-(* Esercitazione di logica 29/10/2008.
-
- Note per gli esercizi:
-
- http://www.cs.unibo.it/~tassi/exercise-duality.ma.html
-
-*)
-
(* Esercizio 0
===========
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Account2: ...
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-
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-
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*)
-(*DOCBEGIN
-
-Il teorema di dualità
-=====================
-
-Il teorema di dualizzazione dice che date due formule `F1` ed `F2`,
-se le due formule sono equivalenti (`F1 ≡ F2`) allora anche le
-loro dualizzate lo sono (`dualize F1 ≡ dualize F2`).
-
-L'ingrediente principale è la funzione di dualizzazione di una formula `F`:
-
- * Scambia FTop con FBot e viceversa
-
- * Scambia il connettivo FAnd con FOr e viceversa
-
- * Sostituisce il connettivo FImpl con FAnd e nega la
- prima sottoformula.
-
- Ad esempio la formula `A → (B ∧ ⊥)` viene dualizzata in
- `¬A ∧ (B ∨ ⊤)`.
-
-Per dimostrare il teorema di dualizzazione in modo agevole è necessario
-definire altre nozioni:
-
-* La funzione `negate` che presa una formula `F` ne nega gli atomi.
- Ad esempio la formula `(A ∨ (⊤ → B))` deve diventare `¬A ∨ (⊤ → ¬B)`.
-
-* La funzione `invert` permette di invertire un mondo `v`.
- Ovvero, per ogni indice di atomo `i`, se `v i` restituisce
- `1` allora `(invert v) i` restituisce `0` e viceversa.
-
-DOCEND*)
-
(* ATTENZIONE
==========
| FNot: Formula → Formula
.
-(* Esercizio 1
- ===========
-
- Modificare la funzione `sem` scritta nella precedente
- esercitazione in modo che valga solo 0 oppure 1 nel caso degli
- atomi, anche nel caso in cui il mondo `v` restituisca un numero
- maggiore di 1.
-
- Suggerimento: non è necessario usare il costrutto if_then_else
- e tantomento il predicato di maggiore o uguale. È invece possibile
- usare la funzione `min`.
-*)
let rec sem (v: nat → nat) (F: Formula) on F : nat ≝
match F with
[ FBot ⇒ 0
| FTop ⇒ 1
- | FAtom n ⇒ (*BEGIN*)min (v n) 1(*END*)
+ | FAtom n ⇒ min (v n) 1
| FAnd F1 F2 ⇒ min (sem v F1) (sem v F2)
| FOr F1 F2 ⇒ max (sem v F1) (sem v F2)
| FImpl F1 F2 ⇒ max (1 - sem v F1) (sem v F2)
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.
- if eqb x 0 then 2
- else if eqb x 1 then 1
- else 0.
-
-(* Test 1
- ======
-
- La semantica della formula `(A ∨ C)` nel mondo `v20` in cui
- `A` vale `2` e `C` vale `0` deve valere `1`.
-
-*)
-eval normalize on [[FOr (FAtom 0) (FAtom 2)]]_v20.
-
(*DOCBEGIN
La libreria di Matita
* 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 `equiv_rewrite` : `∀F1,F2,F3. F1 ≡ F2 → F1 ≡ F3 → F2 ≡ F3`
+* lemma `decidable_eq_nat` : `∀x,y.x = y ∨ x ≠ y`
+
DOCEND*)
interpretation "equivalence for Formulas" 'equivF a b = (equiv a b).
theorem shannon :
- ∀F,x,v. [[ if eqb [[FAtom x]]_v 0 then F[FBot/x] else (F[FTop/x]) ]]_v = [[F]]_v.
-intros; elim F;
-[1,2: cases (eqb [[FAtom x]]_v 0); reflexivity;
-|4,5,6: cases (eqb [[FAtom x]]_v 0) in H H1; simplify; intros; rewrite > H; rewrite > H1; reflexivity;
-|7: cases (eqb [[FAtom x]]_v 0) in H; simplify; intros; rewrite > H; reflexivity;
-| cases (sem_bool (FAtom x) v); rewrite > H; simplify;
- cases (decidable_eq_nat n x); destruct H1;
- [1,3: rewrite > eqb_n_n; simplify; rewrite >H;reflexivity;.
- |*:simplify in H; rewrite > (not_eq_to_eqb_false ?? H1); simplify; reflexivity;]]
+ ∀F,x,v. [[ if eqb [[FAtom x]]_v 0 then F[FBot/x] else (F[FTop/x]) ]]_v = [[F]]_v.
+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).
+case FBot.
+ 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.
+ 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.
+ 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.
+ 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 H3.
+ = [[ (FAtom n)[ FBot/x ] ]]_v.
+ = [[ if eqb n x then FBot else (FAtom n) ]]_v.
+ = [[ if false then FBot else (FAtom n) ]]_v by (not_eq_to_eqb_false ?? H2).
+ = [[ 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 H3.
+ = [[ FAtom n[ FTop/x ] ]]_v.
+ = [[ if eqb n x then FTop else (FAtom n) ]]_v.
+ = [[ if false then FTop else (FAtom n) ]]_v by (not_eq_to_eqb_false ?? H2).
+ = [[ FAtom n ]]_v.
+ done.
+case FAnd.
+ 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).
+ using H4 we proved ([[ f[FBot/x ] ]]_v = [[ f ]]_v) (H5).
+ by H3, 1 we proved
+ ([[ if eqb 0 O then f1[ FBot/x ] else (f1[ FTop/x ]) ]]_v = [[ f1 ]]_v) (H6).
+ using 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).
+ using H4 we proved ([[ f[FTop/x ] ]]_v = [[ f ]]_v) (H5).
+ by H3, 1 we proved
+ ([[ if eqb 1 O then f1[ FBot/x ] else (f1[ FTop/x ]) ]]_v = [[ f1 ]]_v) (H6).
+ using 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.
+ 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).
+ using H4 we proved ([[ f[FBot/x ] ]]_v = [[ f ]]_v) (H5).
+ by H3, 1 we proved
+ ([[ if eqb 0 O then f1[ FBot/x ] else (f1[ FTop/x ]) ]]_v = [[ f1 ]]_v) (H6).
+ using 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).
+ using H4 we proved ([[ f[FTop/x ] ]]_v = [[ f ]]_v) (H5).
+ by H3, 1 we proved
+ ([[ if eqb 1 O then f1[ FBot/x ] else (f1[ FTop/x ]) ]]_v = [[ f1 ]]_v) (H6).
+ using 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.
+ 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).
+ using H4 we proved ([[ f[FBot/x ] ]]_v = [[ f ]]_v) (H5).
+ by H3, 1 we proved
+ ([[ if eqb 0 O then f1[ FBot/x ] else (f1[ FTop/x ]) ]]_v = [[ f1 ]]_v) (H6).
+ using 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).
+ using H4 we proved ([[ f[FTop/x ] ]]_v = [[ f ]]_v) (H5).
+ by H3, 1 we proved
+ ([[ if eqb 1 O then f1[ FBot/x ] else (f1[ FTop/x ]) ]]_v = [[ f1 ]]_v) (H6).
+ using 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.
+ 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).
+ using 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).
+ change with (1 - [[ f[ FBot/x ] ]]_v = [[ FNot f ]]_v).
+ = (1 - [[ f ]]_v) by H5.
+ change with ([[ FNot f ]]_v = [[ 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).
+ using 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).
+ change with (1 - [[ f[ FTop/x ] ]]_v = [[ FNot f ]]_v) .
+ = (1 - [[ f ]]_v) by H5.
+ change with ([[ FNot f ]]_v = [[ FNot f ]]_v).
+ done.
qed.
-
-
-
-
\ No newline at end of file
projects / helm.git / commitdiff