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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 "Ground-1/preamble.ma".
20 \forall (n1: nat).(\forall (n2: nat).(or (eq nat n1 n2) ((eq nat n1 n2) \to
21 (\forall (P: Prop).P))))
23 \lambda (n1: nat).(nat_ind (\lambda (n: nat).(\forall (n2: nat).(or (eq nat
24 n n2) ((eq nat n n2) \to (\forall (P: Prop).P))))) (\lambda (n2:
25 nat).(nat_ind (\lambda (n: nat).(or (eq nat O n) ((eq nat O n) \to (\forall
26 (P: Prop).P)))) (or_introl (eq nat O O) ((eq nat O O) \to (\forall (P:
27 Prop).P)) (refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (eq nat O n)
28 ((eq nat O n) \to (\forall (P: Prop).P)))).(or_intror (eq nat O (S n)) ((eq
29 nat O (S n)) \to (\forall (P: Prop).P)) (\lambda (H0: (eq nat O (S
30 n))).(\lambda (P: Prop).(let H1 \def (eq_ind nat O (\lambda (ee: nat).(match
31 ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _)
32 \Rightarrow False])) I (S n) H0) in (False_ind P H1))))))) n2)) (\lambda (n:
33 nat).(\lambda (H: ((\forall (n2: nat).(or (eq nat n n2) ((eq nat n n2) \to
34 (\forall (P: Prop).P)))))).(\lambda (n2: nat).(nat_ind (\lambda (n0: nat).(or
35 (eq nat (S n) n0) ((eq nat (S n) n0) \to (\forall (P: Prop).P)))) (or_intror
36 (eq nat (S n) O) ((eq nat (S n) O) \to (\forall (P: Prop).P)) (\lambda (H0:
37 (eq nat (S n) O)).(\lambda (P: Prop).(let H1 \def (eq_ind nat (S n) (\lambda
38 (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow
39 False | (S _) \Rightarrow True])) I O H0) in (False_ind P H1))))) (\lambda
40 (n0: nat).(\lambda (H0: (or (eq nat (S n) n0) ((eq nat (S n) n0) \to (\forall
41 (P: Prop).P)))).(or_ind (eq nat n n0) ((eq nat n n0) \to (\forall (P:
42 Prop).P)) (or (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to (\forall (P:
43 Prop).P))) (\lambda (H1: (eq nat n n0)).(let H2 \def (eq_ind_r nat n0
44 (\lambda (n3: nat).(or (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P:
45 Prop).P)))) H0 n H1) in (eq_ind nat n (\lambda (n3: nat).(or (eq nat (S n) (S
46 n3)) ((eq nat (S n) (S n3)) \to (\forall (P: Prop).P)))) (or_introl (eq nat
47 (S n) (S n)) ((eq nat (S n) (S n)) \to (\forall (P: Prop).P)) (refl_equal nat
48 (S n))) n0 H1))) (\lambda (H1: (((eq nat n n0) \to (\forall (P:
49 Prop).P)))).(or_intror (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to
50 (\forall (P: Prop).P)) (\lambda (H2: (eq nat (S n) (S n0))).(\lambda (P:
51 Prop).(let H3 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return
52 (\lambda (_: nat).nat) with [O \Rightarrow n | (S n3) \Rightarrow n3])) (S n)
53 (S n0) H2) in (let H4 \def (eq_ind_r nat n0 (\lambda (n3: nat).((eq nat n n3)
54 \to (\forall (P0: Prop).P0))) H1 n H3) in (let H5 \def (eq_ind_r nat n0
55 (\lambda (n3: nat).(or (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P0:
56 Prop).P0)))) H0 n H3) in (H4 (refl_equal nat n) P)))))))) (H n0)))) n2))))
63 \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus m n)
64 (plus p n)) \to (eq nat m p))))
66 \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (eq nat
67 (plus m n) (plus p n))).(simpl_plus_l n m p (eq_ind_r nat (plus m n) (\lambda
68 (n0: nat).(eq nat n0 (plus n p))) (eq_ind_r nat (plus p n) (\lambda (n0:
69 nat).(eq nat n0 (plus n p))) (sym_eq nat (plus n p) (plus p n) (plus_sym n
70 p)) (plus m n) H) (plus n m) (plus_sym n m)))))).
76 \forall (x: nat).(\forall (y: nat).(eq nat (minus (S x) (S y)) (minus x y)))
78 \lambda (x: nat).(\lambda (y: nat).(refl_equal nat (minus x y))).
84 \forall (m: nat).(\forall (n: nat).(eq nat (minus (plus m n) n) m))
86 \lambda (m: nat).(\lambda (n: nat).(eq_ind_r nat (plus n m) (\lambda (n0:
87 nat).(eq nat (minus n0 n) m)) (minus_plus n m) (plus m n) (plus_sym m n))).
92 theorem plus_permute_2_in_3:
93 \forall (x: nat).(\forall (y: nat).(\forall (z: nat).(eq nat (plus (plus x
94 y) z) (plus (plus x z) y))))
96 \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(eq_ind_r nat (plus x
97 (plus y z)) (\lambda (n: nat).(eq nat n (plus (plus x z) y))) (eq_ind_r nat
98 (plus z y) (\lambda (n: nat).(eq nat (plus x n) (plus (plus x z) y))) (eq_ind
99 nat (plus (plus x z) y) (\lambda (n: nat).(eq nat n (plus (plus x z) y)))
100 (refl_equal nat (plus (plus x z) y)) (plus x (plus z y)) (plus_assoc_r x z
101 y)) (plus y z) (plus_sym y z)) (plus (plus x y) z) (plus_assoc_r x y z)))).
106 theorem plus_permute_2_in_3_assoc:
107 \forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq nat (plus (plus n
108 h) k) (plus n (plus k h)))))
110 \lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind_r nat (plus
111 (plus n k) h) (\lambda (n0: nat).(eq nat n0 (plus n (plus k h)))) (eq_ind_r
112 nat (plus (plus n k) h) (\lambda (n0: nat).(eq nat (plus (plus n k) h) n0))
113 (refl_equal nat (plus (plus n k) h)) (plus n (plus k h)) (plus_assoc_l n k
114 h)) (plus (plus n h) k) (plus_permute_2_in_3 n h k)))).
120 \forall (x: nat).(\forall (y: nat).((eq nat (plus x y) O) \to (land (eq nat
123 \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq nat (plus
124 n y) O) \to (land (eq nat n O) (eq nat y O))))) (\lambda (y: nat).(\lambda
125 (H: (eq nat (plus O y) O)).(conj (eq nat O O) (eq nat y O) (refl_equal nat O)
126 H))) (\lambda (n: nat).(\lambda (_: ((\forall (y: nat).((eq nat (plus n y) O)
127 \to (land (eq nat n O) (eq nat y O)))))).(\lambda (y: nat).(\lambda (H0: (eq
128 nat (plus (S n) y) O)).(let H1 \def (match H0 in eq return (\lambda (n0:
129 nat).(\lambda (_: (eq ? ? n0)).((eq nat n0 O) \to (land (eq nat (S n) O) (eq
130 nat y O))))) with [refl_equal \Rightarrow (\lambda (H1: (eq nat (plus (S n)
131 y) O)).(let H2 \def (eq_ind nat (plus (S n) y) (\lambda (e: nat).(match e in
132 nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
133 \Rightarrow True])) I O H1) in (False_ind (land (eq nat (S n) O) (eq nat y
134 O)) H2)))]) in (H1 (refl_equal nat O))))))) x).
140 \forall (x: nat).(eq nat (minus (S x) (S O)) x)
142 \lambda (x: nat).(eq_ind nat x (\lambda (n: nat).(eq nat n x)) (refl_equal
143 nat x) (minus x O) (minus_n_O x)).
149 \forall (i: nat).(\forall (j: nat).(or (not (eq nat i j)) (eq nat i j)))
151 \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (j: nat).(or (not (eq
152 nat n j)) (eq nat n j)))) (\lambda (j: nat).(nat_ind (\lambda (n: nat).(or
153 (not (eq nat O n)) (eq nat O n))) (or_intror (not (eq nat O O)) (eq nat O O)
154 (refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (not (eq nat O n)) (eq
155 nat O n))).(or_introl (not (eq nat O (S n))) (eq nat O (S n)) (O_S n)))) j))
156 (\lambda (n: nat).(\lambda (H: ((\forall (j: nat).(or (not (eq nat n j)) (eq
157 nat n j))))).(\lambda (j: nat).(nat_ind (\lambda (n0: nat).(or (not (eq nat
158 (S n) n0)) (eq nat (S n) n0))) (or_introl (not (eq nat (S n) O)) (eq nat (S
159 n) O) (sym_not_eq nat O (S n) (O_S n))) (\lambda (n0: nat).(\lambda (_: (or
160 (not (eq nat (S n) n0)) (eq nat (S n) n0))).(or_ind (not (eq nat n n0)) (eq
161 nat n n0) (or (not (eq nat (S n) (S n0))) (eq nat (S n) (S n0))) (\lambda
162 (H1: (not (eq nat n n0))).(or_introl (not (eq nat (S n) (S n0))) (eq nat (S
163 n) (S n0)) (not_eq_S n n0 H1))) (\lambda (H1: (eq nat n n0)).(or_intror (not
164 (eq nat (S n) (S n0))) (eq nat (S n) (S n0)) (f_equal nat nat S n n0 H1))) (H
171 \forall (i: nat).(\forall (j: nat).(\forall (P: Prop).((((not (eq nat i j))
172 \to P)) \to ((((eq nat i j) \to P)) \to P))))
174 \lambda (i: nat).(\lambda (j: nat).(\lambda (P: Prop).(\lambda (H: (((not
175 (eq nat i j)) \to P))).(\lambda (H0: (((eq nat i j) \to P))).(let o \def
176 (eq_nat_dec i j) in (or_ind (not (eq nat i j)) (eq nat i j) P H H0 o)))))).
182 \forall (m: nat).(\forall (n: nat).(\forall (P: Prop).((le m n) \to ((le (S
185 \lambda (m: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).(\forall (P:
186 Prop).((le n n0) \to ((le (S n0) n) \to P))))) (\lambda (n: nat).(\lambda (P:
187 Prop).(\lambda (_: (le O n)).(\lambda (H0: (le (S n) O)).(let H1 \def (match
188 H0 in le return (\lambda (n0: nat).(\lambda (_: (le ? n0)).((eq nat n0 O) \to
189 P))) with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def
190 (eq_ind nat (S n) (\lambda (e: nat).(match e in nat return (\lambda (_:
191 nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in
192 (False_ind P H2))) | (le_S m0 H1) \Rightarrow (\lambda (H2: (eq nat (S m0)
193 O)).((let H3 \def (eq_ind nat (S m0) (\lambda (e: nat).(match e in nat return
194 (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True]))
195 I O H2) in (False_ind ((le (S n) m0) \to P) H3)) H1))]) in (H1 (refl_equal
196 nat O))))))) (\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(\forall (P:
197 Prop).((le n n0) \to ((le (S n0) n) \to P)))))).(\lambda (n0: nat).(nat_ind
198 (\lambda (n1: nat).(\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n))
199 \to P)))) (\lambda (P: Prop).(\lambda (H0: (le (S n) O)).(\lambda (_: (le (S
200 O) (S n))).(let H2 \def (match H0 in le return (\lambda (n1: nat).(\lambda
201 (_: (le ? n1)).((eq nat n1 O) \to P))) with [le_n \Rightarrow (\lambda (H2:
202 (eq nat (S n) O)).(let H3 \def (eq_ind nat (S n) (\lambda (e: nat).(match e
203 in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
204 \Rightarrow True])) I O H2) in (False_ind P H3))) | (le_S m0 H2) \Rightarrow
205 (\lambda (H3: (eq nat (S m0) O)).((let H4 \def (eq_ind nat (S m0) (\lambda
206 (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow
207 False | (S _) \Rightarrow True])) I O H3) in (False_ind ((le (S n) m0) \to P)
208 H4)) H2))]) in (H2 (refl_equal nat O)))))) (\lambda (n1: nat).(\lambda (_:
209 ((\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n)) \to P))))).(\lambda
210 (P: Prop).(\lambda (H1: (le (S n) (S n1))).(\lambda (H2: (le (S (S n1)) (S
211 n))).(H n1 P (le_S_n n n1 H1) (le_S_n (S n1) n H2))))))) n0)))) m).
217 \forall (x: nat).((le (S x) x) \to (\forall (P: Prop).P))
219 \lambda (x: nat).(\lambda (H: (le (S x) x)).(\lambda (P: Prop).(let H0 \def
220 le_Sn_n in (False_ind P (H0 x H))))).
226 \forall (n: nat).(\forall (m: nat).((le n m) \to (le (pred n) (pred m))))
228 \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda
229 (n0: nat).(le (pred n) (pred n0))) (le_n (pred n)) (\lambda (m0:
230 nat).(\lambda (_: (le n m0)).(\lambda (H1: (le (pred n) (pred m0))).(le_trans
231 (pred n) (pred m0) m0 H1 (le_pred_n m0))))) m H))).
237 \forall (x: nat).(\forall (y: nat).(le (minus x y) x))
239 \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).(le (minus n
240 y) n))) (\lambda (_: nat).(le_n O)) (\lambda (n: nat).(\lambda (H: ((\forall
241 (y: nat).(le (minus n y) n)))).(\lambda (y: nat).(nat_ind (\lambda (n0:
242 nat).(le (minus (S n) n0) (S n))) (le_n (S n)) (\lambda (n0: nat).(\lambda
243 (_: (le (match n0 with [O \Rightarrow (S n) | (S l) \Rightarrow (minus n l)])
244 (S n))).(le_S (minus n n0) n (H n0)))) y)))) x).
249 theorem le_plus_minus_sym:
250 \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus (minus m n)
253 \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(eq_ind_r nat
254 (plus n (minus m n)) (\lambda (n0: nat).(eq nat m n0)) (le_plus_minus n m H)
255 (plus (minus m n) n) (plus_sym (minus m n) n)))).
260 theorem le_minus_minus:
261 \forall (x: nat).(\forall (y: nat).((le x y) \to (\forall (z: nat).((le y z)
262 \to (le (minus y x) (minus z x))))))
264 \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (z:
265 nat).(\lambda (H0: (le y z)).(simpl_le_plus_l x (minus y x) (minus z x)
266 (eq_ind_r nat y (\lambda (n: nat).(le n (plus x (minus z x)))) (eq_ind_r nat
267 z (\lambda (n: nat).(le y n)) H0 (plus x (minus z x)) (le_plus_minus_r x z
268 (le_trans x y z H H0))) (plus x (minus y x)) (le_plus_minus_r x y H))))))).
273 theorem le_minus_plus:
274 \forall (z: nat).(\forall (x: nat).((le z x) \to (\forall (y: nat).(eq nat
275 (minus (plus x y) z) (plus (minus x z) y)))))
277 \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((le n x) \to
278 (\forall (y: nat).(eq nat (minus (plus x y) n) (plus (minus x n) y))))))
279 (\lambda (x: nat).(\lambda (H: (le O x)).(let H0 \def (match H in le return
280 (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n x) \to (\forall (y:
281 nat).(eq nat (minus (plus x y) O) (plus (minus x O) y)))))) with [le_n
282 \Rightarrow (\lambda (H0: (eq nat O x)).(eq_ind nat O (\lambda (n:
283 nat).(\forall (y: nat).(eq nat (minus (plus n y) O) (plus (minus n O) y))))
284 (\lambda (y: nat).(sym_eq nat (plus (minus O O) y) (minus (plus O y) O)
285 (minus_n_O (plus O y)))) x H0)) | (le_S m H0) \Rightarrow (\lambda (H1: (eq
286 nat (S m) x)).(eq_ind nat (S m) (\lambda (n: nat).((le O m) \to (\forall (y:
287 nat).(eq nat (minus (plus n y) O) (plus (minus n O) y))))) (\lambda (_: (le O
288 m)).(\lambda (y: nat).(refl_equal nat (plus (minus (S m) O) y)))) x H1 H0))])
289 in (H0 (refl_equal nat x))))) (\lambda (z0: nat).(\lambda (H: ((\forall (x:
290 nat).((le z0 x) \to (\forall (y: nat).(eq nat (minus (plus x y) z0) (plus
291 (minus x z0) y))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).((le (S
292 z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n
293 (S z0)) y))))) (\lambda (H0: (le (S z0) O)).(\lambda (y: nat).(let H1 \def
294 (match H0 in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O)
295 \to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))))) with
296 [le_n \Rightarrow (\lambda (H1: (eq nat (S z0) O)).(let H2 \def (eq_ind nat
297 (S z0) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with
298 [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind (eq
299 nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y)) H2))) | (le_S m H1)
300 \Rightarrow (\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m)
301 (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O
302 \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind ((le (S
303 z0) m) \to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))) H3))
304 H1))]) in (H1 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: (((le (S
305 z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n
306 (S z0)) y)))))).(\lambda (H1: (le (S z0) (S n))).(\lambda (y: nat).(H n
307 (le_S_n z0 n H1) y))))) x)))) z).
313 \forall (x: nat).(\forall (z: nat).(\forall (y: nat).((le (plus x y) z) \to
314 (le x (minus z y)))))
316 \lambda (x: nat).(\lambda (z: nat).(\lambda (y: nat).(\lambda (H: (le (plus
317 x y) z)).(eq_ind nat (minus (plus x y) y) (\lambda (n: nat).(le n (minus z
318 y))) (le_minus_minus y (plus x y) (le_plus_r x y) z H) x (minus_plus_r x
324 theorem le_trans_plus_r:
325 \forall (x: nat).(\forall (y: nat).(\forall (z: nat).((le (plus x y) z) \to
328 \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(\lambda (H: (le (plus
329 x y) z)).(le_trans y (plus x y) z (le_plus_r x y) H)))).
335 \forall (x: nat).((lt x O) \to (\forall (P: Prop).P))
337 \lambda (x: nat).(\lambda (H: (le (S x) O)).(\lambda (P: Prop).(let H_y \def
338 (le_n_O_eq (S x) H) in (let H0 \def (eq_ind nat O (\lambda (ee: nat).(match
339 ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _)
340 \Rightarrow False])) I (S x) H_y) in (False_ind P H0))))).
346 \forall (m: nat).(\forall (x: nat).((le (S m) x) \to (ex2 nat (\lambda (n:
347 nat).(eq nat x (S n))) (\lambda (n: nat).(le m n)))))
349 \lambda (m: nat).(\lambda (x: nat).(\lambda (H: (le (S m) x)).(let H0 \def
350 (match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n x)
351 \to (ex2 nat (\lambda (n0: nat).(eq nat x (S n0))) (\lambda (n0: nat).(le m
352 n0)))))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S m) x)).(eq_ind nat
353 (S m) (\lambda (n: nat).(ex2 nat (\lambda (n0: nat).(eq nat n (S n0)))
354 (\lambda (n0: nat).(le m n0)))) (ex_intro2 nat (\lambda (n: nat).(eq nat (S
355 m) (S n))) (\lambda (n: nat).(le m n)) m (refl_equal nat (S m)) (le_n m)) x
356 H0)) | (le_S m0 H0) \Rightarrow (\lambda (H1: (eq nat (S m0) x)).(eq_ind nat
357 (S m0) (\lambda (n: nat).((le (S m) m0) \to (ex2 nat (\lambda (n0: nat).(eq
358 nat n (S n0))) (\lambda (n0: nat).(le m n0))))) (\lambda (H2: (le (S m)
359 m0)).(ex_intro2 nat (\lambda (n: nat).(eq nat (S m0) (S n))) (\lambda (n:
360 nat).(le m n)) m0 (refl_equal nat (S m0)) (le_S_n m m0 (le_S (S m) m0 H2))))
361 x H1 H0))]) in (H0 (refl_equal nat x))))).
366 theorem lt_x_plus_x_Sy:
367 \forall (x: nat).(\forall (y: nat).(lt x (plus x (S y))))
369 \lambda (x: nat).(\lambda (y: nat).(eq_ind_r nat (plus (S y) x) (\lambda (n:
370 nat).(lt x n)) (le_S_n (S x) (S (plus y x)) (le_n_S (S x) (S (plus y x))
371 (le_n_S x (plus y x) (le_plus_r y x)))) (plus x (S y)) (plus_sym x (S y)))).
376 theorem simpl_lt_plus_r:
377 \forall (p: nat).(\forall (n: nat).(\forall (m: nat).((lt (plus n p) (plus m
380 \lambda (p: nat).(\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (plus
381 n p) (plus m p))).(simpl_lt_plus_l n m p (let H0 \def (eq_ind nat (plus n p)
382 (\lambda (n0: nat).(lt n0 (plus m p))) H (plus p n) (plus_sym n p)) in (let
383 H1 \def (eq_ind nat (plus m p) (\lambda (n0: nat).(lt (plus p n) n0)) H0
384 (plus p m) (plus_sym m p)) in H1)))))).
390 \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq nat (minus x y) (S
393 \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to
394 (eq nat (minus n y) (S (minus n (S y))))))) (\lambda (y: nat).(\lambda (H:
395 (lt y O)).(let H0 \def (match H in le return (\lambda (n: nat).(\lambda (_:
396 (le ? n)).((eq nat n O) \to (eq nat (minus O y) (S (minus O (S y))))))) with
397 [le_n \Rightarrow (\lambda (H0: (eq nat (S y) O)).(let H1 \def (eq_ind nat (S
398 y) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O
399 \Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (eq nat
400 (minus O y) (S (minus O (S y)))) H1))) | (le_S m H0) \Rightarrow (\lambda
401 (H1: (eq nat (S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e:
402 nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False
403 | (S _) \Rightarrow True])) I O H1) in (False_ind ((le (S y) m) \to (eq nat
404 (minus O y) (S (minus O (S y))))) H2)) H0))]) in (H0 (refl_equal nat O)))))
405 (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to (eq nat
406 (minus n y) (S (minus n (S y)))))))).(\lambda (y: nat).(nat_ind (\lambda (n0:
407 nat).((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus (S n) (S n0))))))
408 (\lambda (_: (lt O (S n))).(eq_ind nat n (\lambda (n0: nat).(eq nat (S n) (S
409 n0))) (refl_equal nat (S n)) (minus n O) (minus_n_O n))) (\lambda (n0:
410 nat).(\lambda (_: (((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus (S
411 n) (S n0))))))).(\lambda (H1: (lt (S n0) (S n))).(let H2 \def (le_S_n (S n0)
412 n H1) in (H n0 H2))))) y)))) x).
417 theorem lt_plus_minus:
418 \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus x (minus
421 \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_plus_minus (S
427 theorem lt_plus_minus_r:
428 \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus (minus y
431 \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(eq_ind_r nat
432 (plus x (minus y (S x))) (\lambda (n: nat).(eq nat y (S n))) (lt_plus_minus x
433 y H) (plus (minus y (S x)) x) (plus_sym (minus y (S x)) x)))).
439 \forall (x: nat).((lt O x) \to (eq nat x (S (minus x (S O)))))
441 \lambda (x: nat).(\lambda (H: (lt O x)).(eq_ind nat (minus x O) (\lambda (n:
442 nat).(eq nat x n)) (eq_ind nat x (\lambda (n: nat).(eq nat x n)) (refl_equal
443 nat x) (minus x O) (minus_n_O x)) (S (minus x (S O))) (minus_x_Sy x O H))).
449 \forall (y: nat).(\forall (x: nat).((lt x y) \to (le x (pred y))))
451 \lambda (y: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((lt x n) \to
452 (le x (pred n))))) (\lambda (x: nat).(\lambda (H: (lt x O)).(let H0 \def
453 (match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O)
454 \to (le x O)))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S x) O)).(let
455 H1 \def (eq_ind nat (S x) (\lambda (e: nat).(match e in nat return (\lambda
456 (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H0)
457 in (False_ind (le x O) H1))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat
458 (S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e: nat).(match e in nat
459 return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
460 True])) I O H1) in (False_ind ((le (S x) m) \to (le x O)) H2)) H0))]) in (H0
461 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: ((\forall (x: nat).((lt
462 x n) \to (le x (pred n)))))).(\lambda (x: nat).(\lambda (H0: (lt x (S
463 n))).(le_S_n x n H0))))) y).
469 \forall (x: nat).(\forall (y: nat).((lt x y) \to (le x (minus y (S O)))))
471 \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_minus x y (S
472 O) (eq_ind_r nat (plus (S O) x) (\lambda (n: nat).(le n y)) H (plus x (S O))
473 (plus_sym x (S O)))))).
479 \forall (n: nat).(\forall (d: nat).(\forall (P: Prop).((((lt n d) \to P))
480 \to ((((le d n) \to P)) \to P))))
482 \lambda (n: nat).(\lambda (d: nat).(\lambda (P: Prop).(\lambda (H: (((lt n
483 d) \to P))).(\lambda (H0: (((le d n) \to P))).(let H1 \def (le_or_lt d n) in
484 (or_ind (le d n) (lt n d) P H0 H H1)))))).
490 \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P))
491 \to ((((eq nat x y) \to P)) \to ((le x y) \to P)))))
493 \lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x
494 y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (le x
495 y)).(or_ind (lt x y) (eq nat x y) P H H0 (le_lt_or_eq x y H1))))))).
501 \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P))
502 \to ((((eq nat x y) \to P)) \to ((((lt y x) \to P)) \to P)))))
504 \lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x
505 y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (((lt y x)
506 \to P))).(lt_le_e x y P H (\lambda (H2: (le y x)).(lt_eq_e y x P H1 (\lambda
507 (H3: (eq nat y x)).(H0 (sym_eq nat y x H3))) H2)))))))).
513 \forall (x: nat).(\forall (n: nat).((lt x (S n)) \to (or (eq nat x O) (ex2
514 nat (\lambda (m: nat).(eq nat x (S m))) (\lambda (m: nat).(lt m n))))))
516 \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((lt n (S
517 n0)) \to (or (eq nat n O) (ex2 nat (\lambda (m: nat).(eq nat n (S m)))
518 (\lambda (m: nat).(lt m n0))))))) (\lambda (n: nat).(\lambda (_: (lt O (S
519 n))).(or_introl (eq nat O O) (ex2 nat (\lambda (m: nat).(eq nat O (S m)))
520 (\lambda (m: nat).(lt m n))) (refl_equal nat O)))) (\lambda (n: nat).(\lambda
521 (_: ((\forall (n0: nat).((lt n (S n0)) \to (or (eq nat n O) (ex2 nat (\lambda
522 (m: nat).(eq nat n (S m))) (\lambda (m: nat).(lt m n0)))))))).(\lambda (n0:
523 nat).(\lambda (H0: (lt (S n) (S n0))).(or_intror (eq nat (S n) O) (ex2 nat
524 (\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt m n0)))
525 (ex_intro2 nat (\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt
526 m n0)) n (refl_equal nat (S n)) (le_S_n (S n) n0 H0))))))) x).
532 \forall (x: nat).(\forall (y: nat).((le x y) \to ((lt y x) \to (\forall (P:
535 \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (H0: (lt
536 y x)).(\lambda (P: Prop).(False_ind P (le_not_lt x y H H0)))))).
542 \forall (x: nat).(\forall (y: nat).((lt x y) \to (not (eq nat x y))))
544 \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(\lambda (H0: (eq
545 nat x y)).(let H1 \def (eq_ind nat x (\lambda (n: nat).(lt n y)) H y H0) in
552 \forall (h2: nat).(\forall (d2: nat).(\forall (n: nat).((le (plus d2 h2) n)
553 \to (\forall (h1: nat).(le (plus d2 h1) (minus (plus n h1) h2))))))
555 \lambda (h2: nat).(\lambda (d2: nat).(\lambda (n: nat).(\lambda (H: (le
556 (plus d2 h2) n)).(\lambda (h1: nat).(eq_ind nat (minus (plus h2 (plus d2 h1))
557 h2) (\lambda (n0: nat).(le n0 (minus (plus n h1) h2))) (le_minus_minus h2
558 (plus h2 (plus d2 h1)) (le_plus_l h2 (plus d2 h1)) (plus n h1) (eq_ind_r nat
559 (plus (plus h2 d2) h1) (\lambda (n0: nat).(le n0 (plus n h1))) (eq_ind_r nat
560 (plus d2 h2) (\lambda (n0: nat).(le (plus n0 h1) (plus n h1))) (le_S_n (plus
561 (plus d2 h2) h1) (plus n h1) (le_n_S (plus (plus d2 h2) h1) (plus n h1)
562 (le_plus_plus (plus d2 h2) n h1 h1 H (le_n h1)))) (plus h2 d2) (plus_sym h2
563 d2)) (plus h2 (plus d2 h1)) (plus_assoc_l h2 d2 h1))) (plus d2 h1)
564 (minus_plus h2 (plus d2 h1))))))).
570 \forall (x: nat).(\forall (y: nat).((le x y) \to (eq nat (minus x y) O)))
572 \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to
573 (eq nat (minus n y) O)))) (\lambda (y: nat).(\lambda (_: (le O
574 y)).(refl_equal nat O))) (\lambda (x0: nat).(\lambda (H: ((\forall (y:
575 nat).((le x0 y) \to (eq nat (minus x0 y) O))))).(\lambda (y: nat).(nat_ind
576 (\lambda (n: nat).((le (S x0) n) \to (eq nat (match n with [O \Rightarrow (S
577 x0) | (S l) \Rightarrow (minus x0 l)]) O))) (\lambda (H0: (le (S x0)
578 O)).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le x0
579 n)) (eq nat (S x0) O) (\lambda (x1: nat).(\lambda (H1: (eq nat O (S
580 x1))).(\lambda (_: (le x0 x1)).(let H3 \def (eq_ind nat O (\lambda (ee:
581 nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True
582 | (S _) \Rightarrow False])) I (S x1) H1) in (False_ind (eq nat (S x0) O)
583 H3))))) (le_gen_S x0 O H0))) (\lambda (n: nat).(\lambda (_: (((le (S x0) n)
584 \to (eq nat (match n with [O \Rightarrow (S x0) | (S l) \Rightarrow (minus x0
585 l)]) O)))).(\lambda (H1: (le (S x0) (S n))).(H n (le_S_n x0 n H1))))) y))))
592 \forall (z: nat).(\forall (x: nat).(\forall (y: nat).((le z x) \to ((le z y)
593 \to ((eq nat (minus x z) (minus y z)) \to (eq nat x y))))))
595 \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).(\forall (y:
596 nat).((le n x) \to ((le n y) \to ((eq nat (minus x n) (minus y n)) \to (eq
597 nat x y))))))) (\lambda (x: nat).(\lambda (y: nat).(\lambda (_: (le O
598 x)).(\lambda (_: (le O y)).(\lambda (H1: (eq nat (minus x O) (minus y
599 O))).(let H2 \def (eq_ind_r nat (minus x O) (\lambda (n: nat).(eq nat n
600 (minus y O))) H1 x (minus_n_O x)) in (let H3 \def (eq_ind_r nat (minus y O)
601 (\lambda (n: nat).(eq nat x n)) H2 y (minus_n_O y)) in H3))))))) (\lambda
602 (z0: nat).(\lambda (IH: ((\forall (x: nat).(\forall (y: nat).((le z0 x) \to
603 ((le z0 y) \to ((eq nat (minus x z0) (minus y z0)) \to (eq nat x
604 y)))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le
605 (S z0) n) \to ((le (S z0) y) \to ((eq nat (minus n (S z0)) (minus y (S z0)))
606 \to (eq nat n y)))))) (\lambda (y: nat).(\lambda (H: (le (S z0) O)).(\lambda
607 (_: (le (S z0) y)).(\lambda (_: (eq nat (minus O (S z0)) (minus y (S
608 z0)))).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le
609 z0 n)) (eq nat O y) (\lambda (x0: nat).(\lambda (H2: (eq nat O (S
610 x0))).(\lambda (_: (le z0 x0)).(let H4 \def (eq_ind nat O (\lambda (ee:
611 nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True
612 | (S _) \Rightarrow False])) I (S x0) H2) in (False_ind (eq nat O y) H4)))))
613 (le_gen_S z0 O H)))))) (\lambda (x0: nat).(\lambda (_: ((\forall (y:
614 nat).((le (S z0) x0) \to ((le (S z0) y) \to ((eq nat (minus x0 (S z0)) (minus
615 y (S z0))) \to (eq nat x0 y))))))).(\lambda (y: nat).(nat_ind (\lambda (n:
616 nat).((le (S z0) (S x0)) \to ((le (S z0) n) \to ((eq nat (minus (S x0) (S
617 z0)) (minus n (S z0))) \to (eq nat (S x0) n))))) (\lambda (H: (le (S z0) (S
618 x0))).(\lambda (H0: (le (S z0) O)).(\lambda (_: (eq nat (minus (S x0) (S z0))
619 (minus O (S z0)))).(let H_y \def (le_S_n z0 x0 H) in (ex2_ind nat (\lambda
620 (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le z0 n)) (eq nat (S x0) O)
621 (\lambda (x1: nat).(\lambda (H2: (eq nat O (S x1))).(\lambda (_: (le z0
622 x1)).(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return
623 (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False]))
624 I (S x1) H2) in (False_ind (eq nat (S x0) O) H4))))) (le_gen_S z0 O H0))))))
625 (\lambda (y0: nat).(\lambda (_: (((le (S z0) (S x0)) \to ((le (S z0) y0) \to
626 ((eq nat (minus (S x0) (S z0)) (minus y0 (S z0))) \to (eq nat (S x0)
627 y0)))))).(\lambda (H: (le (S z0) (S x0))).(\lambda (H0: (le (S z0) (S
628 y0))).(\lambda (H1: (eq nat (minus (S x0) (S z0)) (minus (S y0) (S
629 z0)))).(f_equal nat nat S x0 y0 (IH x0 y0 (le_S_n z0 x0 H) (le_S_n z0 y0 H0)
630 H1))))))) y)))) x)))) z).
636 \forall (z: nat).(\forall (x1: nat).(\forall (x2: nat).(\forall (y1:
637 nat).(\forall (y2: nat).((le x1 z) \to ((le x2 z) \to ((eq nat (plus (minus z
638 x1) y1) (plus (minus z x2) y2)) \to (eq nat (plus x1 y2) (plus x2 y1)))))))))
640 \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x1: nat).(\forall (x2:
641 nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 n) \to ((le x2 n) \to ((eq
642 nat (plus (minus n x1) y1) (plus (minus n x2) y2)) \to (eq nat (plus x1 y2)
643 (plus x2 y1)))))))))) (\lambda (x1: nat).(\lambda (x2: nat).(\lambda (y1:
644 nat).(\lambda (y2: nat).(\lambda (H: (le x1 O)).(\lambda (H0: (le x2
645 O)).(\lambda (H1: (eq nat y1 y2)).(eq_ind nat y1 (\lambda (n: nat).(eq nat
646 (plus x1 n) (plus x2 y1))) (let H_y \def (le_n_O_eq x2 H0) in (eq_ind nat O
647 (\lambda (n: nat).(eq nat (plus x1 y1) (plus n y1))) (let H_y0 \def
648 (le_n_O_eq x1 H) in (eq_ind nat O (\lambda (n: nat).(eq nat (plus n y1) (plus
649 O y1))) (refl_equal nat (plus O y1)) x1 H_y0)) x2 H_y)) y2 H1))))))))
650 (\lambda (z0: nat).(\lambda (IH: ((\forall (x1: nat).(\forall (x2:
651 nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 z0) \to ((le x2 z0) \to
652 ((eq nat (plus (minus z0 x1) y1) (plus (minus z0 x2) y2)) \to (eq nat (plus
653 x1 y2) (plus x2 y1))))))))))).(\lambda (x1: nat).(nat_ind (\lambda (n:
654 nat).(\forall (x2: nat).(\forall (y1: nat).(\forall (y2: nat).((le n (S z0))
655 \to ((le x2 (S z0)) \to ((eq nat (plus (minus (S z0) n) y1) (plus (minus (S
656 z0) x2) y2)) \to (eq nat (plus n y2) (plus x2 y1))))))))) (\lambda (x2:
657 nat).(nat_ind (\lambda (n: nat).(\forall (y1: nat).(\forall (y2: nat).((le O
658 (S z0)) \to ((le n (S z0)) \to ((eq nat (plus (minus (S z0) O) y1) (plus
659 (minus (S z0) n) y2)) \to (eq nat (plus O y2) (plus n y1)))))))) (\lambda
660 (y1: nat).(\lambda (y2: nat).(\lambda (_: (le O (S z0))).(\lambda (_: (le O
661 (S z0))).(\lambda (H1: (eq nat (S (plus z0 y1)) (S (plus z0 y2)))).(let H_y
662 \def (IH O O) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n:
663 nat).(\forall (y3: nat).(\forall (y4: nat).((le O z0) \to ((le O z0) \to ((eq
664 nat (plus n y3) (plus n y4)) \to (eq nat y4 y3))))))) H_y z0 (minus_n_O z0))
665 in (H2 y1 y2 (le_O_n z0) (le_O_n z0) (eq_add_S (plus z0 y1) (plus z0 y2)
666 H1))))))))) (\lambda (x3: nat).(\lambda (_: ((\forall (y1: nat).(\forall (y2:
667 nat).((le O (S z0)) \to ((le x3 (S z0)) \to ((eq nat (S (plus z0 y1)) (plus
668 (match x3 with [O \Rightarrow (S z0) | (S l) \Rightarrow (minus z0 l)]) y2))
669 \to (eq nat y2 (plus x3 y1))))))))).(\lambda (y1: nat).(\lambda (y2:
670 nat).(\lambda (_: (le O (S z0))).(\lambda (H0: (le (S x3) (S z0))).(\lambda
671 (H1: (eq nat (S (plus z0 y1)) (plus (minus z0 x3) y2))).(let H_y \def (IH O
672 x3 (S y1)) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n:
673 nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (plus n (S
674 y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H_y z0
675 (minus_n_O z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y1)) (\lambda (n:
676 nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat n (plus
677 (minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H2 (S (plus z0 y1))
678 (plus_n_Sm z0 y1)) in (let H4 \def (eq_ind_r nat (plus x3 (S y1)) (\lambda
679 (n: nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (S (plus
680 z0 y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 n)))))) H3 (S (plus x3 y1))
681 (plus_n_Sm x3 y1)) in (H4 y2 (le_O_n z0) (le_S_n x3 z0 H0) H1))))))))))))
682 x2)) (\lambda (x2: nat).(\lambda (_: ((\forall (x3: nat).(\forall (y1:
683 nat).(\forall (y2: nat).((le x2 (S z0)) \to ((le x3 (S z0)) \to ((eq nat
684 (plus (minus (S z0) x2) y1) (plus (minus (S z0) x3) y2)) \to (eq nat (plus x2
685 y2) (plus x3 y1)))))))))).(\lambda (x3: nat).(nat_ind (\lambda (n:
686 nat).(\forall (y1: nat).(\forall (y2: nat).((le (S x2) (S z0)) \to ((le n (S
687 z0)) \to ((eq nat (plus (minus (S z0) (S x2)) y1) (plus (minus (S z0) n) y2))
688 \to (eq nat (plus (S x2) y2) (plus n y1)))))))) (\lambda (y1: nat).(\lambda
689 (y2: nat).(\lambda (H: (le (S x2) (S z0))).(\lambda (_: (le O (S
690 z0))).(\lambda (H1: (eq nat (plus (minus z0 x2) y1) (S (plus z0 y2)))).(let
691 H_y \def (IH x2 O y1 (S y2)) in (let H2 \def (eq_ind_r nat (minus z0 O)
692 (\lambda (n: nat).((le x2 z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2)
693 y1) (plus n (S y2))) \to (eq nat (plus x2 (S y2)) y1))))) H_y z0 (minus_n_O
694 z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y2)) (\lambda (n: nat).((le x2
695 z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) y1) n) \to (eq nat (plus
696 x2 (S y2)) y1))))) H2 (S (plus z0 y2)) (plus_n_Sm z0 y2)) in (let H4 \def
697 (eq_ind_r nat (plus x2 (S y2)) (\lambda (n: nat).((le x2 z0) \to ((le O z0)
698 \to ((eq nat (plus (minus z0 x2) y1) (S (plus z0 y2))) \to (eq nat n y1)))))
699 H3 (S (plus x2 y2)) (plus_n_Sm x2 y2)) in (H4 (le_S_n x2 z0 H) (le_O_n z0)
700 H1)))))))))) (\lambda (x4: nat).(\lambda (_: ((\forall (y1: nat).(\forall
701 (y2: nat).((le (S x2) (S z0)) \to ((le x4 (S z0)) \to ((eq nat (plus (minus
702 z0 x2) y1) (plus (match x4 with [O \Rightarrow (S z0) | (S l) \Rightarrow
703 (minus z0 l)]) y2)) \to (eq nat (S (plus x2 y2)) (plus x4
704 y1))))))))).(\lambda (y1: nat).(\lambda (y2: nat).(\lambda (H: (le (S x2) (S
705 z0))).(\lambda (H0: (le (S x4) (S z0))).(\lambda (H1: (eq nat (plus (minus z0
706 x2) y1) (plus (minus z0 x4) y2))).(f_equal nat nat S (plus x2 y2) (plus x4
707 y1) (IH x2 x4 y1 y2 (le_S_n x2 z0 H) (le_S_n x4 z0 H0) H1))))))))) x3))))
714 \forall (d: nat).(\forall (h: nat).(\forall (n: nat).((le (plus d h) n) \to
715 (le d (S (minus n h))))))
717 \lambda (d: nat).(\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le (plus
718 d h) n)).(let H0 \def (le_trans d (plus d h) n (le_plus_l d h) H) in (let H1
719 \def (eq_ind nat n (\lambda (n0: nat).(le d n0)) H0 (plus (minus n h) h)
720 (le_plus_minus_sym h n (le_trans h (plus d h) n (le_plus_r d h) H))) in (le_S
721 d (minus n h) (le_minus d n h H))))))).
727 \forall (x: nat).(\forall (y: nat).((lt x (pred y)) \to (lt (S x) y)))
729 \lambda (x: nat).(\lambda (y: nat).(nat_ind (\lambda (n: nat).((lt x (pred
730 n)) \to (lt (S x) n))) (\lambda (H: (lt x O)).(lt_x_O x H (lt (S x) O)))
731 (\lambda (n: nat).(\lambda (_: (((lt x (pred n)) \to (lt (S x) n)))).(\lambda
732 (H0: (lt x n)).(le_S_n (S (S x)) (S n) (le_n_S (S (S x)) (S n) (le_n_S (S x)