From f848433620419e71cd19ec0b4f58c717ac50f85e Mon Sep 17 00:00:00 2001
From: matitaweb <claudio.sacerdoticoen@unibo.it>
Date: Tue, 28 Feb 2012 08:23:11 +0000
Subject: [PATCH] commit by user andrea

---
 weblib/basics/logic.ma      | 180 ++++++++++++++++++------------------
 weblib/tutorial/chapter2.ma |   5 +-
 weblib/tutorial/chapter4.ma | 108 +++++++++++++---------
 3 files changed, 155 insertions(+), 138 deletions(-)

diff --git a/weblib/basics/logic.ma b/weblib/basics/logic.ma
index 8f981d469..b9dfed28c 100644
--- a/weblib/basics/logic.ma
+++ b/weblib/basics/logic.ma
@@ -21,58 +21,58 @@ interpretation "leibnitz's equality" 'eq t x y = (eq t x y).
 interpretation "leibniz reflexivity" 'refl = refl.
 
 lemma eq_rect_r:
- ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → Type[3]. P a (refl A a) → P x p.
+ ∀A.∀a,x.∀p:a href="cic:/matita/basics/logic/eq.ind(1,0,2)"eq/a ? x a.∀P: ∀x:A. a href="cic:/matita/basics/logic/eq.ind(1,0,2)"eq/a ? x a → Type[3]. P a (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a A a) → P x p.
  #A #a #x #p (cases p) // qed.
 
 lemma eq_ind_r :
- ∀A.∀a.∀P: ∀x:A. x = a → Prop. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
- #A #a #P #p #x0 #p0; @(eq_rect_r ? ? ? p0) //; qed.
+ ∀A.∀a.∀P: ∀x:A. x a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a → Prop. P a (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a A a) → ∀x.∀p:a href="cic:/matita/basics/logic/eq.ind(1,0,2)"eq/a ? x a.P x p.
+ #A #a #P #p #x0 #p0; @(a href="cic:/matita/basics/logic/eq_rect_r.def(1)"eq_rect_r/a ? ? ? p0) //; qed.
 
 lemma 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: ∀x:A. a href="cic:/matita/basics/logic/eq.ind(1,0,2)"eq/a ? x a → Type[0]. P a (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a A a) → ∀x.∀p:a href="cic:/matita/basics/logic/eq.ind(1,0,2)"eq/a ? x a.P x p.
   #A #a #P #H #x #p lapply H lapply P
   cases p; //; qed.
 
 lemma eq_rect_Type1_r:
-  ∀A.∀a.∀P: ∀x:A. eq ? x a → Type[1]. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
+  ∀A.∀a.∀P: ∀x:A. a href="cic:/matita/basics/logic/eq.ind(1,0,2)"eq/a ? x a → Type[1]. P a (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a A a) → ∀x.∀p:a href="cic:/matita/basics/logic/eq.ind(1,0,2)"eq/a ? x a.P x p.
   #A #a #P #H #x #p lapply H lapply P
   cases p; //; qed.
 
 lemma eq_rect_Type2_r:
-  ∀A.∀a.∀P: ∀x:A. eq ? x a → Type[2]. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
+  ∀A.∀a.∀P: ∀x:A. a href="cic:/matita/basics/logic/eq.ind(1,0,2)"eq/a ? x a → Type[2]. P a (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a A a) → ∀x.∀p:a href="cic:/matita/basics/logic/eq.ind(1,0,2)"eq/a ? x a.P x p.
   #A #a #P #H #x #p lapply H lapply P
   cases p; //; qed.
 
 lemma eq_rect_Type3_r:
-  ∀A.∀a.∀P: ∀x:A. eq ? x a → Type[3]. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
+  ∀A.∀a.∀P: ∀x:A. a href="cic:/matita/basics/logic/eq.ind(1,0,2)"eq/a ? x a → Type[3]. P a (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a A a) → ∀x.∀p:a href="cic:/matita/basics/logic/eq.ind(1,0,2)"eq/a ? x a.P x p.
   #A #a #P #H #x #p lapply H lapply P
   cases p; //; qed.
 
-theorem rewrite_l: ∀A:Type[2].∀x.∀P:A → Type[2]. P x → ∀y. x = y → P y.
+theorem rewrite_l: ∀A:Type[2].∀x.∀P:A → Type[2]. P x → ∀y. x a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a y → P y.
 #A #x #P #Hx #y #Heq (cases Heq); //; qed.
 
-theorem sym_eq: ∀A.∀x,y:A. x = y → y = x.
-#A #x #y #Heq @(rewrite_l A x (λz.z=x)) // qed.
+theorem sym_eq: ∀A.∀x,y:A. x a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a y → y a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a x.
+#A #x #y #Heq @(a href="cic:/matita/basics/logic/rewrite_l.def(1)"rewrite_l/a A x (λz.za title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/ax)) // qed.
 
-theorem rewrite_r: ∀A:Type[2].∀x.∀P:A → Type[2]. P x → ∀y. y = x → P y.
-#A #x #P #Hx #y #Heq (cases (sym_eq ? ? ? Heq)); //; qed.
+theorem rewrite_r: ∀A:Type[2].∀x.∀P:A → Type[2]. P x → ∀y. y a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a x → P y.
+#A #x #P #Hx #y #Heq (cases (a href="cic:/matita/basics/logic/sym_eq.def(2)"sym_eq/a ? ? ? Heq)); //; qed.
 
-theorem eq_coerc: ∀A,B:Type[0].A→(A=B)→B.
+theorem eq_coerc: ∀A,B:Type[0].A→(Aa title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/aB)→B.
 #A #B #Ha #Heq (elim Heq); //; qed.
 
-theorem trans_eq : ∀A.∀x,y,z:A. x = y → y = z → x = z.
+theorem trans_eq : ∀A.∀x,y,z:A. x a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a y → y a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a z → x a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a z.
 #A #x #y #z #H1 #H2 >H1; //; qed.
 
-theorem eq_f: ∀A,B.∀f:A→B.∀x,y:A. x=y → f x = f y.
+theorem eq_f: ∀A,B.∀f:A→B.∀x,y:A. xa title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/ay → f x a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a 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=x2 → y1=y2 → f x1 y1 = f x2 y2.
+∀x1,x2:A.∀y1,y2:B. x1a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/ax2 → y1a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/ay2 → f x1 y1 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a f x2 y2.
 #A #B #C #f #x1 #x2 #y1 #y2 #E1 #E2 >E1; >E2; //; qed. 
 
 lemma eq_f3: ∀A,B,C,D.∀f:A→B→C->D.
-∀x1,x2:A.∀y1,y2:B. ∀z1,z2:C. x1=x2 → y1=y2 → z1=z2 → f x1 y1 z1 = f x2 y2 z2.
+∀x1,x2:A.∀y1,y2:B. ∀z1,z2:C. x1a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/ax2 → y1a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/ay2 → z1a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/az2 → f x1 y1 z1 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a f x2 y2 z2.
 #A #B #C #D #f #x1 #x2 #y1 #y2 #z1 #z2 #E1 #E2 #E3 >E1; >E2; >E3 //; qed.
 
 (* hint to genereric equality 
@@ -97,26 +97,26 @@ inductive False: Prop ≝ .
 λA. A → False. *)
 
 inductive Not (A:Prop): Prop ≝
-nmk: (A → False) → Not A.
+nmk: (A → a href="cic:/matita/basics/logic/False.ind(1,0,0)"False/a) → Not A.
 
 interpretation "logical not" 'not x = (Not x).
 
-theorem absurd : ∀A:Prop. A → ¬A → False.
-#A #H #Hn (elim Hn); /2/; qed.
+theorem absurd : ∀A:Prop. A → a title="logical not" href="cic:/fakeuri.def(1)"¬/aA → a href="cic:/matita/basics/logic/False.ind(1,0,0)"False/a.
+#A #H #Hn (elim Hn); /span class="autotactic"2span class="autotrace" trace /span/span/; qed.
 
 (*
 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) → ¬B →¬A.
-/4/; qed.
+theorem not_to_not : ∀A,B:Prop. (A → B) → a title="logical not" href="cic:/fakeuri.def(1)"¬/aB →a title="logical not" href="cic:/fakeuri.def(1)"¬/aA.
+/span class="autotactic"4span class="autotrace" trace a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a, a href="cic:/matita/basics/logic/Not.con(0,1,1)"nmk/a/span/span/; 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 ≠ y → y ≠ x.
-/3/; qed.
+theorem sym_not_eq: ∀A.∀x,y:A. x a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"≠/a y → y a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"≠/a x.
+/span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a, a href="cic:/matita/basics/logic/Not.con(0,1,1)"nmk/a/span/span/; qed.
 
 (* and *)
 inductive And (A,B:Prop) : Prop ≝
@@ -124,10 +124,10 @@ inductive And (A,B:Prop) : Prop ≝
 
 interpretation "logical and" 'and x y = (And x y).
 
-theorem proj1: ∀A,B:Prop. A ∧ B → A.
+theorem proj1: ∀A,B:Prop. A a title="logical and" href="cic:/fakeuri.def(1)"∧/a B → A.
 #A #B #AB (elim AB) //; qed.
 
-theorem proj2: ∀ A,B:Prop. A ∧ B → B.
+theorem proj2: ∀ A,B:Prop. A a title="logical and" href="cic:/fakeuri.def(1)"∧/a B → B.
 #A #B #AB (elim AB) //; qed.
 
 (* or *)
@@ -138,7 +138,7 @@ inductive Or (A,B:Prop) : Prop ≝
 interpretation "logical or" 'or x y = (Or x y).
 
 definition decidable : Prop → Prop ≝ 
-λ A:Prop. A ∨ ¬ A.
+λ A:Prop. A a title="logical or" href="cic:/fakeuri.def(1)"∨/a a title="logical not" href="cic:/fakeuri.def(1)"¬/a A.
 
 (* exists *)
 inductive ex (A:Type[0]) (P:A → Prop) : Prop ≝
@@ -151,122 +151,122 @@ inductive ex2 (A:Type[0]) (P,Q:A →Prop) : Prop ≝
 
 (* iff *)
 definition iff :=
- λ A,B. (A → B) ∧ (B → A).
+ λ A,B. (A → B) a title="logical and" href="cic:/fakeuri.def(1)"∧/a (B → A).
 
 interpretation "iff" 'iff a b = (iff a b).
 
-lemma iff_sym: ∀A,B. A ↔ B → B ↔ A.
-#A #B * /3/ qed.
+lemma iff_sym: ∀A,B. A a title="iff" href="cic:/fakeuri.def(1)"↔/a B → B a title="iff" href="cic:/fakeuri.def(1)"↔/a A.
+#A #B * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/ qed.
 
-lemma iff_trans:∀A,B,C. A ↔ B → B ↔ C → A ↔ C.
-#A #B #C * #H1 #H2 * #H3 #H4 % /3/ qed.
+lemma iff_trans:∀A,B,C. A a title="iff" href="cic:/fakeuri.def(1)"↔/a B → B a title="iff" href="cic:/fakeuri.def(1)"↔/a C → A a title="iff" href="cic:/fakeuri.def(1)"↔/a C.
+#A #B #C * #H1 #H2 * #H3 #H4 % /span class="autotactic"3span class="autotrace" trace /span/span/ qed.
 
-lemma iff_not: ∀A,B. A ↔ B → ¬A ↔ ¬B.
-#A #B * #H1 #H2 % /3/ qed.
+lemma iff_not: ∀A,B. A a title="iff" href="cic:/fakeuri.def(1)"↔/a B → a title="logical not" href="cic:/fakeuri.def(1)"¬/aA a title="iff" href="cic:/fakeuri.def(1)"↔/a a title="logical not" href="cic:/fakeuri.def(1)"¬/aB.
+#A #B * #H1 #H2 % /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/not_to_not.def(3)"not_to_not/a/span/span/ qed.
 
-lemma iff_and_l: ∀A,B,C. A ↔ B → C ∧ A ↔ C ∧ B.
-#A #B #C * #H1 #H2 % * /3/ qed.  
+lemma iff_and_l: ∀A,B,C. A a title="iff" href="cic:/fakeuri.def(1)"↔/a B → C a title="logical and" href="cic:/fakeuri.def(1)"∧/a A a title="iff" href="cic:/fakeuri.def(1)"↔/a C a title="logical and" href="cic:/fakeuri.def(1)"∧/a B.
+#A #B #C * #H1 #H2 % * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/ qed.  
 
-lemma iff_and_r: ∀A,B,C. A ↔ B → A ∧ C ↔ B ∧ C.
-#A #B #C * #H1 #H2 % * /3/ qed.  
+lemma iff_and_r: ∀A,B,C. A a title="iff" href="cic:/fakeuri.def(1)"↔/a B → A a title="logical and" href="cic:/fakeuri.def(1)"∧/a C a title="iff" href="cic:/fakeuri.def(1)"↔/a B a title="logical and" href="cic:/fakeuri.def(1)"∧/a C.
+#A #B #C * #H1 #H2 % * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/ qed.  
 
-lemma iff_or_l: ∀A,B,C. A ↔ B → C ∨ A ↔ C ∨ B.
-#A #B #C * #H1 #H2 % * /3/ qed.  
+lemma iff_or_l: ∀A,B,C. A a title="iff" href="cic:/fakeuri.def(1)"↔/a B → C a title="logical or" href="cic:/fakeuri.def(1)"∨/a A a title="iff" href="cic:/fakeuri.def(1)"↔/a C a title="logical or" href="cic:/fakeuri.def(1)"∨/a B.
+#A #B #C * #H1 #H2 % * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/ qed.  
 
-lemma iff_or_r: ∀A,B,C. A ↔ B → A ∨ C ↔ B ∨ C.
-#A #B #C * #H1 #H2 % * /3/ qed.  
+lemma iff_or_r: ∀A,B,C. A a title="iff" href="cic:/fakeuri.def(1)"↔/a B → A a title="logical or" href="cic:/fakeuri.def(1)"∨/a C a title="iff" href="cic:/fakeuri.def(1)"↔/a B a title="logical or" href="cic:/fakeuri.def(1)"∨/a C.
+#A #B #C * #H1 #H2 % * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/ qed.  
 
 (* cose per destruct: da rivedere *) 
 
 definition R0 ≝ λT:Type[0].λt:T.t.
   
-definition R1 ≝ eq_rect_Type0.
+definition R1 ≝ a href="cic:/matita/basics/logic/eq_rect_Type0.fix(0,5,1)"eq_rect_Type0/a.
 
 (* used for lambda-delta *)
 definition R2 :
   ∀T0:Type[0].
   ∀a0:T0.
-  ∀T1:∀x0:T0. a0=x0 → Type[0].
-  ∀a1:T1 a0 (refl ? a0).
-  ∀T2:∀x0:T0. ∀p0:a0=x0. ∀x1:T1 x0 p0. R1 ?? T1 a1 ? p0 = x1 → Type[0].
-  ∀a2:T2 a0 (refl ? a0) a1 (refl ? a1).
+  ∀T1:∀x0:T0. a0a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/ax0 → Type[0].
+  ∀a1:T1 a0 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a ? a0).
+  ∀T2:∀x0:T0. ∀p0:a0a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/ax0. ∀x1:T1 x0 p0. a href="cic:/matita/basics/logic/R1.def(2)"R1/a ?? T1 a1 ? p0 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a x1 → Type[0].
+  ∀a2:T2 a0 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a ? a0) a1 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a ? a1).
   ∀b0:T0.
-  ∀e0:a0 = b0.
+  ∀e0:a0 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a b0.
   ∀b1: T1 b0 e0.
-  ∀e1:R1 ?? T1 a1 ? e0 = b1.
+  ∀e1:a href="cic:/matita/basics/logic/R1.def(2)"R1/a ?? T1 a1 ? e0 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a b1.
   T2 b0 e0 b1 e1.
 #T0 #a0 #T1 #a1 #T2 #a2 #b0 #e0 #b1 #e1 
-@(eq_rect_Type0 ????? e1) 
-@(R1 ?? ? ?? e0) 
+@(a href="cic:/matita/basics/logic/eq_rect_Type0.fix(0,5,1)"eq_rect_Type0/a ????? e1) 
+@(a href="cic:/matita/basics/logic/R1.def(2)"R1/a ?? ? ?? e0) 
 @a2 
 qed.
 
 definition R3 :
   ∀T0:Type[0].
   ∀a0:T0.
-  ∀T1:∀x0:T0. a0=x0 → Type[0].
-  ∀a1:T1 a0 (refl ? a0).
-  ∀T2:∀x0:T0. ∀p0:a0=x0. ∀x1:T1 x0 p0. R1 ?? T1 a1 ? p0 = x1 → Type[0].
-  ∀a2:T2 a0 (refl ? a0) a1 (refl ? a1).
-  ∀T3:∀x0:T0. ∀p0:a0=x0. ∀x1:T1 x0 p0.∀p1:R1 ?? T1 a1 ? p0 = x1.
-      ∀x2:T2 x0 p0 x1 p1.R2 ???? T2 a2 x0 p0 ? p1 = x2 → Type[0].
-  ∀a3:T3 a0 (refl ? a0) a1 (refl ? a1) a2 (refl ? a2).
+  ∀T1:∀x0:T0. a0a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/ax0 → Type[0].
+  ∀a1:T1 a0 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a ? a0).
+  ∀T2:∀x0:T0. ∀p0:a0a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/ax0. ∀x1:T1 x0 p0. a href="cic:/matita/basics/logic/R1.def(2)"R1/a ?? T1 a1 ? p0 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a x1 → Type[0].
+  ∀a2:T2 a0 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a ? a0) a1 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a ? a1).
+  ∀T3:∀x0:T0. ∀p0:a0a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/ax0. ∀x1:T1 x0 p0.∀p1:a href="cic:/matita/basics/logic/R1.def(2)"R1/a ?? T1 a1 ? p0 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a x1.
+      ∀x2:T2 x0 p0 x1 p1.a href="cic:/matita/basics/logic/R2.def(3)"R2/a ???? T2 a2 x0 p0 ? p1 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a x2 → Type[0].
+  ∀a3:T3 a0 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a ? a0) a1 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a ? a1) a2 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a ? a2).
   ∀b0:T0.
-  ∀e0:a0 = b0.
+  ∀e0:a0 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a b0.
   ∀b1: T1 b0 e0.
-  ∀e1:R1 ?? T1 a1 ? e0 = b1.
+  ∀e1:a href="cic:/matita/basics/logic/R1.def(2)"R1/a ?? T1 a1 ? e0 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a b1.
   ∀b2: T2 b0 e0 b1 e1.
-  ∀e2:R2 ???? T2 a2 b0 e0 ? e1 = b2.
+  ∀e2:a href="cic:/matita/basics/logic/R2.def(3)"R2/a ???? T2 a2 b0 e0 ? e1 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a b2.
   T3 b0 e0 b1 e1 b2 e2.
 #T0 #a0 #T1 #a1 #T2 #a2 #T3 #a3 #b0 #e0 #b1 #e1 #b2 #e2 
-@(eq_rect_Type0 ????? e2) 
-@(R2 ?? ? ???? e0 ? e1) 
+@(a href="cic:/matita/basics/logic/eq_rect_Type0.fix(0,5,1)"eq_rect_Type0/a ????? e2) 
+@(a href="cic:/matita/basics/logic/R2.def(3)"R2/a ?? ? ???? e0 ? e1) 
 @a3 
 qed.
 
 definition R4 :
   ∀T0:Type[0].
   ∀a0:T0.
-  ∀T1:∀x0:T0. eq T0 a0 x0 → Type[0].
-  ∀a1:T1 a0 (refl T0 a0).
-  ∀T2:∀x0:T0. ∀p0:eq (T0 …) a0 x0. ∀x1:T1 x0 p0.eq (T1 …) (R1 T0 a0 T1 a1 x0 p0) x1 → Type[0].
-  ∀a2:T2 a0 (refl T0 a0) a1 (refl (T1 a0 (refl T0 a0)) a1).
-  ∀T3:∀x0:T0. ∀p0:eq (T0 …) a0 x0. ∀x1:T1 x0 p0.∀p1:eq (T1 …) (R1 T0 a0 T1 a1 x0 p0) x1.
-      ∀x2:T2 x0 p0 x1 p1.eq (T2 …) (R2 T0 a0 T1 a1 T2 a2 x0 p0 x1 p1) x2 → Type[0].
-  ∀a3:T3 a0 (refl T0 a0) a1 (refl (T1 a0 (refl T0 a0)) a1) 
-      a2 (refl (T2 a0 (refl T0 a0) a1 (refl (T1 a0 (refl T0 a0)) a1)) a2). 
-  ∀T4:∀x0:T0. ∀p0:eq (T0 …) a0 x0. ∀x1:T1 x0 p0.∀p1:eq (T1 …) (R1 T0 a0 T1 a1 x0 p0) x1.
-      ∀x2:T2 x0 p0 x1 p1.∀p2:eq (T2 …) (R2 T0 a0 T1 a1 T2 a2 x0 p0 x1 p1) x2.
-      ∀x3:T3 x0 p0 x1 p1 x2 p2.∀p3:eq (T3 …) (R3 T0 a0 T1 a1 T2 a2 T3 a3 x0 p0 x1 p1 x2 p2) x3. 
+  ∀T1:∀x0:T0. a href="cic:/matita/basics/logic/eq.ind(1,0,2)"eq/a T0 a0 x0 → Type[0].
+  ∀a1:T1 a0 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a T0 a0).
+  ∀T2:∀x0:T0. ∀p0:a href="cic:/matita/basics/logic/eq.ind(1,0,2)"eq/a (T0 …) a0 x0. ∀x1:T1 x0 p0.a href="cic:/matita/basics/logic/eq.ind(1,0,2)"eq/a (T1 …) (a href="cic:/matita/basics/logic/R1.def(2)"R1/a T0 a0 T1 a1 x0 p0) x1 → Type[0].
+  ∀a2:T2 a0 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a T0 a0) a1 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a (T1 a0 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a T0 a0)) a1).
+  ∀T3:∀x0:T0. ∀p0:a href="cic:/matita/basics/logic/eq.ind(1,0,2)"eq/a (T0 …) a0 x0. ∀x1:T1 x0 p0.∀p1:a href="cic:/matita/basics/logic/eq.ind(1,0,2)"eq/a (T1 …) (a href="cic:/matita/basics/logic/R1.def(2)"R1/a T0 a0 T1 a1 x0 p0) x1.
+      ∀x2:T2 x0 p0 x1 p1.a href="cic:/matita/basics/logic/eq.ind(1,0,2)"eq/a (T2 …) (a href="cic:/matita/basics/logic/R2.def(3)"R2/a T0 a0 T1 a1 T2 a2 x0 p0 x1 p1) x2 → Type[0].
+  ∀a3:T3 a0 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a T0 a0) a1 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a (T1 a0 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a T0 a0)) a1) 
+      a2 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a (T2 a0 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a T0 a0) a1 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a (T1 a0 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a T0 a0)) a1)) a2). 
+  ∀T4:∀x0:T0. ∀p0:a href="cic:/matita/basics/logic/eq.ind(1,0,2)"eq/a (T0 …) a0 x0. ∀x1:T1 x0 p0.∀p1:a href="cic:/matita/basics/logic/eq.ind(1,0,2)"eq/a (T1 …) (a href="cic:/matita/basics/logic/R1.def(2)"R1/a T0 a0 T1 a1 x0 p0) x1.
+      ∀x2:T2 x0 p0 x1 p1.∀p2:a href="cic:/matita/basics/logic/eq.ind(1,0,2)"eq/a (T2 …) (a href="cic:/matita/basics/logic/R2.def(3)"R2/a T0 a0 T1 a1 T2 a2 x0 p0 x1 p1) x2.
+      ∀x3:T3 x0 p0 x1 p1 x2 p2.∀p3:a href="cic:/matita/basics/logic/eq.ind(1,0,2)"eq/a (T3 …) (a href="cic:/matita/basics/logic/R3.def(4)"R3/a T0 a0 T1 a1 T2 a2 T3 a3 x0 p0 x1 p1 x2 p2) x3. 
       Type[0].
-  ∀a4:T4 a0 (refl T0 a0) a1 (refl (T1 a0 (refl T0 a0)) a1) 
-      a2 (refl (T2 a0 (refl T0 a0) a1 (refl (T1 a0 (refl T0 a0)) a1)) a2) 
-      a3 (refl (T3 a0 (refl T0 a0) a1 (refl (T1 a0 (refl T0 a0)) a1) 
-                   a2 (refl (T2 a0 (refl T0 a0) a1 (refl (T1 a0 (refl T0 a0)) a1)) a2))
+  ∀a4:T4 a0 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a T0 a0) a1 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a (T1 a0 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a T0 a0)) a1) 
+      a2 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a (T2 a0 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a T0 a0) a1 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a (T1 a0 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a T0 a0)) a1)) a2) 
+      a3 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a (T3 a0 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a T0 a0) a1 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a (T1 a0 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a T0 a0)) a1) 
+                   a2 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a (T2 a0 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a T0 a0) a1 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a (T1 a0 (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a T0 a0)) a1)) a2))
                    a3).
   ∀b0:T0.
-  ∀e0:eq (T0 …) a0 b0.
+  ∀e0:a href="cic:/matita/basics/logic/eq.ind(1,0,2)"eq/a (T0 …) a0 b0.
   ∀b1: T1 b0 e0.
-  ∀e1:eq (T1 …) (R1 T0 a0 T1 a1 b0 e0) b1.
+  ∀e1:a href="cic:/matita/basics/logic/eq.ind(1,0,2)"eq/a (T1 …) (a href="cic:/matita/basics/logic/R1.def(2)"R1/a T0 a0 T1 a1 b0 e0) b1.
   ∀b2: T2 b0 e0 b1 e1.
-  ∀e2:eq (T2 …) (R2 T0 a0 T1 a1 T2 a2 b0 e0 b1 e1) b2.
+  ∀e2:a href="cic:/matita/basics/logic/eq.ind(1,0,2)"eq/a (T2 …) (a href="cic:/matita/basics/logic/R2.def(3)"R2/a T0 a0 T1 a1 T2 a2 b0 e0 b1 e1) b2.
   ∀b3: T3 b0 e0 b1 e1 b2 e2.
-  ∀e3:eq (T3 …) (R3 T0 a0 T1 a1 T2 a2 T3 a3 b0 e0 b1 e1 b2 e2) b3.
+  ∀e3:a href="cic:/matita/basics/logic/eq.ind(1,0,2)"eq/a (T3 …) (a href="cic:/matita/basics/logic/R3.def(4)"R3/a 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 
-@(eq_rect_Type0 ????? e3) 
-@(R3 ????????? e0 ? e1 ? e2) 
+@(a href="cic:/matita/basics/logic/eq_rect_Type0.fix(0,5,1)"eq_rect_Type0/a ????? e3) 
+@(a href="cic:/matita/basics/logic/R3.def(4)"R3/a ????????? e0 ? e1 ? e2) 
 @a4 
 qed.
 
-definition eqProp ≝ λA:Prop.eq A.
+definition eqProp ≝ λA:Prop.a href="cic:/matita/basics/logic/eq.ind(1,0,2)"eq/a A.
 
 (* Example to avoid indexing and the consequential creation of ill typed
    terms during paramodulation *)
-example lemmaK : ∀A.∀x:A.∀h:x=x. eqProp ? h (refl A x).
-#A #x #h @(refl ? h: eqProp ? ? ?).
+example lemmaK : ∀A.∀x:A.∀h:xa title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/ax. a href="cic:/matita/basics/logic/eqProp.def(1)"eqProp/a ? h (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a A x).
+#A #x #h @(a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a ? h: a href="cic:/matita/basics/logic/eqProp.def(1)"eqProp/a ? ? ?).
 qed.
 
-theorem streicherK : ∀T:Type[2].∀t:T.∀P:t = t → Type[3].P (refl ? t) → ∀p.P p.
- #T #t #P #H #p >(lemmaK T t p) @H
-qed.
+theorem streicherK : ∀T:Type[2].∀t:T.∀P:t a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a t → Type[3].P (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a ? t) → ∀p.P p.
+ #T #t #P #H #p >(a href="cic:/matita/basics/logic/lemmaK.def(2)"lemmaK/a T t p) @H
+qed.
\ No newline at end of file
diff --git a/weblib/tutorial/chapter2.ma b/weblib/tutorial/chapter2.ma
index 804ba58ba..ab758c124 100644
--- a/weblib/tutorial/chapter2.ma
+++ b/weblib/tutorial/chapter2.ma
@@ -217,7 +217,7 @@ lemma div2_ok: ∀n,q,r. a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"
   |#a #Hind #q #r 
    cut (a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"div2/a a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/types/fst.def(1)"fst/a … (a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"div2/a a), a href="cic:/matita/basics/types/snd.def(1)"snd/a … (a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"div2/a a)〉) [//] 
    cases (a href="cic:/matita/basics/types/snd.def(1)"snd/a … (a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"div2/a a))
-    [#H >(a href="cic:/matita/tutorial/chapter2/div2S1.def(3)"div2S1/a … H) #H1 destruct @a href="cic:/matita/basics/logic/eq_f.def(3)"eq_f/a span style="text-decoration: underline;">/spana href="cic:/matita/tutorial/chapter2/add_S.def(2)"add_S/a <a href="cic:/matita/tutorial/chapter2/add_S.def(2)"add_S/a @(Hind … H) 
+    [#H >(a href="cic:/matita/tutorial/chapter2/div2S1.def(3)"div2S1/a … H) #H1 destruct @a href="cic:/matita/basics/logic/eq_f.def(3)"eq_f/a span style="text-decoration: underline;">/spana href="cic:/matita/tutorial/chapter2/add_S.def(2)"add_S/a whd in ⊢ (???%); <a href="cic:/matita/tutorial/chapter2/add_S.def(2)"add_S/a @(Hind … H) 
     |#H >(a href="cic:/matita/tutorial/chapter2/div2SO.def(3)"div2SO/a … H) #H1 destruct >a href="cic:/matita/tutorial/chapter2/add_S.def(2)"add_S/a @a href="cic:/matita/basics/logic/eq_f.def(3)"eq_f/a @(Hind … H) 
     ]
 qed.
@@ -259,7 +259,8 @@ definition div2Pagain : ∀n.a href="cic:/matita/tutorial/chapter2/Sub.ind(1,0,
   |#a * #p #qrspec 
    cut (p a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/types/fst.def(1)"fst/a … p, a href="cic:/matita/basics/types/snd.def(1)"snd/a … p〉) [//] 
    cases (a href="cic:/matita/basics/types/snd.def(1)"snd/a … p)
-    [#H @(a href="cic:/matita/tutorial/chapter2/Sub.con(0,1,2)"mk_Sub/a … a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"S/a (a href="cic:/matita/basics/types/fst.def(1)"fst/a … p),a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"ff/a〉) whd #q #r #H1 destruct @a href="cic:/matita/basics/logic/eq_f.def(3)"eq_f/a span style="text-decoration: underline;">/spana href="cic:/matita/tutorial/chapter2/add_S.def(2)"add_S/a <a href="cic:/matita/tutorial/chapter2/add_S.def(2)"add_S/a @(qrspec … H)
+    [#H @(a href="cic:/matita/tutorial/chapter2/Sub.con(0,1,2)"mk_Sub/a … a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"S/a (a href="cic:/matita/basics/types/fst.def(1)"fst/a … p),a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"ff/a〉) whd #q #r #H1 destruct @a href="cic:/matita/basics/logic/eq_f.def(3)"eq_f/a span style="text-decoration: underline;">/spana href="cic:/matita/tutorial/chapter2/add_S.def(2)"add_S/a
+     whd in ⊢ (???%); <a href="cic:/matita/tutorial/chapter2/add_S.def(2)"add_S/a @(qrspec … H)
     |#H @(a href="cic:/matita/tutorial/chapter2/Sub.con(0,1,2)"mk_Sub/a … a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/types/fst.def(1)"fst/a … p,a href="cic:/matita/tutorial/chapter2/bool.con(0,1,0)"tt/a〉) whd #q #r #H1 destruct >a href="cic:/matita/tutorial/chapter2/add_S.def(2)"add_S/a @a href="cic:/matita/basics/logic/eq_f.def(3)"eq_f/a @(qrspec … H) 
   ]
 qed.
diff --git a/weblib/tutorial/chapter4.ma b/weblib/tutorial/chapter4.ma
index 8457aa484..1be877197 100644
--- a/weblib/tutorial/chapter4.ma
+++ b/weblib/tutorial/chapter4.ma
@@ -1,5 +1,5 @@
 (* In this Chapter we shall develop a naif theory of sets represented as characteristic
-predicates over some universe codeA/code, that is as objects of type A→Prop *)
+predicates over some universe codeA/code, that is as objects of type A→Prop. *)
 
 include "basics/logic.ma".
 
@@ -16,7 +16,9 @@ definition singleton ≝ λA.λx,a:A.xa title="leibnitz's equality" href="cic:/
 (* notation "{x}" non associative with precedence 90 for @{'sing_lang $x}. *)
 interpretation "singleton" 'singl x = (singleton ? x).
 
-(* The operations of union, intersection, complement and substraction 
+(* The membership relation between an element of type A and a set S:A →Prop is
+simply the predicate resulting from the application of S to a.
+The operations of union, intersection, complement and substraction 
 are easily defined in terms of the propositional connectives of dijunction,
 conjunction and negation *)
 
@@ -46,88 +48,102 @@ notation "A =1 B" non associative with precedence 45 for @{'eqP $A $B}.
 interpretation "extensional equality" 'eqP a b = (eqP ? a b).
 
 (* This notion of equality is different from the intensional equality of
-fucntions, hence we have to prove the usual properties: *)
+functions; the fact it defines an equivalence relation must be explicitly 
+proved: *)
 
 lemma eqP_sym: ∀U.∀A,B:U →Prop. 
   A a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 B → B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A.
-#U #A #B #eqAB #a @iff_sym @eqAB qed.
+#U #A #B #eqAB #a @a href="cic:/matita/basics/logic/iff_sym.def(2)"iff_sym/a @eqAB qed.
  
 lemma eqP_trans: ∀U.∀A,B,C:U →Prop. 
-  A =1 B → B =1 C → A =1 C.
-#U #A #B #C #eqAB #eqBC #a @iff_trans // qed.
+  A a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 B → B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C → A a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C.
+#U #A #B #C #eqAB #eqBC #a @a href="cic:/matita/basics/logic/iff_trans.def(2)"iff_trans/a // qed.
 
-(* For the same reason, we must also prove that all the operations we have
- lemma eqP_union_r: ∀U.∀A,B,C:U →Prop. 
-  A =1 C  → A ∪ B =1 C ∪ B.
-#U #A #B #C #eqAB #a @iff_or_r @eqAB qed.
+(* For the same reason, we must also prove that all the operations behave well
+with respect to eqP: *)
+
+lemma eqP_union_r: ∀U.∀A,B,C:U →Prop. 
+  A a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C  → A a title="union" href="cic:/fakeuri.def(1)"∪/a B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C a title="union" href="cic:/fakeuri.def(1)"∪/a B.
+#U #A #B #C #eqAB #a @a href="cic:/matita/basics/logic/iff_or_r.def(2)"iff_or_r/a @eqAB qed.
   
 lemma eqP_union_l: ∀U.∀A,B,C:U →Prop. 
-  B =1 C  → A ∪ B =1 A ∪ C.
-#U #A #B #C #eqBC #a @iff_or_l @eqBC qed.
+  B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C  → A a title="union" href="cic:/fakeuri.def(1)"∪/a B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A a title="union" href="cic:/fakeuri.def(1)"∪/a C.
+#U #A #B #C #eqBC #a @a href="cic:/matita/basics/logic/iff_or_l.def(2)"iff_or_l/a @eqBC qed.
   
 lemma eqP_intersect_r: ∀U.∀A,B,C:U →Prop. 
-  A =1 C  → A ∩ B =1 C ∩ B.
-#U #A #B #C #eqAB #a @iff_and_r @eqAB qed.
+  A a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C  → A a title="intersection" href="cic:/fakeuri.def(1)"∩/a B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C a title="intersection" href="cic:/fakeuri.def(1)"∩/a B.
+#U #A #B #C #eqAB #a @a href="cic:/matita/basics/logic/iff_and_r.def(2)"iff_and_r/a @eqAB qed.
   
 lemma eqP_intersect_l: ∀U.∀A,B,C:U →Prop. 
-  B =1 C  → A ∩ B =1 A ∩ C.
-#U #A #B #C #eqBC #a @iff_and_l @eqBC qed.
+  B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C  → A a title="intersection" href="cic:/fakeuri.def(1)"∩/a B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A a title="intersection" href="cic:/fakeuri.def(1)"∩/a C.
+#U #A #B #C #eqBC #a @a href="cic:/matita/basics/logic/iff_and_l.def(2)"iff_and_l/a @eqBC qed.
 
 lemma eqP_substract_r: ∀U.∀A,B,C:U →Prop. 
-  A =1 C  → A - B =1 C - B.
-#U #A #B #C #eqAB #a @iff_and_r @eqAB qed.
+  A a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C  → A a title="substraction" href="cic:/fakeuri.def(1)"-/a B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C a title="substraction" href="cic:/fakeuri.def(1)"-/a B.
+#U #A #B #C #eqAB #a @a href="cic:/matita/basics/logic/iff_and_r.def(2)"iff_and_r/a @eqAB qed.
   
 lemma eqP_substract_l: ∀U.∀A,B,C:U →Prop. 
-  B =1 C  → A - B =1 A - C.
-#U #A #B #C #eqBC #a @iff_and_l /2/ qed.
+  B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C  → A a title="substraction" href="cic:/fakeuri.def(1)"-/a B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A a title="substraction" href="cic:/fakeuri.def(1)"-/a C.
+#U #A #B #C #eqBC #a @a href="cic:/matita/basics/logic/iff_and_l.def(2)"iff_and_l/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/iff_not.def(4)"iff_not/a/span/span/ qed.
+
+(* We can now prove several properties of the previous set-theoretic 
+operations. In particular, union is commutative and associative, and 
+the empty set is an identity element: *)
 
-(* set equalities *)
 lemma union_empty_r: ∀U.∀A:U→Prop. 
-  A ∪ ∅ =1 A.
-#U #A #w % [* // normalize #abs @False_ind /2/ | /2/]
+  A a title="union" href="cic:/fakeuri.def(1)"∪/a a title="empty set" href="cic:/fakeuri.def(1)"∅/a a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A.
+#U #A #w % [* // normalize #abs @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a /span class="autotactic"2span class="autotrace" trace /span/span/ | /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a/span/span/]
 qed.
 
 lemma union_comm : ∀U.∀A,B:U →Prop. 
-  A ∪ B =1 B ∪ A.
-#U #A #B #a % * /2/ qed. 
+  A a title="union" href="cic:/fakeuri.def(1)"∪/a B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 B a title="union" href="cic:/fakeuri.def(1)"∪/a A.
+#U #A #B #a % * /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/ qed. 
 
 lemma union_assoc: ∀U.∀A,B,C:U → Prop. 
-  A ∪ B ∪ C =1 A ∪ (B ∪ C).
-#S #A #B #C #w % [* [* /3/ | /3/] | * [/3/ | * /3/]
+  A a title="union" href="cic:/fakeuri.def(1)"∪/a B a title="union" href="cic:/fakeuri.def(1)"∪/a C a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A a title="union" href="cic:/fakeuri.def(1)"∪/a (B a title="union" href="cic:/fakeuri.def(1)"∪/a C).
+#S #A #B #C #w % [* [* /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/ | /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/] | * [/span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a/span/span/ | * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/]
 qed.   
 
+(* In the same way we prove commutativity and associativity for set
+interesection *)
+
 lemma cap_comm : ∀U.∀A,B:U →Prop. 
-  A ∩ B =1 B ∩ A.
-#U #A #B #a % * /2/ qed. 
+  A a title="intersection" href="cic:/fakeuri.def(1)"∩/a B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 B a title="intersection" href="cic:/fakeuri.def(1)"∩/a A.
+#U #A #B #a % * /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/ qed. 
+
+lemma cap_assoc: ∀U.∀A,B,C:U→Prop.
+  A a title="intersection" href="cic:/fakeuri.def(1)"∩/a (B a title="intersection" href="cic:/fakeuri.def(1)"∩/a C) a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 (A a title="intersection" href="cic:/fakeuri.def(1)"∩/a B) a title="intersection" href="cic:/fakeuri.def(1)"∩/a C.
+#U #A #B #C #w % [ * #Aw * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/ span class="autotactic"span class="autotrace"/span/span| * * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/ ]
+qed.
+
+(* We can also easily prove idempotency for union and intersection *)
 
 lemma union_idemp: ∀U.∀A:U →Prop. 
-  A ∪ A =1 A.
-#U #A #a % [* // | /2/] qed. 
+  A  a title="union" href="cic:/fakeuri.def(1)"∪/a A a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A.
+#U #A #a % [* // | /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/] qed. 
 
 lemma cap_idemp: ∀U.∀A:U →Prop. 
-  A ∩ A =1 A.
-#U #A #a % [* // | /2/] qed. 
+  A a title="intersection" href="cic:/fakeuri.def(1)"∩/a A a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A.
+#U #A #a % [* // | /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/] qed. 
 
-(*distributivities *)
+(* We conclude our examples with a couple of distributivity theorems,
+and a characterization of substraction in terms of interesection and
+complementation. *)
 
 lemma distribute_intersect : ∀U.∀A,B,C:U→Prop. 
-  (A ∪ B) ∩ C =1 (A ∩ C) ∪ (B ∩ C).
-#U #A #B #C #w % [* * /3/ | * * /3/] 
+  (A a title="union" href="cic:/fakeuri.def(1)"∪/a B) a title="intersection" href="cic:/fakeuri.def(1)"∩/a C a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 (A a title="intersection" href="cic:/fakeuri.def(1)"∩/a C) a title="union" href="cic:/fakeuri.def(1)"∪/a (B a title="intersection" href="cic:/fakeuri.def(1)"∩/a C).
+#U #A #B #C #w % [* * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a, a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/ | * * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a, a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/] 
 qed.
   
 lemma distribute_substract : ∀U.∀A,B,C:U→Prop. 
-  (A ∪ B) - C =1 (A - C) ∪ (B - C).
-#U #A #B #C #w % [* * /3/ | * * /3/] 
+  (A a title="union" href="cic:/fakeuri.def(1)"∪/a B) a title="substraction" href="cic:/fakeuri.def(1)"-/a C a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 (A a title="substraction" href="cic:/fakeuri.def(1)"-/a C) a title="union" href="cic:/fakeuri.def(1)"∪/a (B a title="substraction" href="cic:/fakeuri.def(1)"-/a C).
+#U #A #B #C #w % [* * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a, a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/ | * * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a, a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/] 
 qed.
 
-(* substraction *)
-lemma substract_def:∀U.∀A,B:U→Prop. A-B =1 A ∩ ¬B.
-#U #A #B #w normalize /2/
+lemma substract_def:∀U.∀A,B:U→Prop. Aa title="substraction" href="cic:/fakeuri.def(1)"-/aB a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A a title="intersection" href="cic:/fakeuri.def(1)"∩/a a title="complement" href="cic:/fakeuri.def(1)"¬/aB.
+#U #A #B #w normalize /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/
 qed.
 
-include "basics/types.ma".
-include "basics/bool.ma".
-
 (****** DeqSet: a set with a decidbale equality ******)
 
 record DeqSet : Type[1] ≝ { carr :> Type[0];
@@ -168,7 +184,7 @@ definition beqb ≝ λb1,b2.
 
 notation < "a == b" non associative with precedence 45 for @{beqb $a $b }.
 lemma beqb_true: ∀b1,b2. iff (beqb b1 b2 = true) (b1 = b2).
-#b1 #b2 cases b1 cases b2 normalize /2/
+#b1 #b2 cases b1 cases b2 normalize /span class="autotactic"2span class="autotrace" trace conj/span/span/
 qed. 
 
 definition DeqBool ≝ mk_DeqSet bool beqb beqb_true.
@@ -194,7 +210,7 @@ definition eq_pairs ≝
 lemma eq_pairs_true: ∀A,B:DeqSet.∀p1,p2:A×B.
   eq_pairs A B p1 p2 = true ↔ p1 = p2.
 #A #B * #a1 #b1 * #a2 #b2 %
-  [#H cases (andb_true …H) #eqa #eqb >(\P eqa) >(\P eqb) //
+  [#H cases (andb_true …H) normalize #eqa #eqb >(\P eqa) >(\P eqb) //
   |#H destruct normalize >(\b (refl … a2)) >(\b (refl … b2)) //
   ]
 qed.
-- 
2.39.2