From c2fe13e8a63b2b91a5eac3c5e03c0df86310e5ef Mon Sep 17 00:00:00 2001 From: Enrico Tassi Date: Thu, 6 Nov 2008 19:42:37 +0000 Subject: [PATCH] almost there --- .../matita/contribs/didactic/shannon.ma | 632 +++++++----------- 1 file changed, 256 insertions(+), 376 deletions(-) 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. - -- 2.39.2