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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 *********************)
19 include "preamble.ma".
22 \forall (n1: nat).(\forall (n2: nat).(or (eq nat n1 n2) ((eq nat n1 n2) \to
23 (\forall (P: Prop).P))))
25 \lambda (n1: nat).(nat_ind (\lambda (n: nat).(\forall (n2: nat).(or (eq nat
26 n n2) ((eq nat n n2) \to (\forall (P: Prop).P))))) (\lambda (n2:
27 nat).(nat_ind (\lambda (n: nat).(or (eq nat O n) ((eq nat O n) \to (\forall
28 (P: Prop).P)))) (or_introl (eq nat O O) ((eq nat O O) \to (\forall (P:
29 Prop).P)) (refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (eq nat O n)
30 ((eq nat O n) \to (\forall (P: Prop).P)))).(or_intror (eq nat O (S n)) ((eq
31 nat O (S n)) \to (\forall (P: Prop).P)) (\lambda (H0: (eq nat O (S
32 n))).(\lambda (P: Prop).(let H1 \def (eq_ind nat O (\lambda (ee: nat).(match
33 ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _)
34 \Rightarrow False])) I (S n) H0) in (False_ind P H1))))))) n2)) (\lambda (n:
35 nat).(\lambda (H: ((\forall (n2: nat).(or (eq nat n n2) ((eq nat n n2) \to
36 (\forall (P: Prop).P)))))).(\lambda (n2: nat).(nat_ind (\lambda (n0: nat).(or
37 (eq nat (S n) n0) ((eq nat (S n) n0) \to (\forall (P: Prop).P)))) (or_intror
38 (eq nat (S n) O) ((eq nat (S n) O) \to (\forall (P: Prop).P)) (\lambda (H0:
39 (eq nat (S n) O)).(\lambda (P: Prop).(let H1 \def (eq_ind nat (S n) (\lambda
40 (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow
41 False | (S _) \Rightarrow True])) I O H0) in (False_ind P H1))))) (\lambda
42 (n0: nat).(\lambda (H0: (or (eq nat (S n) n0) ((eq nat (S n) n0) \to (\forall
43 (P: Prop).P)))).(or_ind (eq nat n n0) ((eq nat n n0) \to (\forall (P:
44 Prop).P)) (or (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to (\forall (P:
45 Prop).P))) (\lambda (H1: (eq nat n n0)).(let H2 \def (eq_ind_r nat n0
46 (\lambda (n3: nat).(or (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P:
47 Prop).P)))) H0 n H1) in (eq_ind nat n (\lambda (n3: nat).(or (eq nat (S n) (S
48 n3)) ((eq nat (S n) (S n3)) \to (\forall (P: Prop).P)))) (or_introl (eq nat
49 (S n) (S n)) ((eq nat (S n) (S n)) \to (\forall (P: Prop).P)) (refl_equal nat
50 (S n))) n0 H1))) (\lambda (H1: (((eq nat n n0) \to (\forall (P:
51 Prop).P)))).(or_intror (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to
52 (\forall (P: Prop).P)) (\lambda (H2: (eq nat (S n) (S n0))).(\lambda (P:
53 Prop).(let H3 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return
54 (\lambda (_: nat).nat) with [O \Rightarrow n | (S n3) \Rightarrow n3])) (S n)
55 (S n0) H2) in (let H4 \def (eq_ind_r nat n0 (\lambda (n3: nat).((eq nat n n3)
56 \to (\forall (P0: Prop).P0))) H1 n H3) in (let H5 \def (eq_ind_r nat n0
57 (\lambda (n3: nat).(or (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P0:
58 Prop).P0)))) H0 n H3) in (H4 (refl_equal nat n) P)))))))) (H n0)))) n2))))
62 \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus m n)
63 (plus p n)) \to (eq nat m p))))
65 \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (eq nat
66 (plus m n) (plus p n))).(plus_reg_l n m p (eq_ind_r nat (plus m n) (\lambda
67 (n0: nat).(eq nat n0 (plus n p))) (eq_ind_r nat (plus p n) (\lambda (n0:
68 nat).(eq nat n0 (plus n p))) (sym_eq nat (plus n p) (plus p n) (plus_comm n
69 p)) (plus m n) H) (plus n m) (plus_comm n m)))))).
72 \forall (m: nat).(\forall (n: nat).(eq nat (minus (plus m n) n) m))
74 \lambda (m: nat).(\lambda (n: nat).(eq_ind_r nat (plus n m) (\lambda (n0:
75 nat).(eq nat (minus n0 n) m)) (minus_plus n m) (plus m n) (plus_comm m n))).
77 theorem plus_permute_2_in_3:
78 \forall (x: nat).(\forall (y: nat).(\forall (z: nat).(eq nat (plus (plus x
79 y) z) (plus (plus x z) y))))
81 \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(eq_ind_r nat (plus x
82 (plus y z)) (\lambda (n: nat).(eq nat n (plus (plus x z) y))) (eq_ind_r nat
83 (plus z y) (\lambda (n: nat).(eq nat (plus x n) (plus (plus x z) y))) (eq_ind
84 nat (plus (plus x z) y) (\lambda (n: nat).(eq nat n (plus (plus x z) y)))
85 (refl_equal nat (plus (plus x z) y)) (plus x (plus z y)) (plus_assoc_reverse
86 x z y)) (plus y z) (plus_comm y z)) (plus (plus x y) z) (plus_assoc_reverse x
89 theorem plus_permute_2_in_3_assoc:
90 \forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq nat (plus (plus n
91 h) k) (plus n (plus k h)))))
93 \lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind_r nat (plus
94 (plus n k) h) (\lambda (n0: nat).(eq nat n0 (plus n (plus k h)))) (eq_ind_r
95 nat (plus (plus n k) h) (\lambda (n0: nat).(eq nat (plus (plus n k) h) n0))
96 (refl_equal nat (plus (plus n k) h)) (plus n (plus k h)) (plus_assoc n k h))
97 (plus (plus n h) k) (plus_permute_2_in_3 n h k)))).
100 \forall (x: nat).(\forall (y: nat).((eq nat (plus x y) O) \to (land (eq nat
103 \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq nat (plus
104 n y) O) \to (land (eq nat n O) (eq nat y O))))) (\lambda (y: nat).(\lambda
105 (H: (eq nat (plus O y) O)).(conj (eq nat O O) (eq nat y O) (refl_equal nat O)
106 H))) (\lambda (n: nat).(\lambda (_: ((\forall (y: nat).((eq nat (plus n y) O)
107 \to (land (eq nat n O) (eq nat y O)))))).(\lambda (y: nat).(\lambda (H0: (eq
108 nat (plus (S n) y) O)).(let H1 \def (match H0 in eq return (\lambda (n0:
109 nat).(\lambda (_: (eq ? ? n0)).((eq nat n0 O) \to (land (eq nat (S n) O) (eq
110 nat y O))))) with [refl_equal \Rightarrow (\lambda (H1: (eq nat (plus (S n)
111 y) O)).(let H2 \def (eq_ind nat (plus (S n) y) (\lambda (e: nat).(match e in
112 nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
113 \Rightarrow True])) I O H1) in (False_ind (land (eq nat (S n) O) (eq nat y
114 O)) H2)))]) in (H1 (refl_equal nat O))))))) x).
117 \forall (x: nat).(eq nat (minus (S x) (S O)) x)
119 \lambda (x: nat).(eq_ind nat x (\lambda (n: nat).(eq nat n x)) (refl_equal
120 nat x) (minus x O) (minus_n_O x)).
123 \forall (i: nat).(\forall (j: nat).(or (not (eq nat i j)) (eq nat i j)))
125 \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (j: nat).(or (not (eq
126 nat n j)) (eq nat n j)))) (\lambda (j: nat).(nat_ind (\lambda (n: nat).(or
127 (not (eq nat O n)) (eq nat O n))) (or_intror (not (eq nat O O)) (eq nat O O)
128 (refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (not (eq nat O n)) (eq
129 nat O n))).(or_introl (not (eq nat O (S n))) (eq nat O (S n)) (O_S n)))) j))
130 (\lambda (n: nat).(\lambda (H: ((\forall (j: nat).(or (not (eq nat n j)) (eq
131 nat n j))))).(\lambda (j: nat).(nat_ind (\lambda (n0: nat).(or (not (eq nat
132 (S n) n0)) (eq nat (S n) n0))) (or_introl (not (eq nat (S n) O)) (eq nat (S
133 n) O) (sym_not_eq nat O (S n) (O_S n))) (\lambda (n0: nat).(\lambda (_: (or
134 (not (eq nat (S n) n0)) (eq nat (S n) n0))).(or_ind (not (eq nat n n0)) (eq
135 nat n n0) (or (not (eq nat (S n) (S n0))) (eq nat (S n) (S n0))) (\lambda
136 (H1: (not (eq nat n n0))).(or_introl (not (eq nat (S n) (S n0))) (eq nat (S
137 n) (S n0)) (not_eq_S n n0 H1))) (\lambda (H1: (eq nat n n0)).(or_intror (not
138 (eq nat (S n) (S n0))) (eq nat (S n) (S n0)) (f_equal nat nat S n n0 H1))) (H
142 \forall (i: nat).(\forall (j: nat).(\forall (P: Prop).((((not (eq nat i j))
143 \to P)) \to ((((eq nat i j) \to P)) \to P))))
145 \lambda (i: nat).(\lambda (j: nat).(\lambda (P: Prop).(\lambda (H: (((not
146 (eq nat i j)) \to P))).(\lambda (H0: (((eq nat i j) \to P))).(let o \def
147 (eq_nat_dec i j) in (or_ind (not (eq nat i j)) (eq nat i j) P H H0 o)))))).
150 \forall (m: nat).(\forall (n: nat).(\forall (P: Prop).((le m n) \to ((le (S
153 \lambda (m: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).(\forall (P:
154 Prop).((le n n0) \to ((le (S n0) n) \to P))))) (\lambda (n: nat).(\lambda (P:
155 Prop).(\lambda (_: (le O n)).(\lambda (H0: (le (S n) O)).(let H1 \def (match
156 H0 in le return (\lambda (n0: nat).(\lambda (_: (le ? n0)).((eq nat n0 O) \to
157 P))) with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def
158 (eq_ind nat (S n) (\lambda (e: nat).(match e in nat return (\lambda (_:
159 nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in
160 (False_ind P H2))) | (le_S m0 H1) \Rightarrow (\lambda (H2: (eq nat (S m0)
161 O)).((let H3 \def (eq_ind nat (S m0) (\lambda (e: nat).(match e in nat return
162 (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True]))
163 I O H2) in (False_ind ((le (S n) m0) \to P) H3)) H1))]) in (H1 (refl_equal
164 nat O))))))) (\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(\forall (P:
165 Prop).((le n n0) \to ((le (S n0) n) \to P)))))).(\lambda (n0: nat).(nat_ind
166 (\lambda (n1: nat).(\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n))
167 \to P)))) (\lambda (P: Prop).(\lambda (H0: (le (S n) O)).(\lambda (_: (le (S
168 O) (S n))).(let H2 \def (match H0 in le return (\lambda (n1: nat).(\lambda
169 (_: (le ? n1)).((eq nat n1 O) \to P))) with [le_n \Rightarrow (\lambda (H2:
170 (eq nat (S n) O)).(let H3 \def (eq_ind nat (S n) (\lambda (e: nat).(match e
171 in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
172 \Rightarrow True])) I O H2) in (False_ind P H3))) | (le_S m0 H2) \Rightarrow
173 (\lambda (H3: (eq nat (S m0) O)).((let H4 \def (eq_ind nat (S m0) (\lambda
174 (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow
175 False | (S _) \Rightarrow True])) I O H3) in (False_ind ((le (S n) m0) \to P)
176 H4)) H2))]) in (H2 (refl_equal nat O)))))) (\lambda (n1: nat).(\lambda (_:
177 ((\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n)) \to P))))).(\lambda
178 (P: Prop).(\lambda (H1: (le (S n) (S n1))).(\lambda (H2: (le (S (S n1)) (S
179 n))).(H n1 P (le_S_n n n1 H1) (le_S_n (S n1) n H2))))))) n0)))) m).
182 \forall (x: nat).((le (S x) x) \to (\forall (P: Prop).P))
184 \lambda (x: nat).(\lambda (H: (le (S x) x)).(\lambda (P: Prop).(let H0 \def
185 le_Sn_n in (False_ind P (H0 x H))))).
188 \forall (n: nat).(\forall (m: nat).((le n m) \to (le (pred n) (pred m))))
190 \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda
191 (n0: nat).(le (pred n) (pred n0))) (le_n (pred n)) (\lambda (m0:
192 nat).(\lambda (_: (le n m0)).(\lambda (H1: (le (pred n) (pred m0))).(le_trans
193 (pred n) (pred m0) m0 H1 (le_pred_n m0))))) m H))).
196 \forall (x: nat).(\forall (y: nat).(le (minus x y) x))
198 \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).(le (minus n
199 y) n))) (\lambda (_: nat).(le_n O)) (\lambda (n: nat).(\lambda (H: ((\forall
200 (y: nat).(le (minus n y) n)))).(\lambda (y: nat).(nat_ind (\lambda (n0:
201 nat).(le (minus (S n) n0) (S n))) (le_n (S n)) (\lambda (n0: nat).(\lambda
202 (_: (le (match n0 with [O \Rightarrow (S n) | (S l) \Rightarrow (minus n l)])
203 (S n))).(le_S (minus n n0) n (H n0)))) y)))) x).
205 theorem le_plus_minus_sym:
206 \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus (minus m n)
209 \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(eq_ind_r nat
210 (plus n (minus m n)) (\lambda (n0: nat).(eq nat m n0)) (le_plus_minus n m H)
211 (plus (minus m n) n) (plus_comm (minus m n) n)))).
213 theorem le_minus_minus:
214 \forall (x: nat).(\forall (y: nat).((le x y) \to (\forall (z: nat).((le y z)
215 \to (le (minus y x) (minus z x))))))
217 \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (z:
218 nat).(\lambda (H0: (le y z)).(plus_le_reg_l x (minus y x) (minus z x)
219 (eq_ind_r nat y (\lambda (n: nat).(le n (plus x (minus z x)))) (eq_ind_r nat
220 z (\lambda (n: nat).(le y n)) H0 (plus x (minus z x)) (le_plus_minus_r x z
221 (le_trans x y z H H0))) (plus x (minus y x)) (le_plus_minus_r x y H))))))).
223 theorem le_minus_plus:
224 \forall (z: nat).(\forall (x: nat).((le z x) \to (\forall (y: nat).(eq nat
225 (minus (plus x y) z) (plus (minus x z) y)))))
227 \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((le n x) \to
228 (\forall (y: nat).(eq nat (minus (plus x y) n) (plus (minus x n) y))))))
229 (\lambda (x: nat).(\lambda (H: (le O x)).(let H0 \def (match H in le return
230 (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n x) \to (\forall (y:
231 nat).(eq nat (minus (plus x y) O) (plus (minus x O) y)))))) with [le_n
232 \Rightarrow (\lambda (H0: (eq nat O x)).(eq_ind nat O (\lambda (n:
233 nat).(\forall (y: nat).(eq nat (minus (plus n y) O) (plus (minus n O) y))))
234 (\lambda (y: nat).(sym_eq nat (plus (minus O O) y) (minus (plus O y) O)
235 (minus_n_O (plus O y)))) x H0)) | (le_S m H0) \Rightarrow (\lambda (H1: (eq
236 nat (S m) x)).(eq_ind nat (S m) (\lambda (n: nat).((le O m) \to (\forall (y:
237 nat).(eq nat (minus (plus n y) O) (plus (minus n O) y))))) (\lambda (_: (le O
238 m)).(\lambda (y: nat).(refl_equal nat (plus (minus (S m) O) y)))) x H1 H0))])
239 in (H0 (refl_equal nat x))))) (\lambda (z0: nat).(\lambda (H: ((\forall (x:
240 nat).((le z0 x) \to (\forall (y: nat).(eq nat (minus (plus x y) z0) (plus
241 (minus x z0) y))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).((le (S
242 z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n
243 (S z0)) y))))) (\lambda (H0: (le (S z0) O)).(\lambda (y: nat).(let H1 \def
244 (match H0 in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O)
245 \to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))))) with
246 [le_n \Rightarrow (\lambda (H1: (eq nat (S z0) O)).(let H2 \def (eq_ind nat
247 (S z0) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with
248 [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind (eq
249 nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y)) H2))) | (le_S m H1)
250 \Rightarrow (\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m)
251 (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O
252 \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind ((le (S
253 z0) m) \to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))) H3))
254 H1))]) in (H1 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: (((le (S
255 z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n
256 (S z0)) y)))))).(\lambda (H1: (le (S z0) (S n))).(\lambda (y: nat).(H n
257 (le_S_n z0 n H1) y))))) x)))) z).
260 \forall (x: nat).(\forall (z: nat).(\forall (y: nat).((le (plus x y) z) \to
261 (le x (minus z y)))))
263 \lambda (x: nat).(\lambda (z: nat).(\lambda (y: nat).(\lambda (H: (le (plus
264 x y) z)).(eq_ind nat (minus (plus x y) y) (\lambda (n: nat).(le n (minus z
265 y))) (le_minus_minus y (plus x y) (le_plus_r x y) z H) x (minus_plus_r x
268 theorem le_trans_plus_r:
269 \forall (x: nat).(\forall (y: nat).(\forall (z: nat).((le (plus x y) z) \to
272 \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(\lambda (H: (le (plus
273 x y) z)).(le_trans y (plus x y) z (le_plus_r x y) H)))).
276 \forall (x: nat).((lt x O) \to (\forall (P: Prop).P))
278 \lambda (x: nat).(\lambda (H: (le (S x) O)).(\lambda (P: Prop).(let H_y \def
279 (le_n_O_eq (S x) H) in (let H0 \def (eq_ind nat O (\lambda (ee: nat).(match
280 ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _)
281 \Rightarrow False])) I (S x) H_y) in (False_ind P H0))))).
284 \forall (m: nat).(\forall (x: nat).((le (S m) x) \to (ex2 nat (\lambda (n:
285 nat).(eq nat x (S n))) (\lambda (n: nat).(le m n)))))
287 \lambda (m: nat).(\lambda (x: nat).(\lambda (H: (le (S m) x)).(let H0 \def
288 (match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n x)
289 \to (ex2 nat (\lambda (n0: nat).(eq nat x (S n0))) (\lambda (n0: nat).(le m
290 n0)))))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S m) x)).(eq_ind nat
291 (S m) (\lambda (n: nat).(ex2 nat (\lambda (n0: nat).(eq nat n (S n0)))
292 (\lambda (n0: nat).(le m n0)))) (ex_intro2 nat (\lambda (n: nat).(eq nat (S
293 m) (S n))) (\lambda (n: nat).(le m n)) m (refl_equal nat (S m)) (le_n m)) x
294 H0)) | (le_S m0 H0) \Rightarrow (\lambda (H1: (eq nat (S m0) x)).(eq_ind nat
295 (S m0) (\lambda (n: nat).((le (S m) m0) \to (ex2 nat (\lambda (n0: nat).(eq
296 nat n (S n0))) (\lambda (n0: nat).(le m n0))))) (\lambda (H2: (le (S m)
297 m0)).(ex_intro2 nat (\lambda (n: nat).(eq nat (S m0) (S n))) (\lambda (n:
298 nat).(le m n)) m0 (refl_equal nat (S m0)) (le_S_n m m0 (le_S (S m) m0 H2))))
299 x H1 H0))]) in (H0 (refl_equal nat x))))).
301 theorem lt_x_plus_x_Sy:
302 \forall (x: nat).(\forall (y: nat).(lt x (plus x (S y))))
304 \lambda (x: nat).(\lambda (y: nat).(eq_ind_r nat (plus (S y) x) (\lambda (n:
305 nat).(lt x n)) (le_S_n (S x) (S (plus y x)) (le_n_S (S x) (S (plus y x))
306 (le_n_S x (plus y x) (le_plus_r y x)))) (plus x (S y)) (plus_comm x (S y)))).
308 theorem simpl_lt_plus_r:
309 \forall (p: nat).(\forall (n: nat).(\forall (m: nat).((lt (plus n p) (plus m
312 \lambda (p: nat).(\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (plus
313 n p) (plus m p))).(plus_lt_reg_l n m p (let H0 \def (eq_ind nat (plus n p)
314 (\lambda (n0: nat).(lt n0 (plus m p))) H (plus p n) (plus_comm n p)) in (let
315 H1 \def (eq_ind nat (plus m p) (\lambda (n0: nat).(lt (plus p n) n0)) H0
316 (plus p m) (plus_comm m p)) in H1)))))).
319 \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq nat (minus x y) (S
322 \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to
323 (eq nat (minus n y) (S (minus n (S y))))))) (\lambda (y: nat).(\lambda (H:
324 (lt y O)).(let H0 \def (match H in le return (\lambda (n: nat).(\lambda (_:
325 (le ? n)).((eq nat n O) \to (eq nat (minus O y) (S (minus O (S y))))))) with
326 [le_n \Rightarrow (\lambda (H0: (eq nat (S y) O)).(let H1 \def (eq_ind nat (S
327 y) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O
328 \Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (eq nat
329 (minus O y) (S (minus O (S y)))) H1))) | (le_S m H0) \Rightarrow (\lambda
330 (H1: (eq nat (S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e:
331 nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False
332 | (S _) \Rightarrow True])) I O H1) in (False_ind ((le (S y) m) \to (eq nat
333 (minus O y) (S (minus O (S y))))) H2)) H0))]) in (H0 (refl_equal nat O)))))
334 (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to (eq nat
335 (minus n y) (S (minus n (S y)))))))).(\lambda (y: nat).(nat_ind (\lambda (n0:
336 nat).((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus (S n) (S n0))))))
337 (\lambda (_: (lt O (S n))).(eq_ind nat n (\lambda (n0: nat).(eq nat (S n) (S
338 n0))) (refl_equal nat (S n)) (minus n O) (minus_n_O n))) (\lambda (n0:
339 nat).(\lambda (_: (((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus (S
340 n) (S n0))))))).(\lambda (H1: (lt (S n0) (S n))).(let H2 \def (le_S_n (S n0)
341 n H1) in (H n0 H2))))) y)))) x).
343 theorem lt_plus_minus:
344 \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus x (minus
347 \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_plus_minus (S
350 theorem lt_plus_minus_r:
351 \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus (minus y
354 \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(eq_ind_r nat
355 (plus x (minus y (S x))) (\lambda (n: nat).(eq nat y (S n))) (lt_plus_minus x
356 y H) (plus (minus y (S x)) x) (plus_comm (minus y (S x)) x)))).
359 \forall (x: nat).((lt O x) \to (eq nat x (S (minus x (S O)))))
361 \lambda (x: nat).(\lambda (H: (lt O x)).(eq_ind nat (minus x O) (\lambda (n:
362 nat).(eq nat x n)) (eq_ind nat x (\lambda (n: nat).(eq nat x n)) (refl_equal
363 nat x) (minus x O) (minus_n_O x)) (S (minus x (S O))) (minus_x_Sy x O H))).
366 \forall (y: nat).(\forall (x: nat).((lt x y) \to (le x (pred y))))
368 \lambda (y: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((lt x n) \to
369 (le x (pred n))))) (\lambda (x: nat).(\lambda (H: (lt x O)).(let H0 \def
370 (match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O)
371 \to (le x O)))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S x) O)).(let
372 H1 \def (eq_ind nat (S x) (\lambda (e: nat).(match e in nat return (\lambda
373 (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H0)
374 in (False_ind (le x O) H1))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat
375 (S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e: nat).(match e in nat
376 return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
377 True])) I O H1) in (False_ind ((le (S x) m) \to (le x O)) H2)) H0))]) in (H0
378 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: ((\forall (x: nat).((lt
379 x n) \to (le x (pred n)))))).(\lambda (x: nat).(\lambda (H0: (lt x (S
380 n))).(le_S_n x n H0))))) y).
383 \forall (x: nat).(\forall (y: nat).((lt x y) \to (le x (minus y (S O)))))
385 \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_minus x y (S
386 O) (eq_ind_r nat (plus (S O) x) (\lambda (n: nat).(le n y)) H (plus x (S O))
387 (plus_comm x (S O)))))).
390 \forall (n: nat).(\forall (d: nat).(\forall (P: Prop).((((lt n d) \to P))
391 \to ((((le d n) \to P)) \to P))))
393 \lambda (n: nat).(\lambda (d: nat).(\lambda (P: Prop).(\lambda (H: (((lt n
394 d) \to P))).(\lambda (H0: (((le d n) \to P))).(let H1 \def (le_or_lt d n) in
395 (or_ind (le d n) (lt n d) P H0 H H1)))))).
398 \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P))
399 \to ((((eq nat x y) \to P)) \to ((le x y) \to P)))))
401 \lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x
402 y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (le x
403 y)).(or_ind (lt x y) (eq nat x y) P H H0 (le_lt_or_eq x y H1))))))).
406 \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P))
407 \to ((((eq nat x y) \to P)) \to ((((lt y x) \to P)) \to P)))))
409 \lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x
410 y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (((lt y x)
411 \to P))).(lt_le_e x y P H (\lambda (H2: (le y x)).(lt_eq_e y x P H1 (\lambda
412 (H3: (eq nat y x)).(H0 (sym_eq nat y x H3))) H2)))))))).
415 \forall (x: nat).(\forall (n: nat).((lt x (S n)) \to (or (eq nat x O) (ex2
416 nat (\lambda (m: nat).(eq nat x (S m))) (\lambda (m: nat).(lt m n))))))
418 \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((lt n (S
419 n0)) \to (or (eq nat n O) (ex2 nat (\lambda (m: nat).(eq nat n (S m)))
420 (\lambda (m: nat).(lt m n0))))))) (\lambda (n: nat).(\lambda (_: (lt O (S
421 n))).(or_introl (eq nat O O) (ex2 nat (\lambda (m: nat).(eq nat O (S m)))
422 (\lambda (m: nat).(lt m n))) (refl_equal nat O)))) (\lambda (n: nat).(\lambda
423 (_: ((\forall (n0: nat).((lt n (S n0)) \to (or (eq nat n O) (ex2 nat (\lambda
424 (m: nat).(eq nat n (S m))) (\lambda (m: nat).(lt m n0)))))))).(\lambda (n0:
425 nat).(\lambda (H0: (lt (S n) (S n0))).(or_intror (eq nat (S n) O) (ex2 nat
426 (\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt m n0)))
427 (ex_intro2 nat (\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt
428 m n0)) n (refl_equal nat (S n)) (le_S_n (S n) n0 H0))))))) x).
431 \forall (x: nat).(\forall (y: nat).((le x y) \to ((lt y x) \to (\forall (P:
434 \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (H0: (lt
435 y x)).(\lambda (P: Prop).(False_ind P (le_not_lt x y H H0)))))).
438 \forall (x: nat).(\forall (y: nat).((lt x y) \to (not (eq nat x y))))
440 \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(\lambda (H0: (eq
441 nat x y)).(let H1 \def (eq_ind nat x (\lambda (n: nat).(lt n y)) H y H0) in
442 (lt_irrefl y H1))))).
445 \forall (h2: nat).(\forall (d2: nat).(\forall (n: nat).((le (plus d2 h2) n)
446 \to (\forall (h1: nat).(le (plus d2 h1) (minus (plus n h1) h2))))))
448 \lambda (h2: nat).(\lambda (d2: nat).(\lambda (n: nat).(\lambda (H: (le
449 (plus d2 h2) n)).(\lambda (h1: nat).(eq_ind nat (minus (plus h2 (plus d2 h1))
450 h2) (\lambda (n0: nat).(le n0 (minus (plus n h1) h2))) (le_minus_minus h2
451 (plus h2 (plus d2 h1)) (le_plus_l h2 (plus d2 h1)) (plus n h1) (eq_ind_r nat
452 (plus (plus h2 d2) h1) (\lambda (n0: nat).(le n0 (plus n h1))) (eq_ind_r nat
453 (plus d2 h2) (\lambda (n0: nat).(le (plus n0 h1) (plus n h1))) (le_S_n (plus
454 (plus d2 h2) h1) (plus n h1) (lt_le_S (plus (plus d2 h2) h1) (S (plus n h1))
455 (le_lt_n_Sm (plus (plus d2 h2) h1) (plus n h1) (plus_le_compat (plus d2 h2) n
456 h1 h1 H (le_n h1))))) (plus h2 d2) (plus_comm h2 d2)) (plus h2 (plus d2 h1))
457 (plus_assoc h2 d2 h1))) (plus d2 h1) (minus_plus h2 (plus d2 h1))))))).
460 \forall (x: nat).(\forall (y: nat).((le x y) \to (eq nat (minus x y) O)))
462 \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to
463 (eq nat (minus n y) O)))) (\lambda (y: nat).(\lambda (_: (le O
464 y)).(refl_equal nat O))) (\lambda (x0: nat).(\lambda (H: ((\forall (y:
465 nat).((le x0 y) \to (eq nat (minus x0 y) O))))).(\lambda (y: nat).(nat_ind
466 (\lambda (n: nat).((le (S x0) n) \to (eq nat (match n with [O \Rightarrow (S
467 x0) | (S l) \Rightarrow (minus x0 l)]) O))) (\lambda (H0: (le (S x0)
468 O)).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le x0
469 n)) (eq nat (S x0) O) (\lambda (x1: nat).(\lambda (H1: (eq nat O (S
470 x1))).(\lambda (_: (le x0 x1)).(let H3 \def (eq_ind nat O (\lambda (ee:
471 nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True
472 | (S _) \Rightarrow False])) I (S x1) H1) in (False_ind (eq nat (S x0) O)
473 H3))))) (le_gen_S x0 O H0))) (\lambda (n: nat).(\lambda (_: (((le (S x0) n)
474 \to (eq nat (match n with [O \Rightarrow (S x0) | (S l) \Rightarrow (minus x0
475 l)]) O)))).(\lambda (H1: (le (S x0) (S n))).(H n (le_S_n x0 n H1))))) y))))
479 \forall (z: nat).(\forall (x: nat).(\forall (y: nat).((le z x) \to ((le z y)
480 \to ((eq nat (minus x z) (minus y z)) \to (eq nat x y))))))
482 \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).(\forall (y:
483 nat).((le n x) \to ((le n y) \to ((eq nat (minus x n) (minus y n)) \to (eq
484 nat x y))))))) (\lambda (x: nat).(\lambda (y: nat).(\lambda (_: (le O
485 x)).(\lambda (_: (le O y)).(\lambda (H1: (eq nat (minus x O) (minus y
486 O))).(let H2 \def (eq_ind_r nat (minus x O) (\lambda (n: nat).(eq nat n
487 (minus y O))) H1 x (minus_n_O x)) in (let H3 \def (eq_ind_r nat (minus y O)
488 (\lambda (n: nat).(eq nat x n)) H2 y (minus_n_O y)) in H3))))))) (\lambda
489 (z0: nat).(\lambda (IH: ((\forall (x: nat).(\forall (y: nat).((le z0 x) \to
490 ((le z0 y) \to ((eq nat (minus x z0) (minus y z0)) \to (eq nat x
491 y)))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le
492 (S z0) n) \to ((le (S z0) y) \to ((eq nat (minus n (S z0)) (minus y (S z0)))
493 \to (eq nat n y)))))) (\lambda (y: nat).(\lambda (H: (le (S z0) O)).(\lambda
494 (_: (le (S z0) y)).(\lambda (_: (eq nat (minus O (S z0)) (minus y (S
495 z0)))).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le
496 z0 n)) (eq nat O y) (\lambda (x0: nat).(\lambda (H2: (eq nat O (S
497 x0))).(\lambda (_: (le z0 x0)).(let H4 \def (eq_ind nat O (\lambda (ee:
498 nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True
499 | (S _) \Rightarrow False])) I (S x0) H2) in (False_ind (eq nat O y) H4)))))
500 (le_gen_S z0 O H)))))) (\lambda (x0: nat).(\lambda (_: ((\forall (y:
501 nat).((le (S z0) x0) \to ((le (S z0) y) \to ((eq nat (minus x0 (S z0)) (minus
502 y (S z0))) \to (eq nat x0 y))))))).(\lambda (y: nat).(nat_ind (\lambda (n:
503 nat).((le (S z0) (S x0)) \to ((le (S z0) n) \to ((eq nat (minus (S x0) (S
504 z0)) (minus n (S z0))) \to (eq nat (S x0) n))))) (\lambda (_: (le (S z0) (S
505 x0))).(\lambda (H0: (le (S z0) O)).(\lambda (_: (eq nat (minus (S x0) (S z0))
506 (minus O (S z0)))).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda
507 (n: nat).(le z0 n)) (eq nat (S x0) O) (\lambda (x1: nat).(\lambda (H2: (eq
508 nat O (S x1))).(\lambda (_: (le z0 x1)).(let H4 \def (eq_ind nat O (\lambda
509 (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow
510 True | (S _) \Rightarrow False])) I (S x1) H2) in (False_ind (eq nat (S x0)
511 O) H4))))) (le_gen_S z0 O H0))))) (\lambda (y0: nat).(\lambda (_: (((le (S
512 z0) (S x0)) \to ((le (S z0) y0) \to ((eq nat (minus (S x0) (S z0)) (minus y0
513 (S z0))) \to (eq nat (S x0) y0)))))).(\lambda (H: (le (S z0) (S
514 x0))).(\lambda (H0: (le (S z0) (S y0))).(\lambda (H1: (eq nat (minus (S x0)
515 (S z0)) (minus (S y0) (S z0)))).(f_equal nat nat S x0 y0 (IH x0 y0 (le_S_n z0
516 x0 H) (le_S_n z0 y0 H0) H1))))))) y)))) x)))) z).
519 \forall (z: nat).(\forall (x1: nat).(\forall (x2: nat).(\forall (y1:
520 nat).(\forall (y2: nat).((le x1 z) \to ((le x2 z) \to ((eq nat (plus (minus z
521 x1) y1) (plus (minus z x2) y2)) \to (eq nat (plus x1 y2) (plus x2 y1)))))))))
523 \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x1: nat).(\forall (x2:
524 nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 n) \to ((le x2 n) \to ((eq
525 nat (plus (minus n x1) y1) (plus (minus n x2) y2)) \to (eq nat (plus x1 y2)
526 (plus x2 y1)))))))))) (\lambda (x1: nat).(\lambda (x2: nat).(\lambda (y1:
527 nat).(\lambda (y2: nat).(\lambda (H: (le x1 O)).(\lambda (H0: (le x2
528 O)).(\lambda (H1: (eq nat y1 y2)).(eq_ind nat y1 (\lambda (n: nat).(eq nat
529 (plus x1 n) (plus x2 y1))) (let H_y \def (le_n_O_eq x2 H0) in (eq_ind nat O
530 (\lambda (n: nat).(eq nat (plus x1 y1) (plus n y1))) (let H_y0 \def
531 (le_n_O_eq x1 H) in (eq_ind nat O (\lambda (n: nat).(eq nat (plus n y1) (plus
532 O y1))) (refl_equal nat (plus O y1)) x1 H_y0)) x2 H_y)) y2 H1))))))))
533 (\lambda (z0: nat).(\lambda (IH: ((\forall (x1: nat).(\forall (x2:
534 nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 z0) \to ((le x2 z0) \to
535 ((eq nat (plus (minus z0 x1) y1) (plus (minus z0 x2) y2)) \to (eq nat (plus
536 x1 y2) (plus x2 y1))))))))))).(\lambda (x1: nat).(nat_ind (\lambda (n:
537 nat).(\forall (x2: nat).(\forall (y1: nat).(\forall (y2: nat).((le n (S z0))
538 \to ((le x2 (S z0)) \to ((eq nat (plus (minus (S z0) n) y1) (plus (minus (S
539 z0) x2) y2)) \to (eq nat (plus n y2) (plus x2 y1))))))))) (\lambda (x2:
540 nat).(nat_ind (\lambda (n: nat).(\forall (y1: nat).(\forall (y2: nat).((le O
541 (S z0)) \to ((le n (S z0)) \to ((eq nat (plus (minus (S z0) O) y1) (plus
542 (minus (S z0) n) y2)) \to (eq nat (plus O y2) (plus n y1)))))))) (\lambda
543 (y1: nat).(\lambda (y2: nat).(\lambda (_: (le O (S z0))).(\lambda (_: (le O
544 (S z0))).(\lambda (H1: (eq nat (S (plus z0 y1)) (S (plus z0 y2)))).(let H_y
545 \def (IH O O) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n:
546 nat).(\forall (y3: nat).(\forall (y4: nat).((le O z0) \to ((le O z0) \to ((eq
547 nat (plus n y3) (plus n y4)) \to (eq nat y4 y3))))))) H_y z0 (minus_n_O z0))
548 in (H2 y1 y2 (le_O_n z0) (le_O_n z0) (H2 (plus z0 y2) (plus z0 y1) (le_O_n
549 z0) (le_O_n z0) (f_equal nat nat (plus z0) (plus z0 y2) (plus z0 y1) (sym_eq
550 nat (plus z0 y1) (plus z0 y2) (eq_add_S (plus z0 y1) (plus z0 y2)
551 H1)))))))))))) (\lambda (x3: nat).(\lambda (_: ((\forall (y1: nat).(\forall
552 (y2: nat).((le O (S z0)) \to ((le x3 (S z0)) \to ((eq nat (S (plus z0 y1))
553 (plus (match x3 with [O \Rightarrow (S z0) | (S l) \Rightarrow (minus z0 l)])
554 y2)) \to (eq nat y2 (plus x3 y1))))))))).(\lambda (y1: nat).(\lambda (y2:
555 nat).(\lambda (_: (le O (S z0))).(\lambda (H0: (le (S x3) (S z0))).(\lambda
556 (H1: (eq nat (S (plus z0 y1)) (plus (minus z0 x3) y2))).(let H_y \def (IH O
557 x3 (S y1)) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n:
558 nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (plus n (S
559 y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H_y z0
560 (minus_n_O z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y1)) (\lambda (n:
561 nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat n (plus
562 (minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H2 (S (plus z0 y1))
563 (plus_n_Sm z0 y1)) in (let H4 \def (eq_ind_r nat (plus x3 (S y1)) (\lambda
564 (n: nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (S (plus
565 z0 y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 n)))))) H3 (S (plus x3 y1))
566 (plus_n_Sm x3 y1)) in (H4 y2 (le_O_n z0) (le_S_n x3 z0 H0) H1))))))))))))
567 x2)) (\lambda (x2: nat).(\lambda (_: ((\forall (x3: nat).(\forall (y1:
568 nat).(\forall (y2: nat).((le x2 (S z0)) \to ((le x3 (S z0)) \to ((eq nat
569 (plus (minus (S z0) x2) y1) (plus (minus (S z0) x3) y2)) \to (eq nat (plus x2
570 y2) (plus x3 y1)))))))))).(\lambda (x3: nat).(nat_ind (\lambda (n:
571 nat).(\forall (y1: nat).(\forall (y2: nat).((le (S x2) (S z0)) \to ((le n (S
572 z0)) \to ((eq nat (plus (minus (S z0) (S x2)) y1) (plus (minus (S z0) n) y2))
573 \to (eq nat (plus (S x2) y2) (plus n y1)))))))) (\lambda (y1: nat).(\lambda
574 (y2: nat).(\lambda (H: (le (S x2) (S z0))).(\lambda (_: (le O (S
575 z0))).(\lambda (H1: (eq nat (plus (minus z0 x2) y1) (S (plus z0 y2)))).(let
576 H_y \def (IH x2 O y1 (S y2)) in (let H2 \def (eq_ind_r nat (minus z0 O)
577 (\lambda (n: nat).((le x2 z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2)
578 y1) (plus n (S y2))) \to (eq nat (plus x2 (S y2)) y1))))) H_y z0 (minus_n_O
579 z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y2)) (\lambda (n: nat).((le x2
580 z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) y1) n) \to (eq nat (plus
581 x2 (S y2)) y1))))) H2 (S (plus z0 y2)) (plus_n_Sm z0 y2)) in (let H4 \def
582 (eq_ind_r nat (plus x2 (S y2)) (\lambda (n: nat).((le x2 z0) \to ((le O z0)
583 \to ((eq nat (plus (minus z0 x2) y1) (S (plus z0 y2))) \to (eq nat n y1)))))
584 H3 (S (plus x2 y2)) (plus_n_Sm x2 y2)) in (H4 (le_S_n x2 z0 H) (le_O_n z0)
585 H1)))))))))) (\lambda (x4: nat).(\lambda (_: ((\forall (y1: nat).(\forall
586 (y2: nat).((le (S x2) (S z0)) \to ((le x4 (S z0)) \to ((eq nat (plus (minus
587 z0 x2) y1) (plus (match x4 with [O \Rightarrow (S z0) | (S l) \Rightarrow
588 (minus z0 l)]) y2)) \to (eq nat (S (plus x2 y2)) (plus x4
589 y1))))))))).(\lambda (y1: nat).(\lambda (y2: nat).(\lambda (H: (le (S x2) (S
590 z0))).(\lambda (H0: (le (S x4) (S z0))).(\lambda (H1: (eq nat (plus (minus z0
591 x2) y1) (plus (minus z0 x4) y2))).(f_equal nat nat S (plus x2 y2) (plus x4
592 y1) (IH x2 x4 y1 y2 (le_S_n x2 z0 H) (le_S_n x4 z0 H0) H1))))))))) x3))))
596 \forall (d: nat).(\forall (h: nat).(\forall (n: nat).((le (plus d h) n) \to
597 (le d (S (minus n h))))))
599 \lambda (d: nat).(\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le (plus
600 d h) n)).(let H0 \def (le_trans d (plus d h) n (le_plus_l d h) H) in (let H1
601 \def (eq_ind nat n (\lambda (n0: nat).(le d n0)) H0 (plus (minus n h) h)
602 (le_plus_minus_sym h n (le_trans_plus_r d h n H))) in (le_S d (minus n h)
603 (le_minus d n h H))))))).
606 \forall (x: nat).(\forall (y: nat).((lt x (pred y)) \to (lt (S x) y)))
608 \lambda (x: nat).(\lambda (y: nat).(nat_ind (\lambda (n: nat).((lt x (pred
609 n)) \to (lt (S x) n))) (\lambda (H: (lt x O)).(lt_x_O x H (lt (S x) O)))
610 (\lambda (n: nat).(\lambda (_: (((lt x (pred n)) \to (lt (S x) n)))).(\lambda
611 (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)