From: Enrico Tassi <enrico.tassi@inria.fr>
Date: Thu, 6 Nov 2008 19:42:37 +0000 (+0000)
Subject: almost there
X-Git-Tag: make_still_working~4584
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;ds=inline;h=c2fe13e8a63b2b91a5eac3c5e03c0df86310e5ef;p=helm.git

almost there
---

diff --git a/helm/software/matita/contribs/didactic/shannon.ma b/helm/software/matita/contribs/didactic/shannon.ma
index cfc36f690..13b9b018b 100644
--- a/helm/software/matita/contribs/didactic/shannon.ma
+++ b/helm/software/matita/contribs/didactic/shannon.ma
@@ -117,7 +117,8 @@ 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 min_1_sem: ∀F,v.min 1 [[ F ]]_v = [[ F ]]_v. intros; cases (sem_bool F v); rewrite > H; reflexivity; qed.
+lemma max_0_sem: ∀F,v.max [[ F ]]_v 0 = [[ F ]]_v. intros; cases (sem_bool F v); rewrite > H; reflexivity; qed.
 (*DOCBEGIN
 
 La libreria di Matita
@@ -126,8 +127,10 @@ La libreria di Matita
 Per portare a termine l'esercitazione sono necessari i seguenti lemmi:
 
 * lemma `decidable_eq_nat` : `∀x,y.x = y ∨ x ≠ y`
-* lemma `eqb_n_n` : `∀x.eqb x x = true`
 * lemma `not_eq_to_eqb_false` : `∀x,y.x ≠ y → eqb x y = false`
+* lemma `eq_to_eqb_true` : `∀x,y.x = y → eqb x y = true`
+* lemma `min_1_sem` : `∀F,v.min 1 [[ F ]]_v = [[ F ]]_v`
+* lemma `max_0_sem` : `∀F,v.max [[ F ]]_v 0 = [[ F ]]_v`
 
 Il teorema di espansione di Shannon
 ===================================
@@ -142,18 +145,235 @@ nel mondo `v`, altrimenti lo sostituisco con `FTop`.
 
 DOCEND*)
 
+definition IFTE := λA,B,C:Formula. FOr (FAnd A B) (FAnd (FNot A) C).
+
+lemma shannon_false: 
+  ∀F,x,v. [[ FAtom x ]]_v = 0 → [[ F[FBot/x] ]]_v = [[ F ]]_v.
+assume F : Formula.
+assume x : ℕ.
+assume v : (ℕ → ℕ).
+suppose ([[ FAtom x ]]_v = 0) (H).
+we proceed by induction on F to prove ([[ F[FBot/x] ]]_v = [[ F ]]_v).
+case FBot.
+  the thesis becomes ([[ FBot[FBot/x] ]]_v = [[ FBot ]]_v).
+  the thesis becomes ([[ FBot ]]_v = [[ FBot ]]_v).
+  done. 
+case FTop.
+  the thesis becomes ([[ FTop[FBot/x] ]]_v = [[ FTop ]]_v).
+  the thesis becomes ([[ FTop ]]_v = [[ FTop ]]_v).
+  done.
+case FAtom.
+  assume n : ℕ.
+  the thesis becomes ([[ (FAtom n)[FBot/x] ]]_v = [[ FAtom n ]]_v).
+  the thesis becomes ([[ if eqb n x then FBot else (FAtom n) ]]_v = [[ FAtom n ]]_v).
+  by decidable_eq_nat we proved (n = x ∨ n ≠ x) (H1).
+  we proceed by cases on H1 to prove ([[ if eqb n x then FBot else (FAtom n) ]]_v = [[ FAtom n ]]_v).
+    case Left.
+      by H2, eq_to_eqb_true we proved (eqb n x = true) (H3).
+      conclude
+          ([[ if eqb n x then FBot else (FAtom n) ]]_v)
+        = ([[ if true then FBot else (FAtom n) ]]_v) by H3.
+        = ([[ FBot ]]_v). 
+        = 0.
+        = ([[ FAtom x ]]_v) by H.
+        = ([[ FAtom n ]]_v) by H2.
+    done.
+    case Right.
+      by H2, not_eq_to_eqb_false we proved (eqb n x = false) (H3).
+      conclude
+          ([[ if eqb n x then FBot else (FAtom n) ]]_v)
+        = ([[ if false then FBot else (FAtom n) ]]_v) by H3.
+        = ([[ FAtom n ]]_v). 
+    done.
+case FAnd.
+  assume f1 : Formula.
+  by induction hypothesis we know ([[ f1[FBot/x] ]]_v = [[ f1 ]]_v) (H1).
+  assume f2 : Formula. 
+  by induction hypothesis we know ([[ f2[FBot/x] ]]_v = [[ f2 ]]_v) (H2).
+  the thesis becomes ([[ (FAnd f1 f2)[FBot/x] ]]_v = [[ FAnd f1 f2 ]]_v).
+  conclude 
+      ([[ (FAnd f1 f2)[FBot/x] ]]_v)
+    = ([[ FAnd (f1[FBot/x]) (f2[FBot/x]) ]]_v).
+    = (min [[ f1[FBot/x] ]]_v [[ f2[FBot/x] ]]_v).
+    = (min [[ f1 ]]_v [[ f2[FBot/x] ]]_v) by H1.
+    = (min [[ f1 ]]_v [[ f2 ]]_v) by H2.
+    = ([[ FAnd f1 f2 ]]_v).
+  done. 
+case FOr.
+  assume f1 : Formula.
+  by induction hypothesis we know ([[ f1[FBot/x] ]]_v = [[ f1 ]]_v) (H1).
+  assume f2 : Formula. 
+  by induction hypothesis we know ([[ f2[FBot/x] ]]_v = [[ f2 ]]_v) (H2).
+  the thesis becomes ([[ (FOr f1 f2)[FBot/x] ]]_v = [[ FOr f1 f2 ]]_v).
+  conclude 
+      ([[ (FOr f1 f2)[FBot/x] ]]_v)
+    = ([[ FOr (f1[FBot/x]) (f2[FBot/x]) ]]_v).
+    = (max [[ f1[FBot/x] ]]_v  [[ f2[FBot/x] ]]_v).
+    = (max [[ f1 ]]_v  [[ f2[FBot/x] ]]_v) by H1.
+    = (max [[ f1 ]]_v  [[ f2 ]]_v) by H2.
+    = ([[ FOr f1 f2 ]]_v).
+  done.
+case FImpl.
+  assume f1 : Formula.
+  by induction hypothesis we know ([[ f1[FBot/x] ]]_v = [[ f1 ]]_v) (H1).
+  assume f2 : Formula. 
+  by induction hypothesis we know ([[ f2[FBot/x] ]]_v = [[ f2 ]]_v) (H2).
+  the thesis becomes ([[ (FImpl f1 f2)[FBot/x] ]]_v = [[ FImpl f1 f2 ]]_v).
+  conclude
+      ([[ (FImpl f1 f2)[FBot/x] ]]_v)
+    = ([[ FImpl (f1[FBot/x]) (f2[FBot/x]) ]]_v).
+    = (max (1 - [[ f1[FBot/x] ]]_v) [[ f2[FBot/x] ]]_v).
+    = (max (1 - [[ f1 ]]_v) [[ f2[FBot/x] ]]_v) by H1.
+    = (max (1 - [[ f1 ]]_v) [[ f2 ]]_v) by H2.
+    = ([[ FImpl f1 f2 ]]_v).
+  done.
+case FNot.
+  assume f : Formula.
+  by induction hypothesis we know ([[ f[FBot/x] ]]_v = [[ f ]]_v) (H1).
+  the thesis becomes ([[ (FNot f)[FBot/x] ]]_v = [[ FNot f ]]_v).
+  conclude 
+      ([[ (FNot f)[FBot/x] ]]_v)
+    = ([[ FNot (f[FBot/x]) ]]_v).
+    = (1 - [[ f[FBot/x] ]]_v).
+    = (1 - [[ f ]]_v) by H1.
+    = ([[ FNot f ]]_v).
+  done.
+qed. 
+
+lemma shannon_true: 
+  ∀F,x,v. [[ FAtom x ]]_v = 1 → [[ F[FTop/x] ]]_v = [[ F ]]_v.
+assume F : Formula.
+assume x : ℕ.
+assume v : (ℕ → ℕ).
+suppose ([[ FAtom x ]]_v = 1) (H).
+we proceed by induction on F to prove ([[ F[FTop/x] ]]_v = [[ F ]]_v).
+case FBot.
+  the thesis becomes ([[ FBot[FTop/x] ]]_v = [[ FBot ]]_v).
+  the thesis becomes ([[ FBot ]]_v = [[ FBot ]]_v).
+  done. 
+case FTop.
+  the thesis becomes ([[ FTop[FTop/x] ]]_v = [[ FTop ]]_v).
+  the thesis becomes ([[ FTop ]]_v = [[ FTop ]]_v).
+  done.
+case FAtom.
+  assume n : ℕ.
+  the thesis becomes ([[ (FAtom n)[FTop/x] ]]_v = [[ FAtom n ]]_v).
+  the thesis becomes ([[ if eqb n x then FTop else (FAtom n) ]]_v = [[ FAtom n ]]_v).
+  by decidable_eq_nat we proved (n = x ∨ n ≠ x) (H1).
+  we proceed by cases on H1 to prove ([[ if eqb n x then FTop else (FAtom n) ]]_v = [[ FAtom n ]]_v).
+    case Left.
+      by H2, eq_to_eqb_true we proved (eqb n x = true) (H3).
+      conclude
+          ([[ if eqb n x then FTop else (FAtom n) ]]_v)
+        = ([[ if true then FTop else (FAtom n) ]]_v) by H3.
+        = ([[ FTop ]]_v). 
+        = 1.
+        = ([[ FAtom x ]]_v) by H.
+        = ([[ FAtom n ]]_v) by H2.
+    done.
+    case Right.
+      by H2, not_eq_to_eqb_false we proved (eqb n x = false) (H3).
+      conclude
+          ([[ if eqb n x then FTop else (FAtom n) ]]_v)
+        = ([[ if false then FTop else (FAtom n) ]]_v) by H3.
+        = ([[ FAtom n ]]_v). 
+    done.
+case FAnd.
+  assume f1 : Formula.
+  by induction hypothesis we know ([[ f1[FTop/x] ]]_v = [[ f1 ]]_v) (H1).
+  assume f2 : Formula. 
+  by induction hypothesis we know ([[ f2[FTop/x] ]]_v = [[ f2 ]]_v) (H2).
+  the thesis becomes ([[ (FAnd f1 f2)[FTop/x] ]]_v = [[ FAnd f1 f2 ]]_v).
+  conclude 
+      ([[ (FAnd f1 f2)[FTop/x] ]]_v)
+    = ([[ FAnd (f1[FTop/x]) (f2[FTop/x]) ]]_v).
+    = (min [[ f1[FTop/x] ]]_v [[ f2[FTop/x] ]]_v).
+    = (min [[ f1 ]]_v [[ f2[FTop/x] ]]_v) by H1.
+    = (min [[ f1 ]]_v [[ f2 ]]_v) by H2.
+    = ([[ FAnd f1 f2 ]]_v).
+  done. 
+case FOr.
+  assume f1 : Formula.
+  by induction hypothesis we know ([[ f1[FTop/x] ]]_v = [[ f1 ]]_v) (H1).
+  assume f2 : Formula. 
+  by induction hypothesis we know ([[ f2[FTop/x] ]]_v = [[ f2 ]]_v) (H2).
+  the thesis becomes ([[ (FOr f1 f2)[FTop/x] ]]_v = [[ FOr f1 f2 ]]_v).
+  conclude 
+      ([[ (FOr f1 f2)[FTop/x] ]]_v)
+    = ([[ FOr (f1[FTop/x]) (f2[FTop/x]) ]]_v).
+    = (max [[ f1[FTop/x] ]]_v  [[ f2[FTop/x] ]]_v).
+    = (max [[ f1 ]]_v  [[ f2[FTop/x] ]]_v) by H1.
+    = (max [[ f1 ]]_v  [[ f2 ]]_v) by H2.
+    = ([[ FOr f1 f2 ]]_v).
+  done.
+case FImpl.
+  assume f1 : Formula.
+  by induction hypothesis we know ([[ f1[FTop/x] ]]_v = [[ f1 ]]_v) (H1).
+  assume f2 : Formula. 
+  by induction hypothesis we know ([[ f2[FTop/x] ]]_v = [[ f2 ]]_v) (H2).
+  the thesis becomes ([[ (FImpl f1 f2)[FTop/x] ]]_v = [[ FImpl f1 f2 ]]_v).
+  conclude
+      ([[ (FImpl f1 f2)[FTop/x] ]]_v)
+    = ([[ FImpl (f1[FTop/x]) (f2[FTop/x]) ]]_v).
+    = (max (1 - [[ f1[FTop/x] ]]_v) [[ f2[FTop/x] ]]_v).
+    = (max (1 - [[ f1 ]]_v) [[ f2[FTop/x] ]]_v) by H1.
+    = (max (1 - [[ f1 ]]_v) [[ f2 ]]_v) by H2.
+    = ([[ FImpl f1 f2 ]]_v).
+  done.
+case FNot.
+  assume f : Formula.
+  by induction hypothesis we know ([[ f[FTop/x] ]]_v = [[ f ]]_v) (H1).
+  the thesis becomes ([[ (FNot f)[FTop/x] ]]_v = [[ FNot f ]]_v).
+  conclude 
+      ([[ (FNot f)[FTop/x] ]]_v)
+    = ([[ FNot (f[FTop/x]) ]]_v).
+    = (1 - [[ f[FTop/x] ]]_v).
+    = (1 - [[ f ]]_v) by H1.
+    = ([[ FNot f ]]_v).
+  done.
+qed. 
+
 theorem shannon : 
-  ∀F,x,v. [[ if eqb [[FAtom x]]_v 0 then F[FBot/x] else (F[FTop/x]) ]]_v = [[F]]_v.
+  ∀F,x. IFTE (FAtom x) (F[FTop/x]) (F[FBot/x]) ≡ F.
 assume F : Formula.
 assume x : ℕ.
 assume v : (ℕ → ℕ).
-we proceed by induction on F to prove ([[ if eqb [[FAtom x]]_v 0 then F[FBot/x] else (F[FTop/x]) ]]_v = [[F]]_v).
+the thesis becomes ([[ IFTE (FAtom x) (F[FTop/x]) (F[FBot/x])]]_v = [[ F ]]_v).
+by sem_bool we proved ([[ FAtom x]]_v = 0 ∨ [[ FAtom x]]_v = 1) (H).
+we proceed by cases on H to prove ([[ IFTE (FAtom x) (F[FTop/x]) (F[FBot/x])]]_v = [[ F ]]_v).
+case Left.
+  conclude 
+      ([[ IFTE (FAtom x) (F[FTop/x]) (F[FBot/x])]]_v)
+    = ([[ FOr (FAnd (FAtom x) (F[FTop/x])) (FAnd (FNot (FAtom x)) (F[FBot/x]))]]_v).
+    = (max [[ (FAnd (FAtom x) (F[FTop/x])) ]]_v [[ (FAnd (FNot (FAtom x)) (F[FBot/x]))]]_v).
+    = (max (min  [[ FAtom x ]]_v [[ F[FTop/x] ]]_v) (min (1 - [[ FAtom x ]]_v) [[ F[FBot/x] ]]_v)).
+    = (max (min  0 [[ F[FTop/x] ]]_v) (min (1 - 0) [[ F[FBot/x] ]]_v)) by H.    
+    = (max 0 (min 1 [[ F[FBot/x] ]]_v)).
+    = (max 0 [[ F[FBot/x] ]]_v) by min_1_sem.
+    = ([[ F[FBot/x] ]]_v).
+    = ([[ F ]]_v) by H1, shannon_false.
+  done.
+case Right.
+  conclude 
+      ([[ IFTE (FAtom x) (F[FTop/x]) (F[FBot/x])]]_v)
+    = ([[ FOr (FAnd (FAtom x) (F[FTop/x])) (FAnd (FNot (FAtom x)) (F[FBot/x]))]]_v).
+    = (max [[ (FAnd (FAtom x) (F[FTop/x])) ]]_v [[ (FAnd (FNot (FAtom x)) (F[FBot/x]))]]_v).
+    = (max (min  [[ FAtom x ]]_v [[ F[FTop/x] ]]_v) (min (1 - [[ FAtom x ]]_v) [[ F[FBot/x] ]]_v)).
+    = (max (min  1 [[ F[FTop/x] ]]_v) (min (1 - 1) [[ F[FBot/x] ]]_v)) by H.    
+    = (max (min  1 [[ F[FTop/x] ]]_v) (min 0 [[ F[FBot/x] ]]_v)).
+    = (max [[ F[FTop/x] ]]_v (min 0 [[ F[FBot/x] ]]_v)) by min_1_sem.
+    = (max [[ F[FTop/x] ]]_v 0).
+    = ([[ F[FTop/x] ]]_v) by max_0_sem.
+    = ([[ F ]]_v) by H1, shannon_true.
+  done.
+qed.
+
 (*DOCBEGIN
 
-La dimostrazione 
-================
+Note generali
+=============
 
-La dimostrazione procede per induzione sulla formula `F`. Si ricorda che:
+Si ricorda che:
 
 1. Ogni volta che nella finestra di destra compare un simbolo `∀` oppure un 
    simbolo `→` è opportuno usare il comando `assume` oppure `suppose` (che
@@ -183,75 +403,10 @@ La dimostrazione procede per induzione sulla formula `F`. Si ricorda che:
    avrà tante ipotesi induttive quante sono le sue sottoformule e tali
    ipotesi sono necessarie per portare a termine la dimostrazione.
 
-DOCEND*)
-case FBot.
-(*DOCBEGIN
-
-Il caso FBot
-------------
-
-La tesi è 
-
-        [[ if eqb [[FAtom x]]_v 0 then FBot[FBot/x] else (FBot[FTop/x]) ]]_v = [[FBot]]_v
-
-Si procede per casi su `eqb [[FAtom x]]_v 0`. In entrambi i casi basta 
-espandere piano piano le definizioni.
-
-### I comandi da utilizzare
-
-* `the thesis becomes (...).`
-
-  Afferma quale sia la tesi da dimostrare. Se ripetuto
-  permette di espandere le definizioni.
-        
-* `we proceed by cases on (...) to prove (...).`
-
-  Permette di andare per casi su una ipotesi (quando essa è della forma
-  `A ∨ B`) oppure su una espressione come `eqb n m`.
-   
-  Esempio: `we proceed by cases on H to prove Q.`
-  
-  Esempio: `we proceed by cases on (eqb x 0) to prove Q.`
-        
-* `case ... .`
-
-  Nelle dimostrazioni per casi o per induzioni si utulizza tale comando
-  per inizia la sotto prova relativa a un caso. Esempio: `case Fbot.`
-  
-* `done.`
-
-  Ogni caso di una dimostrazione deve essere terminato con il comando
-  `done.`
-  
-DOCEND*)
-  the thesis becomes ([[ if eqb [[FAtom x]]_v 0 then FBot[FBot/x] else (FBot[FTop/x]) ]]_v = [[FBot]]_v). 
-  we proceed by cases on (eqb [[ FAtom x ]]_v 0) 
-    to prove ([[ if eqb [[FAtom x]]_v 0 then FBot[FBot/x] else (FBot[FTop/x]) ]]_v = [[FBot]]_v).
-  case true.
-    the thesis becomes ([[ if true then FBot[FBot/x] else (FBot[FTop/x]) ]]_v = [[FBot]]_v).
-    the thesis becomes ([[ FBot[FBot/x]]]_v = [[FBot]]_v).
-    the thesis becomes ([[ FBot ]]_v = [[FBot]]_v).
-    the thesis becomes (0 = 0).
-    done.
-  case false.
-    done.
-case FTop.
-(*DOCBEGIN
-
-Il caso FTop
-------------
+La dimostrazione 
+================
 
-Analogo al caso FBot
-
-DOCEND*)
-  we proceed by cases on (eqb [[ FAtom x ]]_v 0)
-    to prove ([[ if eqb [[FAtom x]]_v 0 then FTop[FBot/x] else (FTop[FTop/x]) ]]_v = [[FTop]]_v).
-  case true.
-    done.
-  case false.
-    done.     
-case FAtom.
-(*DOCBEGIN
+...
 
 Il caso (FAtom n)
 -----------------
@@ -302,7 +457,32 @@ Abbiamo quindi quattro casi, in tutti si procede con un comando `conclude`:
 
    Analogo al caso precedente. 
 
-### I comandi da usare
+I comandi da utilizzare
+=======================
+
+* `the thesis becomes (...).`
+
+  Afferma quale sia la tesi da dimostrare. Se ripetuto
+  permette di espandere le definizioni.
+        
+* `we proceed by cases on (...) to prove (...).`
+
+  Permette di andare per casi su una ipotesi (quando essa è della forma
+  `A ∨ B`) oppure su una espressione come `eqb n m`.
+   
+  Esempio: `we proceed by cases on H to prove Q.`
+  
+  Esempio: `we proceed by cases on (eqb x 0) to prove Q.`
+        
+* `case ... .`
+
+  Nelle dimostrazioni per casi o per induzioni si utulizza tale comando
+  per inizia la sotto prova relativa a un caso. Esempio: `case Fbot.`
+  
+* `done.`
+
+  Ogni caso di una dimostrazione deve essere terminato con il comando
+  `done.`  
 
 * `assume ... : (...) .`
         
@@ -330,303 +510,3 @@ Abbiamo quindi quattro casi, in tutti si procede con un comando `conclude`:
   tesi.
 
 DOCEND*)
-  assume n : ℕ.
-  the thesis becomes ([[ if eqb [[ FAtom x ]]_v O then (FAtom n)[ FBot/x ] else (FAtom n[ FTop/x ]) ]]_v = [[ FAtom n ]]_ v).
-  by decidable_eq_nat we proved (n = x ∨ n ≠ x) (H).
-  by sem_bool we proved ([[ FAtom x ]]_v = 0 ∨ [[ FAtom x ]]_v = 1) (H1).
-  we proceed by cases on H to prove 
-    ([[ if eqb [[ FAtom x ]]_v O then (FAtom n)[ FBot/x ] else (FAtom n[ FTop/x ]) ]]_v = [[ FAtom n ]]_ v).
-  case Left. (* H2 : n = x *)
-    we proceed by cases on H1 to prove 
-      ([[ if eqb [[ FAtom x ]]_v O then (FAtom n)[ FBot/x ] else (FAtom n[ FTop/x ]) ]]_v = [[ FAtom n ]]_ v).
-    case Left. (* H3 : [[ FAtom x ]]_v = 0 *)
-      conclude 
-          ([[ if eqb [[ FAtom x ]]_v O then (FAtom n)[ FBot/x ] else (FAtom n[ FTop/x ]) ]]_v)
-        = ([[ if eqb 0 0 then (FAtom n)[ FBot/x ] else (FAtom n[ FTop/x ]) ]]_v) by H3.
-        = ([[ if true then (FAtom n)[ FBot/x ] else (FAtom n[ FTop/x ]) ]]_v).
-        = ([[ (FAtom n)[ FBot/x ] ]]_v).
-        = ([[ if eqb n x then FBot else (FAtom  n) ]]_v).
-        = ([[ if eqb n n then FBot else (FAtom  n) ]]_v) by H2.
-        = ([[ if true then FBot else (FAtom  n) ]]_v) by eqb_n_n.
-        = ([[ FBot ]]_v).
-        = 0.
-        = [[ FAtom x ]]_v by H3.
-        = [[ FAtom n ]]_v by H2.
-      done.
-    case Right.
-      conclude 
-          ([[ if eqb [[ FAtom x ]]_v O then (FAtom n)[ FBot/x ] else (FAtom n[ FTop/x ]) ]]_v)
-        = ([[ if eqb 1 0 then (FAtom n)[ FBot/x ] else (FAtom n[ FTop/x ]) ]]_v) by H3.
-        = ([[ if false then (FAtom n)[ FBot/x ] else (FAtom n[ FTop/x ]) ]]_v).
-        = ([[ (FAtom n)[ FTop/x ] ]]_v).
-        = ([[ if eqb n x then FTop else (FAtom  n) ]]_v).
-        = ([[ if eqb n n then FTop else (FAtom  n) ]]_v) by H2.
-        = ([[ if true then FTop else (FAtom  n) ]]_v) by eqb_n_n.
-        = ([[ FTop ]]_v).
-        = 1.
-        = [[ FAtom x ]]_v by H3.
-        = [[ FAtom n ]]_v by H2.
-      done.
-  case Right.
-    by not_eq_to_eqb_false, H2 we proved (eqb n x = false) (H3).
-    we proceed by cases on H1 to prove 
-      ([[ if eqb [[ FAtom x ]]_v O then (FAtom n)[ FBot/x ] else (FAtom n[ FTop/x ]) ]]_v = [[ FAtom n ]]_ v).
-    case Left.
-      conclude 
-          ([[ if eqb [[ FAtom x ]]_v O then (FAtom n)[ FBot/x ] else (FAtom n[ FTop/x ]) ]]_v)
-        = ([[ if eqb 0 O then (FAtom n)[ FBot/x ] else (FAtom n[ FTop/x ]) ]]_v) by H4.  
-        = [[ (FAtom n)[ FBot/x ] ]]_v.
-        = [[ if eqb n x then FBot else (FAtom n) ]]_v.
-        = [[ if false then FBot else (FAtom n) ]]_v by H3.
-        = [[ FAtom n ]]_v. 
-      done.
-    case Right.
-      conclude 
-          ([[ if eqb [[ FAtom x ]]_v O then (FAtom n)[ FBot/x ] else (FAtom n[ FTop/x ]) ]]_v)
-        = ([[ if eqb 1 O then (FAtom n)[ FBot/x ] else (FAtom n[ FTop/x ]) ]]_v) by H4.  
-        = [[ FAtom n[ FTop/x ] ]]_v.
-        = [[ if eqb n x then FTop else (FAtom n) ]]_v.
-        = [[ if false then FTop else (FAtom n) ]]_v by H3.
-        = [[ FAtom n ]]_v. 
-      done.
-case FAnd.
-(*DOCBEGIN
-
-Il caso FAnd
-------------
-
-Una volta assunte eventuali sottoformule (che chiameremo f ed f1) e 
-relative ipotesi induttive 
-la tesi diventa `([[ if eqb [[ FAtom x ]]_v O then ((FAnd f f1)[ FBot/x ]) else ((FAnd f f1)[ FTop/x ]) ]]_v = [[ FAnd f f1 ]]_v)`.
-
-Utilizzando il lemma `sem_bool` si ottiene l'ipotesi aggiuntiva
-`([[ FAtom x ]]_v = 0 ∨ [[ FAtom x ]]_v = 1)`. Si procede poi per casi
-su di essa. 
-
-1. caso in cui vale `[[ FAtom x ]]_v = 0`.
-
-   Componendo le ipotesi induttive con `[[ FAtom x ]]_v = 0` e
-   espandendo alcune definizioni si ottengono
-   `([[ f[FBot/x ] ]]_v = [[ f ]]_v)` e 
-   `([[ f1[FBot/x ] ]]_v = [[ f1 ]]_v)`.
-   
-   La sotto prova termina con una catena di uguaglianze che
-   lavora sul lato sinistro della tesi.
-   Espandendo alcune definizioni, utilizzando 
-   `[[ FAtom x ]]_v = 0` e le nuove ipotesi appena ottenute
-   si arriva a `(min [[ f ]]_v [[ f1 ]]_v)`.
-   Tale espressione è uguale alla parte destra della conclusione.
-
-1. caso in cui vale `[[ FAtom x ]]_v = 1`.
-
-   Analogo al precedente.
-
-DOCEND*)
-  assume f : Formula.
-  by induction hypothesis we know ([[ if eqb [[ FAtom x ]]_v O then f[ FBot/x ] else (f[ FTop/x ])  ]]_v = [[ f ]]_v) (H).
-  assume f1 : Formula.
-  by induction hypothesis we know ([[ if eqb [[ FAtom x ]]_v O then f1[ FBot/x ] else (f1[ FTop/x ])  ]]_v = [[ f1 ]]_v) (H1).
-  the thesis becomes 
-    ([[ if eqb [[ FAtom x ]]_v O then ((FAnd f f1)[ FBot/x ]) else ((FAnd f f1)[ FTop/x ]) ]]_v = [[ FAnd f f1 ]]_v).
-  by sem_bool we proved ([[ FAtom x ]]_v = 0 ∨ [[ FAtom x ]]_v = 1) (H2).
-  we proceed by cases on H2 to prove 
-    ([[ if eqb [[ FAtom x ]]_v O then ((FAnd f f1)[ FBot/x ]) else ((FAnd f f1)[ FTop/x ]) ]]_v = [[ FAnd f f1 ]]_v).
-  case Left.
-    by H3, H we proved 
-      ([[ if eqb 0 O then f[ FBot/x ] else (f[ FTop/x ])  ]]_v = [[ f ]]_v) (H4).
-    by H4 we proved ([[ f[FBot/x ] ]]_v = [[ f ]]_v) (H5).
-    by H3, H1 we proved 
-      ([[ if eqb 0 O then f1[ FBot/x ] else (f1[ FTop/x ])  ]]_v = [[ f1 ]]_v) (H6).
-    by H6 we proved ([[ f1[FBot/x ] ]]_v = [[ f1 ]]_v) (H7).
-    conclude
-        ([[ if eqb [[ FAtom x ]]_v O then ((FAnd f f1)[ FBot/x ]) else ((FAnd f f1)[ FTop/x ]) ]]_v)
-      = ([[ if eqb 0 O then ((FAnd f f1)[ FBot/x ]) else ((FAnd f f1)[ FTop/x ]) ]]_v) by H3.
-      = ([[ if true then ((FAnd f f1)[ FBot/x ]) else ((FAnd f f1)[ FTop/x ]) ]]_v).
-      = ([[ (FAnd f f1)[ FBot/x ] ]]_v).
-      = ([[ FAnd (f[ FBot/x ]) (f1[ FBot/x ]) ]]_v).
-      = (min [[ f[ FBot/x ] ]]_v [[ f1[ FBot/x ] ]]_v).
-      = (min [[ f ]]_v [[ f1[ FBot/x ] ]]_v) by H5.
-      = (min [[ f ]]_v [[ f1 ]]_v) by H6.
-      = ([[ FAnd f f1 ]]_v).
-    done.
-  case Right.
-    by H3, H we proved 
-      ([[ if eqb 1 O then f[ FBot/x ] else (f[ FTop/x ])  ]]_v = [[ f ]]_v) (H4).
-    by H4 we proved ([[ f[FTop/x ] ]]_v = [[ f ]]_v) (H5).
-    by H3, H1 we proved 
-      ([[ if eqb 1 O then f1[ FBot/x ] else (f1[ FTop/x ])  ]]_v = [[ f1 ]]_v) (H6).
-    by H6 we proved ([[ f1[FTop/x ] ]]_v = [[ f1 ]]_v) (H7).
-    conclude
-        ([[ if eqb [[ FAtom x ]]_v O then ((FAnd f f1)[ FBot/x ]) else ((FAnd f f1)[ FTop/x ]) ]]_v)
-      = ([[ if eqb 1 O then ((FAnd f f1)[ FBot/x ]) else ((FAnd f f1)[ FTop/x ]) ]]_v) by H3.
-      = ([[ if false then ((FAnd f f1)[ FBot/x ]) else ((FAnd f f1)[ FTop/x ]) ]]_v).
-      = ([[ (FAnd f f1)[ FTop/x ] ]]_v).
-      = ([[ FAnd (f[ FTop/x ]) (f1[ FTop/x ]) ]]_v).
-      = (min [[ f[ FTop/x ] ]]_v [[ f1[ FTop/x ] ]]_v).
-      = (min [[ f ]]_v [[ f1[ FTop/x ] ]]_v) by H5.
-      = (min [[ f ]]_v [[ f1 ]]_v) by H6.
-      = ([[ FAnd f f1 ]]_v).
-    done.
-case FOr.
-(*DOCBEGIN
-
-Il caso FOr
------------
-
-Una volta assunte eventuali sottoformule e ipotesi induttive 
-la tesi diventa `([[ if eqb [[ FAtom x ]]_v O then ((FOr f f1)[ FBot/x ]) else ((FOr f f1)[ FTop/x ]) ]]_v = [[ FOr f f1 ]]_v)`.
-
-Analogo al caso FAnd.
-
-DOCEND*)
-  assume f : Formula.
-  by induction hypothesis we know ([[ if eqb [[ FAtom x ]]_v O then f[ FBot/x ] else (f[ FTop/x ])  ]]_v = [[ f ]]_v) (H).
-  assume f1 : Formula.
-  by induction hypothesis we know ([[ if eqb [[ FAtom x ]]_v O then f1[ FBot/x ] else (f1[ FTop/x ])  ]]_v = [[ f1 ]]_v) (H1).
-  the thesis becomes 
-    ([[ if eqb [[ FAtom x ]]_v O then ((FOr f f1)[ FBot/x ]) else ((FOr f f1)[ FTop/x ]) ]]_v = [[ FOr f f1 ]]_v).
-  by sem_bool we proved ([[ FAtom x ]]_v = 0 ∨ [[ FAtom x ]]_v = 1) (H2).
-  we proceed by cases on H2 to prove 
-    ([[ if eqb [[ FAtom x ]]_v O then ((FOr f f1)[ FBot/x ]) else ((FOr f f1)[ FTop/x ]) ]]_v = [[ FOr f f1 ]]_v).
-  case Left.
-    by H3, H we proved 
-      ([[ if eqb 0 O then f[ FBot/x ] else (f[ FTop/x ])  ]]_v = [[ f ]]_v) (H4).
-    by H4 we proved ([[ f[FBot/x ] ]]_v = [[ f ]]_v) (H5).
-    by H3, H1 we proved 
-      ([[ if eqb 0 O then f1[ FBot/x ] else (f1[ FTop/x ])  ]]_v = [[ f1 ]]_v) (H6).
-    by H6 we proved ([[ f1[FBot/x ] ]]_v = [[ f1 ]]_v) (H7).
-    conclude
-        ([[ if eqb [[ FAtom x ]]_v O then ((FOr f f1)[ FBot/x ]) else ((FOr f f1)[ FTop/x ]) ]]_v)
-      = ([[ if eqb 0 O then ((FOr f f1)[ FBot/x ]) else ((FOr f f1)[ FTop/x ]) ]]_v) by H3.
-      = ([[ if true then ((FOr f f1)[ FBot/x ]) else ((FOr f f1)[ FTop/x ]) ]]_v).
-      = ([[ (FOr f f1)[ FBot/x ] ]]_v).
-      = ([[ FOr (f[ FBot/x ]) (f1[ FBot/x ]) ]]_v).
-      = (max [[ f[ FBot/x ] ]]_v [[ f1[ FBot/x ] ]]_v).
-      = (max [[ f ]]_v [[ f1[ FBot/x ] ]]_v) by H5.
-      = (max [[ f ]]_v [[ f1 ]]_v) by H6.
-      = ([[ FOr f f1 ]]_v).
-    done.
-  case Right.
-    by H3, H we proved 
-      ([[ if eqb 1 O then f[ FBot/x ] else (f[ FTop/x ])  ]]_v = [[ f ]]_v) (H4).
-    by H4 we proved ([[ f[FTop/x ] ]]_v = [[ f ]]_v) (H5).
-    by H3, H1 we proved 
-      ([[ if eqb 1 O then f1[ FBot/x ] else (f1[ FTop/x ])  ]]_v = [[ f1 ]]_v) (H6).
-    by H6 we proved ([[ f1[FTop/x ] ]]_v = [[ f1 ]]_v) (H7).
-    conclude
-        ([[ if eqb [[ FAtom x ]]_v O then ((FOr f f1)[ FBot/x ]) else ((FOr f f1)[ FTop/x ]) ]]_v)
-      = ([[ if eqb 1 O then ((FOr f f1)[ FBot/x ]) else ((FOr f f1)[ FTop/x ]) ]]_v) by H3.
-      = ([[ if false then ((FOr f f1)[ FBot/x ]) else ((FOr f f1)[ FTop/x ]) ]]_v).
-      = ([[ (FOr f f1)[ FTop/x ] ]]_v).
-      = ([[ FOr (f[ FTop/x ]) (f1[ FTop/x ]) ]]_v).
-      = (max [[ f[ FTop/x ] ]]_v [[ f1[ FTop/x ] ]]_v).
-      = (max [[ f ]]_v [[ f1[ FTop/x ] ]]_v) by H5.
-      = (max [[ f ]]_v [[ f1 ]]_v) by H6.
-      = ([[ FOr f f1 ]]_v).
-    done.
-case FImpl.
-(*DOCBEGIN
-
-Il caso FImpl
--------------
- 
-Una volta assunte eventuali sottoformule e ipotesi induttive 
-la tesi diventa `([[ if eqb [[ FAtom x ]]_v O then ((FImpl f f1)[ FBot/x ]) else ((FImpl f f1)[ FTop/x ]) ]]_v = [[ FImpl f f1 ]]_v)`.
-
-Analogo al caso FAnd.
-
-DOCEND*)
-  assume f : Formula.
-  by induction hypothesis we know ([[ if eqb [[ FAtom x ]]_v O then f[ FBot/x ] else (f[ FTop/x ])  ]]_v = [[ f ]]_v) (H).
-  assume f1 : Formula.
-  by induction hypothesis we know ([[ if eqb [[ FAtom x ]]_v O then f1[ FBot/x ] else (f1[ FTop/x ])  ]]_v = [[ f1 ]]_v) (H1).
-  the thesis becomes 
-    ([[ if eqb [[ FAtom x ]]_v O then ((FImpl f f1)[ FBot/x ]) else ((FImpl f f1)[ FTop/x ]) ]]_v = [[ FImpl f f1 ]]_v).
-  by sem_bool we proved ([[ FAtom x ]]_v = 0 ∨ [[ FAtom x ]]_v = 1) (H2).
-  we proceed by cases on H2 to prove 
-    ([[ if eqb [[ FAtom x ]]_v O then ((FImpl f f1)[ FBot/x ]) else ((FImpl f f1)[ FTop/x ]) ]]_v = [[ FImpl f f1 ]]_v).
-  case Left.
-    by H3, H we proved 
-      ([[ if eqb 0 O then f[ FBot/x ] else (f[ FTop/x ])  ]]_v = [[ f ]]_v) (H4).
-    by H4 we proved ([[ f[FBot/x ] ]]_v = [[ f ]]_v) (H5).
-    by H3, H1 we proved 
-      ([[ if eqb 0 O then f1[ FBot/x ] else (f1[ FTop/x ])  ]]_v = [[ f1 ]]_v) (H6).
-    by H6 we proved ([[ f1[FBot/x ] ]]_v = [[ f1 ]]_v) (H7).
-    conclude
-        ([[ if eqb [[ FAtom x ]]_v O then ((FImpl f f1)[ FBot/x ]) else ((FImpl f f1)[ FTop/x ]) ]]_v)
-      = ([[ if eqb 0 O then ((FImpl f f1)[ FBot/x ]) else ((FImpl f f1)[ FTop/x ]) ]]_v) by H3.
-      = ([[ if true then ((FImpl f f1)[ FBot/x ]) else ((FImpl f f1)[ FTop/x ]) ]]_v).
-      = ([[ (FImpl f f1)[ FBot/x ] ]]_v).
-      = ([[ FImpl (f[ FBot/x ]) (f1[ FBot/x ]) ]]_v).
-      = (max (1 - [[ f[ FBot/x ] ]]_v) [[ f1[ FBot/x ] ]]_v).
-      = (max (1 -  [[ f ]]_v) [[ f1[ FBot/x ] ]]_v) by H5.
-      = (max (1 -  [[ f ]]_v) [[ f1 ]]_v) by H6.
-      = ([[ FImpl f f1 ]]_v).
-    done.
-  case Right.
-    by H3, H we proved 
-      ([[ if eqb 1 O then f[ FBot/x ] else (f[ FTop/x ])  ]]_v = [[ f ]]_v) (H4).
-    by H4 we proved ([[ f[FTop/x ] ]]_v = [[ f ]]_v) (H5).
-    by H3, H1 we proved 
-      ([[ if eqb 1 O then f1[ FBot/x ] else (f1[ FTop/x ])  ]]_v = [[ f1 ]]_v) (H6).
-    by H6 we proved ([[ f1[FTop/x ] ]]_v = [[ f1 ]]_v) (H7).
-    conclude
-        ([[ if eqb [[ FAtom x ]]_v O then ((FImpl f f1)[ FBot/x ]) else ((FImpl f f1)[ FTop/x ]) ]]_v)
-      = ([[ if eqb 1 O then ((FImpl f f1)[ FBot/x ]) else ((FImpl f f1)[ FTop/x ]) ]]_v) by H3.
-      = ([[ if false then ((FImpl f f1)[ FBot/x ]) else ((FImpl f f1)[ FTop/x ]) ]]_v).
-      = ([[ (FImpl f f1)[ FTop/x ] ]]_v).
-      = ([[ FImpl (f[ FTop/x ]) (f1[ FTop/x ]) ]]_v).
-      = (max (1 - [[ f[ FTop/x ] ]]_v) [[ f1[ FTop/x ] ]]_v).
-      = (max (1 - [[ f ]]_v) [[ f1[ FTop/x ] ]]_v) by H5.
-      = (max (1 - [[ f ]]_v) [[ f1 ]]_v) by H6.
-      = ([[ FImpl f f1 ]]_v).
-    done.
-case FNot.
-(*DOCBEGIN
-
-Il caso FNot 
-------------
-
-Una volta assunte eventuali sottoformule e ipotesi induttive 
-la tesi diventa `([[ if eqb [[ FAtom x ]]_v O then ((FNot f)[ FBot/x ]) else ((FNot f)[ FTop/x ]) ]]_v = [[ FNot f ]]_v)`.
-
-Analogo al caso FAnd.
-
-DOCEND*)
-  assume f : Formula.
-  by induction hypothesis we know ([[ if eqb [[ FAtom x ]]_v O then f[ FBot/x ] else (f[ FTop/x ])  ]]_v = [[ f ]]_v) (H).
-  the thesis becomes 
-    ([[ if eqb [[ FAtom x ]]_v O then ((FNot f)[ FBot/x ]) else ((FNot f)[ FTop/x ]) ]]_v = [[ FNot f ]]_v).
-  by sem_bool we proved ([[ FAtom x ]]_v = 0 ∨ [[ FAtom x ]]_v = 1) (H2).
-  we proceed by cases on H2 to prove 
-    ([[ if eqb [[ FAtom x ]]_v O then ((FNot f)[ FBot/x ]) else ((FNot f)[ FTop/x ]) ]]_v = [[ FNot f ]]_v).
-  case Left.
-    by H1, H we proved 
-      ([[ if eqb 0 O then f[ FBot/x ] else (f[ FTop/x ])  ]]_v = [[ f ]]_v) (H4).
-    by H4 we proved ([[ f[FBot/x ] ]]_v = [[ f ]]_v) (H5).
-    conclude
-        ([[ if eqb [[ FAtom x ]]_v O then ((FNot f)[ FBot/x ]) else ((FNot f)[ FTop/x ]) ]]_v)
-      = ([[ if eqb 0 O then ((FNot f)[ FBot/x ]) else ((FNot f)[ FTop/x ]) ]]_v) by H1.
-      = ([[ if true then ((FNot f)[ FBot/x ]) else ((FNot f)[ FTop/x ]) ]]_v).
-      = ([[ (FNot f)[ FBot/x ] ]]_v).
-      = ([[ FNot (f[ FBot/x ]) ]]_v).
-      = (1 - [[ f[ FBot/x ] ]]_v).
-      = (1 - [[ f ]]_v) by H5.
-      = ([[ FNot f ]]_v).
-    done.
-  case Right.
-    by H1, H we proved 
-      ([[ if eqb 1 O then f[ FBot/x ] else (f[ FTop/x ])  ]]_v = [[ f ]]_v) (H4).
-    by H4 we proved ([[ f[FTop/x ] ]]_v = [[ f ]]_v) (H5).
-    conclude
-        ([[ if eqb [[ FAtom x ]]_v O then ((FNot f)[ FBot/x ]) else ((FNot f)[ FTop/x ]) ]]_v)
-      = ([[ if eqb 1 O then ((FNot f)[ FBot/x ]) else ((FNot f)[ FTop/x ]) ]]_v) by H1.
-      = ([[ if false then ((FNot f)[ FBot/x ]) else ((FNot f)[ FTop/x ]) ]]_v).
-      = ([[ (FNot f)[ FTop/x ] ]]_v).
-      = ([[ FNot (f[ FTop/x ]) ]]_v).
-      = (1 - [[ f[ FTop/x ] ]]_v).
-      = (1 - [[ f ]]_v) by H5.
-      = ([[ FNot f ]]_v).
-    done.
-qed.
-