--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* ********************************************************************** *)
+(* Progetto FreeScale *)
+(* *)
+(* Sviluppato da: *)
+(* Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* *)
+(* Questo materiale fa parte della tesi: *)
+(* "Formalizzazione Interattiva dei Microcontroller a 8bit FreeScale" *)
+(* *)
+(* data ultima modifica 15/11/2007 *)
+(* ********************************************************************** *)
+
+include "nat/div_and_mod.ma".
+include "nat/primes.ma".
+include "list/list.ma".
+include "datatypes/constructors.ma".
+include "logic/connectives.ma".
+
+(* BOOLEANI *)
+
+(* ridefinizione degli operatori booleani, per evitare l'overloading di quelli normali *)
+definition not_bool ≝
+λb:bool.match b with [ true ⇒ false | false ⇒ true ].
+
+definition and_bool ≝
+λb1,b2:bool.match b1 with
+ [ true ⇒ b2 | false ⇒ false ].
+
+definition or_bool ≝
+λb1,b2:bool.match b1 with
+ [ true ⇒ true | false ⇒ b2 ].
+
+definition xor_bool ≝
+λb1,b2:bool.match b1 with
+ [ true ⇒ not_bool b2
+ | false ⇒ b2 ].
+
+definition eq_bool ≝
+λb1,b2:bool.match b1 with
+ [ true ⇒ b2
+ | false ⇒ not_bool b2 ].
+
+(* \ominus *)
+notation "hvbox(⊖ a)" non associative with precedence 36
+ for @{ 'not_bool $a }.
+interpretation "not_bool" 'not_bool x =
+ (cic:/matita/freescale/extra/not_bool.con x).
+
+(* \otimes *)
+notation "hvbox(a break ⊗ b)" left associative with precedence 35
+ for @{ 'and_bool $a $b }.
+interpretation "and_bool" 'and_bool x y =
+ (cic:/matita/freescale/extra/and_bool.con x y).
+
+(* \oplus *)
+notation "hvbox(a break ⊕ b)" left associative with precedence 34
+ for @{ 'or_bool $a $b }.
+interpretation "or_bool" 'or_bool x y =
+ (cic:/matita/freescale/extra/or_bool.con x y).
+
+(* \odot *)
+notation "hvbox(a break ⊙ b)" left associative with precedence 33
+ for @{ 'xor_bool $a $b }.
+interpretation "xor_bool" 'xor_bool x y =
+ (cic:/matita/freescale/extra/xor_bool.con x y).
+
+(* ProdT e' gia' definito, aggiungo Prod3T e Prod4T e Prod5T *)
+
+inductive Prod3T (T1:Type) (T2:Type) (T3:Type) : Type ≝
+tripleT : T1 → T2 → T3 → Prod3T T1 T2 T3.
+
+definition fst3T ≝
+λT1.λT2.λT3.λp:Prod3T T1 T2 T3.match p with [ tripleT x _ _ ⇒ x ].
+
+definition snd3T ≝
+λT1.λT2.λT3.λp:Prod3T T1 T2 T3.match p with [ tripleT _ x _ ⇒ x ].
+
+definition thd3T ≝
+λT1.λT2.λT3.λp:Prod3T T1 T2 T3.match p with [ tripleT _ _ x ⇒ x ].
+
+inductive Prod4T (T1:Type) (T2:Type) (T3:Type) (T4:Type) : Type ≝
+quadrupleT : T1 → T2 → T3 → T4 → Prod4T T1 T2 T3 T4.
+
+definition fst4T ≝
+λT1.λT2.λT3.λT4.λp:Prod4T T1 T2 T3 T4.match p with [ quadrupleT x _ _ _ ⇒ x ].
+
+definition snd4T ≝
+λT1.λT2.λT3.λT4.λp:Prod4T T1 T2 T3 T4.match p with [ quadrupleT _ x _ _ ⇒ x ].
+
+definition thd4T ≝
+λT1.λT2.λT3.λT4.λp:Prod4T T1 T2 T3 T4.match p with [ quadrupleT _ _ x _ ⇒ x ].
+
+definition fth4T ≝
+λT1.λT2.λT3.λT4.λp:Prod4T T1 T2 T3 T4.match p with [ quadrupleT _ _ _ x ⇒ x ].
+
+inductive Prod5T (T1:Type) (T2:Type) (T3:Type) (T4:Type) (T5:Type) : Type ≝
+quintupleT : T1 → T2 → T3 → T4 → T5 → Prod5T T1 T2 T3 T4 T5.
+
+definition fst5T ≝
+λT1.λT2.λT3.λT4.λT5.λp:Prod5T T1 T2 T3 T4 T5.match p with [ quintupleT x _ _ _ _ ⇒ x ].
+
+definition snd5T ≝
+λT1.λT2.λT3.λT4.λT5.λp:Prod5T T1 T2 T3 T4 T5.match p with [ quintupleT _ x _ _ _ ⇒ x ].
+
+definition thd5T ≝
+λT1.λT2.λT3.λT4.λT5.λp:Prod5T T1 T2 T3 T4 T5.match p with [ quintupleT _ _ x _ _ ⇒ x ].
+
+definition frth5T ≝
+λT1.λT2.λT3.λT4.λT5.λp:Prod5T T1 T2 T3 T4 T5.match p with [ quintupleT _ _ _ x _ ⇒ x ].
+
+definition ffth5T ≝
+λT1.λT2.λT3.λT4.λT5.λp:Prod5T T1 T2 T3 T4 T5.match p with [ quintupleT _ _ _ _ x ⇒ x ].
+
+(* OPTIOTN MAP *)
+
+(* option map = match ... with [ None ⇒ None ? | Some .. ⇒ .. ] *)
+definition opt_map ≝
+λT1,T2:Type.λt:option T1.λf:T1 → option T2.
+ match t with [ None ⇒ None ? | Some x ⇒ (f x) ].
+
+(* ********************** *)
+(* TEOREMI/LEMMMI/ASSIOMI *)
+(* ********************** *)
+
+axiom mod_plus: ∀a,b,m. (a + b) \mod m = (a \mod m + b \mod m) \mod m.
+axiom mod_mod: ∀a,n,m. n∣m → a \mod n = a \mod n \mod m.
+axiom eq_mod_times_n_m_m_O: ∀n,m. O < m → n * m \mod m = O.
+axiom eq_mod_to_eq_plus_mod: ∀a,b,c,m. a \mod m = b \mod m → (a+c) \mod m = (b+c) \mod m.
+axiom eq_mod_times_times_mod: ∀a,b,n,m. m = a*n → (a*b) \mod m = a * (b \mod n).
+axiom divides_to_eq_mod_mod_mod: ∀a,n,m. n∣m → a \mod m \mod n = a \mod n.
+axiom le_to_le_plus_to_le : ∀a,b,c,d.b\leq d\rarr a+b\leq c+d\rarr a\leq c.
+axiom or_lt_le : ∀n,m. n < m ∨ m ≤ n.
+
+lemma le_to_lt: ∀n,m. n ≤ m → n < S m.
+ intros;
+ autobatch.
+qed.
+
+alias num (instance 0) = "natural number".
+definition nat_of_bool ≝
+ λb:bool.match b return λ_.nat with [ true ⇒ 1 | false ⇒ 0 ].
+
+theorem lt_trans: ∀x,y,z. x < y → y < z → x < z.
+ unfold lt;
+ intros;
+ autobatch.
+qed.
+
+lemma leq_m_n_to_eq_div_n_m_S: ∀n,m:nat. 0 < m → m ≤ n → ∃z. n/m = S z.
+ intros;
+ unfold div;
+ apply (ex_intro ? ? (div_aux (pred n) (n-m) (pred m)));
+ cut (∃w.m = S w);
+ [ elim Hcut;
+ rewrite > H2;
+ rewrite > H2 in H1;
+ clear Hcut; clear H2; clear H;
+ simplify;
+ unfold in ⊢ (? ? % ?);
+ cut (∃z.n = S z);
+ [ elim Hcut; clear Hcut;
+ rewrite > H in H1;
+ rewrite > H; clear m;
+ change in ⊢ (? ? % ?) with
+ (match leb (S a1) a with
+ [ true ⇒ O
+ | false ⇒ S (div_aux a1 ((S a1) - S a) a)]);
+ cut (S a1 ≰ a);
+ [ apply (leb_elim (S a1) a);
+ [ intro;
+ elim (Hcut H2)
+ | intro;
+ simplify;
+ reflexivity
+ ]
+ | intro;
+ autobatch
+ ]
+ | elim H1; autobatch
+ ]
+ | autobatch
+ ].
+qed.
+
+axiom daemon: False.
projects / helm.git / blobdiff