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15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/aplus/props".
19 include "aplus/defs.ma".
22 \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (h1: nat).(\forall
23 (h2: nat).((eq A (aplus g a1 h1) (aplus g a2 h2)) \to (\forall (h: nat).(eq A
24 (aplus g a1 (plus h h1)) (aplus g a2 (plus h h2)))))))))
26 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (h1: nat).(\lambda
27 (h2: nat).(\lambda (H: (eq A (aplus g a1 h1) (aplus g a2 h2))).(\lambda (h:
28 nat).(nat_ind (\lambda (n: nat).(eq A (aplus g a1 (plus n h1)) (aplus g a2
29 (plus n h2)))) H (\lambda (n: nat).(\lambda (H0: (eq A (aplus g a1 (plus n
30 h1)) (aplus g a2 (plus n h2)))).(sym_equal A (asucc g (aplus g a2 (plus n
31 h2))) (asucc g (aplus g a1 (plus n h1))) (sym_equal A (asucc g (aplus g a1
32 (plus n h1))) (asucc g (aplus g a2 (plus n h2))) (sym_equal A (asucc g (aplus
33 g a2 (plus n h2))) (asucc g (aplus g a1 (plus n h1))) (f_equal2 G A A asucc g
34 g (aplus g a2 (plus n h2)) (aplus g a1 (plus n h1)) (refl_equal G g) (sym_eq
35 A (aplus g a1 (plus n h1)) (aplus g a2 (plus n h2)) H0))))))) h))))))).
38 \forall (g: G).(\forall (a: A).(\forall (h1: nat).(\forall (h2: nat).(eq A
39 (aplus g (aplus g a h1) h2) (aplus g a (plus h1 h2))))))
41 \lambda (g: G).(\lambda (a: A).(\lambda (h1: nat).(nat_ind (\lambda (n:
42 nat).(\forall (h2: nat).(eq A (aplus g (aplus g a n) h2) (aplus g a (plus n
43 h2))))) (\lambda (h2: nat).(refl_equal A (aplus g a h2))) (\lambda (n:
44 nat).(\lambda (_: ((\forall (h2: nat).(eq A (aplus g (aplus g a n) h2) (aplus
45 g a (plus n h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(eq A
46 (aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n n0)))))
47 (eq_ind nat n (\lambda (n0: nat).(eq A (asucc g (aplus g a n)) (asucc g
48 (aplus g a n0)))) (refl_equal A (asucc g (aplus g a n))) (plus n O) (plus_n_O
49 n)) (\lambda (n0: nat).(\lambda (H0: (eq A (aplus g (asucc g (aplus g a n))
50 n0) (asucc g (aplus g a (plus n n0))))).(eq_ind nat (S (plus n n0)) (\lambda
51 (n1: nat).(eq A (asucc g (aplus g (asucc g (aplus g a n)) n0)) (asucc g
52 (aplus g a n1)))) (sym_equal A (asucc g (asucc g (aplus g a (plus n n0))))
53 (asucc g (aplus g (asucc g (aplus g a n)) n0)) (sym_equal A (asucc g (aplus g
54 (asucc g (aplus g a n)) n0)) (asucc g (asucc g (aplus g a (plus n n0))))
55 (sym_equal A (asucc g (asucc g (aplus g a (plus n n0)))) (asucc g (aplus g
56 (asucc g (aplus g a n)) n0)) (f_equal2 G A A asucc g g (asucc g (aplus g a
57 (plus n n0))) (aplus g (asucc g (aplus g a n)) n0) (refl_equal G g) (sym_eq A
58 (aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n n0)))
59 H0))))) (plus n (S n0)) (plus_n_Sm n n0)))) h2)))) h1))).
62 \forall (g: G).(\forall (h: nat).(\forall (a: A).(eq A (aplus g (asucc g a)
63 h) (asucc g (aplus g a h)))))
65 \lambda (g: G).(\lambda (h: nat).(\lambda (a: A).(eq_ind_r A (aplus g a
66 (plus (S O) h)) (\lambda (a0: A).(eq A a0 (asucc g (aplus g a h))))
67 (refl_equal A (asucc g (aplus g a h))) (aplus g (aplus g a (S O)) h)
68 (aplus_assoc g a (S O) h)))).
70 theorem aplus_sort_O_S_simpl:
71 \forall (g: G).(\forall (n: nat).(\forall (k: nat).(eq A (aplus g (ASort O
72 n) (S k)) (aplus g (ASort O (next g n)) k))))
74 \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(eq_ind A (aplus g (asucc
75 g (ASort O n)) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n)) k)))
76 (refl_equal A (aplus g (ASort O (next g n)) k)) (asucc g (aplus g (ASort O n)
77 k)) (aplus_asucc g k (ASort O n))))).
79 theorem aplus_sort_S_S_simpl:
80 \forall (g: G).(\forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq A
81 (aplus g (ASort (S h) n) (S k)) (aplus g (ASort h n) k)))))
83 \lambda (g: G).(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind
84 A (aplus g (asucc g (ASort (S h) n)) k) (\lambda (a: A).(eq A a (aplus g
85 (ASort h n) k))) (refl_equal A (aplus g (ASort h n) k)) (asucc g (aplus g
86 (ASort (S h) n) k)) (aplus_asucc g k (ASort (S h) n)))))).
88 alias id "next_plus_next" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/next_plus/props/next_plus_next.con".
89 alias id "next_plus" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/next_plus/defs/next_plus.con".
90 theorem aplus_asort_O_simpl:
91 \forall (g: G).(\forall (h: nat).(\forall (n: nat).(eq A (aplus g (ASort O
92 n) h) (ASort O (next_plus g n h)))))
94 \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (n0:
95 nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0 n))))) (\lambda
96 (n: nat).(refl_equal A (ASort O n))) (\lambda (n: nat).(\lambda (H: ((\forall
97 (n0: nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0
98 n)))))).(\lambda (n0: nat).(eq_ind A (aplus g (asucc g (ASort O n0)) n)
99 (\lambda (a: A).(eq A a (ASort O (next g (next_plus g n0 n))))) (eq_ind nat
100 (next_plus g (next g n0) n) (\lambda (n1: nat).(eq A (aplus g (ASort O (next
101 g n0)) n) (ASort O n1))) (H (next g n0)) (next g (next_plus g n0 n))
102 (next_plus_next g n0 n)) (asucc g (aplus g (ASort O n0) n)) (aplus_asucc g n
103 (ASort O n0)))))) h)).
105 theorem aplus_asort_le_simpl:
106 \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).((le h
107 k) \to (eq A (aplus g (ASort k n) h) (ASort (minus k h) n))))))
109 \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (k:
110 nat).(\forall (n0: nat).((le n k) \to (eq A (aplus g (ASort k n0) n) (ASort
111 (minus k n) n0)))))) (\lambda (k: nat).(\lambda (n: nat).(\lambda (_: (le O
112 k)).(eq_ind nat k (\lambda (n0: nat).(eq A (ASort k n) (ASort n0 n)))
113 (refl_equal A (ASort k n)) (minus k O) (minus_n_O k))))) (\lambda (h0:
114 nat).(\lambda (H: ((\forall (k: nat).(\forall (n: nat).((le h0 k) \to (eq A
115 (aplus g (ASort k n) h0) (ASort (minus k h0) n))))))).(\lambda (k:
116 nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((le (S h0) n) \to (eq A
117 (asucc g (aplus g (ASort n n0) h0)) (ASort (minus n (S h0)) n0))))) (\lambda
118 (n: nat).(\lambda (H0: (le (S h0) O)).(ex2_ind nat (\lambda (n0: nat).(eq nat
119 O (S n0))) (\lambda (n0: nat).(le h0 n0)) (eq A (asucc g (aplus g (ASort O n)
120 h0)) (ASort (minus O (S h0)) n)) (\lambda (x: nat).(\lambda (H1: (eq nat O (S
121 x))).(\lambda (_: (le h0 x)).(let H3 \def (eq_ind nat O (\lambda (ee:
122 nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True
123 | (S _) \Rightarrow False])) I (S x) H1) in (False_ind (eq A (asucc g (aplus
124 g (ASort O n) h0)) (ASort (minus O (S h0)) n)) H3))))) (le_gen_S h0 O H0))))
125 (\lambda (n: nat).(\lambda (_: ((\forall (n0: nat).((le (S h0) n) \to (eq A
126 (asucc g (aplus g (ASort n n0) h0)) (ASort (minus n (S h0)) n0)))))).(\lambda
127 (n0: nat).(\lambda (H1: (le (S h0) (S n))).(eq_ind A (aplus g (asucc g (ASort
128 (S n) n0)) h0) (\lambda (a: A).(eq A a (ASort (minus (S n) (S h0)) n0))) (H n
129 n0 (le_S_n h0 n H1)) (asucc g (aplus g (ASort (S n) n0) h0)) (aplus_asucc g
130 h0 (ASort (S n) n0))))))) k)))) h)).
132 alias id "minus_n_n" = "cic:/Coq/Arith/Minus/minus_n_n.con".
133 theorem aplus_asort_simpl:
134 \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).(eq A
135 (aplus g (ASort k n) h) (ASort (minus k h) (next_plus g n (minus h k)))))))
137 \lambda (g: G).(\lambda (h: nat).(\lambda (k: nat).(\lambda (n:
138 nat).(lt_le_e k h (eq A (aplus g (ASort k n) h) (ASort (minus k h) (next_plus
139 g n (minus h k)))) (\lambda (H: (lt k h)).(eq_ind_r nat (plus k (minus h k))
140 (\lambda (n0: nat).(eq A (aplus g (ASort k n) n0) (ASort (minus k h)
141 (next_plus g n (minus h k))))) (eq_ind A (aplus g (aplus g (ASort k n) k)
142 (minus h k)) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g n (minus
143 h k))))) (eq_ind_r A (ASort (minus k k) n) (\lambda (a: A).(eq A (aplus g a
144 (minus h k)) (ASort (minus k h) (next_plus g n (minus h k))))) (eq_ind nat O
145 (\lambda (n0: nat).(eq A (aplus g (ASort n0 n) (minus h k)) (ASort (minus k
146 h) (next_plus g n (minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A
147 (aplus g (ASort O n) (minus h k)) (ASort n0 (next_plus g n (minus h k)))))
148 (aplus_asort_O_simpl g (minus h k) n) (minus k h) (O_minus k h (le_S_n k h
149 (le_S (S k) h H)))) (minus k k) (minus_n_n k)) (aplus g (ASort k n) k)
150 (aplus_asort_le_simpl g k k n (le_n k))) (aplus g (ASort k n) (plus k (minus
151 h k))) (aplus_assoc g (ASort k n) k (minus h k))) h (le_plus_minus k h
152 (le_S_n k h (le_S (S k) h H))))) (\lambda (H: (le h k)).(eq_ind_r A (ASort
153 (minus k h) n) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g n
154 (minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A (ASort (minus k h)
155 n) (ASort (minus k h) (next_plus g n n0)))) (refl_equal A (ASort (minus k h)
156 (next_plus g n O))) (minus h k) (O_minus h k H)) (aplus g (ASort k n) h)
157 (aplus_asort_le_simpl g h k n H))))))).
159 theorem aplus_ahead_simpl:
160 \forall (g: G).(\forall (h: nat).(\forall (a1: A).(\forall (a2: A).(eq A
161 (aplus g (AHead a1 a2) h) (AHead a1 (aplus g a2 h))))))
163 \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (a1:
164 A).(\forall (a2: A).(eq A (aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2
165 n)))))) (\lambda (a1: A).(\lambda (a2: A).(refl_equal A (AHead a1 a2))))
166 (\lambda (n: nat).(\lambda (H: ((\forall (a1: A).(\forall (a2: A).(eq A
167 (aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2 n))))))).(\lambda (a1:
168 A).(\lambda (a2: A).(eq_ind A (aplus g (asucc g (AHead a1 a2)) n) (\lambda
169 (a: A).(eq A a (AHead a1 (asucc g (aplus g a2 n))))) (eq_ind A (aplus g
170 (asucc g a2) n) (\lambda (a: A).(eq A (aplus g (asucc g (AHead a1 a2)) n)
171 (AHead a1 a))) (H a1 (asucc g a2)) (asucc g (aplus g a2 n)) (aplus_asucc g n
172 a2)) (asucc g (aplus g (AHead a1 a2) n)) (aplus_asucc g n (AHead a1 a2)))))))
175 alias id "next_plus_lt" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/next_plus/props/next_plus_lt.con".
176 theorem aplus_asucc_false:
177 \forall (g: G).(\forall (a: A).(\forall (h: nat).((eq A (aplus g (asucc g a)
178 h) a) \to (\forall (P: Prop).P))))
180 \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (h:
181 nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: Prop).P))))
182 (\lambda (n: nat).(\lambda (n0: nat).(\lambda (h: nat).(\lambda (H: (eq A
183 (aplus g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h)
184 \Rightarrow (ASort h n0)]) h) (ASort n n0))).(\lambda (P: Prop).((match n in
185 nat return (\lambda (n1: nat).((eq A (aplus g (match n1 with [O \Rightarrow
186 (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) h) (ASort n1 n0))
187 \to P)) with [O \Rightarrow (\lambda (H0: (eq A (aplus g (ASort O (next g
188 n0)) h) (ASort O n0))).(let H1 \def (eq_ind A (aplus g (ASort O (next g n0))
189 h) (\lambda (a: A).(eq A a (ASort O n0))) H0 (ASort (minus O h) (next_plus g
190 (next g n0) (minus h O))) (aplus_asort_simpl g h O (next g n0))) in (let H2
191 \def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat)
192 with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow ((let rec next_plus
193 (g: G) (n: nat) (i: nat) on i: nat \def (match i with [O \Rightarrow n | (S
194 i0) \Rightarrow (next g (next_plus g n i0))]) in next_plus) g (next g n0)
195 (minus h O))])) (ASort (minus O h) (next_plus g (next g n0) (minus h O)))
196 (ASort O n0) H1) in (let H3 \def (eq_ind_r nat (minus h O) (\lambda (n:
197 nat).(eq nat (next_plus g (next g n0) n) n0)) H2 h (minus_n_O h)) in
198 (le_lt_false (next_plus g (next g n0) h) n0 (eq_ind nat (next_plus g (next g
199 n0) h) (\lambda (n1: nat).(le (next_plus g (next g n0) h) n1)) (le_n
200 (next_plus g (next g n0) h)) n0 H3) (next_plus_lt g h n0) P))))) | (S n1)
201 \Rightarrow (\lambda (H0: (eq A (aplus g (ASort n1 n0) h) (ASort (S n1)
202 n0))).(let H1 \def (eq_ind A (aplus g (ASort n1 n0) h) (\lambda (a: A).(eq A
203 a (ASort (S n1) n0))) H0 (ASort (minus n1 h) (next_plus g n0 (minus h n1)))
204 (aplus_asort_simpl g h n1 n0)) in (let H2 \def (f_equal A nat (\lambda (e:
205 A).(match e in A return (\lambda (_: A).nat) with [(ASort n _) \Rightarrow n
206 | (AHead _ _) \Rightarrow ((let rec minus (n: nat) on n: (nat \to nat) \def
207 (\lambda (m: nat).(match n with [O \Rightarrow O | (S k) \Rightarrow (match m
208 with [O \Rightarrow (S k) | (S l) \Rightarrow (minus k l)])])) in minus) n1
209 h)])) (ASort (minus n1 h) (next_plus g n0 (minus h n1))) (ASort (S n1) n0)
210 H1) in ((let H3 \def (f_equal A nat (\lambda (e: A).(match e in A return
211 (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _)
212 \Rightarrow ((let rec next_plus (g: G) (n: nat) (i: nat) on i: nat \def
213 (match i with [O \Rightarrow n | (S i0) \Rightarrow (next g (next_plus g n
214 i0))]) in next_plus) g n0 (minus h n1))])) (ASort (minus n1 h) (next_plus g
215 n0 (minus h n1))) (ASort (S n1) n0) H1) in (\lambda (H4: (eq nat (minus n1 h)
216 (S n1))).(le_Sx_x n1 (eq_ind nat (minus n1 h) (\lambda (n2: nat).(le n2 n1))
217 (minus_le n1 h) (S n1) H4) P))) H2))))]) H)))))) (\lambda (a0: A).(\lambda
218 (_: ((\forall (h: nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P:
219 Prop).P))))).(\lambda (a1: A).(\lambda (H0: ((\forall (h: nat).((eq A (aplus
220 g (asucc g a1) h) a1) \to (\forall (P: Prop).P))))).(\lambda (h:
221 nat).(\lambda (H1: (eq A (aplus g (AHead a0 (asucc g a1)) h) (AHead a0
222 a1))).(\lambda (P: Prop).(let H2 \def (eq_ind A (aplus g (AHead a0 (asucc g
223 a1)) h) (\lambda (a: A).(eq A a (AHead a0 a1))) H1 (AHead a0 (aplus g (asucc
224 g a1) h)) (aplus_ahead_simpl g h a0 (asucc g a1))) in (let H3 \def (f_equal A
225 A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _)
226 \Rightarrow ((let rec aplus (g: G) (a: A) (n: nat) on n: A \def (match n with
227 [O \Rightarrow a | (S n0) \Rightarrow (asucc g (aplus g a n0))]) in aplus) g
228 (asucc g a1) h) | (AHead _ a) \Rightarrow a])) (AHead a0 (aplus g (asucc g
229 a1) h)) (AHead a0 a1) H2) in (H0 h H3 P)))))))))) a)).
232 \forall (g: G).(\forall (h1: nat).(\forall (h2: nat).(\forall (a: A).((eq A
233 (aplus g a h1) (aplus g a h2)) \to (eq nat h1 h2)))))
235 \lambda (g: G).(\lambda (h1: nat).(nat_ind (\lambda (n: nat).(\forall (h2:
236 nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n
237 h2))))) (\lambda (h2: nat).(nat_ind (\lambda (n: nat).(\forall (a: A).((eq A
238 (aplus g a O) (aplus g a n)) \to (eq nat O n)))) (\lambda (a: A).(\lambda (_:
239 (eq A a a)).(refl_equal nat O))) (\lambda (n: nat).(\lambda (_: ((\forall (a:
240 A).((eq A a (aplus g a n)) \to (eq nat O n))))).(\lambda (a: A).(\lambda (H0:
241 (eq A a (asucc g (aplus g a n)))).(let H1 \def (eq_ind_r A (asucc g (aplus g
242 a n)) (\lambda (a0: A).(eq A a a0)) H0 (aplus g (asucc g a) n) (aplus_asucc g
243 n a)) in (aplus_asucc_false g a n (sym_eq A a (aplus g (asucc g a) n) H1) (eq
244 nat O (S n)))))))) h2)) (\lambda (n: nat).(\lambda (H: ((\forall (h2:
245 nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n
246 h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(\forall (a: A).((eq
247 A (aplus g a (S n)) (aplus g a n0)) \to (eq nat (S n) n0)))) (\lambda (a:
248 A).(\lambda (H0: (eq A (asucc g (aplus g a n)) a)).(let H1 \def (eq_ind_r A
249 (asucc g (aplus g a n)) (\lambda (a0: A).(eq A a0 a)) H0 (aplus g (asucc g a)
250 n) (aplus_asucc g n a)) in (aplus_asucc_false g a n H1 (eq nat (S n) O)))))
251 (\lambda (n0: nat).(\lambda (_: ((\forall (a: A).((eq A (asucc g (aplus g a
252 n)) (aplus g a n0)) \to (eq nat (S n) n0))))).(\lambda (a: A).(\lambda (H1:
253 (eq A (asucc g (aplus g a n)) (asucc g (aplus g a n0)))).(let H2 \def
254 (eq_ind_r A (asucc g (aplus g a n)) (\lambda (a0: A).(eq A a0 (asucc g (aplus
255 g a n0)))) H1 (aplus g (asucc g a) n) (aplus_asucc g n a)) in (let H3 \def
256 (eq_ind_r A (asucc g (aplus g a n0)) (\lambda (a0: A).(eq A (aplus g (asucc g
257 a) n) a0)) H2 (aplus g (asucc g a) n0) (aplus_asucc g n0 a)) in (f_equal nat
258 nat S n n0 (H n0 (asucc g a) H3)))))))) h2)))) h1)).