X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fdidactic%2Fexercises%2Fduality.ma;h=14927653e2a08093452ebe02c5dd7ce1cab1a335;hb=e78cf74f8976cf0ca554f64baa9979d0423ee927;hp=ea3a987d389937410b4d4f9fb3e05be42847bf9d;hpb=ff6a1396270493dcf0e1f673f32a8213a8ce3751;p=helm.git

diff --git a/helm/software/matita/library/didactic/exercises/duality.ma b/helm/software/matita/library/didactic/exercises/duality.ma
index ea3a987d3..14927653e 100644
--- a/helm/software/matita/library/didactic/exercises/duality.ma
+++ b/helm/software/matita/library/didactic/exercises/duality.ma
@@ -1,3 +1,17 @@
+(**************************************************************************)
+(*       ___	                                                          *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||       A.Asperti, C.Sacerdoti Coen,                          *)
+(*      ||A||       E.Tassi, S.Zacchiroli                                 *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU Lesser General Public License Version 2.1         *)
+(*                                                                        *)
+(**************************************************************************)
+
 (* Esercizio 0
    ===========
 
@@ -65,7 +79,7 @@ include "nat/minus.ma".
 definition if_then_else ≝ λT:Type.λe,t,f.match e return λ_.T with [ true ⇒ t | false ⇒ f].
 notation > "'if' term 19 e 'then' term 19 t 'else' term 90 f" non associative with precedence 19 for @{ 'if_then_else $e $t $f }.
 notation < "'if' \nbsp term 19 e \nbsp 'then' \nbsp term 19 t \nbsp 'else' \nbsp term 90 f \nbsp" non associative with precedence 19 for @{ 'if_then_else $e $t $f }.
-interpretation "Formula if_then_else" 'if_then_else e t f = (if_then_else _ e t f).
+interpretation "Formula if_then_else" 'if_then_else e t f = (if_then_else ? e t f).
 definition max ≝ λn,m. if eqb (n - m) 0 then m else n. 
 definition min ≝ λn,m. if eqb (n - m) 0 then n else m. 
 
@@ -116,7 +130,7 @@ let rec sem (v: nat → nat) (F: Formula) on F : nat ≝
 *)
 notation < "[[ \nbsp term 19 a \nbsp ]] \nbsp \sub term 90 v" non associative with precedence 90 for @{ 'semantics $v $a }.
 notation > "[[ term 19 a ]] \sub term 90 v" non associative with precedence 90 for @{ 'semantics $v $a }.
-notation > "[[ term 19 a ]]_ term 90 v" non associative with precedence 90 for @{ sem $v $a }.
+notation > "[[ term 19 a ]] term 90 v" non associative with precedence 90 for @{ sem $v $a }.
 interpretation "Semantic of Formula" 'semantics v a = (sem v a).
 
 definition v20 ≝ λx.
@@ -130,8 +144,10 @@ definition v20 ≝ λx.
    La semantica della formula `(A ∨ C)` nel mondo `v20` in cui 
    `A` vale `2` e `C` vale `0` deve valere `1`.
    
+   Decommenta ed esegui.
 *)    
-eval normalize on [[FOr (FAtom 0) (FAtom 2)]]_v20. 
+
+(* eval normalize on [[FOr (FAtom 0) (FAtom 2)]]v20. *) 
 
 (*DOCBEGIN
 
@@ -142,12 +158,13 @@ Gli strumenti per la dimostrazione assistita sono corredati da
 librerie di teoremi già dimostrati. Per portare a termine l'esercitazione
 sono necessari i seguenti lemmi:
 
-* lemma `sem_le_1` : `∀F,v. [[ F ]]_v ≤ 1`
+* lemma `sem_le_1` : `∀F,v. [[ F ]]v ≤ 1`
 * lemma `min_1_1` : `∀x. x ≤ 1 → 1 - (1 - x) = x`
 * lemma `min_bool` : `∀n. min n 1 = 0 ∨ min n 1 = 1`
-* lemma `min_max` : `∀F,G,v.min (1 - [[F]]_v) (1 - [[G]]_v) = 1 - max [[F]]_v [[G]]_v`
-* lemma `max_min` : `∀F,G,v.max (1 - [[F]]_v) (1 - [[G]]_v) = 1 - min [[F]]_v [[G]]_v`
+* lemma `min_max` : `∀F,G,v.min (1 - [[F]]v) (1 - [[G]]v) = 1 - max [[F]]v [[G]]v`
+* lemma `max_min` : `∀F,G,v.max (1 - [[F]]v) (1 - [[G]]v) = 1 - min [[F]]v [[G]]v`
 * lemma `equiv_rewrite` : `∀F1,F2,F3. F1 ≡ F2 → F1 ≡ F3 → F2 ≡ F3`
+* lemma `equiv_sym` : `∀F1,F2. F1 ≡ F2 → F2 ≡ F1`
 
 DOCEND*)
 
@@ -156,12 +173,12 @@ DOCEND*)
    
    Non modificare quanto segue.
 *)
-lemma sem_bool : ∀F,v. [[ F ]]_v = 0 ∨ [[ F ]]_v = 1.  intros; elim F; simplify; [left;reflexivity; |right;reflexivity; |cases (v n);[left;|cases n1;right;]reflexivity; |4,5,6: cases H; cases H1; rewrite > H2; rewrite > H3; simplify; first [ left;reflexivity | right; reflexivity ].  |cases H; rewrite > H1; simplify;[right|left]reflexivity;] qed.
+lemma sem_bool : ∀F,v. [[ F ]]v = 0 ∨ [[ F ]]v = 1.  intros; elim F; simplify; [left;reflexivity; |right;reflexivity; |cases (v n);[left;|cases n1;right;]reflexivity; |4,5,6: cases H; cases H1; rewrite > H2; rewrite > H3; simplify; first [ left;reflexivity | right; reflexivity ].  |cases H; rewrite > H1; simplify;[right|left]reflexivity;] qed.
 lemma min_bool : ∀n. min n 1 = 0 ∨ min n 1 = 1.  intros; cases n; [left;reflexivity] cases n1; right; reflexivity; qed.  
-lemma min_max : ∀F,G,v.  min (1 - [[F]]_v) (1 - [[G]]_v) = 1 - max [[F]]_v [[G]]_v.  intros; cases (sem_bool F v);cases (sem_bool G v); rewrite > H; rewrite >H1; simplify; reflexivity; qed.
-lemma max_min : ∀F,G,v.  max (1 - [[F]]_v) (1 - [[G]]_v) = 1 - min [[F]]_v [[G]]_v.  intros; cases (sem_bool F v);cases (sem_bool G v); rewrite > H; rewrite >H1; simplify; reflexivity; qed.
+lemma min_max : ∀F,G,v.  min (1 - [[F]]v) (1 - [[G]]v) = 1 - max [[F]]v [[G]]v.  intros; cases (sem_bool F v);cases (sem_bool G v); rewrite > H; rewrite >H1; simplify; reflexivity; qed.
+lemma max_min : ∀F,G,v.  max (1 - [[F]]v) (1 - [[G]]v) = 1 - min [[F]]v [[G]]v.  intros; cases (sem_bool F v);cases (sem_bool G v); rewrite > H; rewrite >H1; simplify; reflexivity; qed.
 lemma min_1_1 : ∀x.x ≤ 1 → 1 - (1 - x) = x. intros; inversion H; intros; destruct; [reflexivity;] rewrite < (le_n_O_to_eq ? H1); reflexivity;qed.
-lemma sem_le_1 : ∀F,v.[[F]]_v ≤ 1. intros; cases (sem_bool F v); rewrite > H; [apply le_O_n|apply le_n]qed.
+lemma sem_le_1 : ∀F,v.[[F]]v ≤ 1. intros; cases (sem_bool F v); rewrite > H; [apply le_O_n|apply le_n]qed.
 
 (* Esercizio 2
    ===========
@@ -188,21 +205,24 @@ let rec negate (F: Formula) on F : Formula ≝
   
    Testare la funzione `negate`. Il risultato atteso è:
    
-       FOr (FNot (FAtom O)) (FImpl FTop (FNot (FAtom 1))) 
+       FOr (FNot (FAtom O)) (FImpl FTop (FNot (FAtom 1)))
+       
+   Decommenta ed esegui 
 *)
 
-eval normalize on (negate (FOr (FAtom 0) (FImpl FTop (FAtom 1)))).
+(* eval normalize on (negate (FOr (FAtom 0) (FImpl FTop (FAtom 1)))). *)
 
 (* ATTENZIONE
    ==========
    
    Non modificare quanto segue
 *)
-definition equiv ≝ λF1,F2. ∀v.[[ F1 ]]_v = [[ F2 ]]_v.
+definition equiv ≝ λF1,F2. ∀v.[[ F1 ]]v = [[ F2 ]]v.
 notation "hvbox(a \nbsp break mstyle color #0000ff (≡) \nbsp b)"  non associative with precedence 45 for @{ 'equivF $a $b }.
 notation > "a ≡ b" non associative with precedence 50 for @{ equiv $a $b }.
 interpretation "equivalence for Formulas" 'equivF a b = (equiv a b).
-lemma equiv_rewrite : ∀F1,F2,F3. F1 ≡ F2 → F1 ≡ F3 → F2 ≡ F3. intros; intro; autobatch. qed.
+lemma equiv_rewrite : ∀F1,F2,F3. F1 ≡ F2 → F1 ≡ F3 → F2 ≡ F3. intros; intro; rewrite < H; rewrite < H1; reflexivity. qed.
+lemma equiv_sym : ∀a,b.a ≡ b → b ≡ a. intros 4;symmetry;apply H;qed.
 
 (* Esercizio 3
    ===========
@@ -239,9 +259,11 @@ let rec dualize (F : Formula) on F : Formula ≝
    Testare la funzione `dualize`. Il risultato atteso è:
    
        FAnd (FNot (FAtom O)) (FOr (FAtom 1) FTop) 
+       
+   Decommenta ed esegui.
 *)
 
-eval normalize on (dualize (FImpl (FAtom 0) (FAnd (FAtom 1) FBot))).
+(* eval normalize on (dualize (FImpl (FAtom 0) (FAnd (FAtom 1) FBot))). *)
 
 (* Spiegazione
    ===========
@@ -281,23 +303,23 @@ DOCEND*)
    a `F` in un mondo invertito.
 *) 
 lemma negate_invert:
-  ∀F:Formula.∀v:ℕ→ℕ.[[ negate F ]]_v=[[ F ]]_(invert v).
+  ∀F:Formula.∀v:ℕ→ℕ.[[ negate F ]]v=[[ F ]](invert v).
 assume F:Formula.
 assume v:(ℕ→ℕ).
-we proceed by induction on F to prove ([[ negate F ]]_v=[[ F ]]_(invert v)).
+we proceed by induction on F to prove ([[ negate F ]]v=[[ F ]](invert v)).
   case FBot.
     (*BEGIN*)
-    the thesis becomes ([[ negate FBot ]]_v=[[ FBot ]]_(invert v)).
+    the thesis becomes ([[ negate FBot ]]v=[[ FBot ]](invert v)).
     (*END*)
   done.
   case FTop.
     (*BEGIN*)
-    the thesis becomes ([[ negate FTop ]]_v=[[ FTop ]]_(invert v)).
+    the thesis becomes ([[ negate FTop ]]v=[[ FTop ]](invert v)).
     (*END*)
   done.
   case FAtom.
     assume n : ℕ.
-    the thesis becomes ((*BEGIN*)[[ negate (FAtom n) ]]_v=[[ FAtom n ]]_(invert v)(*END*)).
+    the thesis becomes ((*BEGIN*)[[ negate (FAtom n) ]]v=[[ FAtom n ]](invert v)(*END*)).
     the thesis becomes ((*BEGIN*)1 - (min (v n) 1)= min (invert v n) 1(*END*)).
     the thesis becomes (1 - (min (v n) 1)= min (if eqb (min (v n) 1) 0 then 1 else 0) 1).
     by min_bool we proved (min (v n) 1 = 0 ∨ min (v n) 1 = 1) (H1);
@@ -326,65 +348,65 @@ we proceed by induction on F to prove ([[ negate F ]]_v=[[ F ]]_(invert v)).
   case FAnd.
     assume f : Formula.
     by induction hypothesis we know
-      ((*BEGIN*)[[ negate f ]]_v=[[ f ]]_(invert v)(*END*)) (H).
+      ((*BEGIN*)[[ negate f ]]v=[[ f ]](invert v)(*END*)) (H).
     assume f1 : Formula.
     by induction hypothesis we know
-     ((*BEGIN*)[[ negate f1 ]]_v=[[ f1 ]]_(invert v)(*END*)) (H1).
+     ((*BEGIN*)[[ negate f1 ]]v=[[ f1 ]](invert v)(*END*)) (H1).
     the thesis becomes
-     ([[ negate (FAnd f f1) ]]_v=[[ FAnd f f1 ]]_(invert v)).
+     ([[ negate (FAnd f f1) ]]v=[[ FAnd f f1 ]](invert v)).
     the thesis becomes
-     (min [[ negate f ]]_v [[ negate f1]]_v = [[ FAnd f f1 ]]_(invert v)).
+     (min [[ negate f ]]v [[ negate f1]]v = [[ FAnd f f1 ]](invert v)).
     conclude 
-        (min [[ negate f ]]_v [[ negate f1]]_v)
-      = (min [[ f ]]_(invert v) [[ negate f1]]_v) by (*BEGIN*)H(*END*).
-      = (min [[ f ]]_(invert v) [[ f1]]_(invert v)) by (*BEGIN*)H1(*END*).
+        (min [[ negate f ]]v [[ negate f1]]v)
+      = (min [[ f ]](invert v) [[ negate f1]]v) by (*BEGIN*)H(*END*).
+      = (min [[ f ]](invert v) [[ f1]](invert v)) by (*BEGIN*)H1(*END*).
   done.
   case FOr.
     (*BEGIN*)
     assume f : Formula.
     by induction hypothesis we know
-      ([[ negate f ]]_v=[[ f ]]_(invert v)) (H).
+      ([[ negate f ]]v=[[ f ]](invert v)) (H).
     assume f1 : Formula.
     by induction hypothesis we know
-     ([[ negate f1 ]]_v=[[ f1 ]]_(invert v)) (H1).
+     ([[ negate f1 ]]v=[[ f1 ]](invert v)) (H1).
     the thesis becomes
-     ([[ negate (FOr f f1) ]]_v=[[ FOr f f1 ]]_(invert v)).
+     ([[ negate (FOr f f1) ]]v=[[ FOr f f1 ]](invert v)).
     the thesis becomes
-     (max [[ negate f ]]_v [[ negate f1]]_v = [[ FOr f f1 ]]_(invert v)).
+     (max [[ negate f ]]v [[ negate f1]]v = [[ FOr f f1 ]](invert v)).
     conclude 
-        (max [[ negate f ]]_v [[ negate f1]]_v)
-      = (max [[ f ]]_(invert v) [[ negate f1]]_v) by H.
-      = (max [[ f ]]_(invert v) [[ f1]]_(invert v)) by H1.
+        (max [[ negate f ]]v [[ negate f1]]v)
+      = (max [[ f ]](invert v) [[ negate f1]]v) by H.
+      = (max [[ f ]](invert v) [[ f1]](invert v)) by H1.
     (*END*)
   done.
   case FImpl.
     (*BEGIN*)
     assume f : Formula.
     by induction hypothesis we know
-      ([[ negate f ]]_v=[[ f ]]_(invert v)) (H).
+      ([[ negate f ]]v=[[ f ]](invert v)) (H).
     assume f1 : Formula.
     by induction hypothesis we know
-     ([[ negate f1 ]]_v=[[ f1 ]]_(invert v)) (H1).
+     ([[ negate f1 ]]v=[[ f1 ]](invert v)) (H1).
     the thesis becomes
-     ([[ negate (FImpl f f1) ]]_v=[[ FImpl f f1 ]]_(invert v)).
+     ([[ negate (FImpl f f1) ]]v=[[ FImpl f f1 ]](invert v)).
     the thesis becomes
-     (max (1 - [[ negate f ]]_v) [[ negate f1]]_v = [[ FImpl f f1 ]]_(invert v)).
+     (max (1 - [[ negate f ]]v) [[ negate f1]]v = [[ FImpl f f1 ]](invert v)).
     conclude 
-        (max (1 - [[ negate f ]]_v) [[ negate f1]]_v)
-      = (max (1 - [[ f ]]_(invert v)) [[ negate f1]]_v) by H.
-      = (max (1 - [[ f ]]_(invert v)) [[ f1]]_(invert v)) by H1.
+        (max (1 - [[ negate f ]]v) [[ negate f1]]v)
+      = (max (1 - [[ f ]](invert v)) [[ negate f1]]v) by H.
+      = (max (1 - [[ f ]](invert v)) [[ f1]](invert v)) by H1.
     (*END*)
   done.
   case FNot.
     (*BEGIN*)
     assume f : Formula.
     by induction hypothesis we know
-      ([[ negate f ]]_v=[[ f ]]_(invert v)) (H).
+      ([[ negate f ]]v=[[ f ]](invert v)) (H).
     the thesis becomes
-      ([[ negate (FNot f) ]]_v=[[ FNot f ]]_(invert v)).
+      ([[ negate (FNot f) ]]v=[[ FNot f ]](invert v)).
     the thesis becomes
-      (1 - [[ negate f ]]_v=[[ FNot f ]]_(invert v)).
-    conclude (1 - [[ negate f ]]_v) = (1 - [[f]]_(invert v)) by H.
+      (1 - [[ negate f ]]v=[[ FNot f ]](invert v)).
+    conclude (1 - [[ negate f ]]v) = (1 - [[f]](invert v)) by H.
     (*END*)
   done.  
 qed.   
@@ -400,13 +422,13 @@ lemma negate_fun:
  assume (*BEGIN*)G:Formula(*END*).
  suppose (*BEGIN*)(F ≡ G) (H)(*END*).
  the thesis becomes (*BEGIN*)(negate F ≡ negate G)(*END*).
- the thesis becomes (*BEGIN*)(∀v:ℕ→ℕ.[[ negate F ]]_v=[[ negate G ]]_v)(*END*).
+ the thesis becomes (*BEGIN*)(∀v:ℕ→ℕ.[[ negate F ]]v=[[ negate G ]]v)(*END*).
  assume v:(ℕ→ℕ).
  conclude 
-     [[ negate F ]]_v
-   = [[ F ]]_(invert v) by (*BEGIN*)negate_invert(*END*).
-   = [[ G ]]_((*BEGIN*)invert v(*BEGIN*)) by (*BEGIN*)H(*BEGIN*).
-   = [[ negate G ]]_(*BEGIN*)v(*BEGIN*) by (*BEGIN*)negate_invert(*END*).
+     [[ negate F ]]v
+   = [[ F ]](invert v) by (*BEGIN*)negate_invert(*END*).
+   = [[ G ]]((*BEGIN*)invert v(*BEGIN*)) by (*BEGIN*)H(*BEGIN*).
+   = [[ negate G ]](*BEGIN*)v(*BEGIN*) by (*BEGIN*)negate_invert(*END*).
  done.  
 qed.
 
@@ -420,94 +442,94 @@ lemma not_dualize_eq_negate:
  ∀F:Formula.negate F ≡ FNot (dualize F).
  (*BEGIN*)
  assume F:Formula.
- the thesis becomes (∀v:ℕ→ℕ.[[negate F]]_v=[[FNot (dualize F)]]_v).
+ the thesis becomes (∀v:ℕ→ℕ.[[negate F]]v=[[FNot (dualize F)]]v).
  (*END*)
  assume v:(ℕ→ℕ).
- we proceed by induction on F to prove ([[negate F]]_v=[[FNot (dualize F)]]_v).
+ we proceed by induction on F to prove ([[negate F]]v=[[FNot (dualize F)]]v).
  case FBot.
    (*BEGIN*)
-   the thesis becomes ([[ negate FBot ]]_v=[[ FNot (dualize FBot) ]]_v).
+   the thesis becomes ([[ negate FBot ]]v=[[ FNot (dualize FBot) ]]v).
    (*END*)
  done.
  case FTop.
    (*BEGIN*)
-   the thesis becomes ([[ negate FTop ]]_v=[[ FNot (dualize FTop) ]]_v).
+   the thesis becomes ([[ negate FTop ]]v=[[ FNot (dualize FTop) ]]v).
    (*END*)
  done.
  case FAtom.
    (*BEGIN*)
    assume n : ℕ.
-   the thesis becomes ([[ negate (FAtom n) ]]_v=[[ FNot (dualize (FAtom n)) ]]_v).
+   the thesis becomes ([[ negate (FAtom n) ]]v=[[ FNot (dualize (FAtom n)) ]]v).
    (*END*)
  done.
  case FAnd.
    assume f : Formula.
    by induction hypothesis we know
-     ([[ negate f ]]_v=[[ FNot (dualize f) ]]_v) (H).
+     ([[ negate f ]]v=[[ FNot (dualize f) ]]v) (H).
    assume f1 : Formula.
    by induction hypothesis we know
-    ([[ negate f1 ]]_v=[[ FNot (dualize f1) ]]_v) (H1).
+    ([[ negate f1 ]]v=[[ FNot (dualize f1) ]]v) (H1).
    the thesis becomes
-    ([[ negate (FAnd f f1) ]]_v=[[ FNot (dualize (FAnd f f1)) ]]_v).
+    ([[ negate (FAnd f f1) ]]v=[[ FNot (dualize (FAnd f f1)) ]]v).
    the thesis becomes
-    (min [[ negate f ]]_v [[ negate f1 ]]_v=[[ FNot (dualize (FAnd f f1)) ]]_v).
+    (min [[ negate f ]]v [[ negate f1 ]]v=[[ FNot (dualize (FAnd f f1)) ]]v).
    conclude 
-       (min (*BEGIN*)[[ negate f ]]_v(*END*) (*BEGIN*)[[ negate f1 ]]_v(*END*))
-     = (min (*BEGIN*)[[ FNot (dualize f) ]]_v(*END*) (*BEGIN*)[[ negate f1 ]]_v(*END*)) by H.    
-     = (min (*BEGIN*)[[ FNot (dualize f) ]]_v(*END*) (*BEGIN*)[[ FNot (dualize f1) ]]_v(*END*)) by H1.
-     = (min (1 - [[ dualize f ]]_v) (1 - [[ dualize f1 ]]_v)).
-     = (1 - (max [[ dualize f ]]_v [[ dualize f1 ]]_v)) by min_max.
+       (min (*BEGIN*)[[ negate f ]]v(*END*) (*BEGIN*)[[ negate f1 ]]v(*END*))
+     = (min (*BEGIN*)[[ FNot (dualize f) ]]v(*END*) (*BEGIN*)[[ negate f1 ]]v(*END*)) by H.    
+     = (min (*BEGIN*)[[ FNot (dualize f) ]]v(*END*) (*BEGIN*)[[ FNot (dualize f1) ]]v(*END*)) by H1.
+     = (min (1 - [[ dualize f ]]v) (1 - [[ dualize f1 ]]v)).
+     = (1 - (max [[ dualize f ]]v [[ dualize f1 ]]v)) by min_max.
  done.
  case FOr.
    (*BEGIN*)
    assume f : Formula.
    by induction hypothesis we know
-     ([[ negate f ]]_v=[[ FNot (dualize f) ]]_v) (H).
+     ([[ negate f ]]v=[[ FNot (dualize f) ]]v) (H).
    assume f1 : Formula.
    by induction hypothesis we know
-    ([[ negate f1 ]]_v=[[ FNot (dualize f1) ]]_v) (H1).
+    ([[ negate f1 ]]v=[[ FNot (dualize f1) ]]v) (H1).
    the thesis becomes
-    ([[ negate (FOr f f1) ]]_v=[[ FNot (dualize (FOr f f1)) ]]_v).
+    ([[ negate (FOr f f1) ]]v=[[ FNot (dualize (FOr f f1)) ]]v).
    the thesis becomes
-    (max [[ negate f ]]_v [[ negate f1 ]]_v=[[ FNot (dualize (FOr f f1)) ]]_v).
+    (max [[ negate f ]]v [[ negate f1 ]]v=[[ FNot (dualize (FOr f f1)) ]]v).
    conclude 
-       (max [[ negate f ]]_v [[ negate f1 ]]_v)
-     = (max [[ FNot (dualize f) ]]_v [[ negate f1 ]]_v) by H.    
-     = (max [[ FNot (dualize f) ]]_v [[ FNot (dualize f1) ]]_v) by H1.
-     = (max (1 - [[ dualize f ]]_v) (1 - [[ dualize f1 ]]_v)).
-     = (1 - (min [[ dualize f ]]_v [[ dualize f1 ]]_v)) by max_min.
+       (max [[ negate f ]]v [[ negate f1 ]]v)
+     = (max [[ FNot (dualize f) ]]v [[ negate f1 ]]v) by H.    
+     = (max [[ FNot (dualize f) ]]v [[ FNot (dualize f1) ]]v) by H1.
+     = (max (1 - [[ dualize f ]]v) (1 - [[ dualize f1 ]]v)).
+     = (1 - (min [[ dualize f ]]v [[ dualize f1 ]]v)) by max_min.
    (*END*)
  done.
  case FImpl.
    (*BEGIN*)
    assume f : Formula.
    by induction hypothesis we know
-     ([[ negate f ]]_v=[[ FNot (dualize f) ]]_v) (H).
+     ([[ negate f ]]v=[[ FNot (dualize f) ]]v) (H).
    assume f1 : Formula.
    by induction hypothesis we know
-    ([[ negate f1 ]]_v=[[ FNot (dualize f1) ]]_v) (H1).
+    ([[ negate f1 ]]v=[[ FNot (dualize f1) ]]v) (H1).
    the thesis becomes
-    ([[ negate (FImpl f f1) ]]_v=[[ FNot (dualize (FImpl f f1)) ]]_v).
+    ([[ negate (FImpl f f1) ]]v=[[ FNot (dualize (FImpl f f1)) ]]v).
    the thesis becomes
-    (max (1 - [[ negate f ]]_v) [[ negate f1 ]]_v=[[ FNot (dualize (FImpl f f1)) ]]_v).
+    (max (1 - [[ negate f ]]v) [[ negate f1 ]]v=[[ FNot (dualize (FImpl f f1)) ]]v).
    conclude 
-       (max (1-[[ negate f ]]_v) [[ negate f1 ]]_v)
-     = (max (1-[[ FNot (dualize f) ]]_v) [[ negate f1 ]]_v) by H.    
-     = (max (1-[[ FNot (dualize f) ]]_v) [[ FNot (dualize f1) ]]_v) by H1.
-     = (max (1 - [[ FNot (dualize f) ]]_v) (1 - [[ dualize f1 ]]_v)).
-     = (1 - (min [[ FNot (dualize f) ]]_v [[ dualize f1 ]]_v)) by max_min.
+       (max (1-[[ negate f ]]v) [[ negate f1 ]]v)
+     = (max (1-[[ FNot (dualize f) ]]v) [[ negate f1 ]]v) by H.    
+     = (max (1-[[ FNot (dualize f) ]]v) [[ FNot (dualize f1) ]]v) by H1.
+     = (max (1 - [[ FNot (dualize f) ]]v) (1 - [[ dualize f1 ]]v)).
+     = (1 - (min [[ FNot (dualize f) ]]v [[ dualize f1 ]]v)) by max_min.
    (*END*)
  done.
  case FNot.
    (*BEGIN*) 
    assume f : Formula.
    by induction hypothesis we know
-     ([[ negate f ]]_v=[[ FNot (dualize f) ]]_v) (H).
+     ([[ negate f ]]v=[[ FNot (dualize f) ]]v) (H).
    the thesis becomes
-      ([[ negate (FNot f) ]]_v=[[ FNot (dualize (FNot f)) ]]_v).
+      ([[ negate (FNot f) ]]v=[[ FNot (dualize (FNot f)) ]]v).
    the thesis becomes
-      (1 - [[ negate f ]]_v=[[ FNot (dualize (FNot f)) ]]_v).
-   conclude (1 - [[ negate f ]]_v) = (1 - [[ FNot (dualize f) ]]_v) by H.
+      (1 - [[ negate f ]]v=[[ FNot (dualize (FNot f)) ]]v).
+   conclude (1 - [[ negate f ]]v) = (1 - [[ FNot (dualize f) ]]v) by H.
    (*END*)
  done.
 qed.
@@ -524,20 +546,20 @@ theorem not_inj:
  assume G:Formula.
  suppose (FNot F ≡ FNot G) (H).
  the thesis becomes (F ≡ G).
- the thesis becomes (∀v:ℕ→ℕ.[[ F ]]_v=[[ G ]]_v).
+ the thesis becomes (∀v:ℕ→ℕ.[[ F ]]v=[[ G ]]v).
  (*END*)
  assume v:(ℕ→ℕ).
- by sem_le_1 we proved ([[F]]_v ≤ 1) (H1).
- by (*BEGIN*)sem_le_1(*END*) we proved ([[G]]_v ≤ 1) (H2).
- by min_1_1, H1 we proved (1 - (1 - [[F]]_v) = [[F]]_v) (H3).
- by (*BEGIN*)min_1_1, H2(*END*) we proved ((*BEGIN*)1 - (1 - [[G]]_v)(*END*) = [[G]]_v) (H4).
+ by sem_le_1 we proved ([[F]]v ≤ 1) (H1).
+ by (*BEGIN*)sem_le_1(*END*) we proved ([[G]]v ≤ 1) (H2).
+ by min_1_1, H1 we proved (1 - (1 - [[F]]v) = [[F]]v) (H3).
+ by (*BEGIN*)min_1_1, H2(*END*) we proved ((*BEGIN*)1 - (1 - [[G]]v)(*END*) = [[G]]v) (H4).
  conclude 
-     ([[F]]_v)
-   = (1 - (1 - [[F]]_v)) by (*BEGIN*)H3(*END*).
-   = (1 - [[(*BEGIN*)FNot F(*END*)]]_v).
-   = (1 - [[ FNot G]]_v) by H.
-   = (1 - (*BEGIN*)(1 - [[G]]_v)(*END*)).
-   = [[G]]_v by (*BEGIN*)H4(*END*).
+     ([[F]]v)
+   = (1 - (1 - [[F]]v)) by (*BEGIN*)H3(*END*).
+   = (1 - [[(*BEGIN*)FNot F(*END*)]]v).
+   = (1 - [[ FNot G]]v) by H.
+   = (1 - (*BEGIN*)(1 - [[G]]v)(*END*)).
+   = [[G]]v by (*BEGIN*)H4(*END*).
  done.
 qed.
 
@@ -556,7 +578,7 @@ Per dimostrare tale teorema è bene suddividere la prova in lemmi intermedi
 1. lemma `negate_invert`, dimostrato per induzione su F, utilizzando
    `min_bool`
    
-        ∀F:Formula.∀v:ℕ→ℕ.[[ negate F ]]_v=[[ F ]]_(invert v).
+        ∀F:Formula.∀v:ℕ→ℕ.[[ negate F ]]v=[[ F ]]_(invert v).
 
 2. lemma `negate_fun`, conseguenza di `negate_invert`
 
@@ -605,7 +627,7 @@ theorem duality: ∀F1,F2:Formula.F1 ≡ F2 → dualize F1 ≡ dualize F2.
  the thesis becomes (dualize F1 ≡ dualize F2).
  by (*BEGIN*)negate_fun(*END*), H we proved (negate F1 ≡ negate F2) (H1).
  by (*BEGIN*)not_dualize_eq_negate(*END*), (*BEGIN*)equiv_rewrite(*END*), H1 we proved (FNot (dualize F1) ≡ negate F2) (H2).
- by (*BEGIN*)not_dualize_eq_negate(*END*), (*BEGIN*)equiv_rewrite(*END*), H2 we proved (FNot (dualize F1) ≡ FNot (dualize F2)) (H3).
+ by (*BEGIN*)not_dualize_eq_negate(*END*), (*BEGIN*)equiv_rewrite(*END*), H2, equiv_sym we proved (FNot (dualize F1) ≡ FNot (dualize F2)) (H3).
  by (*BEGIN*)not_inj(*END*), H3 we proved (dualize F1 ≡ dualize F2) (H4).
  by H4 done.
 qed.