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 (* \ / This file is distributed under the terms of the *)
11 (* v GNU Lesser General Public License Version 2.1 *)
13 (**************************************************************************)
16 inductive True: Prop \def
19 inductive False: Prop \def .
21 definition Not: Prop \to Prop \def
22 \lambda A:Prop. (A \to False).
24 theorem absurd : \forall A,C:Prop. A \to Not A \to C.
25 intros.cut False.elim Hcut.apply H1.assumption.
28 inductive And (A,B:Prop) : Prop \def
29 conj : A \to B \to (And A B).
31 theorem proj1: \forall A,B:Prop. (And A B) \to A.
32 intros. elim H. assumption.
35 theorem proj2: \forall A,B:Prop. (And A B) \to A.
36 intros. elim H. assumption.
39 inductive Or (A,B:Prop) : Prop \def
40 or_introl : A \to (Or A B)
41 | or_intror : B \to (Or A B).
43 inductive ex (A:Type) (P:A \to Prop) : Prop \def
44 ex_intro: \forall x:A. P x \to ex A P.
46 inductive ex2 (A:Type) (P,Q:A \to Prop) : Prop \def
47 ex_intro2: \forall x:A. P x \to Q x \to ex2 A P Q.
49 inductive eq (A:Type) (x:A) : A \to Prop \def
50 refl_equal : eq A x x.
52 theorem sym_eq : \forall A:Type.\forall x,y:A. eq A x y \to eq A y x.
53 intros. elim H. apply refl_equal.
56 theorem trans_eq : \forall A:Type.
57 \forall x,y,z:A. eq A x y \to eq A y z \to eq A x z.
58 intros.elim H1.assumption.
61 theorem f_equal: \forall A,B:Type.\forall f:A\to B.
62 \forall x,y:A. eq A x y \to eq B (f x) (f y).
63 intros.elim H.apply refl_equal.
66 theorem f_equal2: \forall A,B,C:Type.\forall f:A\to B \to C.
67 \forall x1,x2:A. \forall y1,y2:B.
68 eq A x1 x2\to eq B y1 y2\to eq C (f x1 y1) (f x2 y2).
69 intros.elim H1.elim H.apply refl_equal.
73 inductive nat : Set \def
77 definition pred: nat \to nat \def
78 \lambda n:nat. match n with
80 | (S u) \Rightarrow u ].
82 theorem pred_Sn : \forall n:nat.
83 (eq nat n (pred (S n))).
84 intros.apply refl_equal.
87 theorem injective_S : \forall n,m:nat.
88 (eq nat (S n) (S m)) \to (eq nat n m).
89 intros.(elim (sym_eq ? ? ? (pred_Sn n))).
90 (elim (sym_eq ? ? ? (pred_Sn m))).
91 apply f_equal. assumption.
94 theorem not_eq_S : \forall n,m:nat.
95 Not (eq nat n m) \to Not (eq nat (S n) (S m)).
96 intros. simplify.intros.
97 apply H.apply injective_S.assumption.
100 definition not_zero : nat \to Prop \def
103 [ O \Rightarrow False
104 | (S p) \Rightarrow True ].
106 theorem O_S : \forall n:nat. Not (eq nat O (S n)).
107 intros.simplify.intros.
108 cut (not_zero O).exact Hcut.elim (sym_eq ? ? ? H).
112 theorem n_Sn : \forall n:nat. Not (eq nat n (S n)).
113 intros.elim n.apply O_S.apply not_eq_S.assumption.
116 definition plus : nat \to nat \to nat \def
117 let rec plus (n,m:nat) \def
120 | (S p) \Rightarrow S (plus p m) ]
124 theorem plus_n_O: \forall n:nat. eq nat n (plus n O).
125 intros.elim n.simplify.apply refl_equal.simplify.apply f_equal.assumption.
128 theorem plus_n_Sm : \forall n,m:nat. eq nat (S (plus n m)) (plus n (S m)).
129 intros.elim n.simplify.apply refl_equal.simplify.apply f_equal.assumption.
132 theorem sym_plus: \forall n,m:nat. eq nat (plus n m) (plus m n).
133 intros.elim n.simplify.apply plus_n_O.
134 simplify.elim (sym_eq ? ? ? H).apply plus_n_Sm.
138 \forall n,m,p:nat. eq nat (plus (plus n m) p) (plus n (plus m p)).
139 intros.elim n.simplify.apply refl_equal.simplify.apply f_equal.assumption.
142 definition times : nat \to nat \to nat \def
143 let rec times (n,m:nat) \def
146 | (S p) \Rightarrow (plus m (times p m)) ]
150 theorem times_n_O: \forall n:nat. eq nat O (times n O).
151 intros.elim n.simplify.apply refl_equal.simplify.assumption.
155 \forall n,m:nat. eq nat (plus n (times n m)) (times n (S m)).
156 intros.elim n.simplify.apply refl_equal.
157 simplify.apply f_equal.elim H.
158 apply trans_eq ? ? (plus (plus e m) (times e m)).apply sym_eq.
159 apply assoc_plus.apply trans_eq ? ? (plus (plus m e) (times e m)).
161 apply sym_plus.apply refl_equal.apply assoc_plus.
165 \forall n,m:nat. eq nat (times n m) (times m n).
166 intros.elim n.simplify.apply times_n_O.
167 simplify.elim (sym_eq ? ? ? H).apply times_n_Sm.
170 definition minus : nat \to nat \to nat \def
171 let rec minus (n,m:nat) \def
172 [\lambda n:nat.nat] match n with
175 [\lambda n:nat.nat] match m with
177 | (S q) \Rightarrow minus p q ]]
182 \forall n:nat.\forall P:nat \to Prop.
183 P O \to (\forall m:nat. P (S m)) \to P n.
184 intros.elim n.assumption.apply H1.
187 theorem nat_double_ind :
188 \forall R:nat \to nat \to Prop.
189 (\forall n:nat. R O n) \to
190 (\forall n:nat. R (S n) O) \to
191 (\forall n,m:nat. R n m \to R (S n) (S m)) \to \forall n,m:nat. R n m.
192 intros.cut \forall m:nat.R n m.apply Hcut.elim n.apply H.
193 apply nat_case m1.apply H1.intros.apply H2. apply H3.
196 inductive bool : Set \def
200 definition notn : bool \to bool \def
203 [ true \Rightarrow false
204 | false \Rightarrow true ].
206 definition andb : bool \to bool \to bool\def
210 match b2 with [true \Rightarrow true | false \Rightarrow false]
211 | false \Rightarrow false ].
213 definition orb : bool \to bool \to bool\def
217 match b2 with [true \Rightarrow true | false \Rightarrow false]
218 | false \Rightarrow false ].
220 definition if_then_else : bool \to Prop \to Prop \to Prop \def
221 \lambda b:bool.\lambda P,Q:Prop.
224 | false \Rightarrow Q].
226 inductive le (n:nat) : nat \to Prop \def
228 | le_S : \forall m:nat. le n m \to le n (S m).
230 theorem trans_le: \forall n,m,p:nat. le n m \to le m p \to le n p.
233 apply le_S.assumption.
236 theorem le_n_S: \forall n,m:nat. le n m \to le (S n) (S m).
238 apply le_n.apply le_S.assumption.
241 theorem le_O_n : \forall n:nat. le O n.
242 intros.elim n.apply le_n.apply le_S. assumption.
245 theorem le_n_Sn : \forall n:nat. le n (S n).
246 intros. apply le_S.apply le_n.
249 theorem le_pred_n : \forall n:nat. le (pred n) n.
250 intros.elim n.simplify.apply le_n.simplify.
254 theorem not_zero_le : \forall n,m:nat. (le (S n) m ) \to not_zero m.
255 intros.elim H.exact I.exact I.
258 theorem le_Sn_O: \forall n:nat. Not (le (S n) O).
259 intros.simplify.intros.apply not_zero_le ? O H.
262 theorem le_n_O_eq : \forall n:nat. (le n O) \to (eq nat O n).
263 intros.cut (le n O) \to (eq nat O n).apply Hcut. assumption.
264 elim n.apply refl_equal.apply False_ind.apply (le_Sn_O ? H2).
267 theorem le_S_n : \forall n,m:nat. le (S n) (S m) \to le n m.
268 intros.cut le (pred (S n)) (pred (S m)).exact Hcut.
269 elim H.apply le_n.apply trans_le ? (pred x).assumption.
273 theorem le_Sn_n : \forall n:nat. Not (le (S n) n).
274 intros.elim n.apply le_Sn_O.simplify.intro.
275 cut le (S e) e.apply H.assumption.apply le_S_n.assumption.
278 theorem le_antisym : \forall n,m:nat. (le n m) \to (le m n) \to (eq nat n m).
279 intros.cut (le n m) \to (le m n) \to (eq nat n m).exact Hcut H H1.
280 apply nat_double_ind (\lambda n,m.((le n m) \to (le m n) \to eq nat n m)).
282 apply le_n_O_eq.assumption.
283 intros.whd.intro.apply sym_eq.apply le_n_O_eq.assumption.
284 intros.whd.intro.apply f_equal.apply H2.
285 apply le_S_n.assumption.
286 apply le_S_n.assumption.
290 definition leb : nat \to nat \to bool \def
291 let rec leb (n,m:nat) \def
292 [\lambda n:nat.bool] match n with
295 [\lambda n:nat.bool] match m with
296 [ O \Rightarrow false
297 | (S q) \Rightarrow leb p q]]
300 theorem le_dec: \forall n,m:nat. if_then_else (leb n m) (le n m) (Not (le n m)).
302 apply (nat_double_ind
303 (\lambda n,m:nat.if_then_else (leb n m) (le n m) (Not (le n m))) ? ? ? n m).
304 simplify.intros.apply le_O_n.
305 simplify.exact le_Sn_O.
306 intros 2.simplify.elim (leb n1 m1).
307 simplify.apply le_n_S.apply H.
308 simplify.intro.apply H.apply le_S_n.assumption.