1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| A.Asperti, C.Sacerdoti Coen, *)
8 (* ||A|| E.Tassi, S.Zacchiroli *)
10 (* \ / Matita is distributed under the terms of the *)
11 (* v GNU Lesser General Public License Version 2.1 *)
13 (**************************************************************************)
15 set "baseuri" "cic:/matita/nat/minimization".
17 include "nat/minus.ma".
18 include "datatypes/bool.ma".
26 | (S j) \Rightarrow max j f ]].
28 theorem max_O_f : \forall f: nat \to bool. max O f = O.
35 theorem max_S_max : \forall f: nat \to bool. \forall n:nat.
36 (f (S n) = true \land max (S n) f = (S n)) \lor
37 (f (S n) = false \land max (S n) f = max n f).
38 intros.simplify.elim (f (S n)).
39 simplify.left.split.reflexivity.reflexivity.
40 simplify.right.split.reflexivity.reflexivity.
43 definition max_spec \def \lambda f:nat \to bool.\lambda n: nat.
44 ex nat (\lambda i:nat. (le i n) \land (f i = true)) \to
45 (f n) = true \land (\forall i. i < n \to (f i = false)).
47 (* perche' si blocca per mezzo minuto qui ??? *)
48 theorem f_max_true : \forall f:nat \to bool. \forall n:nat.
49 ex nat (\lambda i:nat. (le i n) \land (f i = true)) \to f (max n f) = true.
51 elim n.elim H.elim H1.generalize in match H3.
52 apply le_n_O_elim a H2.intro.simplify.rewrite > H4.
55 apply bool_ind (\lambda b:bool.
56 (f (S n1) = b) \to (f ([\lambda b:bool.nat] match b in bool with
57 [ true \Rightarrow (S n1)
58 | false \Rightarrow (max n1 f)])) = true) ? ? ?.
60 simplify.intro.assumption.
61 simplify.intro.apply H.
62 elim H1.elim H3.generalize in match H5.
63 apply le_n_Sm_elim a n1 H4.
65 apply ex_intro nat ? a.
66 split.apply le_S_S_to_le.assumption.assumption.
67 intros.apply False_ind.apply not_eq_true_false ?.
68 rewrite < H2.rewrite < H7.rewrite > H6. reflexivity.
71 theorem lt_max_to_false : \forall f:nat \to bool.
72 \forall n,m:nat. (max n f) < m \to m \leq n \to f m = false.
74 elim n.absurd le m O.assumption.
75 cut O < m.apply lt_O_n_elim m Hcut.exact not_le_Sn_O.
76 rewrite < max_O_f f.assumption.
77 generalize in match H1.
78 (* ?? non posso generalizzare su un goal implicativo ?? *)
81 absurd m \le S n1.assumption.
82 apply lt_to_not_le.rewrite < H6.assumption.
84 apply le_n_Sm_elim m n1 H2.
86 apply H.rewrite < H6.assumption.
87 apply le_S_S_to_le.assumption.
88 intro.rewrite > H7.assumption.
91 let rec min_aux off n f \def
93 [ true \Rightarrow (n-off)
97 | (S p) \Rightarrow min_aux p n f]].
99 definition min : nat \to (nat \to bool) \to nat \def
100 \lambda n.\lambda f. min_aux n n f.
102 theorem min_aux_O_f: \forall f:nat \to bool. \forall i :nat.
104 intros.simplify.rewrite < minus_n_O.
106 simplify.reflexivity.
107 simplify.reflexivity.
110 theorem min_O_f : \forall f:nat \to bool.
112 intro.apply min_aux_O_f f O.
115 theorem min_aux_S : \forall f: nat \to bool. \forall i,n:nat.
116 (f (n -(S i)) = true \land min_aux (S i) n f = (n - (S i))) \lor
117 (f (n -(S i)) = false \land min_aux (S i) n f = min_aux i n f).
118 intros.simplify.elim (f (n - (S i))).
119 simplify.left.split.reflexivity.reflexivity.
120 simplify.right.split.reflexivity.reflexivity.
123 theorem f_min_aux_true: \forall f:nat \to bool. \forall off,m:nat.
124 ex nat (\lambda i:nat. (le (m-off) i) \land (le i m) \land (f i = true)) \to
125 f (min_aux off m f) = true.
127 elim off.elim H.elim H1.elim H2.
129 rewrite > min_aux_O_f f.rewrite < Hcut.assumption.
130 apply antisym_le a m .assumption.rewrite > minus_n_O m.assumption.
132 apply bool_ind (\lambda b:bool.
133 (f (m-(S n)) = b) \to (f ([\lambda b:bool.nat] match b in bool with
134 [ true \Rightarrow m-(S n)
135 | false \Rightarrow (min_aux n m f)])) = true) ? ? ?.
137 simplify.intro.assumption.
138 simplify.intro.apply H.
139 elim H1.elim H3.elim H4.
140 elim (le_to_or_lt_eq (m-(S n)) a H6).
141 apply ex_intro nat ? a.
143 apply lt_minus_S_n_to_le_minus_n.assumption.
144 assumption.assumption.
145 absurd f a = false.rewrite < H8.assumption.
147 apply not_eq_true_false.
150 theorem lt_min_aux_to_false : \forall f:nat \to bool.
151 \forall n,off,m:nat. (n-off) \leq m \to m < (min_aux off n f) \to f m = false.
153 elim off.absurd le n m.rewrite > minus_n_O.assumption.
154 apply lt_to_not_le.rewrite < min_aux_O_f f n.assumption.
155 generalize in match H1.
156 elim min_aux_S f n1 n.
158 absurd n - S n1 \le m.assumption.
159 apply lt_to_not_le.rewrite < H6.assumption.
161 elim le_to_or_lt_eq (n -(S n1)) m.
162 apply H.apply lt_minus_S_n_to_le_minus_n.assumption.
163 rewrite < H6.assumption.assumption.
164 rewrite < H7.assumption.
167 theorem le_min_aux : \forall f:nat \to bool.
168 \forall n,off:nat. (n-off) \leq (min_aux off n f).
170 elim off.rewrite < minus_n_O.
171 rewrite > min_aux_O_f f n.apply le_n.
172 elim min_aux_S f n1 n.
173 elim H1.rewrite > H3.apply le_n.
174 elim H1.rewrite > H3.
175 apply trans_le (n-(S n1)) (n-n1) ?.
176 apply monotonic_le_minus_r.