definition if_then_else ≝ λT:Type.λe,t,f.match e return λ_.T with [ true ⇒ t | false ⇒ f].
notation > "'if' term 19 e 'then' term 19 t 'else' term 90 f" non associative with precedence 19 for @{ 'if_then_else $e $t $f }.
notation < "'if' \nbsp term 19 e \nbsp 'then' \nbsp term 19 t \nbsp 'else' \nbsp term 90 f \nbsp" non associative with precedence 19 for @{ 'if_then_else $e $t $f }.
-interpretation "Formula if_then_else" 'if_then_else e t f = (if_then_else _ e t f).
+interpretation "Formula if_then_else" 'if_then_else e t f = (if_then_else ? e t f).
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 2
=========
- La semantica di una formula `F` in un mondo `v`: `[[ F ]]_v`
+ 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
*)
notation < "[[ \nbsp term 19 a \nbsp ]] \nbsp \sub term 90 v" non associative with precedence 90 for @{ 'semantics $v $a }.
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 }.
+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.
+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.
(* Ripasso 3
=========
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).
-definition equiv ≝ λF1,F2. ∀v.[[ F1 ]]_v = [[ F2 ]]_v.
+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.
+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).
(*DOCBEGIN
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 `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`
+* 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`
---------------------------
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
+ ∀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`.
+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.
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
+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.
+ ∀F,x,v. [[ FAtom x ]]v = 0 → [[ F[FBot/x] ]]v = [[ F ]]v.
(*BEGIN*)
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).
+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).
+ 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).
+ 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).
+ 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).
+ 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).
+ ([[ 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.
+ = ([[ 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).
+ ([[ 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).
+ 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).
+ 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).
+ ([[ (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).
+ 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).
+ 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).
+ ([[ (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).
+ 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).
+ 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).
+ ([[ (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).
+ 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).
+ ([[ (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.
+ ∀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).
+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).
+ 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).
+ 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).
+ 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).
+ 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).
+ ([[ 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.
+ = ([[ 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).
+ ([[ 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).
+ 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).
+ 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).
+ ([[ (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).
+ 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).
+ 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).
+ ([[ (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).
+ 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).
+ 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).
+ ([[ (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).
+ 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).
+ ([[ (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.
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).
+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 H1.
- = (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.
+ ([[ 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 H1.
+ = (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 H1.
- = (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.
+ ([[ 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 H1.
+ = (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.
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.`
+ 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.
projects / helm.git / blobdiff