+(* Esercizio -1
+ ============
+
+ 1. Leggere ATTENTAMENTE, e magari stampare, la documentazione
+ reperibile all'URL seguente:
+
+ http://mowgli.cs.unibo.it/~tassi/exercise-shannon.ma.html
+
+ 2. Questa volta si fa sul serio:
+
+ l'esercizio proposto è MOLTO difficile, occorre la vostra massima
+ concentrazione (leggi: niente cut&paste selvaggio)
+
+*)
+
+
(* Esercizio 0
===========
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.
-(* Ripasso
- =======
+(* Ripasso 1
+ =========
Il linguaggio delle formule, dove gli atomi sono
rapperesentati da un numero naturale
| FNot: Formula → Formula
.
+(* Ripasso 2
+ =========
+
+ La semantica di una formula `F` in un mondo `v`: `[[ F ]]_v`
+*)
let rec sem (v: nat → nat) (F: Formula) on F : nat ≝
match F with
[ FBot ⇒ 0
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 }.
interpretation "Semantic of Formula" 'semantics v a = (sem v a).
+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.
-(*DOCBEGIN
-
-La libreria di Matita
-=====================
-
-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 `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 `decidable_eq_nat` : `∀x,y.x = y ∨ x ≠ y`
-
-
-DOCEND*)
-
-(* ATTENZIONE
- ==========
+(* Ripasso 3
+ =========
- Non modificare quanto segue.
+ L'operazione di sostituzione di una formula `G` al posto dell'atomo
+ `x` in una formula `F`: `F[G/x]`
*)
-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_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.
+
let rec subst (x:nat) (G: Formula) (F: Formula) on F ≝
match F with
[ FBot ⇒ FBot
| FImpl F1 F2 ⇒ FImpl (subst x G F1) (subst x G F2)
| FNot F ⇒ FNot (subst x G F)
].
-
+
+(* ATTENZIONE
+ ==========
+
+ Non modificare quanto segue.
+*)
notation < "t [ \nbsp term 19 a / term 19 b \nbsp ]" non associative with precedence 19 for @{ 'substitution $b $a $t }.
notation > "t [ term 90 a / term 90 b]" non associative with precedence 19 for @{ 'substitution $b $a $t }.
interpretation "Substitution for Formula" 'substitution b a t = (subst b a t).
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.
+definition IFTE := λA,B,C:Formula. FOr (FAnd A B) (FAnd (FNot A) C).
-theorem shannon :
- ∀F,x,v. [[ if eqb [[FAtom x]]_v 0 then F[FBot/x] else (F[FTop/x]) ]]_v = [[F]]_v.
+(*DOCBEGIN
+
+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 `sem_bool` : `∀F,v. [[ F ]]_v = 0 ∨ [[ F ]]_v = 1`
+* 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`
+
+Nota su `x = y` e `eqb x y`
+---------------------------
+
+Se vi siete mai chiesti la differenza tra `x = y` ed `eqb x y`
+quanto segue prova a chiarirla.
+
+Presi due numeri `x` e `y` in ℕ, dire che `x = y` significa i due numeri
+sono lo stesso numero, ovvero che se `x` è il numero `3`,
+anche `y` è il numero `3`.
+
+`eqb` è un funzione, un programma, che confronta due numeri naturali
+e restituisce `true` se sono uguali, `false` se sono diversi. L'utilizzo
+di tale programma è necessario per usare il costrutto (che è a sua volta
+un programma) `if E then A else B`, che lancia il programma `E`,
+e se il suo
+risultato è `true` si comporta come `A` altrimenti come `B`. Come
+ben sapete i programmi possono contenere errori. In particolare anche
+`eqb` potrebbe essere sbagliato, e per esempio restituire sempre `true`.
+I teoremi `eq_to_eqb_true` e
+`not_eq_to_eqb_false` sono la dimostrazione che il programma `eqb` è
+corretto, ovvero che che se `x = y` allora `eqb x y` restituisce `true`,
+se `x ≠ y` allora `eqb x y` restituisce `false`.
+
+Il teorema di espansione di Shannon
+===================================
+
+Si definisce un connettivo logico `IFTE A B C` come
+
+ FOr (FAnd A B) (FAnd (FNot A) C)
+
+Il teorema dice che data una formula `F`, e preso un atomo `x`, la seguente
+formula è equivalente a `F`:
+
+ IFTE (FAtom x) (F[FTop/x]) (F[FBot/x])
+
+Ovvero, fissato un mondo `v`, sostituisco l'atomo `x` con `FBot` se tale
+atomo è falso, lo sostituisco con `FTop` se è vero.
+
+La dimostrazione è composta da due lemmi, `shannon_false` e `shannon_true`.
+
+Vediamo solo la dimostrazione del primo, essendo il secondo del tutto analogo.
+Il lemma asserisce quanto segue:
+
+ ∀F,x,v. [[ FAtom x ]]_v = 0 → [[ F[FBot/x] ]]_v = [[ F ]]_v
+
+Una volta assunte la formula `F`, l'atomo `x`, il mondo `v` e aver
+supposto che `[[ FAtom x ]]_v = 0` si procede per induzione su `F`.
+I casi `FTop` e `FBot` sono banali. Nei casi `FAnd/FOr/FImpl/FNot`,
+una volta assunte le sottoformule e le relative ipotesi induttive,
+si conclude con una catena di uguaglianze.
+
+Il caso `FAtom` richiede maggiore cura. Assunto l'indice dell'atomo `n`,
+occorre utilizzare il lemma `decidable_eq_nat` per ottenere l'ipotesi
+aggiuntiva `n = x ∨ n ≠ x` (dove `x` è l'atomo su cui predica il teorema).
+Si procede per casi sull'ipotesi appena ottenuta.
+In entrambi i casi, usando i lemmi `eq_to_eqb_true` oppure `not_eq_to_eqb_false`
+si ottengolo le ipotesi aggiuntive `(eqb n x = true)` oppure `(eqb n x = false)`.
+Entrambi i casi si concludono con una catena di uguaglianze.
+
+Il teorema principale si dimostra utilizzando il lemma `sem_bool` per
+ottenre l'ipotesi `[[ FAtom x ]]_v = 0 ∨ [[ FAtom x ]]_v = 1` su cui
+si procede poi per casi. Entrambi i casi si concludono con
+una catena di uguaglianze che utilizza i lemmi dimostrati in precedenza
+e i lemmi `min_1_sem` oppure `max_0_sem`.
+
+DOCEND*)
+
+lemma shannon_false:
+ ∀F,x,v. [[ FAtom x ]]_v = 0 → [[ F[FBot/x] ]]_v = [[ F ]]_v.
+(*BEGIN*)
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).
+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 ([[ 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.
+ the thesis becomes ([[ FBot[FBot/x] ]]_v = [[ FBot ]]_v).
+ the thesis becomes ([[ FBot ]]_v = [[ FBot ]]_v).
+ 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.
+ 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 ([[ 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).
+ 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.
- 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.
+ 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.
- 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.
+ 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 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.
+ 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 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.
+ 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 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.
+ 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 ([[ 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).
+ 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.
+(*END*)
+qed.
+
+lemma shannon_true:
+ ∀F,x,v. [[ FAtom x ]]_v = 1 → [[ F[FTop/x] ]]_v = [[ F ]]_v.
+(*BEGIN*)
+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 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).
+ 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.
+(*END*)
+qed.
+
+theorem shannon :
+ ∀F,x. IFTE (FAtom x) (F[FTop/x]) (F[FBot/x]) ≡ F.
+(*BEGIN*)
+assume F : Formula.
+assume x : ℕ.
+assume v : (ℕ → ℕ).
+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.
+(*END*)
qed.
-let rec maxatom (F : Formula) on F : ℕ ≝
- match F with
- [ FTop ⇒ 0
- | FBot ⇒ 0
- | FAtom n ⇒ n
- | FAnd F1 F2 ⇒ max (maxatom F1) (maxatom F2)
- | FOr F1 F2 ⇒ max (maxatom F1) (maxatom F2)
- | FImpl F1 F2 ⇒ max (maxatom F1) (maxatom F2)
- | FNot F1 ⇒ maxatom F1
- ]
-.
+(*DOCBEGIN
-let rec expandall (F : Formula) (v : ℕ → ℕ) (n : nat) on n: Formula ≝
- match n with
- [ O ⇒ F
- | S m ⇒
- if eqb [[ FAtom n ]]_v 0
- then (expandall F v m)[FBot/n]
- else ((expandall F v m)[FTop/n])
- ]
-.
+Note generali
+=============
-lemma BDD : ∀F,n,v. [[ expandall F v n ]]_v = [[ F ]]_v.
-intros; elim n; simplify; [reflexivity]
-cases (sem_bool (FAtom (S n1)) v); simplify in H1; rewrite > H1; simplify;
-[ lapply (shannon (expandall F v n1) (S n1) v);
- simplify in Hletin; rewrite > H1 in Hletin; simplify in Hletin;
- rewrite > Hletin; assumption;
-| lapply (shannon (expandall F v n1) (S n1) v);
- simplify in Hletin; rewrite > H1 in Hletin; simplify in Hletin;
- rewrite > Hletin; assumption;]
-qed.
+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`
+ oppure (se si è in un caso di una dimostrazione per induzione) il comando
+ `by induction hypothesis we know` (che vengono nuovamente spiegati in seguito).
+
+2. Ogni caso (o sotto caso) della dimostrazione:
+
+ 1. Inizia con una sequenza di comandi `assume` o `suppose` oppure
+ `by induction hypothesis we know`. Tale sequenza di comandi può anche
+ essere vuota.
+
+ 2. Continua poi con almeno un comando `the thesis becomes`.
+
+ 3. Eventualmente seguono vari comandi `by ... we proved` per
+ utilizzare i teoremi già disponibili per generare nuove
+ ipotesi.
+
+ 4. Eventualmente uno o più comandi `we proceed by cases on (...) to prove (...)`.
+
+ 5. Se necessario un comando `conclude` seguito da un numero anche
+ molto lungo di passi `= (...) by ... .` per rendere la parte
+ sinistra della vostra tesi uguale alla parte destra.
+
+ 6. Ogni caso termina con `done`.
+
+3. Ogni caso corrispondente a un nodo con sottoformule (FAnd/For/FNot)
+ avrà tante ipotesi induttive quante sono le sue sottoformule e tali
+ ipotesi sono necessarie per portare a termine la dimostrazione.
+
+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`).
+
+ Esempio: `we proceed by cases on H 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 ... : (...) .`
+
+ Assume una formula o un numero, ad esempio `assume n : (ℕ).` assume
+ un numero naturale `n`.
+
+* `by ..., ..., ..., we proved (...) (...).`
+
+ Permette di comporre lemmi e ipotesi per ottenere nuove ipotesi.
+ Ad esempio `by H, H1 we prove (F ≡ G) (H2).` ottiene una nuova ipotesi
+ `H2` che dice che `F ≡ G` componendo insieme `H` e `H1`.
+
+* `conclude (...) = (...) by ... .`
+
+ Il comando conclude lavora SOLO sulla parte sinistra della tesi. È il comando
+ con cui si inizia una catena di uguaglianze. La prima formula che si
+ scrive deve essere esattamente uguale alla parte sinistra della conclusione
+ originale. Esempio `conclude ([[ FAtom x ]]_v) = ([[ FAtom n ]]_v) by H.`
+ Se la giustificazione non è un lemma o una ipotesi ma la semplice espansione
+ di una definizione, la parte `by ...` deve essere omessa.
+
+* `= (...) by ... .`
+
+ Continua un comando `conclude`, lavorando sempre sulla parte sinistra della
+ tesi.
+
+DOCEND*)
projects / helm.git / blobdiff