1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 include "basic_1/aplus/defs.ma".
19 include "basic_1/A/fwd.ma".
21 include "basic_1/next_plus/props.ma".
24 \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (h1: nat).(\forall
25 (h2: nat).((eq A (aplus g a1 h1) (aplus g a2 h2)) \to (\forall (h: nat).(eq A
26 (aplus g a1 (plus h h1)) (aplus g a2 (plus h h2)))))))))
28 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (h1: nat).(\lambda
29 (h2: nat).(\lambda (H: (eq A (aplus g a1 h1) (aplus g a2 h2))).(\lambda (h:
30 nat).(let TMP_5 \def (\lambda (n: nat).(let TMP_1 \def (plus n h1) in (let
31 TMP_2 \def (aplus g a1 TMP_1) in (let TMP_3 \def (plus n h2) in (let TMP_4
32 \def (aplus g a2 TMP_3) in (eq A TMP_2 TMP_4)))))) in (let TMP_11 \def
33 (\lambda (n: nat).(\lambda (H0: (eq A (aplus g a1 (plus n h1)) (aplus g a2
34 (plus n h2)))).(let TMP_6 \def (plus n h1) in (let TMP_7 \def (aplus g a1
35 TMP_6) in (let TMP_8 \def (plus n h2) in (let TMP_9 \def (aplus g a2 TMP_8)
36 in (let TMP_10 \def (refl_equal G g) in (f_equal2 G A A asucc g g TMP_7 TMP_9
37 TMP_10 H0)))))))) in (nat_ind TMP_5 H TMP_11 h))))))))).
40 \forall (g: G).(\forall (a: A).(\forall (h1: nat).(\forall (h2: nat).(eq A
41 (aplus g (aplus g a h1) h2) (aplus g a (plus h1 h2))))))
43 \lambda (g: G).(\lambda (a: A).(\lambda (h1: nat).(let TMP_5 \def (\lambda
44 (n: nat).(\forall (h2: nat).(let TMP_1 \def (aplus g a n) in (let TMP_2 \def
45 (aplus g TMP_1 h2) in (let TMP_3 \def (plus n h2) in (let TMP_4 \def (aplus g
46 a TMP_3) in (eq A TMP_2 TMP_4))))))) in (let TMP_7 \def (\lambda (h2:
47 nat).(let TMP_6 \def (aplus g a h2) in (refl_equal A TMP_6))) in (let TMP_47
48 \def (\lambda (n: nat).(\lambda (_: ((\forall (h2: nat).(eq A (aplus g (aplus
49 g a n) h2) (aplus g a (plus n h2)))))).(\lambda (h2: nat).(let TMP_14 \def
50 (\lambda (n0: nat).(let TMP_8 \def (aplus g a n) in (let TMP_9 \def (asucc g
51 TMP_8) in (let TMP_10 \def (aplus g TMP_9 n0) in (let TMP_11 \def (plus n n0)
52 in (let TMP_12 \def (aplus g a TMP_11) in (let TMP_13 \def (asucc g TMP_12)
53 in (eq A TMP_10 TMP_13)))))))) in (let TMP_19 \def (\lambda (n0: nat).(let
54 TMP_15 \def (aplus g a n) in (let TMP_16 \def (asucc g TMP_15) in (let TMP_17
55 \def (aplus g a n0) in (let TMP_18 \def (asucc g TMP_17) in (eq A TMP_16
56 TMP_18)))))) in (let TMP_20 \def (aplus g a n) in (let TMP_21 \def (asucc g
57 TMP_20) in (let TMP_22 \def (refl_equal A TMP_21) in (let TMP_23 \def (plus n
58 O) in (let TMP_24 \def (plus_n_O n) in (let TMP_25 \def (eq_ind nat n TMP_19
59 TMP_22 TMP_23 TMP_24) in (let TMP_46 \def (\lambda (n0: nat).(\lambda (H0:
60 (eq A (aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n
61 n0))))).(let TMP_26 \def (plus n n0) in (let TMP_27 \def (S TMP_26) in (let
62 TMP_34 \def (\lambda (n1: nat).(let TMP_28 \def (aplus g a n) in (let TMP_29
63 \def (asucc g TMP_28) in (let TMP_30 \def (aplus g TMP_29 n0) in (let TMP_31
64 \def (asucc g TMP_30) in (let TMP_32 \def (aplus g a n1) in (let TMP_33 \def
65 (asucc g TMP_32) in (eq A TMP_31 TMP_33)))))))) in (let TMP_35 \def (aplus g
66 a n) in (let TMP_36 \def (asucc g TMP_35) in (let TMP_37 \def (aplus g TMP_36
67 n0) in (let TMP_38 \def (plus n n0) in (let TMP_39 \def (aplus g a TMP_38) in
68 (let TMP_40 \def (asucc g TMP_39) in (let TMP_41 \def (refl_equal G g) in
69 (let TMP_42 \def (f_equal2 G A A asucc g g TMP_37 TMP_40 TMP_41 H0) in (let
70 TMP_43 \def (S n0) in (let TMP_44 \def (plus n TMP_43) in (let TMP_45 \def
71 (plus_n_Sm n n0) in (eq_ind nat TMP_27 TMP_34 TMP_42 TMP_44
72 TMP_45))))))))))))))))) in (nat_ind TMP_14 TMP_25 TMP_46 h2))))))))))))) in
73 (nat_ind TMP_5 TMP_7 TMP_47 h1)))))).
76 \forall (g: G).(\forall (h: nat).(\forall (a: A).(eq A (aplus g (asucc g a)
77 h) (asucc g (aplus g a h)))))
79 \lambda (g: G).(\lambda (h: nat).(\lambda (a: A).(let TMP_1 \def (S O) in
80 (let TMP_2 \def (plus TMP_1 h) in (let TMP_3 \def (aplus g a TMP_2) in (let
81 TMP_6 \def (\lambda (a0: A).(let TMP_4 \def (aplus g a h) in (let TMP_5 \def
82 (asucc g TMP_4) in (eq A a0 TMP_5)))) in (let TMP_7 \def (aplus g a h) in
83 (let TMP_8 \def (asucc g TMP_7) in (let TMP_9 \def (refl_equal A TMP_8) in
84 (let TMP_10 \def (S O) in (let TMP_11 \def (aplus g a TMP_10) in (let TMP_12
85 \def (aplus g TMP_11 h) in (let TMP_13 \def (S O) in (let TMP_14 \def
86 (aplus_assoc g a TMP_13 h) in (eq_ind_r A TMP_3 TMP_6 TMP_9 TMP_12
87 TMP_14))))))))))))))).
89 theorem aplus_sort_O_S_simpl:
90 \forall (g: G).(\forall (n: nat).(\forall (k: nat).(eq A (aplus g (ASort O
91 n) (S k)) (aplus g (ASort O (next g n)) k))))
93 \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(let TMP_1 \def (ASort O
94 n) in (let TMP_2 \def (asucc g TMP_1) in (let TMP_3 \def (aplus g TMP_2 k) in
95 (let TMP_7 \def (\lambda (a: A).(let TMP_4 \def (next g n) in (let TMP_5 \def
96 (ASort O TMP_4) in (let TMP_6 \def (aplus g TMP_5 k) in (eq A a TMP_6))))) in
97 (let TMP_8 \def (next g n) in (let TMP_9 \def (ASort O TMP_8) in (let TMP_10
98 \def (aplus g TMP_9 k) in (let TMP_11 \def (refl_equal A TMP_10) in (let
99 TMP_12 \def (ASort O n) in (let TMP_13 \def (aplus g TMP_12 k) in (let TMP_14
100 \def (asucc g TMP_13) in (let TMP_15 \def (ASort O n) in (let TMP_16 \def
101 (aplus_asucc g k TMP_15) in (eq_ind A TMP_3 TMP_7 TMP_11 TMP_14
102 TMP_16)))))))))))))))).
104 theorem aplus_sort_S_S_simpl:
105 \forall (g: G).(\forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq A
106 (aplus g (ASort (S h) n) (S k)) (aplus g (ASort h n) k)))))
108 \lambda (g: G).(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(let
109 TMP_1 \def (S h) in (let TMP_2 \def (ASort TMP_1 n) in (let TMP_3 \def (asucc
110 g TMP_2) in (let TMP_4 \def (aplus g TMP_3 k) in (let TMP_7 \def (\lambda (a:
111 A).(let TMP_5 \def (ASort h n) in (let TMP_6 \def (aplus g TMP_5 k) in (eq A
112 a TMP_6)))) in (let TMP_8 \def (ASort h n) in (let TMP_9 \def (aplus g TMP_8
113 k) in (let TMP_10 \def (refl_equal A TMP_9) in (let TMP_11 \def (S h) in (let
114 TMP_12 \def (ASort TMP_11 n) in (let TMP_13 \def (aplus g TMP_12 k) in (let
115 TMP_14 \def (asucc g TMP_13) in (let TMP_15 \def (S h) in (let TMP_16 \def
116 (ASort TMP_15 n) in (let TMP_17 \def (aplus_asucc g k TMP_16) in (eq_ind A
117 TMP_4 TMP_7 TMP_10 TMP_14 TMP_17))))))))))))))))))).
119 theorem aplus_asort_O_simpl:
120 \forall (g: G).(\forall (h: nat).(\forall (n: nat).(eq A (aplus g (ASort O
121 n) h) (ASort O (next_plus g n h)))))
123 \lambda (g: G).(\lambda (h: nat).(let TMP_5 \def (\lambda (n: nat).(\forall
124 (n0: nat).(let TMP_1 \def (ASort O n0) in (let TMP_2 \def (aplus g TMP_1 n)
125 in (let TMP_3 \def (next_plus g n0 n) in (let TMP_4 \def (ASort O TMP_3) in
126 (eq A TMP_2 TMP_4))))))) in (let TMP_7 \def (\lambda (n: nat).(let TMP_6 \def
127 (ASort O n) in (refl_equal A TMP_6))) in (let TMP_33 \def (\lambda (n:
128 nat).(\lambda (H: ((\forall (n0: nat).(eq A (aplus g (ASort O n0) n) (ASort O
129 (next_plus g n0 n)))))).(\lambda (n0: nat).(let TMP_8 \def (ASort O n0) in
130 (let TMP_9 \def (asucc g TMP_8) in (let TMP_10 \def (aplus g TMP_9 n) in (let
131 TMP_14 \def (\lambda (a: A).(let TMP_11 \def (next_plus g n0 n) in (let
132 TMP_12 \def (next g TMP_11) in (let TMP_13 \def (ASort O TMP_12) in (eq A a
133 TMP_13))))) in (let TMP_15 \def (next g n0) in (let TMP_16 \def (next_plus g
134 TMP_15 n) in (let TMP_21 \def (\lambda (n1: nat).(let TMP_17 \def (next g n0)
135 in (let TMP_18 \def (ASort O TMP_17) in (let TMP_19 \def (aplus g TMP_18 n)
136 in (let TMP_20 \def (ASort O n1) in (eq A TMP_19 TMP_20)))))) in (let TMP_22
137 \def (next g n0) in (let TMP_23 \def (H TMP_22) in (let TMP_24 \def
138 (next_plus g n0 n) in (let TMP_25 \def (next g TMP_24) in (let TMP_26 \def
139 (next_plus_next g n0 n) in (let TMP_27 \def (eq_ind nat TMP_16 TMP_21 TMP_23
140 TMP_25 TMP_26) in (let TMP_28 \def (ASort O n0) in (let TMP_29 \def (aplus g
141 TMP_28 n) in (let TMP_30 \def (asucc g TMP_29) in (let TMP_31 \def (ASort O
142 n0) in (let TMP_32 \def (aplus_asucc g n TMP_31) in (eq_ind A TMP_10 TMP_14
143 TMP_27 TMP_30 TMP_32)))))))))))))))))))))) in (nat_ind TMP_5 TMP_7 TMP_33
146 theorem aplus_asort_le_simpl:
147 \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).((le h
148 k) \to (eq A (aplus g (ASort k n) h) (ASort (minus k h) n))))))
150 \lambda (g: G).(\lambda (h: nat).(let TMP_5 \def (\lambda (n: nat).(\forall
151 (k: nat).(\forall (n0: nat).((le n k) \to (let TMP_1 \def (ASort k n0) in
152 (let TMP_2 \def (aplus g TMP_1 n) in (let TMP_3 \def (minus k n) in (let
153 TMP_4 \def (ASort TMP_3 n0) in (eq A TMP_2 TMP_4))))))))) in (let TMP_13 \def
154 (\lambda (k: nat).(\lambda (n: nat).(\lambda (_: (le O k)).(let TMP_8 \def
155 (\lambda (n0: nat).(let TMP_6 \def (ASort k n) in (let TMP_7 \def (ASort n0
156 n) in (eq A TMP_6 TMP_7)))) in (let TMP_9 \def (ASort k n) in (let TMP_10
157 \def (refl_equal A TMP_9) in (let TMP_11 \def (minus k O) in (let TMP_12 \def
158 (minus_n_O k) in (eq_ind nat k TMP_8 TMP_10 TMP_11 TMP_12))))))))) in (let
159 TMP_62 \def (\lambda (h0: nat).(\lambda (H: ((\forall (k: nat).(\forall (n:
160 nat).((le h0 k) \to (eq A (aplus g (ASort k n) h0) (ASort (minus k h0)
161 n))))))).(\lambda (k: nat).(let TMP_20 \def (\lambda (n: nat).(\forall (n0:
162 nat).((le (S h0) n) \to (let TMP_14 \def (ASort n n0) in (let TMP_15 \def
163 (aplus g TMP_14 h0) in (let TMP_16 \def (asucc g TMP_15) in (let TMP_17 \def
164 (S h0) in (let TMP_18 \def (minus n TMP_17) in (let TMP_19 \def (ASort TMP_18
165 n0) in (eq A TMP_16 TMP_19)))))))))) in (let TMP_42 \def (\lambda (n:
166 nat).(\lambda (H0: (le (S h0) O)).(let TMP_22 \def (\lambda (n0: nat).(let
167 TMP_21 \def (S n0) in (eq nat O TMP_21))) in (let TMP_23 \def (\lambda (n0:
168 nat).(le h0 n0)) in (let TMP_24 \def (ASort O n) in (let TMP_25 \def (aplus g
169 TMP_24 h0) in (let TMP_26 \def (asucc g TMP_25) in (let TMP_27 \def (S h0) in
170 (let TMP_28 \def (minus O TMP_27) in (let TMP_29 \def (ASort TMP_28 n) in
171 (let TMP_30 \def (eq A TMP_26 TMP_29) in (let TMP_40 \def (\lambda (x:
172 nat).(\lambda (H1: (eq nat O (S x))).(\lambda (_: (le h0 x)).(let TMP_31 \def
173 (\lambda (ee: nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow
174 False])) in (let TMP_32 \def (S x) in (let H3 \def (eq_ind nat O TMP_31 I
175 TMP_32 H1) in (let TMP_33 \def (ASort O n) in (let TMP_34 \def (aplus g
176 TMP_33 h0) in (let TMP_35 \def (asucc g TMP_34) in (let TMP_36 \def (S h0) in
177 (let TMP_37 \def (minus O TMP_36) in (let TMP_38 \def (ASort TMP_37 n) in
178 (let TMP_39 \def (eq A TMP_35 TMP_38) in (False_ind TMP_39 H3))))))))))))))
179 in (let TMP_41 \def (le_gen_S h0 O H0) in (ex2_ind nat TMP_22 TMP_23 TMP_30
180 TMP_40 TMP_41)))))))))))))) in (let TMP_61 \def (\lambda (n: nat).(\lambda
181 (_: ((\forall (n0: nat).((le (S h0) n) \to (eq A (asucc g (aplus g (ASort n
182 n0) h0)) (ASort (minus n (S h0)) n0)))))).(\lambda (n0: nat).(\lambda (H1:
183 (le (S h0) (S n))).(let TMP_43 \def (S n) in (let TMP_44 \def (ASort TMP_43
184 n0) in (let TMP_45 \def (asucc g TMP_44) in (let TMP_46 \def (aplus g TMP_45
185 h0) in (let TMP_51 \def (\lambda (a: A).(let TMP_47 \def (S n) in (let TMP_48
186 \def (S h0) in (let TMP_49 \def (minus TMP_47 TMP_48) in (let TMP_50 \def
187 (ASort TMP_49 n0) in (eq A a TMP_50)))))) in (let TMP_52 \def (le_S_n h0 n
188 H1) in (let TMP_53 \def (H n n0 TMP_52) in (let TMP_54 \def (S n) in (let
189 TMP_55 \def (ASort TMP_54 n0) in (let TMP_56 \def (aplus g TMP_55 h0) in (let
190 TMP_57 \def (asucc g TMP_56) in (let TMP_58 \def (S n) in (let TMP_59 \def
191 (ASort TMP_58 n0) in (let TMP_60 \def (aplus_asucc g h0 TMP_59) in (eq_ind A
192 TMP_46 TMP_51 TMP_53 TMP_57 TMP_60))))))))))))))))))) in (nat_ind TMP_20
193 TMP_42 TMP_61 k))))))) in (nat_ind TMP_5 TMP_13 TMP_62 h))))).
195 theorem aplus_asort_simpl:
196 \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).(eq A
197 (aplus g (ASort k n) h) (ASort (minus k h) (next_plus g n (minus h k)))))))
199 \lambda (g: G).(\lambda (h: nat).(\lambda (k: nat).(\lambda (n: nat).(let
200 TMP_1 \def (ASort k n) in (let TMP_2 \def (aplus g TMP_1 h) in (let TMP_3
201 \def (minus k h) in (let TMP_4 \def (minus h k) in (let TMP_5 \def (next_plus
202 g n TMP_4) in (let TMP_6 \def (ASort TMP_3 TMP_5) in (let TMP_7 \def (eq A
203 TMP_2 TMP_6) in (let TMP_92 \def (\lambda (H: (lt k h)).(let TMP_8 \def
204 (minus h k) in (let TMP_9 \def (plus k TMP_8) in (let TMP_16 \def (\lambda
205 (n0: nat).(let TMP_10 \def (ASort k n) in (let TMP_11 \def (aplus g TMP_10
206 n0) in (let TMP_12 \def (minus k h) in (let TMP_13 \def (minus h k) in (let
207 TMP_14 \def (next_plus g n TMP_13) in (let TMP_15 \def (ASort TMP_12 TMP_14)
208 in (eq A TMP_11 TMP_15)))))))) in (let TMP_17 \def (ASort k n) in (let TMP_18
209 \def (aplus g TMP_17 k) in (let TMP_19 \def (minus h k) in (let TMP_20 \def
210 (aplus g TMP_18 TMP_19) in (let TMP_25 \def (\lambda (a: A).(let TMP_21 \def
211 (minus k h) in (let TMP_22 \def (minus h k) in (let TMP_23 \def (next_plus g
212 n TMP_22) in (let TMP_24 \def (ASort TMP_21 TMP_23) in (eq A a TMP_24))))))
213 in (let TMP_26 \def (minus k k) in (let TMP_27 \def (ASort TMP_26 n) in (let
214 TMP_34 \def (\lambda (a: A).(let TMP_28 \def (minus h k) in (let TMP_29 \def
215 (aplus g a TMP_28) in (let TMP_30 \def (minus k h) in (let TMP_31 \def (minus
216 h k) in (let TMP_32 \def (next_plus g n TMP_31) in (let TMP_33 \def (ASort
217 TMP_30 TMP_32) in (eq A TMP_29 TMP_33)))))))) in (let TMP_42 \def (\lambda
218 (n0: nat).(let TMP_35 \def (ASort n0 n) in (let TMP_36 \def (minus h k) in
219 (let TMP_37 \def (aplus g TMP_35 TMP_36) in (let TMP_38 \def (minus k h) in
220 (let TMP_39 \def (minus h k) in (let TMP_40 \def (next_plus g n TMP_39) in
221 (let TMP_41 \def (ASort TMP_38 TMP_40) in (eq A TMP_37 TMP_41))))))))) in
222 (let TMP_49 \def (\lambda (n0: nat).(let TMP_43 \def (ASort O n) in (let
223 TMP_44 \def (minus h k) in (let TMP_45 \def (aplus g TMP_43 TMP_44) in (let
224 TMP_46 \def (minus h k) in (let TMP_47 \def (next_plus g n TMP_46) in (let
225 TMP_48 \def (ASort n0 TMP_47) in (eq A TMP_45 TMP_48)))))))) in (let TMP_50
226 \def (minus h k) in (let TMP_51 \def (aplus_asort_O_simpl g TMP_50 n) in (let
227 TMP_52 \def (minus k h) in (let TMP_53 \def (S k) in (let TMP_54 \def (S h)
228 in (let TMP_55 \def (S k) in (let TMP_56 \def (S TMP_55) in (let TMP_57 \def
229 (S h) in (let TMP_58 \def (S k) in (let TMP_59 \def (le_n_S TMP_58 h H) in
230 (let TMP_60 \def (le_S TMP_56 TMP_57 TMP_59) in (let TMP_61 \def (le_S_n
231 TMP_53 TMP_54 TMP_60) in (let TMP_62 \def (le_S_n k h TMP_61) in (let TMP_63
232 \def (O_minus k h TMP_62) in (let TMP_64 \def (eq_ind_r nat O TMP_49 TMP_51
233 TMP_52 TMP_63) in (let TMP_65 \def (minus k k) in (let TMP_66 \def (minus_n_n
234 k) in (let TMP_67 \def (eq_ind nat O TMP_42 TMP_64 TMP_65 TMP_66) in (let
235 TMP_68 \def (ASort k n) in (let TMP_69 \def (aplus g TMP_68 k) in (let TMP_70
236 \def (le_n k) in (let TMP_71 \def (aplus_asort_le_simpl g k k n TMP_70) in
237 (let TMP_72 \def (eq_ind_r A TMP_27 TMP_34 TMP_67 TMP_69 TMP_71) in (let
238 TMP_73 \def (ASort k n) in (let TMP_74 \def (minus h k) in (let TMP_75 \def
239 (plus k TMP_74) in (let TMP_76 \def (aplus g TMP_73 TMP_75) in (let TMP_77
240 \def (ASort k n) in (let TMP_78 \def (minus h k) in (let TMP_79 \def
241 (aplus_assoc g TMP_77 k TMP_78) in (let TMP_80 \def (eq_ind A TMP_20 TMP_25
242 TMP_72 TMP_76 TMP_79) in (let TMP_81 \def (S k) in (let TMP_82 \def (S h) in
243 (let TMP_83 \def (S k) in (let TMP_84 \def (S TMP_83) in (let TMP_85 \def (S
244 h) in (let TMP_86 \def (S k) in (let TMP_87 \def (le_n_S TMP_86 h H) in (let
245 TMP_88 \def (le_S TMP_84 TMP_85 TMP_87) in (let TMP_89 \def (le_S_n TMP_81
246 TMP_82 TMP_88) in (let TMP_90 \def (le_S_n k h TMP_89) in (let TMP_91 \def
247 (le_plus_minus k h TMP_90) in (eq_ind_r nat TMP_9 TMP_16 TMP_80 h
248 TMP_91))))))))))))))))))))))))))))))))))))))))))))))))))))))))) in (let
249 TMP_116 \def (\lambda (H: (le h k)).(let TMP_93 \def (minus k h) in (let
250 TMP_94 \def (ASort TMP_93 n) in (let TMP_99 \def (\lambda (a: A).(let TMP_95
251 \def (minus k h) in (let TMP_96 \def (minus h k) in (let TMP_97 \def
252 (next_plus g n TMP_96) in (let TMP_98 \def (ASort TMP_95 TMP_97) in (eq A a
253 TMP_98)))))) in (let TMP_105 \def (\lambda (n0: nat).(let TMP_100 \def (minus
254 k h) in (let TMP_101 \def (ASort TMP_100 n) in (let TMP_102 \def (minus k h)
255 in (let TMP_103 \def (next_plus g n n0) in (let TMP_104 \def (ASort TMP_102
256 TMP_103) in (eq A TMP_101 TMP_104))))))) in (let TMP_106 \def (minus k h) in
257 (let TMP_107 \def (next_plus g n O) in (let TMP_108 \def (ASort TMP_106
258 TMP_107) in (let TMP_109 \def (refl_equal A TMP_108) in (let TMP_110 \def
259 (minus h k) in (let TMP_111 \def (O_minus h k H) in (let TMP_112 \def
260 (eq_ind_r nat O TMP_105 TMP_109 TMP_110 TMP_111) in (let TMP_113 \def (ASort
261 k n) in (let TMP_114 \def (aplus g TMP_113 h) in (let TMP_115 \def
262 (aplus_asort_le_simpl g h k n H) in (eq_ind_r A TMP_94 TMP_99 TMP_112 TMP_114
263 TMP_115)))))))))))))))) in (lt_le_e k h TMP_7 TMP_92 TMP_116))))))))))))).
265 theorem aplus_ahead_simpl:
266 \forall (g: G).(\forall (h: nat).(\forall (a1: A).(\forall (a2: A).(eq A
267 (aplus g (AHead a1 a2) h) (AHead a1 (aplus g a2 h))))))
269 \lambda (g: G).(\lambda (h: nat).(let TMP_5 \def (\lambda (n: nat).(\forall
270 (a1: A).(\forall (a2: A).(let TMP_1 \def (AHead a1 a2) in (let TMP_2 \def
271 (aplus g TMP_1 n) in (let TMP_3 \def (aplus g a2 n) in (let TMP_4 \def (AHead
272 a1 TMP_3) in (eq A TMP_2 TMP_4)))))))) in (let TMP_7 \def (\lambda (a1:
273 A).(\lambda (a2: A).(let TMP_6 \def (AHead a1 a2) in (refl_equal A TMP_6))))
274 in (let TMP_33 \def (\lambda (n: nat).(\lambda (H: ((\forall (a1: A).(\forall
275 (a2: A).(eq A (aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2
276 n))))))).(\lambda (a1: A).(\lambda (a2: A).(let TMP_8 \def (AHead a1 a2) in
277 (let TMP_9 \def (asucc g TMP_8) in (let TMP_10 \def (aplus g TMP_9 n) in (let
278 TMP_14 \def (\lambda (a: A).(let TMP_11 \def (aplus g a2 n) in (let TMP_12
279 \def (asucc g TMP_11) in (let TMP_13 \def (AHead a1 TMP_12) in (eq A a
280 TMP_13))))) in (let TMP_15 \def (asucc g a2) in (let TMP_16 \def (aplus g
281 TMP_15 n) in (let TMP_21 \def (\lambda (a: A).(let TMP_17 \def (AHead a1 a2)
282 in (let TMP_18 \def (asucc g TMP_17) in (let TMP_19 \def (aplus g TMP_18 n)
283 in (let TMP_20 \def (AHead a1 a) in (eq A TMP_19 TMP_20)))))) in (let TMP_22
284 \def (asucc g a2) in (let TMP_23 \def (H a1 TMP_22) in (let TMP_24 \def
285 (aplus g a2 n) in (let TMP_25 \def (asucc g TMP_24) in (let TMP_26 \def
286 (aplus_asucc g n a2) in (let TMP_27 \def (eq_ind A TMP_16 TMP_21 TMP_23
287 TMP_25 TMP_26) in (let TMP_28 \def (AHead a1 a2) in (let TMP_29 \def (aplus g
288 TMP_28 n) in (let TMP_30 \def (asucc g TMP_29) in (let TMP_31 \def (AHead a1
289 a2) in (let TMP_32 \def (aplus_asucc g n TMP_31) in (eq_ind A TMP_10 TMP_14
290 TMP_27 TMP_30 TMP_32))))))))))))))))))))))) in (nat_ind TMP_5 TMP_7 TMP_33
293 theorem aplus_asucc_false:
294 \forall (g: G).(\forall (a: A).(\forall (h: nat).((eq A (aplus g (asucc g a)
295 h) a) \to (\forall (P: Prop).P))))
297 \lambda (g: G).(\lambda (a: A).(let TMP_1 \def (\lambda (a0: A).(\forall (h:
298 nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: Prop).P)))) in (let
299 TMP_70 \def (\lambda (n: nat).(\lambda (n0: nat).(\lambda (h: nat).(\lambda
300 (H: (eq A (aplus g (match n with [O \Rightarrow (ASort O (next g n0)) | (S
301 h0) \Rightarrow (ASort h0 n0)]) h) (ASort n n0))).(\lambda (P: Prop).(let
302 TMP_2 \def (\lambda (n1: nat).((eq A (aplus g (match n1 with [O \Rightarrow
303 (ASort O (next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0))
304 \to P)) in (let TMP_36 \def (\lambda (H0: (eq A (aplus g (ASort O (next g
305 n0)) h) (ASort O n0))).(let TMP_3 \def (next g n0) in (let TMP_4 \def (ASort
306 O TMP_3) in (let TMP_5 \def (aplus g TMP_4 h) in (let TMP_7 \def (\lambda
307 (a0: A).(let TMP_6 \def (ASort O n0) in (eq A a0 TMP_6))) in (let TMP_8 \def
308 (minus O h) in (let TMP_9 \def (next g n0) in (let TMP_10 \def (minus h O) in
309 (let TMP_11 \def (next_plus g TMP_9 TMP_10) in (let TMP_12 \def (ASort TMP_8
310 TMP_11) in (let TMP_13 \def (next g n0) in (let TMP_14 \def
311 (aplus_asort_simpl g h O TMP_13) in (let H1 \def (eq_ind A TMP_5 TMP_7 H0
312 TMP_12 TMP_14) in (let TMP_18 \def (\lambda (e: A).(match e with [(ASort _
313 n1) \Rightarrow n1 | (AHead _ _) \Rightarrow (let TMP_16 \def (next g n0) in
314 (let TMP_17 \def (minus h O) in (next_plus g TMP_16 TMP_17)))])) in (let
315 TMP_19 \def (minus O h) in (let TMP_20 \def (next g n0) in (let TMP_21 \def
316 (minus h O) in (let TMP_22 \def (next_plus g TMP_20 TMP_21) in (let TMP_23
317 \def (ASort TMP_19 TMP_22) in (let TMP_24 \def (ASort O n0) in (let H2 \def
318 (f_equal A nat TMP_18 TMP_23 TMP_24 H1) in (let TMP_25 \def (minus h O) in
319 (let TMP_28 \def (\lambda (n1: nat).(let TMP_26 \def (next g n0) in (let
320 TMP_27 \def (next_plus g TMP_26 n1) in (eq nat TMP_27 n0)))) in (let TMP_29
321 \def (minus_n_O h) in (let H3 \def (eq_ind_r nat TMP_25 TMP_28 H2 h TMP_29)
322 in (let TMP_30 \def (le_n n0) in (let TMP_31 \def (next g n0) in (let TMP_32
323 \def (next_plus g TMP_31 h) in (let TMP_33 \def (\lambda (n1: nat).(lt n0
324 n1)) in (let TMP_34 \def (next_plus_lt g h n0) in (let TMP_35 \def (eq_ind
325 nat TMP_32 TMP_33 TMP_34 n0 H3) in (le_lt_false n0 n0 TMP_30 TMP_35
326 P)))))))))))))))))))))))))))))))) in (let TMP_69 \def (\lambda (n1:
327 nat).(\lambda (_: (((eq A (aplus g (match n1 with [O \Rightarrow (ASort O
328 (next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) \to
329 P))).(\lambda (H0: (eq A (aplus g (ASort n1 n0) h) (ASort (S n1) n0))).(let
330 TMP_37 \def (ASort n1 n0) in (let TMP_38 \def (aplus g TMP_37 h) in (let
331 TMP_41 \def (\lambda (a0: A).(let TMP_39 \def (S n1) in (let TMP_40 \def
332 (ASort TMP_39 n0) in (eq A a0 TMP_40)))) in (let TMP_42 \def (minus n1 h) in
333 (let TMP_43 \def (minus h n1) in (let TMP_44 \def (next_plus g n0 TMP_43) in
334 (let TMP_45 \def (ASort TMP_42 TMP_44) in (let TMP_46 \def (aplus_asort_simpl
335 g h n1 n0) in (let H1 \def (eq_ind A TMP_38 TMP_41 H0 TMP_45 TMP_46) in (let
336 TMP_47 \def (\lambda (e: A).(match e with [(ASort n2 _) \Rightarrow n2 |
337 (AHead _ _) \Rightarrow (minus n1 h)])) in (let TMP_48 \def (minus n1 h) in
338 (let TMP_49 \def (minus h n1) in (let TMP_50 \def (next_plus g n0 TMP_49) in
339 (let TMP_51 \def (ASort TMP_48 TMP_50) in (let TMP_52 \def (S n1) in (let
340 TMP_53 \def (ASort TMP_52 n0) in (let H2 \def (f_equal A nat TMP_47 TMP_51
341 TMP_53 H1) in (let TMP_56 \def (\lambda (e: A).(match e with [(ASort _ n2)
342 \Rightarrow n2 | (AHead _ _) \Rightarrow (let TMP_55 \def (minus h n1) in
343 (next_plus g n0 TMP_55))])) in (let TMP_57 \def (minus n1 h) in (let TMP_58
344 \def (minus h n1) in (let TMP_59 \def (next_plus g n0 TMP_58) in (let TMP_60
345 \def (ASort TMP_57 TMP_59) in (let TMP_61 \def (S n1) in (let TMP_62 \def
346 (ASort TMP_61 n0) in (let H3 \def (f_equal A nat TMP_56 TMP_60 TMP_62 H1) in
347 (let TMP_68 \def (\lambda (H4: (eq nat (minus n1 h) (S n1))).(let TMP_63 \def
348 (minus n1 h) in (let TMP_64 \def (\lambda (n2: nat).(le n2 n1)) in (let
349 TMP_65 \def (minus_le n1 h) in (let TMP_66 \def (S n1) in (let TMP_67 \def
350 (eq_ind nat TMP_63 TMP_64 TMP_65 TMP_66 H4) in (le_Sx_x n1 TMP_67 P))))))) in
351 (TMP_68 H2)))))))))))))))))))))))))))))) in (nat_ind TMP_2 TMP_36 TMP_69 n
352 H))))))))) in (let TMP_88 \def (\lambda (a0: A).(\lambda (_: ((\forall (h:
353 nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P:
354 Prop).P))))).(\lambda (a1: A).(\lambda (H0: ((\forall (h: nat).((eq A (aplus
355 g (asucc g a1) h) a1) \to (\forall (P: Prop).P))))).(\lambda (h:
356 nat).(\lambda (H1: (eq A (aplus g (AHead a0 (asucc g a1)) h) (AHead a0
357 a1))).(\lambda (P: Prop).(let TMP_71 \def (asucc g a1) in (let TMP_72 \def
358 (AHead a0 TMP_71) in (let TMP_73 \def (aplus g TMP_72 h) in (let TMP_75 \def
359 (\lambda (a2: A).(let TMP_74 \def (AHead a0 a1) in (eq A a2 TMP_74))) in (let
360 TMP_76 \def (asucc g a1) in (let TMP_77 \def (aplus g TMP_76 h) in (let
361 TMP_78 \def (AHead a0 TMP_77) in (let TMP_79 \def (asucc g a1) in (let TMP_80
362 \def (aplus_ahead_simpl g h a0 TMP_79) in (let H2 \def (eq_ind A TMP_73
363 TMP_75 H1 TMP_78 TMP_80) in (let TMP_83 \def (\lambda (e: A).(match e with
364 [(ASort _ _) \Rightarrow (let TMP_82 \def (asucc g a1) in (aplus g TMP_82 h))
365 | (AHead _ a2) \Rightarrow a2])) in (let TMP_84 \def (asucc g a1) in (let
366 TMP_85 \def (aplus g TMP_84 h) in (let TMP_86 \def (AHead a0 TMP_85) in (let
367 TMP_87 \def (AHead a0 a1) in (let H3 \def (f_equal A A TMP_83 TMP_86 TMP_87
368 H2) in (H0 h H3 P)))))))))))))))))))))))) in (A_ind TMP_1 TMP_70 TMP_88
372 \forall (g: G).(\forall (h1: nat).(\forall (h2: nat).(\forall (a: A).((eq A
373 (aplus g a h1) (aplus g a h2)) \to (eq nat h1 h2)))))
375 \lambda (g: G).(\lambda (h1: nat).(let TMP_1 \def (\lambda (n: nat).(\forall
376 (h2: nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n
377 h2))))) in (let TMP_16 \def (\lambda (h2: nat).(let TMP_2 \def (\lambda (n:
378 nat).(\forall (a: A).((eq A (aplus g a O) (aplus g a n)) \to (eq nat O n))))
379 in (let TMP_3 \def (\lambda (a: A).(\lambda (_: (eq A a a)).(refl_equal nat
380 O))) in (let TMP_15 \def (\lambda (n: nat).(\lambda (_: ((\forall (a: A).((eq
381 A a (aplus g a n)) \to (eq nat O n))))).(\lambda (a: A).(\lambda (H0: (eq A a
382 (asucc g (aplus g a n)))).(let TMP_4 \def (aplus g a n) in (let TMP_5 \def
383 (asucc g TMP_4) in (let TMP_6 \def (\lambda (a0: A).(eq A a a0)) in (let
384 TMP_7 \def (asucc g a) in (let TMP_8 \def (aplus g TMP_7 n) in (let TMP_9
385 \def (aplus_asucc g n a) in (let H1 \def (eq_ind_r A TMP_5 TMP_6 H0 TMP_8
386 TMP_9) in (let TMP_10 \def (asucc g a) in (let TMP_11 \def (aplus g TMP_10 n)
387 in (let TMP_12 \def (sym_eq A a TMP_11 H1) in (let TMP_13 \def (S n) in (let
388 TMP_14 \def (eq nat O TMP_13) in (aplus_asucc_false g a n TMP_12
389 TMP_14))))))))))))))))) in (nat_ind TMP_2 TMP_3 TMP_15 h2))))) in (let TMP_47
390 \def (\lambda (n: nat).(\lambda (H: ((\forall (h2: nat).(\forall (a: A).((eq
391 A (aplus g a n) (aplus g a h2)) \to (eq nat n h2)))))).(\lambda (h2:
392 nat).(let TMP_18 \def (\lambda (n0: nat).(\forall (a: A).((eq A (aplus g a (S
393 n)) (aplus g a n0)) \to (let TMP_17 \def (S n) in (eq nat TMP_17 n0))))) in
394 (let TMP_27 \def (\lambda (a: A).(\lambda (H0: (eq A (asucc g (aplus g a n))
395 a)).(let TMP_19 \def (aplus g a n) in (let TMP_20 \def (asucc g TMP_19) in
396 (let TMP_21 \def (\lambda (a0: A).(eq A a0 a)) in (let TMP_22 \def (asucc g
397 a) in (let TMP_23 \def (aplus g TMP_22 n) in (let TMP_24 \def (aplus_asucc g
398 n a) in (let H1 \def (eq_ind_r A TMP_20 TMP_21 H0 TMP_23 TMP_24) in (let
399 TMP_25 \def (S n) in (let TMP_26 \def (eq nat TMP_25 O) in (aplus_asucc_false
400 g a n H1 TMP_26)))))))))))) in (let TMP_46 \def (\lambda (n0: nat).(\lambda
401 (_: ((\forall (a: A).((eq A (asucc g (aplus g a n)) (aplus g a n0)) \to (eq
402 nat (S n) n0))))).(\lambda (a: A).(\lambda (H1: (eq A (asucc g (aplus g a n))
403 (asucc g (aplus g a n0)))).(let TMP_28 \def (aplus g a n) in (let TMP_29 \def
404 (asucc g TMP_28) in (let TMP_32 \def (\lambda (a0: A).(let TMP_30 \def (aplus
405 g a n0) in (let TMP_31 \def (asucc g TMP_30) in (eq A a0 TMP_31)))) in (let
406 TMP_33 \def (asucc g a) in (let TMP_34 \def (aplus g TMP_33 n) in (let TMP_35
407 \def (aplus_asucc g n a) in (let H2 \def (eq_ind_r A TMP_29 TMP_32 H1 TMP_34
408 TMP_35) in (let TMP_36 \def (aplus g a n0) in (let TMP_37 \def (asucc g
409 TMP_36) in (let TMP_40 \def (\lambda (a0: A).(let TMP_38 \def (asucc g a) in
410 (let TMP_39 \def (aplus g TMP_38 n) in (eq A TMP_39 a0)))) in (let TMP_41
411 \def (asucc g a) in (let TMP_42 \def (aplus g TMP_41 n0) in (let TMP_43 \def
412 (aplus_asucc g n0 a) in (let H3 \def (eq_ind_r A TMP_37 TMP_40 H2 TMP_42
413 TMP_43) in (let TMP_44 \def (asucc g a) in (let TMP_45 \def (H n0 TMP_44 H3)
414 in (f_equal nat nat S n n0 TMP_45))))))))))))))))))))) in (nat_ind TMP_18
415 TMP_27 TMP_46 h2))))))) in (nat_ind TMP_1 TMP_16 TMP_47 h1))))).