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 (* This file was automatically generated: do not edit *********************)
17 include "legacy_1/coq/fwd.ma".
20 \forall (A: Type[0]).(\forall (B: Type[0]).(\forall (f: ((A \to
21 B))).(\forall (x: A).(\forall (y: A).((eq A x y) \to (eq B (f x) (f y)))))))
23 \lambda (A: Type[0]).(\lambda (B: Type[0]).(\lambda (f: ((A \to
24 B))).(\lambda (x: A).(\lambda (y: A).(\lambda (H: (eq A x y)).(let TMP_3 \def
25 (\lambda (a: A).(let TMP_1 \def (f x) in (let TMP_2 \def (f a) in (eq B TMP_1
26 TMP_2)))) in (let TMP_4 \def (f x) in (let TMP_5 \def (refl_equal B TMP_4) in
27 (eq_ind A x TMP_3 TMP_5 y H))))))))).
30 \forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall (B: Type[0]).(\forall
31 (f: ((A1 \to (A2 \to B)))).(\forall (x1: A1).(\forall (y1: A1).(\forall (x2:
32 A2).(\forall (y2: A2).((eq A1 x1 y1) \to ((eq A2 x2 y2) \to (eq B (f x1 x2)
35 \lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda (B: Type[0]).(\lambda
36 (f: ((A1 \to (A2 \to B)))).(\lambda (x1: A1).(\lambda (y1: A1).(\lambda (x2:
37 A2).(\lambda (y2: A2).(\lambda (H: (eq A1 x1 y1)).(let TMP_3 \def (\lambda
38 (a: A1).((eq A2 x2 y2) \to (let TMP_1 \def (f x1 x2) in (let TMP_2 \def (f a
39 y2) in (eq B TMP_1 TMP_2))))) in (let TMP_9 \def (\lambda (H0: (eq A2 x2
40 y2)).(let TMP_6 \def (\lambda (a: A2).(let TMP_4 \def (f x1 x2) in (let TMP_5
41 \def (f x1 a) in (eq B TMP_4 TMP_5)))) in (let TMP_7 \def (f x1 x2) in (let
42 TMP_8 \def (refl_equal B TMP_7) in (eq_ind A2 x2 TMP_6 TMP_8 y2 H0))))) in
43 (eq_ind A1 x1 TMP_3 TMP_9 y1 H))))))))))).
46 \forall (A1: Type[0]).(\forall (A2: Type[0]).(\forall (A3: Type[0]).(\forall
47 (B: Type[0]).(\forall (f: ((A1 \to (A2 \to (A3 \to B))))).(\forall (x1:
48 A1).(\forall (y1: A1).(\forall (x2: A2).(\forall (y2: A2).(\forall (x3:
49 A3).(\forall (y3: A3).((eq A1 x1 y1) \to ((eq A2 x2 y2) \to ((eq A3 x3 y3)
50 \to (eq B (f x1 x2 x3) (f y1 y2 y3)))))))))))))))
52 \lambda (A1: Type[0]).(\lambda (A2: Type[0]).(\lambda (A3: Type[0]).(\lambda
53 (B: Type[0]).(\lambda (f: ((A1 \to (A2 \to (A3 \to B))))).(\lambda (x1:
54 A1).(\lambda (y1: A1).(\lambda (x2: A2).(\lambda (y2: A2).(\lambda (x3:
55 A3).(\lambda (y3: A3).(\lambda (H: (eq A1 x1 y1)).(let TMP_3 \def (\lambda
56 (a: A1).((eq A2 x2 y2) \to ((eq A3 x3 y3) \to (let TMP_1 \def (f x1 x2 x3) in
57 (let TMP_2 \def (f a y2 y3) in (eq B TMP_1 TMP_2)))))) in (let TMP_13 \def
58 (\lambda (H0: (eq A2 x2 y2)).(let TMP_6 \def (\lambda (a: A2).((eq A3 x3 y3)
59 \to (let TMP_4 \def (f x1 x2 x3) in (let TMP_5 \def (f x1 a y3) in (eq B
60 TMP_4 TMP_5))))) in (let TMP_12 \def (\lambda (H1: (eq A3 x3 y3)).(let TMP_9
61 \def (\lambda (a: A3).(let TMP_7 \def (f x1 x2 x3) in (let TMP_8 \def (f x1
62 x2 a) in (eq B TMP_7 TMP_8)))) in (let TMP_10 \def (f x1 x2 x3) in (let
63 TMP_11 \def (refl_equal B TMP_10) in (eq_ind A3 x3 TMP_9 TMP_11 y3 H1))))) in
64 (eq_ind A2 x2 TMP_6 TMP_12 y2 H0)))) in (eq_ind A1 x1 TMP_3 TMP_13 y1
68 \forall (A: Type[0]).(\forall (x: A).(\forall (y: A).((eq A x y) \to (eq A y
71 \lambda (A: Type[0]).(\lambda (x: A).(\lambda (y: A).(\lambda (H: (eq A x
72 y)).(let TMP_1 \def (\lambda (a: A).(eq A a x)) in (let TMP_2 \def
73 (refl_equal A x) in (eq_ind A x TMP_1 TMP_2 y H)))))).
76 \forall (A: Type[0]).(\forall (x: A).(\forall (P: ((A \to Prop))).((P x) \to
77 (\forall (y: A).((eq A y x) \to (P y))))))
79 \lambda (A: Type[0]).(\lambda (x: A).(\lambda (P: ((A \to Prop))).(\lambda
80 (H: (P x)).(\lambda (y: A).(\lambda (H0: (eq A y x)).(match (sym_eq A y x H0)
81 with [refl_equal \Rightarrow H])))))).
84 \forall (A: Type[0]).(\forall (x: A).(\forall (y: A).(\forall (z: A).((eq A
85 x y) \to ((eq A y z) \to (eq A x z))))))
87 \lambda (A: Type[0]).(\lambda (x: A).(\lambda (y: A).(\lambda (z:
88 A).(\lambda (H: (eq A x y)).(\lambda (H0: (eq A y z)).(let TMP_1 \def
89 (\lambda (a: A).(eq A x a)) in (eq_ind A y TMP_1 H z H0))))))).
92 \forall (A: Type[0]).(\forall (x: A).(\forall (y: A).((not (eq A x y)) \to
95 \lambda (A: Type[0]).(\lambda (x: A).(\lambda (y: A).(\lambda (h1: (not (eq
96 A x y))).(\lambda (h2: (eq A y x)).(let TMP_1 \def (\lambda (a: A).(eq A a
97 y)) in (let TMP_2 \def (refl_equal A y) in (let TMP_3 \def (eq_ind A y TMP_1
98 TMP_2 x h2) in (h1 TMP_3)))))))).
100 theorem nat_double_ind:
101 \forall (R: ((nat \to (nat \to Prop)))).(((\forall (n: nat).(R O n))) \to
102 (((\forall (n: nat).(R (S n) O))) \to (((\forall (n: nat).(\forall (m:
103 nat).((R n m) \to (R (S n) (S m)))))) \to (\forall (n: nat).(\forall (m:
106 \lambda (R: ((nat \to (nat \to Prop)))).(\lambda (H: ((\forall (n: nat).(R O
107 n)))).(\lambda (H0: ((\forall (n: nat).(R (S n) O)))).(\lambda (H1: ((\forall
108 (n: nat).(\forall (m: nat).((R n m) \to (R (S n) (S m))))))).(\lambda (n:
109 nat).(let TMP_1 \def (\lambda (n0: nat).(\forall (m: nat).(R n0 m))) in (let
110 TMP_7 \def (\lambda (n0: nat).(\lambda (H2: ((\forall (m: nat).(R n0
111 m)))).(\lambda (m: nat).(let TMP_3 \def (\lambda (n1: nat).(let TMP_2 \def (S
112 n0) in (R TMP_2 n1))) in (let TMP_4 \def (H0 n0) in (let TMP_6 \def (\lambda
113 (n1: nat).(\lambda (_: (R (S n0) n1)).(let TMP_5 \def (H2 n1) in (H1 n0 n1
114 TMP_5)))) in (nat_ind TMP_3 TMP_4 TMP_6 m))))))) in (nat_ind TMP_1 H TMP_7
118 \forall (n: nat).(\forall (m: nat).((eq nat (S n) (S m)) \to (eq nat n m)))
120 \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (eq nat (S n) (S m))).(let
121 TMP_1 \def (S n) in (let TMP_2 \def (S m) in (f_equal nat nat pred TMP_1
125 \forall (n: nat).(not (eq nat O (S n)))
127 \lambda (n: nat).(\lambda (H: (eq nat O (S n))).(let TMP_1 \def (S n) in
128 (let TMP_2 \def (\lambda (n0: nat).(IsSucc n0)) in (let TMP_3 \def (S n) in
129 (let TMP_4 \def (sym_eq nat O TMP_3 H) in (eq_ind nat TMP_1 TMP_2 I O
133 \forall (n: nat).(\forall (m: nat).((not (eq nat n m)) \to (not (eq nat (S
136 \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (not (eq nat n m))).(\lambda
137 (H0: (eq nat (S n) (S m))).(let TMP_1 \def (eq_add_S n m H0) in (H TMP_1))))).
140 \forall (m: nat).(eq nat m (pred (S m)))
142 \lambda (m: nat).(let TMP_1 \def (S m) in (let TMP_2 \def (pred TMP_1) in
143 (refl_equal nat TMP_2))).
146 \forall (n: nat).(\forall (m: nat).((lt m n) \to (eq nat n (S (pred n)))))
148 \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt m n)).(let TMP_1 \def (S
149 m) in (let TMP_4 \def (\lambda (n0: nat).(let TMP_2 \def (pred n0) in (let
150 TMP_3 \def (S TMP_2) in (eq nat n0 TMP_3)))) in (let TMP_5 \def (S m) in (let
151 TMP_6 \def (pred TMP_5) in (let TMP_7 \def (S TMP_6) in (let TMP_8 \def
152 (refl_equal nat TMP_7) in (let TMP_12 \def (\lambda (m0: nat).(\lambda (_:
153 (le (S m) m0)).(\lambda (_: (eq nat m0 (S (pred m0)))).(let TMP_9 \def (S m0)
154 in (let TMP_10 \def (pred TMP_9) in (let TMP_11 \def (S TMP_10) in
155 (refl_equal nat TMP_11))))))) in (le_ind TMP_1 TMP_4 TMP_8 TMP_12 n
159 \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to ((le m p)
162 \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n
163 m)).(\lambda (H0: (le m p)).(let TMP_1 \def (\lambda (n0: nat).(le n n0)) in
164 (let TMP_2 \def (\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle:
165 (le n m0)).(le_S n m0 IHle)))) in (le_ind m TMP_1 H TMP_2 p H0))))))).
168 \forall (n: nat).(\forall (m: nat).((le (S n) m) \to (le n m)))
170 \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le (S n) m)).(let TMP_1
171 \def (S n) in (let TMP_2 \def (le_n n) in (let TMP_3 \def (le_S n n TMP_2) in
172 (le_trans n TMP_1 m TMP_3 H)))))).
175 \forall (n: nat).(\forall (m: nat).((le n m) \to (le (S n) (S m))))
177 \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(let TMP_3 \def
178 (\lambda (n0: nat).(let TMP_1 \def (S n) in (let TMP_2 \def (S n0) in (le
179 TMP_1 TMP_2)))) in (let TMP_4 \def (S n) in (let TMP_5 \def (le_n TMP_4) in
180 (let TMP_8 \def (\lambda (m0: nat).(\lambda (_: (le n m0)).(\lambda (IHle:
181 (le (S n) (S m0))).(let TMP_6 \def (S n) in (let TMP_7 \def (S m0) in (le_S
182 TMP_6 TMP_7 IHle)))))) in (le_ind n TMP_3 TMP_5 TMP_8 m H))))))).
185 \forall (n: nat).(le O n)
187 \lambda (n: nat).(let TMP_1 \def (\lambda (n0: nat).(le O n0)) in (let TMP_2
188 \def (le_n O) in (let TMP_3 \def (\lambda (n0: nat).(\lambda (IHn: (le O
189 n0)).(le_S O n0 IHn))) in (nat_ind TMP_1 TMP_2 TMP_3 n)))).
192 \forall (n: nat).(\forall (m: nat).((le (S n) (S m)) \to (le n m)))
194 \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le (S n) (S m))).(let TMP_1
195 \def (S n) in (let TMP_5 \def (\lambda (n0: nat).(let TMP_2 \def (S n) in
196 (let TMP_3 \def (pred TMP_2) in (let TMP_4 \def (pred n0) in (le TMP_3
197 TMP_4))))) in (let TMP_6 \def (le_n n) in (let TMP_7 \def (\lambda (m0:
198 nat).(\lambda (H0: (le (S n) m0)).(\lambda (_: (le n (pred m0))).(le_trans_S
199 n m0 H0)))) in (let TMP_8 \def (S m) in (le_ind TMP_1 TMP_5 TMP_6 TMP_7 TMP_8
203 \forall (n: nat).(not (le (S n) O))
205 \lambda (n: nat).(\lambda (H: (le (S n) O)).(let TMP_1 \def (S n) in (let
206 TMP_2 \def (\lambda (n0: nat).(IsSucc n0)) in (let TMP_3 \def (\lambda (m:
207 nat).(\lambda (_: (le (S n) m)).(\lambda (_: (IsSucc m)).I))) in (le_ind
208 TMP_1 TMP_2 I TMP_3 O H))))).
211 \forall (n: nat).(not (le (S n) n))
213 \lambda (n: nat).(let TMP_3 \def (\lambda (n0: nat).(let TMP_1 \def (S n0)
214 in (let TMP_2 \def (le TMP_1 n0) in (not TMP_2)))) in (let TMP_4 \def
215 (le_Sn_O O) in (let TMP_7 \def (\lambda (n0: nat).(\lambda (IHn: (not (le (S
216 n0) n0))).(\lambda (H: (le (S (S n0)) (S n0))).(let TMP_5 \def (S n0) in (let
217 TMP_6 \def (le_S_n TMP_5 n0 H) in (IHn TMP_6)))))) in (nat_ind TMP_3 TMP_4
221 \forall (n: nat).(\forall (m: nat).((le n m) \to ((le m n) \to (eq nat n
224 \lambda (n: nat).(\lambda (m: nat).(\lambda (h: (le n m)).(let TMP_1 \def
225 (\lambda (n0: nat).((le n0 n) \to (eq nat n n0))) in (let TMP_2 \def (\lambda
226 (_: (le n n)).(refl_equal nat n)) in (let TMP_8 \def (\lambda (m0:
227 nat).(\lambda (H: (le n m0)).(\lambda (_: (((le m0 n) \to (eq nat n
228 m0)))).(\lambda (H1: (le (S m0) n)).(let TMP_3 \def (S m0) in (let TMP_4 \def
229 (eq nat n TMP_3) in (let TMP_5 \def (S m0) in (let H2 \def (le_trans TMP_5 n
230 m0 H1 H) in (let H3 \def (le_Sn_n m0) in (let TMP_6 \def (\lambda (H4: (le (S
231 m0) m0)).(H3 H4)) in (let TMP_7 \def (TMP_6 H2) in (False_ind TMP_4
232 TMP_7)))))))))))) in (le_ind n TMP_1 TMP_2 TMP_8 m h)))))).
235 \forall (n: nat).((le n O) \to (eq nat O n))
237 \lambda (n: nat).(\lambda (H: (le n O)).(let TMP_1 \def (le_O_n n) in
238 (le_antisym O n TMP_1 H))).
241 \forall (P: ((nat \to (nat \to Prop)))).(((\forall (p: nat).(P O p))) \to
242 (((\forall (p: nat).(\forall (q: nat).((le p q) \to ((P p q) \to (P (S p) (S
243 q))))))) \to (\forall (n: nat).(\forall (m: nat).((le n m) \to (P n m))))))
245 \lambda (P: ((nat \to (nat \to Prop)))).(\lambda (H: ((\forall (p: nat).(P O
246 p)))).(\lambda (H0: ((\forall (p: nat).(\forall (q: nat).((le p q) \to ((P p
247 q) \to (P (S p) (S q)))))))).(\lambda (n: nat).(let TMP_1 \def (\lambda (n0:
248 nat).(\forall (m: nat).((le n0 m) \to (P n0 m)))) in (let TMP_2 \def (\lambda
249 (m: nat).(\lambda (_: (le O m)).(H m))) in (let TMP_14 \def (\lambda (n0:
250 nat).(\lambda (IHn: ((\forall (m: nat).((le n0 m) \to (P n0 m))))).(\lambda
251 (m: nat).(\lambda (Le: (le (S n0) m)).(let TMP_3 \def (S n0) in (let TMP_5
252 \def (\lambda (n1: nat).(let TMP_4 \def (S n0) in (P TMP_4 n1))) in (let
253 TMP_6 \def (le_n n0) in (let TMP_7 \def (le_n n0) in (let TMP_8 \def (IHn n0
254 TMP_7) in (let TMP_9 \def (H0 n0 n0 TMP_6 TMP_8) in (let TMP_13 \def (\lambda
255 (m0: nat).(\lambda (H1: (le (S n0) m0)).(\lambda (_: (P (S n0) m0)).(let
256 TMP_10 \def (le_trans_S n0 m0 H1) in (let TMP_11 \def (le_trans_S n0 m0 H1)
257 in (let TMP_12 \def (IHn m0 TMP_11) in (H0 n0 m0 TMP_10 TMP_12))))))) in
258 (le_ind TMP_3 TMP_5 TMP_9 TMP_13 m Le)))))))))))) in (nat_ind TMP_1 TMP_2
262 \forall (n: nat).(not (lt n n))
267 \forall (n: nat).(\forall (m: nat).((lt n m) \to (lt (S n) (S m))))
269 \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n m)).(let TMP_1 \def (S
270 n) in (le_n_S TMP_1 m H)))).
273 \forall (n: nat).(lt n (S n))
275 \lambda (n: nat).(let TMP_1 \def (S n) in (le_n TMP_1)).
278 \forall (n: nat).(\forall (m: nat).((lt (S n) (S m)) \to (lt n m)))
280 \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (S n) (S m))).(let TMP_1
281 \def (S n) in (le_S_n TMP_1 m H)))).
284 \forall (n: nat).(not (lt n O))
289 \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to ((lt m p)
292 \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n
293 m)).(\lambda (H0: (lt m p)).(let TMP_1 \def (S m) in (let TMP_2 \def (\lambda
294 (n0: nat).(lt n n0)) in (let TMP_3 \def (S n) in (let TMP_4 \def (le_S TMP_3
295 m H) in (let TMP_6 \def (\lambda (m0: nat).(\lambda (_: (le (S m)
296 m0)).(\lambda (IHle: (lt n m0)).(let TMP_5 \def (S n) in (le_S TMP_5 m0
297 IHle))))) in (le_ind TMP_1 TMP_2 TMP_4 TMP_6 p H0)))))))))).
300 \forall (n: nat).(lt O (S n))
302 \lambda (n: nat).(let TMP_1 \def (le_O_n n) in (le_n_S O n TMP_1)).
305 \forall (n: nat).(\forall (p: nat).((lt n p) \to (le (S n) p)))
307 \lambda (n: nat).(\lambda (p: nat).(\lambda (H: (lt n p)).H)).
310 \forall (n: nat).(\forall (m: nat).((le n m) \to (not (lt m n))))
312 \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(let TMP_2 \def
313 (\lambda (n0: nat).(let TMP_1 \def (lt n0 n) in (not TMP_1))) in (let TMP_3
314 \def (lt_n_n n) in (let TMP_6 \def (\lambda (m0: nat).(\lambda (_: (le n
315 m0)).(\lambda (IHle: (not (lt m0 n))).(\lambda (H1: (lt (S m0) n)).(let TMP_4
316 \def (S m0) in (let TMP_5 \def (le_trans_S TMP_4 n H1) in (IHle TMP_5)))))))
317 in (le_ind n TMP_2 TMP_3 TMP_6 m H)))))).
320 \forall (n: nat).(\forall (m: nat).((le n m) \to (lt n (S m))))
322 \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_n_S n m H))).
325 \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to ((lt m p)
328 \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n
329 m)).(\lambda (H0: (lt m p)).(let TMP_1 \def (S m) in (let TMP_2 \def (\lambda
330 (n0: nat).(lt n n0)) in (let TMP_3 \def (le_n_S n m H) in (let TMP_5 \def
331 (\lambda (m0: nat).(\lambda (_: (le (S m) m0)).(\lambda (IHle: (lt n
332 m0)).(let TMP_4 \def (S n) in (le_S TMP_4 m0 IHle))))) in (le_ind TMP_1 TMP_2
333 TMP_3 TMP_5 p H0))))))))).
336 \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to ((le m p)
339 \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n
340 m)).(\lambda (H0: (le m p)).(let TMP_1 \def (\lambda (n0: nat).(lt n n0)) in
341 (let TMP_3 \def (\lambda (m0: nat).(\lambda (_: (le m m0)).(\lambda (IHle:
342 (lt n m0)).(let TMP_2 \def (S n) in (le_S TMP_2 m0 IHle))))) in (le_ind m
343 TMP_1 H TMP_3 p H0))))))).
346 \forall (n: nat).(\forall (m: nat).((lt n m) \to (le n m)))
348 \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n m)).(le_trans_S n m
352 \forall (n: nat).(\forall (m: nat).((lt n (S m)) \to (le n m)))
354 \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt n (S m))).(le_S_n n m
358 \forall (n: nat).(\forall (m: nat).((le n m) \to (or (lt n m) (eq nat n m))))
360 \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(let TMP_3 \def
361 (\lambda (n0: nat).(let TMP_1 \def (lt n n0) in (let TMP_2 \def (eq nat n n0)
362 in (or TMP_1 TMP_2)))) in (let TMP_4 \def (lt n n) in (let TMP_5 \def (eq nat
363 n n) in (let TMP_6 \def (refl_equal nat n) in (let TMP_7 \def (or_intror
364 TMP_4 TMP_5 TMP_6) in (let TMP_13 \def (\lambda (m0: nat).(\lambda (H0: (le n
365 m0)).(\lambda (_: (or (lt n m0) (eq nat n m0))).(let TMP_8 \def (S m0) in
366 (let TMP_9 \def (lt n TMP_8) in (let TMP_10 \def (S m0) in (let TMP_11 \def
367 (eq nat n TMP_10) in (let TMP_12 \def (le_n_S n m0 H0) in (or_introl TMP_9
368 TMP_11 TMP_12))))))))) in (le_ind n TMP_3 TMP_7 TMP_13 m H))))))))).
371 \forall (n: nat).(\forall (m: nat).(or (le n m) (lt m n)))
373 \lambda (n: nat).(\lambda (m: nat).(let TMP_3 \def (\lambda (n0:
374 nat).(\lambda (n1: nat).(let TMP_1 \def (le n0 n1) in (let TMP_2 \def (lt n1
375 n0) in (or TMP_1 TMP_2))))) in (let TMP_7 \def (\lambda (n0: nat).(let TMP_4
376 \def (le O n0) in (let TMP_5 \def (lt n0 O) in (let TMP_6 \def (le_O_n n0) in
377 (or_introl TMP_4 TMP_5 TMP_6))))) in (let TMP_15 \def (\lambda (n0: nat).(let
378 TMP_8 \def (S n0) in (let TMP_9 \def (le TMP_8 O) in (let TMP_10 \def (S n0)
379 in (let TMP_11 \def (lt O TMP_10) in (let TMP_12 \def (S n0) in (let TMP_13
380 \def (lt_O_Sn n0) in (let TMP_14 \def (lt_le_S O TMP_12 TMP_13) in (or_intror
381 TMP_9 TMP_11 TMP_14))))))))) in (let TMP_42 \def (\lambda (n0: nat).(\lambda
382 (m0: nat).(\lambda (H: (or (le n0 m0) (lt m0 n0))).(let TMP_16 \def (le n0
383 m0) in (let TMP_17 \def (lt m0 n0) in (let TMP_18 \def (S n0) in (let TMP_19
384 \def (S m0) in (let TMP_20 \def (le TMP_18 TMP_19) in (let TMP_21 \def (S m0)
385 in (let TMP_22 \def (S n0) in (let TMP_23 \def (lt TMP_21 TMP_22) in (let
386 TMP_24 \def (or TMP_20 TMP_23) in (let TMP_32 \def (\lambda (H0: (le n0
387 m0)).(let TMP_25 \def (S n0) in (let TMP_26 \def (S m0) in (let TMP_27 \def
388 (le TMP_25 TMP_26) in (let TMP_28 \def (S m0) in (let TMP_29 \def (S n0) in
389 (let TMP_30 \def (lt TMP_28 TMP_29) in (let TMP_31 \def (le_n_S n0 m0 H0) in
390 (or_introl TMP_27 TMP_30 TMP_31))))))))) in (let TMP_41 \def (\lambda (H0:
391 (lt m0 n0)).(let TMP_33 \def (S n0) in (let TMP_34 \def (S m0) in (let TMP_35
392 \def (le TMP_33 TMP_34) in (let TMP_36 \def (S m0) in (let TMP_37 \def (S n0)
393 in (let TMP_38 \def (lt TMP_36 TMP_37) in (let TMP_39 \def (S m0) in (let
394 TMP_40 \def (le_n_S TMP_39 n0 H0) in (or_intror TMP_35 TMP_38
395 TMP_40)))))))))) in (or_ind TMP_16 TMP_17 TMP_24 TMP_32 TMP_41
396 H))))))))))))))) in (nat_double_ind TMP_3 TMP_7 TMP_15 TMP_42 n m)))))).
399 \forall (n: nat).(eq nat n (plus n O))
401 \lambda (n: nat).(let TMP_2 \def (\lambda (n0: nat).(let TMP_1 \def (plus n0
402 O) in (eq nat n0 TMP_1))) in (let TMP_3 \def (refl_equal nat O) in (let TMP_5
403 \def (\lambda (n0: nat).(\lambda (H: (eq nat n0 (plus n0 O))).(let TMP_4 \def
404 (plus n0 O) in (f_equal nat nat S n0 TMP_4 H)))) in (nat_ind TMP_2 TMP_3
408 \forall (n: nat).(\forall (m: nat).(eq nat (S (plus n m)) (plus n (S m))))
410 \lambda (m: nat).(\lambda (n: nat).(let TMP_5 \def (\lambda (n0: nat).(let
411 TMP_1 \def (plus n0 n) in (let TMP_2 \def (S TMP_1) in (let TMP_3 \def (S n)
412 in (let TMP_4 \def (plus n0 TMP_3) in (eq nat TMP_2 TMP_4)))))) in (let TMP_6
413 \def (S n) in (let TMP_7 \def (refl_equal nat TMP_6) in (let TMP_12 \def
414 (\lambda (n0: nat).(\lambda (H: (eq nat (S (plus n0 n)) (plus n0 (S
415 n)))).(let TMP_8 \def (plus n0 n) in (let TMP_9 \def (S TMP_8) in (let TMP_10
416 \def (S n) in (let TMP_11 \def (plus n0 TMP_10) in (f_equal nat nat S TMP_9
417 TMP_11 H))))))) in (nat_ind TMP_5 TMP_7 TMP_12 m)))))).
420 \forall (n: nat).(\forall (m: nat).(eq nat (plus n m) (plus m n)))
422 \lambda (n: nat).(\lambda (m: nat).(let TMP_3 \def (\lambda (n0: nat).(let
423 TMP_1 \def (plus n0 m) in (let TMP_2 \def (plus m n0) in (eq nat TMP_1
424 TMP_2)))) in (let TMP_4 \def (plus_n_O m) in (let TMP_16 \def (\lambda (y:
425 nat).(\lambda (H: (eq nat (plus y m) (plus m y))).(let TMP_5 \def (plus m y)
426 in (let TMP_6 \def (S TMP_5) in (let TMP_9 \def (\lambda (n0: nat).(let TMP_7
427 \def (plus y m) in (let TMP_8 \def (S TMP_7) in (eq nat TMP_8 n0)))) in (let
428 TMP_10 \def (plus y m) in (let TMP_11 \def (plus m y) in (let TMP_12 \def
429 (f_equal nat nat S TMP_10 TMP_11 H) in (let TMP_13 \def (S y) in (let TMP_14
430 \def (plus m TMP_13) in (let TMP_15 \def (plus_n_Sm m y) in (eq_ind nat TMP_6
431 TMP_9 TMP_12 TMP_14 TMP_15)))))))))))) in (nat_ind TMP_3 TMP_4 TMP_16 n))))).
433 theorem plus_Snm_nSm:
434 \forall (n: nat).(\forall (m: nat).(eq nat (plus (S n) m) (plus n (S m))))
436 \lambda (n: nat).(\lambda (m: nat).(let TMP_1 \def (plus m n) in (let TMP_5
437 \def (\lambda (n0: nat).(let TMP_2 \def (S n0) in (let TMP_3 \def (S m) in
438 (let TMP_4 \def (plus n TMP_3) in (eq nat TMP_2 TMP_4))))) in (let TMP_6 \def
439 (S m) in (let TMP_7 \def (plus TMP_6 n) in (let TMP_10 \def (\lambda (n0:
440 nat).(let TMP_8 \def (plus m n) in (let TMP_9 \def (S TMP_8) in (eq nat TMP_9
441 n0)))) in (let TMP_11 \def (S m) in (let TMP_12 \def (plus TMP_11 n) in (let
442 TMP_13 \def (refl_equal nat TMP_12) in (let TMP_14 \def (S m) in (let TMP_15
443 \def (plus n TMP_14) in (let TMP_16 \def (S m) in (let TMP_17 \def (plus_sym
444 n TMP_16) in (let TMP_18 \def (eq_ind_r nat TMP_7 TMP_10 TMP_13 TMP_15
445 TMP_17) in (let TMP_19 \def (plus n m) in (let TMP_20 \def (plus_sym n m) in
446 (eq_ind_r nat TMP_1 TMP_5 TMP_18 TMP_19 TMP_20))))))))))))))))).
448 theorem plus_assoc_l:
449 \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(eq nat (plus n (plus m
450 p)) (plus (plus n m) p))))
452 \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(let TMP_5 \def
453 (\lambda (n0: nat).(let TMP_1 \def (plus m p) in (let TMP_2 \def (plus n0
454 TMP_1) in (let TMP_3 \def (plus n0 m) in (let TMP_4 \def (plus TMP_3 p) in
455 (eq nat TMP_2 TMP_4)))))) in (let TMP_6 \def (plus m p) in (let TMP_7 \def
456 (refl_equal nat TMP_6) in (let TMP_12 \def (\lambda (n0: nat).(\lambda (H:
457 (eq nat (plus n0 (plus m p)) (plus (plus n0 m) p))).(let TMP_8 \def (plus m
458 p) in (let TMP_9 \def (plus n0 TMP_8) in (let TMP_10 \def (plus n0 m) in (let
459 TMP_11 \def (plus TMP_10 p) in (f_equal nat nat S TMP_9 TMP_11 H))))))) in
460 (nat_ind TMP_5 TMP_7 TMP_12 n))))))).
462 theorem plus_assoc_r:
463 \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(eq nat (plus (plus n
464 m) p) (plus n (plus m p)))))
466 \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(let TMP_1 \def (plus m
467 p) in (let TMP_2 \def (plus n TMP_1) in (let TMP_3 \def (plus n m) in (let
468 TMP_4 \def (plus TMP_3 p) in (let TMP_5 \def (plus_assoc_l n m p) in (sym_eq
469 nat TMP_2 TMP_4 TMP_5)))))))).
471 theorem simpl_plus_l:
472 \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus n m)
473 (plus n p)) \to (eq nat m p))))
475 \lambda (n: nat).(let TMP_1 \def (\lambda (n0: nat).(\forall (m:
476 nat).(\forall (p: nat).((eq nat (plus n0 m) (plus n0 p)) \to (eq nat m p)))))
477 in (let TMP_2 \def (\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (eq nat m
478 p)).H))) in (let TMP_13 \def (\lambda (n0: nat).(\lambda (IHn: ((\forall (m:
479 nat).(\forall (p: nat).((eq nat (plus n0 m) (plus n0 p)) \to (eq nat m
480 p)))))).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (eq nat (S (plus n0
481 m)) (S (plus n0 p)))).(let TMP_3 \def (plus n0 m) in (let TMP_4 \def (plus n0
482 p) in (let TMP_5 \def (plus n0) in (let TMP_6 \def (plus n0 m) in (let TMP_7
483 \def (plus n0 p) in (let TMP_8 \def (plus n0 m) in (let TMP_9 \def (plus n0
484 p) in (let TMP_10 \def (eq_add_S TMP_8 TMP_9 H) in (let TMP_11 \def (f_equal
485 nat nat TMP_5 TMP_6 TMP_7 TMP_10) in (let TMP_12 \def (IHn TMP_3 TMP_4
486 TMP_11) in (IHn m p TMP_12)))))))))))))))) in (nat_ind TMP_1 TMP_2 TMP_13
490 \forall (n: nat).(eq nat n (minus n O))
492 \lambda (n: nat).(let TMP_2 \def (\lambda (n0: nat).(let TMP_1 \def (minus
493 n0 O) in (eq nat n0 TMP_1))) in (let TMP_3 \def (refl_equal nat O) in (let
494 TMP_5 \def (\lambda (n0: nat).(\lambda (_: (eq nat n0 (minus n0 O))).(let
495 TMP_4 \def (S n0) in (refl_equal nat TMP_4)))) in (nat_ind TMP_2 TMP_3 TMP_5
499 \forall (n: nat).(eq nat O (minus n n))
501 \lambda (n: nat).(let TMP_2 \def (\lambda (n0: nat).(let TMP_1 \def (minus
502 n0 n0) in (eq nat O TMP_1))) in (let TMP_3 \def (refl_equal nat O) in (let
503 TMP_4 \def (\lambda (n0: nat).(\lambda (IHn: (eq nat O (minus n0 n0))).IHn))
504 in (nat_ind TMP_2 TMP_3 TMP_4 n)))).
507 \forall (n: nat).(\forall (m: nat).((le m n) \to (eq nat (S (minus n m))
510 \lambda (n: nat).(\lambda (m: nat).(\lambda (Le: (le m n)).(let TMP_5 \def
511 (\lambda (n0: nat).(\lambda (n1: nat).(let TMP_1 \def (minus n1 n0) in (let
512 TMP_2 \def (S TMP_1) in (let TMP_3 \def (S n1) in (let TMP_4 \def (minus
513 TMP_3 n0) in (eq nat TMP_2 TMP_4))))))) in (let TMP_10 \def (\lambda (p:
514 nat).(let TMP_6 \def (minus p O) in (let TMP_7 \def (minus p O) in (let TMP_8
515 \def (minus_n_O p) in (let TMP_9 \def (sym_eq nat p TMP_7 TMP_8) in (f_equal
516 nat nat S TMP_6 p TMP_9)))))) in (let TMP_11 \def (\lambda (p: nat).(\lambda
517 (q: nat).(\lambda (_: (le p q)).(\lambda (H0: (eq nat (S (minus q p)) (match
518 p with [O \Rightarrow (S q) | (S l) \Rightarrow (minus q l)]))).H0)))) in
519 (le_elim_rel TMP_5 TMP_10 TMP_11 m n Le)))))).
522 \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat n (plus m p))
523 \to (eq nat p (minus n m)))))
525 \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(let TMP_2 \def
526 (\lambda (n0: nat).(\lambda (n1: nat).((eq nat n1 (plus n0 p)) \to (let TMP_1
527 \def (minus n1 n0) in (eq nat p TMP_1))))) in (let TMP_7 \def (\lambda (n0:
528 nat).(\lambda (H: (eq nat n0 p)).(let TMP_3 \def (\lambda (n1: nat).(eq nat p
529 n1)) in (let TMP_4 \def (sym_eq nat n0 p H) in (let TMP_5 \def (minus n0 O)
530 in (let TMP_6 \def (minus_n_O n0) in (eq_ind nat n0 TMP_3 TMP_4 TMP_5
531 TMP_6))))))) in (let TMP_12 \def (\lambda (n0: nat).(\lambda (H: (eq nat O (S
532 (plus n0 p)))).(let TMP_8 \def (eq nat p O) in (let H0 \def H in (let TMP_9
533 \def (plus n0 p) in (let H1 \def (O_S TMP_9) in (let TMP_10 \def (\lambda
534 (H2: (eq nat O (S (plus n0 p)))).(H1 H2)) in (let TMP_11 \def (TMP_10 H0) in
535 (False_ind TMP_8 TMP_11))))))))) in (let TMP_15 \def (\lambda (n0:
536 nat).(\lambda (m0: nat).(\lambda (H: (((eq nat m0 (plus n0 p)) \to (eq nat p
537 (minus m0 n0))))).(\lambda (H0: (eq nat (S m0) (S (plus n0 p)))).(let TMP_13
538 \def (plus n0 p) in (let TMP_14 \def (eq_add_S m0 TMP_13 H0) in (H
539 TMP_14))))))) in (nat_double_ind TMP_2 TMP_7 TMP_12 TMP_15 m n))))))).
542 \forall (n: nat).(\forall (m: nat).(eq nat (minus (plus n m) n) m))
544 \lambda (n: nat).(\lambda (m: nat).(let TMP_1 \def (plus n m) in (let TMP_2
545 \def (minus TMP_1 n) in (let TMP_3 \def (plus n m) in (let TMP_4 \def (plus n
546 m) in (let TMP_5 \def (refl_equal nat TMP_4) in (let TMP_6 \def (plus_minus
547 TMP_3 n m TMP_5) in (sym_eq nat m TMP_2 TMP_6)))))))).
550 \forall (n: nat).(le (pred n) n)
552 \lambda (n: nat).(let TMP_2 \def (\lambda (n0: nat).(let TMP_1 \def (pred
553 n0) in (le TMP_1 n0))) in (let TMP_3 \def (le_n O) in (let TMP_7 \def
554 (\lambda (n0: nat).(\lambda (_: (le (pred n0) n0)).(let TMP_4 \def (S n0) in
555 (let TMP_5 \def (pred TMP_4) in (let TMP_6 \def (le_n n0) in (le_S TMP_5 n0
556 TMP_6)))))) in (nat_ind TMP_2 TMP_3 TMP_7 n)))).
559 \forall (n: nat).(\forall (m: nat).(le n (plus n m)))
561 \lambda (n: nat).(let TMP_2 \def (\lambda (n0: nat).(\forall (m: nat).(let
562 TMP_1 \def (plus n0 m) in (le n0 TMP_1)))) in (let TMP_3 \def (\lambda (m:
563 nat).(le_O_n m)) in (let TMP_6 \def (\lambda (n0: nat).(\lambda (IHn:
564 ((\forall (m: nat).(le n0 (plus n0 m))))).(\lambda (m: nat).(let TMP_4 \def
565 (plus n0 m) in (let TMP_5 \def (IHn m) in (le_n_S n0 TMP_4 TMP_5)))))) in
566 (nat_ind TMP_2 TMP_3 TMP_6 n)))).
569 \forall (n: nat).(\forall (m: nat).(le m (plus n m)))
571 \lambda (n: nat).(\lambda (m: nat).(let TMP_2 \def (\lambda (n0: nat).(let
572 TMP_1 \def (plus n0 m) in (le m TMP_1))) in (let TMP_3 \def (le_n m) in (let
573 TMP_5 \def (\lambda (n0: nat).(\lambda (H: (le m (plus n0 m))).(let TMP_4
574 \def (plus n0 m) in (le_S m TMP_4 H)))) in (nat_ind TMP_2 TMP_3 TMP_5 n))))).
576 theorem simpl_le_plus_l:
577 \forall (p: nat).(\forall (n: nat).(\forall (m: nat).((le (plus p n) (plus p
580 \lambda (p: nat).(let TMP_1 \def (\lambda (n: nat).(\forall (n0:
581 nat).(\forall (m: nat).((le (plus n n0) (plus n m)) \to (le n0 m))))) in (let
582 TMP_2 \def (\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).H))) in
583 (let TMP_6 \def (\lambda (p0: nat).(\lambda (IHp: ((\forall (n: nat).(\forall
584 (m: nat).((le (plus p0 n) (plus p0 m)) \to (le n m)))))).(\lambda (n:
585 nat).(\lambda (m: nat).(\lambda (H: (le (S (plus p0 n)) (S (plus p0
586 m)))).(let TMP_3 \def (plus p0 n) in (let TMP_4 \def (plus p0 m) in (let
587 TMP_5 \def (le_S_n TMP_3 TMP_4 H) in (IHp n m TMP_5))))))))) in (nat_ind
588 TMP_1 TMP_2 TMP_6 p)))).
590 theorem le_plus_trans:
591 \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to (le n
594 \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (le n
595 m)).(let TMP_1 \def (plus m p) in (let TMP_2 \def (le_plus_l m p) in
596 (le_trans n m TMP_1 H TMP_2)))))).
599 \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((le n m) \to (le (plus
602 \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(let TMP_3 \def
603 (\lambda (n0: nat).((le n m) \to (let TMP_1 \def (plus n0 n) in (let TMP_2
604 \def (plus n0 m) in (le TMP_1 TMP_2))))) in (let TMP_4 \def (\lambda (H: (le
605 n m)).H) in (let TMP_8 \def (\lambda (p0: nat).(\lambda (IHp: (((le n m) \to
606 (le (plus p0 n) (plus p0 m))))).(\lambda (H: (le n m)).(let TMP_5 \def (plus
607 p0 n) in (let TMP_6 \def (plus p0 m) in (let TMP_7 \def (IHp H) in (le_n_S
608 TMP_5 TMP_6 TMP_7))))))) in (nat_ind TMP_3 TMP_4 TMP_8 p)))))).
610 theorem le_plus_plus:
611 \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((le
612 n m) \to ((le p q) \to (le (plus n p) (plus m q)))))))
614 \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q:
615 nat).(\lambda (H: (le n m)).(\lambda (H0: (le p q)).(let TMP_3 \def (\lambda
616 (n0: nat).(let TMP_1 \def (plus n p) in (let TMP_2 \def (plus n0 q) in (le
617 TMP_1 TMP_2)))) in (let TMP_4 \def (le_reg_l p q n H0) in (let TMP_7 \def
618 (\lambda (m0: nat).(\lambda (_: (le n m0)).(\lambda (H2: (le (plus n p) (plus
619 m0 q))).(let TMP_5 \def (plus n p) in (let TMP_6 \def (plus m0 q) in (le_S
620 TMP_5 TMP_6 H2)))))) in (le_ind n TMP_3 TMP_4 TMP_7 m H))))))))).
622 theorem le_plus_minus:
623 \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus n (minus m
626 \lambda (n: nat).(\lambda (m: nat).(\lambda (Le: (le n m)).(let TMP_3 \def
627 (\lambda (n0: nat).(\lambda (n1: nat).(let TMP_1 \def (minus n1 n0) in (let
628 TMP_2 \def (plus n0 TMP_1) in (eq nat n1 TMP_2))))) in (let TMP_4 \def
629 (\lambda (p: nat).(minus_n_O p)) in (let TMP_7 \def (\lambda (p:
630 nat).(\lambda (q: nat).(\lambda (_: (le p q)).(\lambda (H0: (eq nat q (plus p
631 (minus q p)))).(let TMP_5 \def (minus q p) in (let TMP_6 \def (plus p TMP_5)
632 in (f_equal nat nat S q TMP_6 H0))))))) in (le_elim_rel TMP_3 TMP_4 TMP_7 n m
635 theorem le_plus_minus_r:
636 \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat (plus n (minus m
639 \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(let TMP_1 \def
640 (minus m n) in (let TMP_2 \def (plus n TMP_1) in (let TMP_3 \def
641 (le_plus_minus n m H) in (sym_eq nat m TMP_2 TMP_3)))))).
643 theorem simpl_lt_plus_l:
644 \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt (plus p n) (plus p
647 \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(let TMP_1 \def
648 (\lambda (n0: nat).((lt (plus n0 n) (plus n0 m)) \to (lt n m))) in (let TMP_2
649 \def (\lambda (H: (lt n m)).H) in (let TMP_7 \def (\lambda (p0: nat).(\lambda
650 (IHp: (((lt (plus p0 n) (plus p0 m)) \to (lt n m)))).(\lambda (H: (lt (S
651 (plus p0 n)) (S (plus p0 m)))).(let TMP_3 \def (plus p0 n) in (let TMP_4 \def
652 (S TMP_3) in (let TMP_5 \def (plus p0 m) in (let TMP_6 \def (le_S_n TMP_4
653 TMP_5 H) in (IHp TMP_6)))))))) in (nat_ind TMP_1 TMP_2 TMP_7 p)))))).
656 \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to (lt (plus
659 \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(let TMP_3 \def
660 (\lambda (n0: nat).((lt n m) \to (let TMP_1 \def (plus n0 n) in (let TMP_2
661 \def (plus n0 m) in (lt TMP_1 TMP_2))))) in (let TMP_4 \def (\lambda (H: (lt
662 n m)).H) in (let TMP_8 \def (\lambda (p0: nat).(\lambda (IHp: (((lt n m) \to
663 (lt (plus p0 n) (plus p0 m))))).(\lambda (H: (lt n m)).(let TMP_5 \def (plus
664 p0 n) in (let TMP_6 \def (plus p0 m) in (let TMP_7 \def (IHp H) in (lt_n_S
665 TMP_5 TMP_6 TMP_7))))))) in (nat_ind TMP_3 TMP_4 TMP_8 p)))))).
668 \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((lt n m) \to (lt (plus
671 \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (lt n
672 m)).(let TMP_1 \def (plus p n) in (let TMP_3 \def (\lambda (n0: nat).(let
673 TMP_2 \def (plus m p) in (lt n0 TMP_2))) in (let TMP_4 \def (plus p m) in
674 (let TMP_6 \def (\lambda (n0: nat).(let TMP_5 \def (plus p n) in (lt TMP_5
675 n0))) in (let TMP_9 \def (\lambda (n0: nat).(let TMP_7 \def (plus n0 n) in
676 (let TMP_8 \def (plus n0 m) in (lt TMP_7 TMP_8)))) in (let TMP_11 \def
677 (\lambda (n0: nat).(\lambda (_: (lt (plus n0 n) (plus n0 m))).(let TMP_10
678 \def (S n0) in (lt_reg_l n m TMP_10 H)))) in (let TMP_12 \def (nat_ind TMP_9
679 H TMP_11 p) in (let TMP_13 \def (plus m p) in (let TMP_14 \def (plus_sym m p)
680 in (let TMP_15 \def (eq_ind_r nat TMP_4 TMP_6 TMP_12 TMP_13 TMP_14) in (let
681 TMP_16 \def (plus n p) in (let TMP_17 \def (plus_sym n p) in (eq_ind_r nat
682 TMP_1 TMP_3 TMP_15 TMP_16 TMP_17)))))))))))))))).
684 theorem le_lt_plus_plus:
685 \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((le
686 n m) \to ((lt p q) \to (lt (plus n p) (plus m q)))))))
688 \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q:
689 nat).(\lambda (H: (le n m)).(\lambda (H0: (le (S p) q)).(let TMP_1 \def (S p)
690 in (let TMP_2 \def (plus n TMP_1) in (let TMP_4 \def (\lambda (n0: nat).(let
691 TMP_3 \def (plus m q) in (le n0 TMP_3))) in (let TMP_5 \def (S p) in (let
692 TMP_6 \def (le_plus_plus n m TMP_5 q H H0) in (let TMP_7 \def (S n) in (let
693 TMP_8 \def (plus TMP_7 p) in (let TMP_9 \def (plus_Snm_nSm n p) in (eq_ind_r
694 nat TMP_2 TMP_4 TMP_6 TMP_8 TMP_9)))))))))))))).
696 theorem lt_le_plus_plus:
697 \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((lt
698 n m) \to ((le p q) \to (lt (plus n p) (plus m q)))))))
700 \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q:
701 nat).(\lambda (H: (le (S n) m)).(\lambda (H0: (le p q)).(let TMP_1 \def (S n)
702 in (le_plus_plus TMP_1 m p q H H0))))))).
704 theorem lt_plus_plus:
705 \forall (n: nat).(\forall (m: nat).(\forall (p: nat).(\forall (q: nat).((lt
706 n m) \to ((lt p q) \to (lt (plus n p) (plus m q)))))))
708 \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (q:
709 nat).(\lambda (H: (lt n m)).(\lambda (H0: (lt p q)).(let TMP_1 \def
710 (lt_le_weak p q H0) in (lt_le_plus_plus n m p q H TMP_1))))))).
712 theorem well_founded_ltof:
713 \forall (A: Type[0]).(\forall (f: ((A \to nat))).(well_founded A (ltof A f)))
715 \lambda (A: Type[0]).(\lambda (f: ((A \to nat))).(let H \def (\lambda (n:
716 nat).(let TMP_2 \def (\lambda (n0: nat).(\forall (a: A).((lt (f a) n0) \to
717 (let TMP_1 \def (ltof A f) in (Acc A TMP_1 a))))) in (let TMP_8 \def (\lambda
718 (a: A).(\lambda (H: (lt (f a) O)).(let TMP_3 \def (ltof A f) in (let TMP_4
719 \def (Acc A TMP_3 a) in (let H0 \def H in (let TMP_5 \def (f a) in (let H1
720 \def (lt_n_O TMP_5) in (let TMP_6 \def (\lambda (H2: (lt (f a) O)).(H1 H2))
721 in (let TMP_7 \def (TMP_6 H0) in (False_ind TMP_4 TMP_7)))))))))) in (let
722 TMP_16 \def (\lambda (n0: nat).(\lambda (IHn: ((\forall (a: A).((lt (f a) n0)
723 \to (Acc A (ltof A f) a))))).(\lambda (a: A).(\lambda (ltSma: (lt (f a) (S
724 n0))).(let TMP_9 \def (ltof A f) in (let TMP_15 \def (\lambda (b: A).(\lambda
725 (ltfafb: (lt (f b) (f a))).(let TMP_10 \def (f b) in (let TMP_11 \def (f a)
726 in (let TMP_12 \def (f a) in (let TMP_13 \def (lt_n_Sm_le TMP_12 n0 ltSma) in
727 (let TMP_14 \def (lt_le_trans TMP_10 TMP_11 n0 ltfafb TMP_13) in (IHn b
728 TMP_14)))))))) in (Acc_intro A TMP_9 a TMP_15))))))) in (nat_ind TMP_2 TMP_8
729 TMP_16 n))))) in (\lambda (a: A).(let TMP_17 \def (f a) in (let TMP_18 \def
730 (S TMP_17) in (let TMP_19 \def (f a) in (let TMP_20 \def (S TMP_19) in (let
731 TMP_21 \def (le_n TMP_20) in (H TMP_18 a TMP_21))))))))).
736 let TMP_1 \def (\lambda (m: nat).m) in (well_founded_ltof nat TMP_1).
739 \forall (p: nat).(\forall (P: ((nat \to Prop))).(((\forall (n:
740 nat).(((\forall (m: nat).((lt m n) \to (P m)))) \to (P n)))) \to (P p)))
742 \lambda (p: nat).(\lambda (P: ((nat \to Prop))).(\lambda (H: ((\forall (n:
743 nat).(((\forall (m: nat).((lt m n) \to (P m)))) \to (P n))))).(let TMP_1 \def
744 (\lambda (n: nat).(P n)) in (let TMP_2 \def (\lambda (x: nat).(\lambda (_:
745 ((\forall (y: nat).((lt y x) \to (Acc nat lt y))))).(\lambda (H1: ((\forall
746 (y: nat).((lt y x) \to (P y))))).(H x H1)))) in (let TMP_3 \def (lt_wf p) in
747 (Acc_ind nat lt TMP_1 TMP_2 p TMP_3)))))).