From 72f8ff3c58c5ac927f572267386f17be39d5a026 Mon Sep 17 00:00:00 2001
From: Enrico Tassi <enrico.tassi@inria.fr>
Date: Sun, 26 Oct 2008 14:02:50 +0000
Subject: [PATCH] duality done

---
 .../matita/contribs/didactic/duality.ma       | 90 +++++++++++++------
 1 file changed, 64 insertions(+), 26 deletions(-)

diff --git a/helm/software/matita/contribs/didactic/duality.ma b/helm/software/matita/contribs/didactic/duality.ma
index 4eaa9c7e3..7fe9082e1 100644
--- a/helm/software/matita/contribs/didactic/duality.ma
+++ b/helm/software/matita/contribs/didactic/duality.ma
@@ -98,7 +98,7 @@ inductive Formula : Type ≝
    Suggerimento: non è necessario usare il costrutto if_then_else
    e tantomento il predicato di maggiore o uguale.
 *) 
-let rec sem (v: nat → nat) (F: Formula) on F ≝
+let rec sem (v: nat → nat) (F: Formula) on F : nat ≝
  match F with
   [ FBot ⇒ 0
   | FTop ⇒ 1
@@ -170,15 +170,15 @@ lemma max_min : ∀F,G,v.  max (1 - [[F]]_v) (1 - [[G]]_v) = 1 - min [[F]]_v [[G
    Ad esempio la formula `(A ∨ (⊤ → B))` deve diventare
    `¬A ∨ (⊤ → ¬B)`.
 *)
-let rec negate (F: Formula) on F ≝
+let rec negate (F: Formula) on F : Formula ≝
  match F with
-  [ FBot ⇒ FBot
+  [ (*BEGIN*)FBot ⇒ FBot
   | FTop ⇒ FTop
   | FAtom n ⇒ FNot (FAtom n)
   | FAnd F1 F2 ⇒ FAnd (negate F1) (negate F2)
   | FOr F1 F2 ⇒ FOr (negate F1) (negate F2)
   | FImpl F1 F2 ⇒ FImpl (negate F1) (negate F2)
-  | FNot F ⇒ FNot (negate F)
+  | FNot F ⇒ FNot (negate F)(*END*)
   ].
 
 (* Test 2
@@ -263,6 +263,12 @@ utilizzare il seguente comando:
 * `symmetry` 
 
   Quando la conclusuine è `a = b` permette di cambiarla in `b = a`.
+  
+* by H1, H2 we proved P (H)
+
+  Il comando `by ... we proved` visto nella scorsa esercitazione
+  permette di utilizzare più ipotesi o lemmi alla volta
+  separandoli con una virgola.
 
 DOCEND*)
 
@@ -279,18 +285,22 @@ lemma negate_invert:
 assume F:Formula.
 assume v:(ℕ→ℕ).
 we proceed by induction on F to prove ([[ negate F ]]_v=[[ F ]]_(invert v)).
-  case FBot .
+  case FBot.
+    (*BEGIN*)
     the thesis becomes ([[ negate FBot ]]_v=[[ FBot ]]_(invert v)).
+    (*END*)
   done.
-  case FTop .
+  case FTop.
+    (*BEGIN*)
     the thesis becomes ([[ negate FTop ]]_v=[[ FTop ]]_(invert v)).
+    (*END*)
   done.
   case FAtom.
     assume n : ℕ.
     the thesis becomes ([[ negate (FAtom n) ]]_v=[[ FAtom n ]]_(invert v)).
     the thesis becomes (1 - (min (v n) 1)= min (invert v n) 1).
     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);
+    by min_bool we proved ((*BEGIN*)min (v n) 1 = 0 ∨ min (v n) 1 = 1(*END*)) (H1);
     we proceed by cases on (H1) to prove (1 - (min (v n) 1)= min (if eqb (min (v n) 1) 0 then 1 else 0) 1).
       case Left.
         conclude 
@@ -305,6 +315,7 @@ we proceed by induction on F to prove ([[ negate F ]]_v=[[ F ]]_(invert v)).
           = 1.
       done.
       case Right.
+        (*BEGIN*)
         conclude 
             (1 - (min (v n) 1)) 
           = (1 - 1) by H.
@@ -315,6 +326,7 @@ we proceed by induction on F to prove ([[ negate F ]]_v=[[ F ]]_(invert v)).
           = (min (if eqb 1 0 then 1 else O) 1) by H.
           = (min 0 1).
           = 0.
+        (*END*)
       done.
   case FAnd.
     assume f : Formula.
@@ -330,9 +342,10 @@ we proceed by induction on F to prove ([[ negate F ]]_v=[[ F ]]_(invert v)).
     conclude 
         (min [[ negate f ]]_v [[ negate f1]]_v)
       = (min [[ f ]]_(invert v) [[ negate f1]]_v) by H.
-      = (min [[ f ]]_(invert v) [[ f1]]_(invert v)) by H1.
+      = (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).
@@ -347,8 +360,10 @@ we proceed by induction on F to prove ([[ negate F ]]_v=[[ F ]]_(invert v)).
         (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).
@@ -363,17 +378,20 @@ we proceed by induction on F to prove ([[ negate F ]]_v=[[ F ]]_(invert v)).
         (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).
-   the thesis becomes
-     ([[ 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.
- done.  
+    the thesis becomes
+      ([[ 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.
+    (*END*)
+  done.  
 qed.   
 
 (* Esercizio 5
@@ -383,17 +401,19 @@ qed.
 *)
 lemma negate_fun:
  ∀F:Formula.∀G:Formula.F ≡ G→negate F ≡ negate G.
+ (*BEGIN*)
  assume F:Formula.
  assume G:Formula.
  suppose (F ≡ G) (H).
  the thesis becomes (negate F ≡ negate G).
  the thesis becomes (∀v:ℕ→ℕ.[[ negate F ]]_v=[[ negate G ]]_v).
+ (*END*)
  assume v:(ℕ→ℕ).
  conclude 
      [[ negate F ]]_v
    = [[ F ]]_(invert v) by negate_invert.
-   = [[ G ]]_(invert v) by H.
-   = [[ negate G ]]_v by negate_invert.
+   = [[ G ]]_((*BEGIN*)invert v(*BEGIN*)) by (*BEGIN*)H(*BEGIN*).
+   = [[ negate G ]]_(*BEGIN*)v(*BEGIN*) by (*BEGIN*)negate_invert(*END*).
  done.  
 qed.
 
@@ -405,19 +425,27 @@ qed.
 *)
 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).
+ (*END*)
  assume v:(ℕ→ℕ).
  we proceed by induction on F to prove ([[negate F]]_v=[[FNot (dualize F)]]_v).
- case FBot .
+ case FBot.
+   (*BEGIN*)
    the thesis becomes ([[ negate FBot ]]_v=[[ FNot (dualize FBot) ]]_v).
+   (*END*)
  done.
- case FTop .
+ case FTop.
+   (*BEGIN*)
    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).
+   (*END*)
  done.
  case FAnd.
    assume f : Formula.
@@ -438,6 +466,7 @@ lemma not_dualize_eq_negate:
      = (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).
@@ -454,8 +483,10 @@ lemma not_dualize_eq_negate:
      = (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).
@@ -472,8 +503,10 @@ lemma not_dualize_eq_negate:
      = (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. 
+ case FNot.
+   (*BEGIN*) 
    assume f : Formula.
    by induction hypothesis we know
      ([[ negate f ]]_v=[[ FNot (dualize f) ]]_v) (H).
@@ -482,6 +515,7 @@ lemma not_dualize_eq_negate:
    the thesis becomes
       (1 - [[ negate f ]]_v=[[ FNot (dualize (FNot f)) ]]_v).
    conclude (1 - [[ negate f ]]_v) = (1 - [[ FNot (dualize f) ]]_v) by H.
+   (*END*)
  done.
 qed.
 
@@ -492,15 +526,17 @@ qed.
 *)
 theorem not_inj:
  ∀F,G:Formula.FNot F ≡ FNot G→F ≡ G.
+ (*BEGIN*)
  assume F:Formula.
  assume G:Formula.
  suppose (FNot F ≡ FNot G) (H).
  the thesis becomes (F ≡ G).
  the thesis becomes (∀v:ℕ→ℕ.[[ F ]]_v=[[ G ]]_v).
+ (*END*)
  assume v:(ℕ→ℕ).
  by H we proved ([[ FNot F ]]_v=[[ FNot G ]]_v) (H1).
- by sem_bool we proved ([[ F ]]_v=O∨[[ F ]]_v=1) (H2).
- by sem_bool we proved ([[ G ]]_v=O∨[[ G ]]_v=1) (H3).
+ by sem_bool we proved ([[ F ]]_v=O ∨ [[ F ]]_v=1) (H2).
+ by (*BEGIN*)sem_bool(*END*) we proved ([[ G ]]_v=(*BEGIN*)O ∨ [[ G ]]_v=1(*END*)) (H3).
  we proceed by cases on H2 to prove ([[ F ]]_v=[[ G ]]_v).
  case Left.
    we proceed by cases on H3 to prove ([[ F ]]_v=[[ G ]]_v).
@@ -519,6 +555,7 @@ theorem not_inj:
        = 1.
      done.
  case Right.
+   (*BEGIN*)
    we proceed by cases on H3 to prove ([[ F ]]_v=[[ G ]]_v).
    case Left.
      conclude 
@@ -533,7 +570,8 @@ theorem not_inj:
        = 0.
      done.
    case Right.
-   done.
+   (*END*)
+     done.
 qed.
 
 (*DOCBEGIN
@@ -598,9 +636,9 @@ theorem duality: ∀F1,F2:Formula.F1 ≡ F2 → dualize F1 ≡ dualize F2.
  assume F2:Formula.
  suppose (F1 ≡ F2) (H).
  the thesis becomes (dualize F1 ≡ dualize F2).
- by negate_fun we proved (negate F1 ≡ negate F2) (H1).
- by not_dualize_eq_negate, equiv_rewrite we proved (FNot (dualize F1) ≡ negate F2) (H2).
- by not_dualize_eq_negate, equiv_rewrite we proved (FNot (dualize F1) ≡ FNot (dualize F2)) (H3).
- by not_inj we proved (dualize F1 ≡ dualize F2) (H4).
+ by (*BEGIN*)negate_fun(*END*) 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_inj(*END*), H3 we proved (dualize F1 ≡ dualize F2) (H4).
  done.
 qed.
-- 
2.39.2