A.∀x,y:A. x \ 5A title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/A\ 6 y → y \ 5A title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/A\ 6 x.
+theorem sym_not_eq: ∀A.∀x,y:A. x \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 y → y \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 x.
 /3/; qed.
 
 (* and *)
@@ -108,10 +108,10 @@ inductive And (A,B:Prop) : Prop ≝
 
 interpretation "logical and" 'and x y = (And x y).
 
-theorem proj1: ∀A,B:Prop. A \ 5A title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/A\ 6 B → A.
+theorem proj1: ∀A,B:Prop. A \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 B → A.
 #A #B #AB (elim AB) //; qed.
 
-theorem proj2: ∀ A,B:Prop. A \ 5A title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/A\ 6 B → B.
+theorem proj2: ∀ A,B:Prop. A \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 B → B.
 #A #B #AB (elim AB) //; qed.
 
 (* or *)
@@ -122,7 +122,7 @@ inductive Or (A,B:Prop) : Prop ≝
 interpretation "logical or" 'or x y = (Or x y).
 
 definition decidable : Prop → Prop ≝ 
-λ A:Prop. A \ 5A title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/A\ 6 \ 5A title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/A\ 6 A.
+λ A:Prop. A \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6 A.
 
 (* exists *)
 inductive ex (A:Type[0]) (P:A → Prop) : Prop ≝
@@ -135,7 +135,7 @@ inductive ex2 (A:Type[0]) (P,Q:A →Prop) : Prop ≝
 
 (* iff *)
 definition iff :=
- λ A,B. (A → B) \ 5A title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/A\ 6 (B → A).
+ λ A,B. (A → B) \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 (B → A).
 
 interpretation "iff" 'iff a b = (iff a b).  
 
@@ -143,84 +143,85 @@ interpretation "iff" 'iff a b = (iff a b).
 
 definition R0 ≝ λT:Type[0].λt:T.t.
   
-definition R1 ≝ \ 5A href="cic:/matita/basics/logic/eq_rect_Type0.fix(0,5,1)"\ 6eq_rect_Type0\ 5/A\ 6.
+definition R1 ≝ \ 5a href="cic:/matita/basics/logic/eq_rect_Type0.fix(0,5,1)"\ 6eq_rect_Type0\ 5/a\ 6.
 
 (* useless stuff *)
 definition R2 :
   ∀T0:Type[0].
   ∀a0:T0.
-  ∀T1:∀x0:T0. a0\ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6x0 → Type[0].
-  ∀a1:T1 a0 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 ? a0).
-  ∀T2:∀x0:T0. ∀p0:a0\ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6x0. ∀x1:T1 x0 p0. \ 5A href="cic:/matita/basics/logic/R1.def(2)"\ 6R1\ 5/A\ 6 ?? T1 a1 ? p0 \ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6 x1 → Type[0].
-  ∀a2:T2 a0 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 ? a0) a1 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 ? a1).
+  ∀T1:∀x0:T0. a0\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6x0 → Type[0].
+  ∀a1:T1 a0 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 ? a0).
+  ∀T2:∀x0:T0. ∀p0:a0\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6x0. ∀x1:T1 x0 p0. \ 5a href="cic:/matita/basics/logic/R1.def(2)"\ 6R1\ 5/a\ 6 ?? T1 a1 ? p0 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 x1 → Type[0].
+  ∀a2:T2 a0 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 ? a0) a1 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 ? a1).
   ∀b0:T0.
-  ∀e0:a0 \ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6 b0.
+  ∀e0:a0 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b0.
   ∀b1: T1 b0 e0.
-  ∀e1:\ 5A href="cic:/matita/basics/logic/R1.def(2)"\ 6R1\ 5/A\ 6 ?? T1 a1 ? e0 \ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6 b1.
+  ∀e1:\ 5a href="cic:/matita/basics/logic/R1.def(2)"\ 6R1\ 5/a\ 6 ?? T1 a1 ? e0 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b1.
   T2 b0 e0 b1 e1.
 #T0 #a0 #T1 #a1 #T2 #a2 #b0 #e0 #b1 #e1 
-@(\ 5A href="cic:/matita/basics/logic/eq_rect_Type0.fix(0,5,1)"\ 6eq_rect_Type0\ 5/A\ 6 ????? e1) 
-@(\ 5A href="cic:/matita/basics/logic/R1.def(2)"\ 6R1\ 5/A\ 6 ?? ? ?? e0) 
+@(\ 5a href="cic:/matita/basics/logic/eq_rect_Type0.fix(0,5,1)"\ 6eq_rect_Type0\ 5/a\ 6 ????? e1) 
+@(\ 5a href="cic:/matita/basics/logic/R1.def(2)"\ 6R1\ 5/a\ 6 ?? ? ?? e0) 
 @a2 
 qed.
 
 definition R3 :
   ∀T0:Type[0].
   ∀a0:T0.
-  ∀T1:∀x0:T0. a0\ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6x0 → Type[0].
-  ∀a1:T1 a0 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 ? a0).
-  ∀T2:∀x0:T0. ∀p0:a0\ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6x0. ∀x1:T1 x0 p0. \ 5A href="cic:/matita/basics/logic/R1.def(2)"\ 6R1\ 5/A\ 6 ?? T1 a1 ? p0 \ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6 x1 → Type[0].
-  ∀a2:T2 a0 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 ? a0) a1 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 ? a1).
-  ∀T3:∀x0:T0. ∀p0:a0\ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6x0. ∀x1:T1 x0 p0.∀p1:\ 5A href="cic:/matita/basics/logic/R1.def(2)"\ 6R1\ 5/A\ 6 ?? T1 a1 ? p0 \ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6 x1.
-      ∀x2:T2 x0 p0 x1 p1.\ 5A href="cic:/matita/basics/logic/R2.def(3)"\ 6R2\ 5/A\ 6 ???? T2 a2 x0 p0 ? p1 \ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6 x2 → Type[0].
-  ∀a3:T3 a0 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 ? a0) a1 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 ? a1) a2 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 ? a2).
+  ∀T1:∀x0:T0. a0\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6x0 → Type[0].
+  ∀a1:T1 a0 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 ? a0).
+  ∀T2:∀x0:T0. ∀p0:a0\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6x0. ∀x1:T1 x0 p0. \ 5a href="cic:/matita/basics/logic/R1.def(2)"\ 6R1\ 5/a\ 6 ?? T1 a1 ? p0 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 x1 → Type[0].
+  ∀a2:T2 a0 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 ? a0) a1 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 ? a1).
+  ∀T3:∀x0:T0. ∀p0:a0\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6x0. ∀x1:T1 x0 p0.∀p1:\ 5a href="cic:/matita/basics/logic/R1.def(2)"\ 6R1\ 5/a\ 6 ?? T1 a1 ? p0 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 x1.
+      ∀x2:T2 x0 p0 x1 p1.\ 5a href="cic:/matita/basics/logic/R2.def(3)"\ 6R2\ 5/a\ 6 ???? T2 a2 x0 p0 ? p1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 x2 → Type[0].
+  ∀a3:T3 a0 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 ? a0) a1 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 ? a1) a2 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 ? a2).
   ∀b0:T0.
-  ∀e0:a0 \ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6 b0.
+  ∀e0:a0 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b0.
   ∀b1: T1 b0 e0.
-  ∀e1:\ 5A href="cic:/matita/basics/logic/R1.def(2)"\ 6R1\ 5/A\ 6 ?? T1 a1 ? e0 \ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6 b1.
+  ∀e1:\ 5a href="cic:/matita/basics/logic/R1.def(2)"\ 6R1\ 5/a\ 6 ?? T1 a1 ? e0 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b1.
   ∀b2: T2 b0 e0 b1 e1.
-  ∀e2:\ 5A href="cic:/matita/basics/logic/R2.def(3)"\ 6R2\ 5/A\ 6 ???? T2 a2 b0 e0 ? e1 \ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6 b2.
+  ∀e2:\ 5a href="cic:/matita/basics/logic/R2.def(3)"\ 6R2\ 5/a\ 6 ???? T2 a2 b0 e0 ? e1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b2.
   T3 b0 e0 b1 e1 b2 e2.
 #T0 #a0 #T1 #a1 #T2 #a2 #T3 #a3 #b0 #e0 #b1 #e1 #b2 #e2 
-@(\ 5A href="cic:/matita/basics/logic/eq_rect_Type0.fix(0,5,1)"\ 6eq_rect_Type0\ 5/A\ 6 ????? e2) 
-@(\ 5A href="cic:/matita/basics/logic/R2.def(3)"\ 6R2\ 5/A\ 6 ?? ? ???? e0 ? e1) 
+@(\ 5a href="cic:/matita/basics/logic/eq_rect_Type0.fix(0,5,1)"\ 6eq_rect_Type0\ 5/a\ 6 ????? e2) 
+@(\ 5a href="cic:/matita/basics/logic/R2.def(3)"\ 6R2\ 5/a\ 6 ?? ? ???? e0 ? e1) 
 @a3 
 qed.
 
 definition R4 :
   ∀T0:Type[0].
   ∀a0:T0.
-  ∀T1:∀x0:T0. \ 5A href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/A\ 6 T0 a0 x0 → Type[0].
-  ∀a1:T1 a0 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 T0 a0).
-  ∀T2:∀x0:T0. ∀p0:\ 5A href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/A\ 6 (T0 …) a0 x0. ∀x1:T1 x0 p0.\ 5A href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/A\ 6 (T1 …) (\ 5A href="cic:/matita/basics/logic/R1.def(2)"\ 6R1\ 5/A\ 6 T0 a0 T1 a1 x0 p0) x1 → Type[0].
-  ∀a2:T2 a0 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 T0 a0) a1 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 (T1 a0 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 T0 a0)) a1).
-  ∀T3:∀x0:T0. ∀p0:\ 5A href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/A\ 6 (T0 …) a0 x0. ∀x1:T1 x0 p0.∀p1:\ 5A href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/A\ 6 (T1 …) (\ 5A href="cic:/matita/basics/logic/R1.def(2)"\ 6R1\ 5/A\ 6 T0 a0 T1 a1 x0 p0) x1.
-      ∀x2:T2 x0 p0 x1 p1.\ 5A href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/A\ 6 (T2 …) (\ 5A href="cic:/matita/basics/logic/R2.def(3)"\ 6R2\ 5/A\ 6 T0 a0 T1 a1 T2 a2 x0 p0 x1 p1) x2 → Type[0].
-  ∀a3:T3 a0 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 T0 a0) a1 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 (T1 a0 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 T0 a0)) a1) 
-      a2 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 (T2 a0 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 T0 a0) a1 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 (T1 a0 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 T0 a0)) a1)) a2). 
-  ∀T4:∀x0:T0. ∀p0:\ 5A href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/A\ 6 (T0 …) a0 x0. ∀x1:T1 x0 p0.∀p1:\ 5A href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/A\ 6 (T1 …) (\ 5A href="cic:/matita/basics/logic/R1.def(2)"\ 6R1\ 5/A\ 6 T0 a0 T1 a1 x0 p0) x1.
-      ∀x2:T2 x0 p0 x1 p1.∀p2:\ 5A href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/A\ 6 (T2 …) (\ 5A href="cic:/matita/basics/logic/R2.def(3)"\ 6R2\ 5/A\ 6 T0 a0 T1 a1 T2 a2 x0 p0 x1 p1) x2.
-      ∀x3:T3 x0 p0 x1 p1 x2 p2.∀p3:\ 5A href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/A\ 6 (T3 …) (\ 5A href="cic:/matita/basics/logic/R3.def(4)"\ 6R3\ 5/A\ 6 T0 a0 T1 a1 T2 a2 T3 a3 x0 p0 x1 p1 x2 p2) x3. 
+  ∀T1:∀x0:T0. \ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 T0 a0 x0 → Type[0].
+  ∀a1:T1 a0 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 T0 a0).
+  ∀T2:∀x0:T0. ∀p0:\ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 (T0 …) a0 x0. ∀x1:T1 x0 p0.\ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 (T1 …) (\ 5a href="cic:/matita/basics/logic/R1.def(2)"\ 6R1\ 5/a\ 6 T0 a0 T1 a1 x0 p0) x1 → Type[0].
+  ∀a2:T2 a0 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 T0 a0) a1 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 (T1 a0 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 T0 a0)) a1).
+  ∀T3:∀x0:T0. ∀p0:\ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 (T0 …) a0 x0. ∀x1:T1 x0 p0.∀p1:\ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 (T1 …) (\ 5a href="cic:/matita/basics/logic/R1.def(2)"\ 6R1\ 5/a\ 6 T0 a0 T1 a1 x0 p0) x1.
+      ∀x2:T2 x0 p0 x1 p1.\ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 (T2 …) (\ 5a href="cic:/matita/basics/logic/R2.def(3)"\ 6R2\ 5/a\ 6 T0 a0 T1 a1 T2 a2 x0 p0 x1 p1) x2 → Type[0].
+  ∀a3:T3 a0 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 T0 a0) a1 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 (T1 a0 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 T0 a0)) a1) 
+      a2 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 (T2 a0 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 T0 a0) a1 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 (T1 a0 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 T0 a0)) a1)) a2). 
+  ∀T4:∀x0:T0. ∀p0:\ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 (T0 …) a0 x0. ∀x1:T1 x0 p0.∀p1:\ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 (T1 …) (\ 5a href="cic:/matita/basics/logic/R1.def(2)"\ 6R1\ 5/a\ 6 T0 a0 T1 a1 x0 p0) x1.
+      ∀x2:T2 x0 p0 x1 p1.∀p2:\ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 (T2 …) (\ 5a href="cic:/matita/basics/logic/R2.def(3)"\ 6R2\ 5/a\ 6 T0 a0 T1 a1 T2 a2 x0 p0 x1 p1) x2.
+      ∀x3:T3 x0 p0 x1 p1 x2 p2.∀p3:\ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 (T3 …) (\ 5a href="cic:/matita/basics/logic/R3.def(4)"\ 6R3\ 5/a\ 6 T0 a0 T1 a1 T2 a2 T3 a3 x0 p0 x1 p1 x2 p2) x3. 
       Type[0].
-  ∀a4:T4 a0 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 T0 a0) a1 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 (T1 a0 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 T0 a0)) a1) 
-      a2 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 (T2 a0 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 T0 a0) a1 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 (T1 a0 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 T0 a0)) a1)) a2) 
-      a3 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 (T3 a0 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 T0 a0) a1 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 (T1 a0 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 T0 a0)) a1) 
-                   a2 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 (T2 a0 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 T0 a0) a1 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 (T1 a0 (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 T0 a0)) a1)) a2))
+  ∀a4:T4 a0 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 T0 a0) a1 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 (T1 a0 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 T0 a0)) a1) 
+      a2 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 (T2 a0 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 T0 a0) a1 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 (T1 a0 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 T0 a0)) a1)) a2) 
+      a3 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 (T3 a0 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 T0 a0) a1 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 (T1 a0 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 T0 a0)) a1) 
+                   a2 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 (T2 a0 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 T0 a0) a1 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 (T1 a0 (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 T0 a0)) a1)) a2))
                    a3).
   ∀b0:T0.
-  ∀e0:\ 5A href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/A\ 6 (T0 …) a0 b0.
+  ∀e0:\ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 (T0 …) a0 b0.
   ∀b1: T1 b0 e0.
-  ∀e1:\ 5A href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/A\ 6 (T1 …) (\ 5A href="cic:/matita/basics/logic/R1.def(2)"\ 6R1\ 5/A\ 6 T0 a0 T1 a1 b0 e0) b1.
+  ∀e1:\ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 (T1 …) (\ 5a href="cic:/matita/basics/logic/R1.def(2)"\ 6R1\ 5/a\ 6 T0 a0 T1 a1 b0 e0) b1.
   ∀b2: T2 b0 e0 b1 e1.
-  ∀e2:\ 5A href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/A\ 6 (T2 …) (\ 5A href="cic:/matita/basics/logic/R2.def(3)"\ 6R2\ 5/A\ 6 T0 a0 T1 a1 T2 a2 b0 e0 b1 e1) b2.
+  ∀e2:\ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 (T2 …) (\ 5a href="cic:/matita/basics/logic/R2.def(3)"\ 6R2\ 5/a\ 6 T0 a0 T1 a1 T2 a2 b0 e0 b1 e1) b2.
   ∀b3: T3 b0 e0 b1 e1 b2 e2.
-  ∀e3:\ 5A href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/A\ 6 (T3 …) (\ 5A href="cic:/matita/basics/logic/R3.def(4)"\ 6R3\ 5/A\ 6 T0 a0 T1 a1 T2 a2 T3 a3 b0 e0 b1 e1 b2 e2) b3.
+  ∀e3:\ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 (T3 …) (\ 5a href="cic:/matita/basics/logic/R3.def(4)"\ 6R3\ 5/a\ 6 T0 a0 T1 a1 T2 a2 T3 a3 b0 e0 b1 e1 b2 e2) b3.
   T4 b0 e0 b1 e1 b2 e2 b3 e3.
 #T0 #a0 #T1 #a1 #T2 #a2 #T3 #a3 #T4 #a4 #b0 #e0 #b1 #e1 #b2 #e2 #b3 #e3 
-@(\ 5A href="cic:/matita/basics/logic/eq_rect_Type0.fix(0,5,1)"\ 6eq_rect_Type0\ 5/A\ 6 ????? e3) 
-@(\ 5A href="cic:/matita/basics/logic/R3.def(4)"\ 6R3\ 5/A\ 6 ????????? e0 ? e1 ? e2)
+@(\ 5a href="cic:/matita/basics/logic/eq_rect_Type0.fix(0,5,1)"\ 6eq_rect_Type0\ 5/a\ 6 ????? e3) 
+@(\ 5a href="cic:/matita/basics/logic/R3.def(4)"\ 6R3\ 5/a\ 6 ????????? e0 ? e1 ? e2)
 @a4 
 qed.
 
 (* TODO concrete definition by means of proof irrelevance *)
-axiom streicherK : ∀T:Type[1].∀t:T.∀P:t \ 5A title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/A\ 6 t → Type[2].P (\ 5A href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/A\ 6 ? t) → ∀p.P p.
+axiom streicherK : ∀T:Type[1].∀t:T.∀P:t \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 t → Type[2].P (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 ? t) → ∀p.P p.
+