1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* ********************************************************************** *)
16 (* Progetto FreeScale *)
19 (* Cosimo Oliboni, oliboni@cs.unibo.it *)
21 (* Questo materiale fa parte della tesi: *)
22 (* "Formalizzazione Interattiva dei Microcontroller a 8bit FreeScale" *)
24 (* data ultima modifica 15/11/2007 *)
25 (* ********************************************************************** *)
27 set "baseuri" "cic:/matita/freescale/extra".
29 include "nat/div_and_mod.ma".
30 include "nat/primes.ma".
31 include "list/list.ma".
32 include "datatypes/constructors.ma".
33 include "logic/connectives.ma".
37 (* ridefinizione degli operatori booleani, per evitare l'overloading di quelli normali *)
39 λb:bool.match b with [ true ⇒ false | false ⇒ true ].
42 λb1,b2:bool.match b1 with
43 [ true ⇒ b2 | false ⇒ false ].
46 λb1,b2:bool.match b1 with
47 [ true ⇒ true | false ⇒ b2 ].
50 λb1,b2:bool.match b1 with
55 λb1,b2:bool.match b1 with
57 | false ⇒ not_bool b2 ].
60 notation "hvbox(⊖ a)" non associative with precedence 36
61 for @{ 'not_bool $a }.
62 interpretation "not_bool" 'not_bool x =
63 (cic:/matita/freescale/extra/not_bool.con x).
66 notation "hvbox(a break ⊗ b)" left associative with precedence 35
67 for @{ 'and_bool $a $b }.
68 interpretation "and_bool" 'and_bool x y =
69 (cic:/matita/freescale/extra/and_bool.con x y).
72 notation "hvbox(a break ⊕ b)" left associative with precedence 34
73 for @{ 'or_bool $a $b }.
74 interpretation "or_bool" 'or_bool x y =
75 (cic:/matita/freescale/extra/or_bool.con x y).
78 notation "hvbox(a break ⊙ b)" left associative with precedence 33
79 for @{ 'xor_bool $a $b }.
80 interpretation "xor_bool" 'xor_bool x y =
81 (cic:/matita/freescale/extra/xor_bool.con x y).
83 (* ProdT e' gia' definito, aggiungo Prod3T e Prod4T e Prod5T *)
85 inductive Prod3T (T1:Type) (T2:Type) (T3:Type) : Type ≝
86 tripleT : T1 → T2 → T3 → Prod3T T1 T2 T3.
89 λT1.λT2.λT3.λp:Prod3T T1 T2 T3.match p with [ tripleT x _ _ ⇒ x ].
92 λT1.λT2.λT3.λp:Prod3T T1 T2 T3.match p with [ tripleT _ x _ ⇒ x ].
95 λT1.λT2.λT3.λp:Prod3T T1 T2 T3.match p with [ tripleT _ _ x ⇒ x ].
97 inductive Prod4T (T1:Type) (T2:Type) (T3:Type) (T4:Type) : Type ≝
98 quadrupleT : T1 → T2 → T3 → T4 → Prod4T T1 T2 T3 T4.
101 λT1.λT2.λT3.λT4.λp:Prod4T T1 T2 T3 T4.match p with [ quadrupleT x _ _ _ ⇒ x ].
104 λT1.λT2.λT3.λT4.λp:Prod4T T1 T2 T3 T4.match p with [ quadrupleT _ x _ _ ⇒ x ].
107 λT1.λT2.λT3.λT4.λp:Prod4T T1 T2 T3 T4.match p with [ quadrupleT _ _ x _ ⇒ x ].
110 λT1.λT2.λT3.λT4.λp:Prod4T T1 T2 T3 T4.match p with [ quadrupleT _ _ _ x ⇒ x ].
112 inductive Prod5T (T1:Type) (T2:Type) (T3:Type) (T4:Type) (T5:Type) : Type ≝
113 quintupleT : T1 → T2 → T3 → T4 → T5 → Prod5T T1 T2 T3 T4 T5.
116 λT1.λT2.λT3.λT4.λT5.λp:Prod5T T1 T2 T3 T4 T5.match p with [ quintupleT x _ _ _ _ ⇒ x ].
119 λT1.λT2.λT3.λT4.λT5.λp:Prod5T T1 T2 T3 T4 T5.match p with [ quintupleT _ x _ _ _ ⇒ x ].
122 λT1.λT2.λT3.λT4.λT5.λp:Prod5T T1 T2 T3 T4 T5.match p with [ quintupleT _ _ x _ _ ⇒ x ].
125 λT1.λT2.λT3.λT4.λT5.λp:Prod5T T1 T2 T3 T4 T5.match p with [ quintupleT _ _ _ x _ ⇒ x ].
128 λT1.λT2.λT3.λT4.λT5.λp:Prod5T T1 T2 T3 T4 T5.match p with [ quintupleT _ _ _ _ x ⇒ x ].
132 (* option map = match ... with [ None ⇒ None ? | Some .. ⇒ .. ] *)
134 λT1,T2:Type.λt:option T1.λf:T1 → option T2.
135 match t with [ None ⇒ None ? | Some x ⇒ (f x) ].
137 (* ********************** *)
138 (* TEOREMI/LEMMMI/ASSIOMI *)
139 (* ********************** *)
141 axiom mod_plus: ∀a,b,m. (a + b) \mod m = (a \mod m + b \mod m) \mod m.
142 axiom mod_mod: ∀a,n,m. n∣m → a \mod n = a \mod n \mod m.
143 axiom eq_mod_times_n_m_m_O: ∀n,m. O < m → n * m \mod m = O.
144 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.
145 axiom eq_mod_times_times_mod: ∀a,b,n,m. m = a*n → (a*b) \mod m = a * (b \mod n).
146 axiom divides_to_eq_mod_mod_mod: ∀a,n,m. n∣m → a \mod m \mod n = a \mod n.
147 axiom le_to_le_plus_to_le : ∀a,b,c,d.b\leq d\rarr a+b\leq c+d\rarr a\leq c.
148 axiom or_lt_le : ∀n,m. n < m ∨ m ≤ n.
150 lemma le_to_lt: ∀n,m. n ≤ m → n < S m.
155 alias num (instance 0) = "natural number".
156 definition nat_of_bool ≝
157 λb:bool.match b return λ_.nat with [ true ⇒ 1 | false ⇒ 0 ].
159 theorem lt_trans: ∀x,y,z. x < y → y < z → x < z.
165 lemma leq_m_n_to_eq_div_n_m_S: ∀n,m:nat. 0 < m → m ≤ n → ∃z. n/m = S z.
168 apply (ex_intro ? ? (div_aux (pred n) (n-m) (pred m)));
173 clear Hcut; clear H2; clear H;
175 unfold in ⊢ (? ? % ?);
177 [ elim Hcut; clear Hcut;
179 rewrite > H; clear m;
180 change in ⊢ (? ? % ?) with
181 (match leb (S a1) a with
183 | false ⇒ S (div_aux a1 ((S a1) - S a) a)]);
185 [ apply (leb_elim (S a1) a);