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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 "Base-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))))
60 \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus m n)
61 (plus p n)) \to (eq nat m p))))
63 \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (eq nat
64 (plus m n) (plus p n))).(simpl_plus_l n m p (eq_ind_r nat (plus m n) (\lambda
65 (n0: nat).(eq nat n0 (plus n p))) (eq_ind_r nat (plus p n) (\lambda (n0:
66 nat).(eq nat n0 (plus n p))) (sym_eq nat (plus n p) (plus p n) (plus_sym n
67 p)) (plus m n) H) (plus n m) (plus_sym n m)))))).
70 \forall (x: nat).(\forall (y: nat).(eq nat (minus (S x) (S y)) (minus x y)))
72 \lambda (x: nat).(\lambda (y: nat).(refl_equal nat (minus x y))).
75 \forall (m: nat).(\forall (n: nat).(eq nat (minus (plus m n) n) m))
77 \lambda (m: nat).(\lambda (n: nat).(eq_ind_r nat (plus n m) (\lambda (n0:
78 nat).(eq nat (minus n0 n) m)) (minus_plus n m) (plus m n) (plus_sym m n))).
80 theorem plus_permute_2_in_3:
81 \forall (x: nat).(\forall (y: nat).(\forall (z: nat).(eq nat (plus (plus x
82 y) z) (plus (plus x z) y))))
84 \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(eq_ind_r nat (plus x
85 (plus y z)) (\lambda (n: nat).(eq nat n (plus (plus x z) y))) (eq_ind_r nat
86 (plus z y) (\lambda (n: nat).(eq nat (plus x n) (plus (plus x z) y))) (eq_ind
87 nat (plus (plus x z) y) (\lambda (n: nat).(eq nat n (plus (plus x z) y)))
88 (refl_equal nat (plus (plus x z) y)) (plus x (plus z y)) (plus_assoc_r x z
89 y)) (plus y z) (plus_sym y z)) (plus (plus x y) z) (plus_assoc_r x y z)))).
91 theorem plus_permute_2_in_3_assoc:
92 \forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq nat (plus (plus n
93 h) k) (plus n (plus k h)))))
95 \lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind_r nat (plus
96 (plus n k) h) (\lambda (n0: nat).(eq nat n0 (plus n (plus k h)))) (eq_ind_r
97 nat (plus (plus n k) h) (\lambda (n0: nat).(eq nat (plus (plus n k) h) n0))
98 (refl_equal nat (plus (plus n k) h)) (plus n (plus k h)) (plus_assoc_l n k
99 h)) (plus (plus n h) k) (plus_permute_2_in_3 n h k)))).
102 \forall (x: nat).(\forall (y: nat).((eq nat (plus x y) O) \to (land (eq nat
105 \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq nat (plus
106 n y) O) \to (land (eq nat n O) (eq nat y O))))) (\lambda (y: nat).(\lambda
107 (H: (eq nat (plus O y) O)).(conj (eq nat O O) (eq nat y O) (refl_equal nat O)
108 H))) (\lambda (n: nat).(\lambda (_: ((\forall (y: nat).((eq nat (plus n y) O)
109 \to (land (eq nat n O) (eq nat y O)))))).(\lambda (y: nat).(\lambda (H0: (eq
110 nat (plus (S n) y) O)).(let H1 \def (match H0 in eq return (\lambda (n0:
111 nat).(\lambda (_: (eq ? ? n0)).((eq nat n0 O) \to (land (eq nat (S n) O) (eq
112 nat y O))))) with [refl_equal \Rightarrow (\lambda (H1: (eq nat (plus (S n)
113 y) O)).(let H2 \def (eq_ind nat (plus (S n) y) (\lambda (e: nat).(match e in
114 nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
115 \Rightarrow True])) I O H1) in (False_ind (land (eq nat (S n) O) (eq nat y
116 O)) H2)))]) in (H1 (refl_equal nat O))))))) x).
119 \forall (x: nat).(eq nat (minus (S x) (S O)) x)
121 \lambda (x: nat).(eq_ind nat x (\lambda (n: nat).(eq nat n x)) (refl_equal
122 nat x) (minus x O) (minus_n_O x)).
125 \forall (i: nat).(\forall (j: nat).(or (not (eq nat i j)) (eq nat i j)))
127 \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (j: nat).(or (not (eq
128 nat n j)) (eq nat n j)))) (\lambda (j: nat).(nat_ind (\lambda (n: nat).(or
129 (not (eq nat O n)) (eq nat O n))) (or_intror (not (eq nat O O)) (eq nat O O)
130 (refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (not (eq nat O n)) (eq
131 nat O n))).(or_introl (not (eq nat O (S n))) (eq nat O (S n)) (O_S n)))) j))
132 (\lambda (n: nat).(\lambda (H: ((\forall (j: nat).(or (not (eq nat n j)) (eq
133 nat n j))))).(\lambda (j: nat).(nat_ind (\lambda (n0: nat).(or (not (eq nat
134 (S n) n0)) (eq nat (S n) n0))) (or_introl (not (eq nat (S n) O)) (eq nat (S
135 n) O) (sym_not_eq nat O (S n) (O_S n))) (\lambda (n0: nat).(\lambda (_: (or
136 (not (eq nat (S n) n0)) (eq nat (S n) n0))).(or_ind (not (eq nat n n0)) (eq
137 nat n n0) (or (not (eq nat (S n) (S n0))) (eq nat (S n) (S n0))) (\lambda
138 (H1: (not (eq nat n n0))).(or_introl (not (eq nat (S n) (S n0))) (eq nat (S
139 n) (S n0)) (not_eq_S n n0 H1))) (\lambda (H1: (eq nat n n0)).(or_intror (not
140 (eq nat (S n) (S n0))) (eq nat (S n) (S n0)) (f_equal nat nat S n n0 H1))) (H
144 \forall (i: nat).(\forall (j: nat).(\forall (P: Prop).((((not (eq nat i j))
145 \to P)) \to ((((eq nat i j) \to P)) \to P))))
147 \lambda (i: nat).(\lambda (j: nat).(\lambda (P: Prop).(\lambda (H: (((not
148 (eq nat i j)) \to P))).(\lambda (H0: (((eq nat i j) \to P))).(let o \def
149 (eq_nat_dec i j) in (or_ind (not (eq nat i j)) (eq nat i j) P H H0 o)))))).
152 \forall (m: nat).(\forall (n: nat).(\forall (P: Prop).((le m n) \to ((le (S
155 \lambda (m: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).(\forall (P:
156 Prop).((le n n0) \to ((le (S n0) n) \to P))))) (\lambda (n: nat).(\lambda (P:
157 Prop).(\lambda (_: (le O n)).(\lambda (H0: (le (S n) O)).(let H1 \def (match
158 H0 in le return (\lambda (n0: nat).(\lambda (_: (le ? n0)).((eq nat n0 O) \to
159 P))) with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def
160 (eq_ind nat (S n) (\lambda (e: nat).(match e in nat return (\lambda (_:
161 nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in
162 (False_ind P H2))) | (le_S m0 H1) \Rightarrow (\lambda (H2: (eq nat (S m0)
163 O)).((let H3 \def (eq_ind nat (S m0) (\lambda (e: nat).(match e in nat return
164 (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True]))
165 I O H2) in (False_ind ((le (S n) m0) \to P) H3)) H1))]) in (H1 (refl_equal
166 nat O))))))) (\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(\forall (P:
167 Prop).((le n n0) \to ((le (S n0) n) \to P)))))).(\lambda (n0: nat).(nat_ind
168 (\lambda (n1: nat).(\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n))
169 \to P)))) (\lambda (P: Prop).(\lambda (H0: (le (S n) O)).(\lambda (_: (le (S
170 O) (S n))).(let H2 \def (match H0 in le return (\lambda (n1: nat).(\lambda
171 (_: (le ? n1)).((eq nat n1 O) \to P))) with [le_n \Rightarrow (\lambda (H2:
172 (eq nat (S n) O)).(let H3 \def (eq_ind nat (S n) (\lambda (e: nat).(match e
173 in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
174 \Rightarrow True])) I O H2) in (False_ind P H3))) | (le_S m0 H2) \Rightarrow
175 (\lambda (H3: (eq nat (S m0) O)).((let H4 \def (eq_ind nat (S m0) (\lambda
176 (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow
177 False | (S _) \Rightarrow True])) I O H3) in (False_ind ((le (S n) m0) \to P)
178 H4)) H2))]) in (H2 (refl_equal nat O)))))) (\lambda (n1: nat).(\lambda (_:
179 ((\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n)) \to P))))).(\lambda
180 (P: Prop).(\lambda (H1: (le (S n) (S n1))).(\lambda (H2: (le (S (S n1)) (S
181 n))).(H n1 P (le_S_n n n1 H1) (le_S_n (S n1) n H2))))))) n0)))) m).
184 \forall (x: nat).((le (S x) x) \to (\forall (P: Prop).P))
186 \lambda (x: nat).(\lambda (H: (le (S x) x)).(\lambda (P: Prop).(let H0 \def
187 le_Sn_n in (False_ind P (H0 x H))))).
190 \forall (n: nat).(\forall (m: nat).((le n m) \to (le (pred n) (pred m))))
192 \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(le_ind n (\lambda
193 (n0: nat).(le (pred n) (pred n0))) (le_n (pred n)) (\lambda (m0:
194 nat).(\lambda (_: (le n m0)).(\lambda (H1: (le (pred n) (pred m0))).(le_trans
195 (pred n) (pred m0) m0 H1 (le_pred_n m0))))) m H))).
198 \forall (x: nat).(\forall (y: nat).(le (minus x y) x))
200 \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).(le (minus n
201 y) n))) (\lambda (_: nat).(le_n O)) (\lambda (n: nat).(\lambda (H: ((\forall
202 (y: nat).(le (minus n y) n)))).(\lambda (y: nat).(nat_ind (\lambda (n0:
203 nat).(le (minus (S n) n0) (S n))) (le_n (S n)) (\lambda (n0: nat).(\lambda
204 (_: (le (match n0 with [O \Rightarrow (S n) | (S l) \Rightarrow (minus n l)])
205 (S n))).(le_S (minus n n0) n (H n0)))) y)))) x).
207 theorem le_plus_minus_sym:
208 \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus (minus m n)
211 \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(eq_ind_r nat
212 (plus n (minus m n)) (\lambda (n0: nat).(eq nat m n0)) (le_plus_minus n m H)
213 (plus (minus m n) n) (plus_sym (minus m n) n)))).
215 theorem le_minus_minus:
216 \forall (x: nat).(\forall (y: nat).((le x y) \to (\forall (z: nat).((le y z)
217 \to (le (minus y x) (minus z x))))))
219 \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (z:
220 nat).(\lambda (H0: (le y z)).(simpl_le_plus_l x (minus y x) (minus z x)
221 (eq_ind_r nat y (\lambda (n: nat).(le n (plus x (minus z x)))) (eq_ind_r nat
222 z (\lambda (n: nat).(le y n)) H0 (plus x (minus z x)) (le_plus_minus_r x z
223 (le_trans x y z H H0))) (plus x (minus y x)) (le_plus_minus_r x y H))))))).
225 theorem le_minus_plus:
226 \forall (z: nat).(\forall (x: nat).((le z x) \to (\forall (y: nat).(eq nat
227 (minus (plus x y) z) (plus (minus x z) y)))))
229 \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((le n x) \to
230 (\forall (y: nat).(eq nat (minus (plus x y) n) (plus (minus x n) y))))))
231 (\lambda (x: nat).(\lambda (H: (le O x)).(let H0 \def (match H in le return
232 (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n x) \to (\forall (y:
233 nat).(eq nat (minus (plus x y) O) (plus (minus x O) y)))))) with [le_n
234 \Rightarrow (\lambda (H0: (eq nat O x)).(eq_ind nat O (\lambda (n:
235 nat).(\forall (y: nat).(eq nat (minus (plus n y) O) (plus (minus n O) y))))
236 (\lambda (y: nat).(sym_eq nat (plus (minus O O) y) (minus (plus O y) O)
237 (minus_n_O (plus O y)))) x H0)) | (le_S m H0) \Rightarrow (\lambda (H1: (eq
238 nat (S m) x)).(eq_ind nat (S m) (\lambda (n: nat).((le O m) \to (\forall (y:
239 nat).(eq nat (minus (plus n y) O) (plus (minus n O) y))))) (\lambda (_: (le O
240 m)).(\lambda (y: nat).(refl_equal nat (plus (minus (S m) O) y)))) x H1 H0))])
241 in (H0 (refl_equal nat x))))) (\lambda (z0: nat).(\lambda (H: ((\forall (x:
242 nat).((le z0 x) \to (\forall (y: nat).(eq nat (minus (plus x y) z0) (plus
243 (minus x z0) y))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).((le (S
244 z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n
245 (S z0)) y))))) (\lambda (H0: (le (S z0) O)).(\lambda (y: nat).(let H1 \def
246 (match H0 in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O)
247 \to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))))) with
248 [le_n \Rightarrow (\lambda (H1: (eq nat (S z0) O)).(let H2 \def (eq_ind nat
249 (S z0) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with
250 [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind (eq
251 nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y)) H2))) | (le_S m H1)
252 \Rightarrow (\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m)
253 (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O
254 \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind ((le (S
255 z0) m) \to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))) H3))
256 H1))]) in (H1 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: (((le (S
257 z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n
258 (S z0)) y)))))).(\lambda (H1: (le (S z0) (S n))).(\lambda (y: nat).(H n
259 (le_S_n z0 n H1) y))))) x)))) z).
262 \forall (x: nat).(\forall (z: nat).(\forall (y: nat).((le (plus x y) z) \to
263 (le x (minus z y)))))
265 \lambda (x: nat).(\lambda (z: nat).(\lambda (y: nat).(\lambda (H: (le (plus
266 x y) z)).(eq_ind nat (minus (plus x y) y) (\lambda (n: nat).(le n (minus z
267 y))) (le_minus_minus y (plus x y) (le_plus_r x y) z H) x (minus_plus_r x
270 theorem le_trans_plus_r:
271 \forall (x: nat).(\forall (y: nat).(\forall (z: nat).((le (plus x y) z) \to
274 \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(\lambda (H: (le (plus
275 x y) z)).(le_trans y (plus x y) z (le_plus_r x y) H)))).
278 \forall (x: nat).((lt x O) \to (\forall (P: Prop).P))
280 \lambda (x: nat).(\lambda (H: (le (S x) O)).(\lambda (P: Prop).(let H_y \def
281 (le_n_O_eq (S x) H) in (let H0 \def (eq_ind nat O (\lambda (ee: nat).(match
282 ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _)
283 \Rightarrow False])) I (S x) H_y) in (False_ind P H0))))).
286 \forall (m: nat).(\forall (x: nat).((le (S m) x) \to (ex2 nat (\lambda (n:
287 nat).(eq nat x (S n))) (\lambda (n: nat).(le m n)))))
289 \lambda (m: nat).(\lambda (x: nat).(\lambda (H: (le (S m) x)).(let H0 \def
290 (match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n x)
291 \to (ex2 nat (\lambda (n0: nat).(eq nat x (S n0))) (\lambda (n0: nat).(le m
292 n0)))))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S m) x)).(eq_ind nat
293 (S m) (\lambda (n: nat).(ex2 nat (\lambda (n0: nat).(eq nat n (S n0)))
294 (\lambda (n0: nat).(le m n0)))) (ex_intro2 nat (\lambda (n: nat).(eq nat (S
295 m) (S n))) (\lambda (n: nat).(le m n)) m (refl_equal nat (S m)) (le_n m)) x
296 H0)) | (le_S m0 H0) \Rightarrow (\lambda (H1: (eq nat (S m0) x)).(eq_ind nat
297 (S m0) (\lambda (n: nat).((le (S m) m0) \to (ex2 nat (\lambda (n0: nat).(eq
298 nat n (S n0))) (\lambda (n0: nat).(le m n0))))) (\lambda (H2: (le (S m)
299 m0)).(ex_intro2 nat (\lambda (n: nat).(eq nat (S m0) (S n))) (\lambda (n:
300 nat).(le m n)) m0 (refl_equal nat (S m0)) (le_S_n m m0 (le_S (S m) m0 H2))))
301 x H1 H0))]) in (H0 (refl_equal nat x))))).
303 theorem lt_x_plus_x_Sy:
304 \forall (x: nat).(\forall (y: nat).(lt x (plus x (S y))))
306 \lambda (x: nat).(\lambda (y: nat).(eq_ind_r nat (plus (S y) x) (\lambda (n:
307 nat).(lt x n)) (le_S_n (S x) (S (plus y x)) (le_n_S (S x) (S (plus y x))
308 (le_n_S x (plus y x) (le_plus_r y x)))) (plus x (S y)) (plus_sym x (S y)))).
310 theorem simpl_lt_plus_r:
311 \forall (p: nat).(\forall (n: nat).(\forall (m: nat).((lt (plus n p) (plus m
314 \lambda (p: nat).(\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (plus
315 n p) (plus m p))).(simpl_lt_plus_l n m p (let H0 \def (eq_ind nat (plus n p)
316 (\lambda (n0: nat).(lt n0 (plus m p))) H (plus p n) (plus_sym n p)) in (let
317 H1 \def (eq_ind nat (plus m p) (\lambda (n0: nat).(lt (plus p n) n0)) H0
318 (plus p m) (plus_sym m p)) in H1)))))).
321 \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq nat (minus x y) (S
324 \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to
325 (eq nat (minus n y) (S (minus n (S y))))))) (\lambda (y: nat).(\lambda (H:
326 (lt y O)).(let H0 \def (match H in le return (\lambda (n: nat).(\lambda (_:
327 (le ? n)).((eq nat n O) \to (eq nat (minus O y) (S (minus O (S y))))))) with
328 [le_n \Rightarrow (\lambda (H0: (eq nat (S y) O)).(let H1 \def (eq_ind nat (S
329 y) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O
330 \Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (eq nat
331 (minus O y) (S (minus O (S y)))) H1))) | (le_S m H0) \Rightarrow (\lambda
332 (H1: (eq nat (S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e:
333 nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False
334 | (S _) \Rightarrow True])) I O H1) in (False_ind ((le (S y) m) \to (eq nat
335 (minus O y) (S (minus O (S y))))) H2)) H0))]) in (H0 (refl_equal nat O)))))
336 (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to (eq nat
337 (minus n y) (S (minus n (S y)))))))).(\lambda (y: nat).(nat_ind (\lambda (n0:
338 nat).((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus (S n) (S n0))))))
339 (\lambda (_: (lt O (S n))).(eq_ind nat n (\lambda (n0: nat).(eq nat (S n) (S
340 n0))) (refl_equal nat (S n)) (minus n O) (minus_n_O n))) (\lambda (n0:
341 nat).(\lambda (_: (((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus (S
342 n) (S n0))))))).(\lambda (H1: (lt (S n0) (S n))).(let H2 \def (le_S_n (S n0)
343 n H1) in (H n0 H2))))) y)))) x).
345 theorem lt_plus_minus:
346 \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus x (minus
349 \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_plus_minus (S
352 theorem lt_plus_minus_r:
353 \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus (minus y
356 \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(eq_ind_r nat
357 (plus x (minus y (S x))) (\lambda (n: nat).(eq nat y (S n))) (lt_plus_minus x
358 y H) (plus (minus y (S x)) x) (plus_sym (minus y (S x)) x)))).
361 \forall (x: nat).((lt O x) \to (eq nat x (S (minus x (S O)))))
363 \lambda (x: nat).(\lambda (H: (lt O x)).(eq_ind nat (minus x O) (\lambda (n:
364 nat).(eq nat x n)) (eq_ind nat x (\lambda (n: nat).(eq nat x n)) (refl_equal
365 nat x) (minus x O) (minus_n_O x)) (S (minus x (S O))) (minus_x_Sy x O H))).
368 \forall (y: nat).(\forall (x: nat).((lt x y) \to (le x (pred y))))
370 \lambda (y: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((lt x n) \to
371 (le x (pred n))))) (\lambda (x: nat).(\lambda (H: (lt x O)).(let H0 \def
372 (match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O)
373 \to (le x O)))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S x) O)).(let
374 H1 \def (eq_ind nat (S x) (\lambda (e: nat).(match e in nat return (\lambda
375 (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H0)
376 in (False_ind (le x O) H1))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat
377 (S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e: nat).(match e in nat
378 return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
379 True])) I O H1) in (False_ind ((le (S x) m) \to (le x O)) H2)) H0))]) in (H0
380 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: ((\forall (x: nat).((lt
381 x n) \to (le x (pred n)))))).(\lambda (x: nat).(\lambda (H0: (lt x (S
382 n))).(le_S_n x n H0))))) y).
385 \forall (x: nat).(\forall (y: nat).((lt x y) \to (le x (minus y (S O)))))
387 \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_minus x y (S
388 O) (eq_ind_r nat (plus (S O) x) (\lambda (n: nat).(le n y)) H (plus x (S O))
389 (plus_sym x (S O)))))).
392 \forall (n: nat).(\forall (d: nat).(\forall (P: Prop).((((lt n d) \to P))
393 \to ((((le d n) \to P)) \to P))))
395 \lambda (n: nat).(\lambda (d: nat).(\lambda (P: Prop).(\lambda (H: (((lt n
396 d) \to P))).(\lambda (H0: (((le d n) \to P))).(let H1 \def (le_or_lt d n) in
397 (or_ind (le d n) (lt n d) P H0 H H1)))))).
400 \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P))
401 \to ((((eq nat x y) \to P)) \to ((le x y) \to P)))))
403 \lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x
404 y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (le x
405 y)).(or_ind (lt x y) (eq nat x y) P H H0 (le_lt_or_eq x y H1))))))).
408 \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P))
409 \to ((((eq nat x y) \to P)) \to ((((lt y x) \to P)) \to P)))))
411 \lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x
412 y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (((lt y x)
413 \to P))).(lt_le_e x y P H (\lambda (H2: (le y x)).(lt_eq_e y x P H1 (\lambda
414 (H3: (eq nat y x)).(H0 (sym_eq nat y x H3))) H2)))))))).
417 \forall (x: nat).(\forall (n: nat).((lt x (S n)) \to (or (eq nat x O) (ex2
418 nat (\lambda (m: nat).(eq nat x (S m))) (\lambda (m: nat).(lt m n))))))
420 \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((lt n (S
421 n0)) \to (or (eq nat n O) (ex2 nat (\lambda (m: nat).(eq nat n (S m)))
422 (\lambda (m: nat).(lt m n0))))))) (\lambda (n: nat).(\lambda (_: (lt O (S
423 n))).(or_introl (eq nat O O) (ex2 nat (\lambda (m: nat).(eq nat O (S m)))
424 (\lambda (m: nat).(lt m n))) (refl_equal nat O)))) (\lambda (n: nat).(\lambda
425 (_: ((\forall (n0: nat).((lt n (S n0)) \to (or (eq nat n O) (ex2 nat (\lambda
426 (m: nat).(eq nat n (S m))) (\lambda (m: nat).(lt m n0)))))))).(\lambda (n0:
427 nat).(\lambda (H0: (lt (S n) (S n0))).(or_intror (eq nat (S n) O) (ex2 nat
428 (\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt m n0)))
429 (ex_intro2 nat (\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt
430 m n0)) n (refl_equal nat (S n)) (le_S_n (S n) n0 H0))))))) x).
433 \forall (x: nat).(\forall (y: nat).((le x y) \to ((lt y x) \to (\forall (P:
436 \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (H0: (lt
437 y x)).(\lambda (P: Prop).(False_ind P (le_not_lt x y H H0)))))).
440 \forall (x: nat).(\forall (y: nat).((lt x y) \to (not (eq nat x y))))
442 \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(\lambda (H0: (eq
443 nat x y)).(let H1 \def (eq_ind nat x (\lambda (n: nat).(lt n y)) H y H0) in
447 \forall (h2: nat).(\forall (d2: nat).(\forall (n: nat).((le (plus d2 h2) n)
448 \to (\forall (h1: nat).(le (plus d2 h1) (minus (plus n h1) h2))))))
450 \lambda (h2: nat).(\lambda (d2: nat).(\lambda (n: nat).(\lambda (H: (le
451 (plus d2 h2) n)).(\lambda (h1: nat).(eq_ind nat (minus (plus h2 (plus d2 h1))
452 h2) (\lambda (n0: nat).(le n0 (minus (plus n h1) h2))) (le_minus_minus h2
453 (plus h2 (plus d2 h1)) (le_plus_l h2 (plus d2 h1)) (plus n h1) (eq_ind_r nat
454 (plus (plus h2 d2) h1) (\lambda (n0: nat).(le n0 (plus n h1))) (eq_ind_r nat
455 (plus d2 h2) (\lambda (n0: nat).(le (plus n0 h1) (plus n h1))) (le_S_n (plus
456 (plus d2 h2) h1) (plus n h1) (le_n_S (plus (plus d2 h2) h1) (plus n h1)
457 (le_plus_plus (plus d2 h2) n h1 h1 H (le_n h1)))) (plus h2 d2) (plus_sym h2
458 d2)) (plus h2 (plus d2 h1)) (plus_assoc_l h2 d2 h1))) (plus d2 h1)
459 (minus_plus h2 (plus d2 h1))))))).
462 \forall (x: nat).(\forall (y: nat).((le x y) \to (eq nat (minus x y) O)))
464 \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to
465 (eq nat (minus n y) O)))) (\lambda (y: nat).(\lambda (_: (le O
466 y)).(refl_equal nat O))) (\lambda (x0: nat).(\lambda (H: ((\forall (y:
467 nat).((le x0 y) \to (eq nat (minus x0 y) O))))).(\lambda (y: nat).(nat_ind
468 (\lambda (n: nat).((le (S x0) n) \to (eq nat (match n with [O \Rightarrow (S
469 x0) | (S l) \Rightarrow (minus x0 l)]) O))) (\lambda (H0: (le (S x0)
470 O)).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le x0
471 n)) (eq nat (S x0) O) (\lambda (x1: nat).(\lambda (H1: (eq nat O (S
472 x1))).(\lambda (_: (le x0 x1)).(let H3 \def (eq_ind nat O (\lambda (ee:
473 nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True
474 | (S _) \Rightarrow False])) I (S x1) H1) in (False_ind (eq nat (S x0) O)
475 H3))))) (le_gen_S x0 O H0))) (\lambda (n: nat).(\lambda (_: (((le (S x0) n)
476 \to (eq nat (match n with [O \Rightarrow (S x0) | (S l) \Rightarrow (minus x0
477 l)]) O)))).(\lambda (H1: (le (S x0) (S n))).(H n (le_S_n x0 n H1))))) y))))
481 \forall (z: nat).(\forall (x: nat).(\forall (y: nat).((le z x) \to ((le z y)
482 \to ((eq nat (minus x z) (minus y z)) \to (eq nat x y))))))
484 \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).(\forall (y:
485 nat).((le n x) \to ((le n y) \to ((eq nat (minus x n) (minus y n)) \to (eq
486 nat x y))))))) (\lambda (x: nat).(\lambda (y: nat).(\lambda (_: (le O
487 x)).(\lambda (_: (le O y)).(\lambda (H1: (eq nat (minus x O) (minus y
488 O))).(let H2 \def (eq_ind_r nat (minus x O) (\lambda (n: nat).(eq nat n
489 (minus y O))) H1 x (minus_n_O x)) in (let H3 \def (eq_ind_r nat (minus y O)
490 (\lambda (n: nat).(eq nat x n)) H2 y (minus_n_O y)) in H3))))))) (\lambda
491 (z0: nat).(\lambda (IH: ((\forall (x: nat).(\forall (y: nat).((le z0 x) \to
492 ((le z0 y) \to ((eq nat (minus x z0) (minus y z0)) \to (eq nat x
493 y)))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le
494 (S z0) n) \to ((le (S z0) y) \to ((eq nat (minus n (S z0)) (minus y (S z0)))
495 \to (eq nat n y)))))) (\lambda (y: nat).(\lambda (H: (le (S z0) O)).(\lambda
496 (_: (le (S z0) y)).(\lambda (_: (eq nat (minus O (S z0)) (minus y (S
497 z0)))).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le
498 z0 n)) (eq nat O y) (\lambda (x0: nat).(\lambda (H2: (eq nat O (S
499 x0))).(\lambda (_: (le z0 x0)).(let H4 \def (eq_ind nat O (\lambda (ee:
500 nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True
501 | (S _) \Rightarrow False])) I (S x0) H2) in (False_ind (eq nat O y) H4)))))
502 (le_gen_S z0 O H)))))) (\lambda (x0: nat).(\lambda (_: ((\forall (y:
503 nat).((le (S z0) x0) \to ((le (S z0) y) \to ((eq nat (minus x0 (S z0)) (minus
504 y (S z0))) \to (eq nat x0 y))))))).(\lambda (y: nat).(nat_ind (\lambda (n:
505 nat).((le (S z0) (S x0)) \to ((le (S z0) n) \to ((eq nat (minus (S x0) (S
506 z0)) (minus n (S z0))) \to (eq nat (S x0) n))))) (\lambda (H: (le (S z0) (S
507 x0))).(\lambda (H0: (le (S z0) O)).(\lambda (_: (eq nat (minus (S x0) (S z0))
508 (minus O (S z0)))).(let H_y \def (le_S_n z0 x0 H) in (ex2_ind nat (\lambda
509 (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le z0 n)) (eq nat (S x0) O)
510 (\lambda (x1: nat).(\lambda (H2: (eq nat O (S x1))).(\lambda (_: (le z0
511 x1)).(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return
512 (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False]))
513 I (S x1) H2) in (False_ind (eq nat (S x0) O) H4))))) (le_gen_S z0 O H0))))))
514 (\lambda (y0: nat).(\lambda (_: (((le (S z0) (S x0)) \to ((le (S z0) y0) \to
515 ((eq nat (minus (S x0) (S z0)) (minus y0 (S z0))) \to (eq nat (S x0)
516 y0)))))).(\lambda (H: (le (S z0) (S x0))).(\lambda (H0: (le (S z0) (S
517 y0))).(\lambda (H1: (eq nat (minus (S x0) (S z0)) (minus (S y0) (S
518 z0)))).(f_equal nat nat S x0 y0 (IH x0 y0 (le_S_n z0 x0 H) (le_S_n z0 y0 H0)
519 H1))))))) y)))) x)))) z).
522 \forall (z: nat).(\forall (x1: nat).(\forall (x2: nat).(\forall (y1:
523 nat).(\forall (y2: nat).((le x1 z) \to ((le x2 z) \to ((eq nat (plus (minus z
524 x1) y1) (plus (minus z x2) y2)) \to (eq nat (plus x1 y2) (plus x2 y1)))))))))
526 \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x1: nat).(\forall (x2:
527 nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 n) \to ((le x2 n) \to ((eq
528 nat (plus (minus n x1) y1) (plus (minus n x2) y2)) \to (eq nat (plus x1 y2)
529 (plus x2 y1)))))))))) (\lambda (x1: nat).(\lambda (x2: nat).(\lambda (y1:
530 nat).(\lambda (y2: nat).(\lambda (H: (le x1 O)).(\lambda (H0: (le x2
531 O)).(\lambda (H1: (eq nat y1 y2)).(eq_ind nat y1 (\lambda (n: nat).(eq nat
532 (plus x1 n) (plus x2 y1))) (let H_y \def (le_n_O_eq x2 H0) in (eq_ind nat O
533 (\lambda (n: nat).(eq nat (plus x1 y1) (plus n y1))) (let H_y0 \def
534 (le_n_O_eq x1 H) in (eq_ind nat O (\lambda (n: nat).(eq nat (plus n y1) (plus
535 O y1))) (refl_equal nat (plus O y1)) x1 H_y0)) x2 H_y)) y2 H1))))))))
536 (\lambda (z0: nat).(\lambda (IH: ((\forall (x1: nat).(\forall (x2:
537 nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 z0) \to ((le x2 z0) \to
538 ((eq nat (plus (minus z0 x1) y1) (plus (minus z0 x2) y2)) \to (eq nat (plus
539 x1 y2) (plus x2 y1))))))))))).(\lambda (x1: nat).(nat_ind (\lambda (n:
540 nat).(\forall (x2: nat).(\forall (y1: nat).(\forall (y2: nat).((le n (S z0))
541 \to ((le x2 (S z0)) \to ((eq nat (plus (minus (S z0) n) y1) (plus (minus (S
542 z0) x2) y2)) \to (eq nat (plus n y2) (plus x2 y1))))))))) (\lambda (x2:
543 nat).(nat_ind (\lambda (n: nat).(\forall (y1: nat).(\forall (y2: nat).((le O
544 (S z0)) \to ((le n (S z0)) \to ((eq nat (plus (minus (S z0) O) y1) (plus
545 (minus (S z0) n) y2)) \to (eq nat (plus O y2) (plus n y1)))))))) (\lambda
546 (y1: nat).(\lambda (y2: nat).(\lambda (_: (le O (S z0))).(\lambda (_: (le O
547 (S z0))).(\lambda (H1: (eq nat (S (plus z0 y1)) (S (plus z0 y2)))).(let H_y
548 \def (IH O O) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n:
549 nat).(\forall (y3: nat).(\forall (y4: nat).((le O z0) \to ((le O z0) \to ((eq
550 nat (plus n y3) (plus n y4)) \to (eq nat y4 y3))))))) H_y z0 (minus_n_O z0))
551 in (H2 y1 y2 (le_O_n z0) (le_O_n z0) (eq_add_S (plus z0 y1) (plus z0 y2)
552 H1))))))))) (\lambda (x3: nat).(\lambda (_: ((\forall (y1: nat).(\forall (y2:
553 nat).((le O (S z0)) \to ((le x3 (S z0)) \to ((eq nat (S (plus z0 y1)) (plus
554 (match x3 with [O \Rightarrow (S z0) | (S l) \Rightarrow (minus z0 l)]) y2))
555 \to (eq nat y2 (plus x3 y1))))))))).(\lambda (y1: nat).(\lambda (y2:
556 nat).(\lambda (_: (le O (S z0))).(\lambda (H0: (le (S x3) (S z0))).(\lambda
557 (H1: (eq nat (S (plus z0 y1)) (plus (minus z0 x3) y2))).(let H_y \def (IH O
558 x3 (S y1)) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n:
559 nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (plus n (S
560 y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H_y z0
561 (minus_n_O z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y1)) (\lambda (n:
562 nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat n (plus
563 (minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H2 (S (plus z0 y1))
564 (plus_n_Sm z0 y1)) in (let H4 \def (eq_ind_r nat (plus x3 (S y1)) (\lambda
565 (n: nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (S (plus
566 z0 y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 n)))))) H3 (S (plus x3 y1))
567 (plus_n_Sm x3 y1)) in (H4 y2 (le_O_n z0) (le_S_n x3 z0 H0) H1))))))))))))
568 x2)) (\lambda (x2: nat).(\lambda (_: ((\forall (x3: nat).(\forall (y1:
569 nat).(\forall (y2: nat).((le x2 (S z0)) \to ((le x3 (S z0)) \to ((eq nat
570 (plus (minus (S z0) x2) y1) (plus (minus (S z0) x3) y2)) \to (eq nat (plus x2
571 y2) (plus x3 y1)))))))))).(\lambda (x3: nat).(nat_ind (\lambda (n:
572 nat).(\forall (y1: nat).(\forall (y2: nat).((le (S x2) (S z0)) \to ((le n (S
573 z0)) \to ((eq nat (plus (minus (S z0) (S x2)) y1) (plus (minus (S z0) n) y2))
574 \to (eq nat (plus (S x2) y2) (plus n y1)))))))) (\lambda (y1: nat).(\lambda
575 (y2: nat).(\lambda (H: (le (S x2) (S z0))).(\lambda (_: (le O (S
576 z0))).(\lambda (H1: (eq nat (plus (minus z0 x2) y1) (S (plus z0 y2)))).(let
577 H_y \def (IH x2 O y1 (S y2)) in (let H2 \def (eq_ind_r nat (minus z0 O)
578 (\lambda (n: nat).((le x2 z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2)
579 y1) (plus n (S y2))) \to (eq nat (plus x2 (S y2)) y1))))) H_y z0 (minus_n_O
580 z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y2)) (\lambda (n: nat).((le x2
581 z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) y1) n) \to (eq nat (plus
582 x2 (S y2)) y1))))) H2 (S (plus z0 y2)) (plus_n_Sm z0 y2)) in (let H4 \def
583 (eq_ind_r nat (plus x2 (S y2)) (\lambda (n: nat).((le x2 z0) \to ((le O z0)
584 \to ((eq nat (plus (minus z0 x2) y1) (S (plus z0 y2))) \to (eq nat n y1)))))
585 H3 (S (plus x2 y2)) (plus_n_Sm x2 y2)) in (H4 (le_S_n x2 z0 H) (le_O_n z0)
586 H1)))))))))) (\lambda (x4: nat).(\lambda (_: ((\forall (y1: nat).(\forall
587 (y2: nat).((le (S x2) (S z0)) \to ((le x4 (S z0)) \to ((eq nat (plus (minus
588 z0 x2) y1) (plus (match x4 with [O \Rightarrow (S z0) | (S l) \Rightarrow
589 (minus z0 l)]) y2)) \to (eq nat (S (plus x2 y2)) (plus x4
590 y1))))))))).(\lambda (y1: nat).(\lambda (y2: nat).(\lambda (H: (le (S x2) (S
591 z0))).(\lambda (H0: (le (S x4) (S z0))).(\lambda (H1: (eq nat (plus (minus z0
592 x2) y1) (plus (minus z0 x4) y2))).(f_equal nat nat S (plus x2 y2) (plus x4
593 y1) (IH x2 x4 y1 y2 (le_S_n x2 z0 H) (le_S_n x4 z0 H0) H1))))))))) x3))))
597 \forall (d: nat).(\forall (h: nat).(\forall (n: nat).((le (plus d h) n) \to
598 (le d (S (minus n h))))))
600 \lambda (d: nat).(\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le (plus
601 d h) n)).(let H0 \def (le_trans d (plus d h) n (le_plus_l d h) H) in (let H1
602 \def (eq_ind nat n (\lambda (n0: nat).(le d n0)) H0 (plus (minus n h) h)
603 (le_plus_minus_sym h n (le_trans h (plus d h) n (le_plus_r d h) H))) in (le_S
604 d (minus n h) (le_minus d n h H))))))).
607 \forall (x: nat).(\forall (y: nat).((lt x (pred y)) \to (lt (S x) y)))
609 \lambda (x: nat).(\lambda (y: nat).(nat_ind (\lambda (n: nat).((lt x (pred
610 n)) \to (lt (S x) n))) (\lambda (H: (lt x O)).(lt_x_O x H (lt (S x) O)))
611 (\lambda (n: nat).(\lambda (_: (((lt x (pred n)) \to (lt (S x) n)))).(\lambda
612 (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)