true : bool
| false : bool.
-ndefinition bool_ind : ΠP:bool → Prop.P true → P false → Πb:bool.P b ≝
-λP:bool → Prop.λp:P true.λp1:P false.λb:bool.
- match b with [ true ⇒ p | false ⇒ p1 ].
-
-ndefinition bool_rec : ΠP:bool → Set.P true → P false → Πb:bool.P b ≝
-λP:bool → Set.λp:P true.λp1:P false.λb:bool.
- match b with [ true ⇒ p | false ⇒ p1 ].
-
-ndefinition bool_rect : ΠP:bool → Type.P true → P false → Πb:bool.P b ≝
-λP:bool → Type.λp:P true.λp1:P false.λb:bool.
- match b with [ true ⇒ p | false ⇒ p1 ].
-
(* operatori booleani *)
ndefinition eq_bool ≝
nelim b2;
nnormalize;
#H;
- ##[ ##2: napply (False_ind ??);
+ ##[ ##2: napply (False_ind (λx.P) ?);
+ (* perche' non napply (False_ind ??); !!! *)
nchange with (match true with [ true ⇒ False | false ⇒ True]);
nrewrite > H;
nnormalize;
napply I
- ##| ##3: napply (False_ind ??);
+ ##| ##3: napply (False_ind (λx.P) ?);
+ (* perche' non napply (False_ind ??); !!! *)
nchange with (match true with [ true ⇒ False | false ⇒ True]);
nrewrite < H;
nnormalize;
napply (refl_eq ??).
nqed.
-nlemma associative_orbool : ∀b1,b2,b3.((b1 ⊙ b2) ⊙ b3) = (b1 ⊙ (b2 ⊙ b3)).
+nlemma associative_xorbool : ∀b1,b2,b3.((b1 ⊙ b2) ⊙ b3) = (b1 ⊙ (b2 ⊙ b3)).
#b1; #b2; #b3;
nelim b1;
nelim b2;
(* ESADECIMALI *)
(* *********** *)
+(* non riesce a terminare l'esecuzione !!! *)
+
ninductive exadecim : Type ≝
x0: exadecim
| x1: exadecim
| xE: exadecim
| xF: exadecim.
-ndefinition exadecim_ind :
- ΠP:exadecim → Prop.P x0 → P x1 → P x2 → P x3 → P x4 → P x5 → P x6 → P x7 →
- P x8 → P x9 → P xA → P xB → P xC → P xD → P xE → P xF → Πe:exadecim.P e ≝
-λP:exadecim → Prop.
-λp:P x0.λp1:P x1.λp2:P x2.λp3:P x3.λp4:P x4.λp5:P x5.λp6:P x6.λp7:P x7.
-λp8:P x8.λp9:P x9.λp10:P xA.λp11:P xB.λp12:P xC.λp13:P xD.λp14:P xE.λp15:P xF.λe:exadecim.
- match e with
- [ x0 ⇒ p | x1 ⇒ p1 | x2 ⇒ p2 | x3 ⇒ p3 | x4 ⇒ p4 | x5 ⇒ p5 | x6 ⇒ p6 | x7 ⇒ p7
- | x8 ⇒ p8 | x9 ⇒ p9 | xA ⇒ p10 |xB ⇒ p11 | xC ⇒ p12 | xD ⇒ p13 | xE ⇒ p14 | xF ⇒ p15 ].
-
-ndefinition exadecim_rec :
- ΠP:exadecim → Set.P x0 → P x1 → P x2 → P x3 → P x4 → P x5 → P x6 → P x7 →
- P x8 → P x9 → P xA → P xB → P xC → P xD → P xE → P xF → Πe:exadecim.P e ≝
-λP:exadecim → Set.
-λp:P x0.λp1:P x1.λp2:P x2.λp3:P x3.λp4:P x4.λp5:P x5.λp6:P x6.λp7:P x7.
-λp8:P x8.λp9:P x9.λp10:P xA.λp11:P xB.λp12:P xC.λp13:P xD.λp14:P xE.λp15:P xF.λe:exadecim.
- match e with
- [ x0 ⇒ p | x1 ⇒ p1 | x2 ⇒ p2 | x3 ⇒ p3 | x4 ⇒ p4 | x5 ⇒ p5 | x6 ⇒ p6 | x7 ⇒ p7
- | x8 ⇒ p8 | x9 ⇒ p9 | xA ⇒ p10 |xB ⇒ p11 | xC ⇒ p12 | xD ⇒ p13 | xE ⇒ p14 | xF ⇒ p15 ].
-
-ndefinition exadecim_rect :
- ΠP:exadecim → Type.P x0 → P x1 → P x2 → P x3 → P x4 → P x5 → P x6 → P x7 →
- P x8 → P x9 → P xA → P xB → P xC → P xD → P xE → P xF → Πe:exadecim.P e ≝
-λP:exadecim → Type.
-λp:P x0.λp1:P x1.λp2:P x2.λp3:P x3.λp4:P x4.λp5:P x5.λp6:P x6.λp7:P x7.
-λp8:P x8.λp9:P x9.λp10:P xA.λp11:P xB.λp12:P xC.λp13:P xD.λp14:P xE.λp15:P xF.λe:exadecim.
- match e with
- [ x0 ⇒ p | x1 ⇒ p1 | x2 ⇒ p2 | x3 ⇒ p3 | x4 ⇒ p4 | x5 ⇒ p5 | x6 ⇒ p6 | x7 ⇒ p7
- | x8 ⇒ p8 | x9 ⇒ p9 | xA ⇒ p10 |xB ⇒ p11 | xC ⇒ p12 | xD ⇒ p13 | xE ⇒ p14 | xF ⇒ p15 ].
-
(* operatore = *)
ndefinition eq_ex ≝
λe1,e2:exadecim.
alias num (instance 0) = "natural number".
-nlet rec nat_ind (P:nat → Prop) (p:P O) (f:Πn:nat.P n → P (S n)) (n:nat) on n : P n ≝
- match n with [ O ⇒ p | S (n1:nat) ⇒ f n1 (nat_ind P p f n1) ].
-
-nlet rec nat_rec (P:nat → Set) (p:P O) (f:Πn:nat.P n → P (S n)) (n:nat) on n : P n ≝
- match n with [ O ⇒ p | S (n1:nat) ⇒ f n1 (nat_rec P p f n1) ].
-
-nlet rec nat_rect (P:nat → Type) (p:P O) (f:Πn:nat.P n → P (S n)) (n:nat) on n : P n ≝
- match n with [ O ⇒ p | S (n1:nat) ⇒ f n1 (nat_rect P p f n1) ].
-
nlet rec eq_nat (n1,n2:nat) on n1 ≝
match n1 with
[ O ⇒ match n2 with [ O ⇒ true | S _ ⇒ false ]
ninductive ProdT (T1:Type) (T2:Type) : Type ≝
pair : T1 → T2 → ProdT T1 T2.
-ndefinition ProdT_ind
- : ΠT1,T2:Type.ΠP:ProdT T1 T2 → Prop.
- (Πt:T1.Πt1:T2.P (pair T1 T2 t t1)) →
- Πp:ProdT T1 T2.P p ≝
-λT1,T2:Type.λP:ProdT T1 T2 → Prop.
-λf:Πt:T1.Πt1:T2.P (pair T1 T2 t t1).
-λp:ProdT T1 T2.
- match p with [ pair t t1 ⇒ f t t1 ].
-
-ndefinition ProdT_rec
- : ΠT1,T2:Type.ΠP:ProdT T1 T2 → Set.
- (Πt:T1.Πt1:T2.P (pair T1 T2 t t1)) →
- Πp:ProdT T1 T2.P p ≝
-λT1,T2:Type.λP:ProdT T1 T2 → Set.
-λf:Πt:T1.Πt1:T2.P (pair T1 T2 t t1).
-λp:ProdT T1 T2.
- match p with [ pair t t1 ⇒ f t t1 ].
-
-ndefinition ProdT_rect
- : ΠT1,T2:Type.ΠP:ProdT T1 T2 → Type.
- (Πt:T1.Πt1:T2.P (pair T1 T2 t t1)) →
- Πp:ProdT T1 T2.P p ≝
-λT1,T2:Type.λP:ProdT T1 T2 → Type.
-λf:Πt:T1.Πt1:T2.P (pair T1 T2 t t1).
-λp:ProdT T1 T2.
- match p with [ pair t t1 ⇒ f t t1 ].
-
ndefinition fst ≝
λT1,T2:Type.λp:ProdT T1 T2.match p with [ pair x _ ⇒ x ].
ninductive Prod3T (T1:Type) (T2:Type) (T3:Type) : Type ≝
triple : T1 → T2 → T3 → Prod3T T1 T2 T3.
-ndefinition Prod3T_ind
- : ΠT1,T2,T3:Type.ΠP:Prod3T T1 T2 T3 → Prop.
- (Πt:T1.Πt1:T2.Πt2:T3.P (triple T1 T2 T3 t t1 t2)) →
- Πp:Prod3T T1 T2 T3.P p ≝
-λT1,T2,T3:Type.λP:Prod3T T1 T2 T3 → Prop.
-λf:Πt:T1.Πt1:T2.Πt2:T3.P (triple T1 T2 T3 t t1 t2).
-λp:Prod3T T1 T2 T3.
- match p with [ triple t t1 t2 ⇒ f t t1 t2 ].
-
-ndefinition Prod3T_rec
- : ΠT1,T2,T3:Type.ΠP:Prod3T T1 T2 T3 → Set.
- (Πt:T1.Πt1:T2.Πt2:T3.P (triple T1 T2 T3 t t1 t2)) →
- Πp:Prod3T T1 T2 T3.P p ≝
-λT1,T2,T3:Type.λP:Prod3T T1 T2 T3 → Set.
-λf:Πt:T1.Πt1:T2.Πt2:T3.P (triple T1 T2 T3 t t1 t2).
-λp:Prod3T T1 T2 T3.
- match p with [ triple t t1 t2 ⇒ f t t1 t2 ].
-
-ndefinition Prod3T_rect
- : ΠT1,T2,T3:Type.ΠP:Prod3T T1 T2 T3 → Type.
- (Πt:T1.Πt1:T2.Πt2:T3.P (triple T1 T2 T3 t t1 t2)) →
- Πp:Prod3T T1 T2 T3.P p ≝
-λT1,T2,T3:Type.λP:Prod3T T1 T2 T3 → Type.
-λf:Πt:T1.Πt1:T2.Πt2:T3.P (triple T1 T2 T3 t t1 t2).
-λp:Prod3T T1 T2 T3.
- match p with [ triple t t1 t2 ⇒ f t t1 t2 ].
-
ndefinition fst3T ≝
λT1.λT2.λT3.λp:Prod3T T1 T2 T3.match p with [ triple x _ _ ⇒ x ].
ninductive Prod4T (T1:Type) (T2:Type) (T3:Type) (T4:Type) : Type ≝
quadruple : T1 → T2 → T3 → T4 → Prod4T T1 T2 T3 T4.
-ndefinition Prod4T_ind
- : ΠT1,T2,T3,T4:Type.ΠP:Prod4T T1 T2 T3 T4 → Prop.
- (Πt:T1.Πt1:T2.Πt2:T3.Πt3:T4.P (quadruple T1 T2 T3 T4 t t1 t2 t3)) →
- Πp:Prod4T T1 T2 T3 T4.P p ≝
-λT1,T2,T3,T4:Type.λP:Prod4T T1 T2 T3 T4 → Prop.
-λf:Πt:T1.Πt1:T2.Πt2:T3.Πt3:T4.P (quadruple T1 T2 T3 T4 t t1 t2 t3).
-λp:Prod4T T1 T2 T3 T4.
- match p with [ quadruple t t1 t2 t3 ⇒ f t t1 t2 t3 ].
-
-ndefinition Prod4T_rec
- : ΠT1,T2,T3,T4:Type.ΠP:Prod4T T1 T2 T3 T4 → Set.
- (Πt:T1.Πt1:T2.Πt2:T3.Πt3:T4.P (quadruple T1 T2 T3 T4 t t1 t2 t3)) →
- Πp:Prod4T T1 T2 T3 T4.P p ≝
-λT1,T2,T3,T4:Type.λP:Prod4T T1 T2 T3 T4 → Set.
-λf:Πt:T1.Πt1:T2.Πt2:T3.Πt3:T4.P (quadruple T1 T2 T3 T4 t t1 t2 t3).
-λp:Prod4T T1 T2 T3 T4.
- match p with [ quadruple t t1 t2 t3 ⇒ f t t1 t2 t3 ].
-
-ndefinition Prod4T_rect
- : ΠT1,T2,T3,T4:Type.ΠP:Prod4T T1 T2 T3 T4 → Type.
- (Πt:T1.Πt1:T2.Πt2:T3.Πt3:T4.P (quadruple T1 T2 T3 T4 t t1 t2 t3)) →
- Πp:Prod4T T1 T2 T3 T4.P p ≝
-λT1,T2,T3,T4:Type.λP:Prod4T T1 T2 T3 T4 → Type.
-λf:Πt:T1.Πt1:T2.Πt2:T3.Πt3:T4.P (quadruple T1 T2 T3 T4 t t1 t2 t3).
-λp:Prod4T T1 T2 T3 T4.
- match p with [ quadruple t t1 t2 t3 ⇒ f t t1 t2 t3 ].
-
ndefinition fst4T ≝
λT1.λT2.λT3.λT4.λp:Prod4T T1 T2 T3 T4.match p with [ quadruple x _ _ _ ⇒ x ].
ninductive Prod5T (T1:Type) (T2:Type) (T3:Type) (T4:Type) (T5:Type) : Type ≝
quintuple : T1 → T2 → T3 → T4 → T5 → Prod5T T1 T2 T3 T4 T5.
-ndefinition Prod5T_ind
- : ΠT1,T2,T3,T4,T5:Type.ΠP:Prod5T T1 T2 T3 T4 T5 → Prop.
- (Πt:T1.Πt1:T2.Πt2:T3.Πt3:T4.Πt4:T5.P (quintuple T1 T2 T3 T4 T5 t t1 t2 t3 t4)) →
- Πp:Prod5T T1 T2 T3 T4 T5.P p ≝
-λT1,T2,T3,T4,T5:Type.λP:Prod5T T1 T2 T3 T4 T5 → Prop.
-λf:Πt:T1.Πt1:T2.Πt2:T3.Πt3:T4.Πt4:T5.P (quintuple T1 T2 T3 T4 T5 t t1 t2 t3 t4).
-λp:Prod5T T1 T2 T3 T4 T5.
- match p with [ quintuple t t1 t2 t3 t4 ⇒ f t t1 t2 t3 t4 ].
-
-ndefinition Prod5T_rec
- : ΠT1,T2,T3,T4,T5:Type.ΠP:Prod5T T1 T2 T3 T4 T5 → Set.
- (Πt:T1.Πt1:T2.Πt2:T3.Πt3:T4.Πt4:T5.P (quintuple T1 T2 T3 T4 T5 t t1 t2 t3 t4)) →
- Πp:Prod5T T1 T2 T3 T4 T5.P p ≝
-λT1,T2,T3,T4,T5:Type.λP:Prod5T T1 T2 T3 T4 T5 → Set.
-λf:Πt:T1.Πt1:T2.Πt2:T3.Πt3:T4.Πt4:T5.P (quintuple T1 T2 T3 T4 T5 t t1 t2 t3 t4).
-λp:Prod5T T1 T2 T3 T4 T5.
- match p with [ quintuple t t1 t2 t3 t4 ⇒ f t t1 t2 t3 t4 ].
-
-ndefinition Prod5T_rect
- : ΠT1,T2,T3,T4,T5:Type.ΠP:Prod5T T1 T2 T3 T4 T5 → Type.
- (Πt:T1.Πt1:T2.Πt2:T3.Πt3:T4.Πt4:T5.P (quintuple T1 T2 T3 T4 T5 t t1 t2 t3 t4)) →
- Πp:Prod5T T1 T2 T3 T4 T5.P p ≝
-λT1,T2,T3,T4,T5:Type.λP:Prod5T T1 T2 T3 T4 T5 → Type.
-λf:Πt:T1.Πt1:T2.Πt2:T3.Πt3:T4.Πt4:T5.P (quintuple T1 T2 T3 T4 T5 t t1 t2 t3 t4).
-λp:Prod5T T1 T2 T3 T4 T5.
- match p with [ quintuple t t1 t2 t3 t4 ⇒ f t t1 t2 t3 t4 ].
-
ndefinition fst5T ≝
λT1.λT2.λT3.λT4.λT5.λp:Prod5T T1 T2 T3 T4 T5.match p with [ quintuple x _ _ _ _ ⇒ x ].
ninductive True: Prop ≝
I : True.
-(*ndefinition True_ind_xx : ΠP:Prop.P → True → P ≝
-λP:Prop.λp:P.λH:True.
- match H with [ I ⇒ p ].
-
-ndefinition True_rec_xx : ΠP:Set.P → True → P ≝
-λP:Set.λp:P.λH:True.
- match H with [ I ⇒ p ].
-
-ndefinition True_rect_xx : ΠP:Type.P → True → P ≝
-λP:Type.λp:P.λH:True.
- match H with [ I ⇒ p ].*)
-
ninductive False: Prop ≝.
-(*ndefinition False_ind_xx : ΠP:Prop.False → P ≝
-λP:Prop.λH:False.
- match H in False return λH1:False.P with [].
-
-ndefinition False_rec_xx : ΠP:Set.False → P ≝
-λP:Set.λH:False.
- match H in False return λH1:False.P with [].
-
-ndefinition False_rect_xx : ΠP:Type.False → P ≝
-λP:Type.λH:False.
- match H in False return λH1:False.P with [].*)
-
ndefinition Not: Prop → Prop ≝
λA.(A → False).
ninductive And (A,B:Prop) : Prop ≝
conj : A → B → (And A B).
-(*ndefinition And_ind_xx : ΠA,B:Prop.ΠP:Prop.(A → B → P) → And A B → P ≝
-λA,B:Prop.λP:Prop.λf:A → B → P.λH:And A B.
- match H with [conj H1 H2 ⇒ f H1 H2 ].
-
-ndefinition And_rec_xx : ΠA,B:Prop.ΠP:Set.(A → B → P) → And A B → P ≝
-λA,B:Prop.λP:Set.λf:A → B → P.λH:And A B.
- match H with [conj H1 H2 ⇒ f H1 H2 ].
-
-ndefinition And_rect_xx : ΠA,B:Prop.ΠP:Type.(A → B → P) → And A B → P ≝
-λA,B:Prop.λP:Type.λf:A → B → P.λH:And A B.
- match H with [conj H1 H2 ⇒ f H1 H2 ].*)
-
interpretation "logical and" 'and x y = (And x y).
nlemma proj1: ∀A,B:Prop.A ∧ B → A.
or_introl : A → (Or A B)
| or_intror : B → (Or A B).
-(*ndefinition Or_ind_xx : ΠA,B:Prop.ΠP:Prop.(A → P) → (B → P) → Or A B → P ≝
-λA,B:Prop.λP:Prop.λf1:A → P.λf2:B → P.λH:Or A B.
- match H with [ or_introl H1 ⇒ f1 H1 | or_intror H1 ⇒ f2 H1 ].*)
-
interpretation "logical or" 'or x y = (Or x y).
ndefinition decidable : Prop → Prop ≝ λA:Prop.A ∨ ¬A.
ninductive ex (A:Type) (Q:A → Prop) : Prop ≝
ex_intro: ∀x:A.Q x → ex A Q.
-(*ndefinition ex_ind_xx : ΠA:Type.ΠQ:A → Prop.ΠP:Prop.(Πa:A.Q a → P) → ex A Q → P ≝
-λA:Type.λQ:A → Prop.λP:Prop.λf:(Πa:A.Q a → P).λH:ex A Q.
- match H with [ ex_intro H1 H2 ⇒ f H1 H2 ].*)
-
interpretation "exists" 'exists x = (ex ? x).
ninductive ex2 (A:Type) (Q,R:A → Prop) : Prop ≝
ex_intro2: ∀x:A.Q x → R x → ex2 A Q R.
-(*ndefinition ex2_ind_xx : ΠA:Type.ΠQ,R:A → Prop.ΠP:Prop.(Πa:A.Q a → R a → P) → ex2 A Q R → P ≝
-λA:Type.λQ,R:A → Prop.λP:Prop.λf:(Πa:A.Q a → R a → P).λH:ex2 A Q R.
- match H with [ ex_intro2 H1 H2 H3 ⇒ f H1 H2 H3 ].*)
-
ndefinition iff ≝
λA,B.(A -> B) ∧ (B -> A).
ninductive eq (A:Type) (x:A) : A → Prop ≝
refl_eq : eq A x x.
-(*ndefinition eq_ind_xx : ΠA:Type.Πx:A.ΠP:A → Prop.P x → Πa:A.eq A x a → P a ≝
-λA:Type.λx:A.λP:A → Prop.λp:P x.λa:A.λH:eq A x a.
- match H with [refl_eq ⇒ p ].
-
-ndefinition eq_rec_xx : ΠA:Type.Πx:A.ΠP:A → Set.P x → Πa:A.eq A x a → P a ≝
-λA:Type.λx:A.λP:A → Set.λp:P x.λa:A.λH:eq A x a.
- match H with [refl_eq ⇒ p ].
-
-ndefinition eq_rect_xx : ΠA:Type.Πx:A.ΠP:A → Type.P x → Πa:A.eq A x a → P a ≝
-λA:Type.λx:A.λP:A → Type.λp:P x.λa:A.λH:eq A x a.
- match H with [refl_eq ⇒ p ].*)
-
interpretation "leibnitz's equality" 'eq t x y = (eq t x y).
interpretation "leibnitz's non-equality" 'neq t x y = (Not (eq t x y)).
nlemma eq_elim_r: ∀A:Type.∀x:A.∀P:A → Prop.P x → ∀y:A.y=x → P y.
#A; #x; #P; #H; #y; #H1;
- nletin H2 ≝ (symmetric_eq ??? H1);
- nrewrite < H2;
- nassumption.
+ nrewrite < (symmetric_eq ??? H1);
+ napply H.
nqed.
ndefinition relationT : Type → Type → Type ≝
nlemma append_nil : ∀T:Type.∀l:list T.(l@[]) = l.
#T; #l;
+ (* perche' non nelim l; !!! *)
napply (list_ind T ??? l);
nnormalize;
##[ ##1: napply (refl_eq ??)
nlemma associative_list : ∀T.associative (list T) (append T).
#T; #x; #y; #z;
+ (* perche' non nelim x; !!! *)
napply (list_ind T ??? x);
nnormalize;
##[ ##1: napply (refl_eq ??)
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