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/subst1/defs.ma".
19 include "basic_1/subst0/fwd.ma".
22 \forall (i: nat).(\forall (v: T).(\forall (t1: T).(\forall (P: ((T \to
23 Prop))).((P t1) \to (((\forall (t2: T).((subst0 i v t1 t2) \to (P t2)))) \to
24 (\forall (t: T).((subst1 i v t1 t) \to (P t))))))))
26 \lambda (i: nat).(\lambda (v: T).(\lambda (t1: T).(\lambda (P: ((T \to
27 Prop))).(\lambda (f: (P t1)).(\lambda (f0: ((\forall (t2: T).((subst0 i v t1
28 t2) \to (P t2))))).(\lambda (t: T).(\lambda (s0: (subst1 i v t1 t)).(match s0
29 with [subst1_refl \Rightarrow f | (subst1_single x x0) \Rightarrow (f0 x
32 theorem subst1_gen_sort:
33 \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1
34 i v (TSort n) x) \to (eq T x (TSort n))))))
36 \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
37 (H: (subst1 i v (TSort n) x)).(let TMP_1 \def (TSort n) in (let TMP_3 \def
38 (\lambda (t: T).(let TMP_2 \def (TSort n) in (eq T t TMP_2))) in (let TMP_4
39 \def (TSort n) in (let TMP_5 \def (refl_equal T TMP_4) in (let TMP_8 \def
40 (\lambda (t2: T).(\lambda (H0: (subst0 i v (TSort n) t2)).(let TMP_6 \def
41 (TSort n) in (let TMP_7 \def (eq T t2 TMP_6) in (subst0_gen_sort v t2 i n H0
42 TMP_7))))) in (subst1_ind i v TMP_1 TMP_3 TMP_5 TMP_8 x H)))))))))).
44 theorem subst1_gen_lref:
45 \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1
46 i v (TLRef n) x) \to (or (eq T x (TLRef n)) (land (eq nat n i) (eq T x (lift
49 \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
50 (H: (subst1 i v (TLRef n) x)).(let TMP_1 \def (TLRef n) in (let TMP_9 \def
51 (\lambda (t: T).(let TMP_2 \def (TLRef n) in (let TMP_3 \def (eq T t TMP_2)
52 in (let TMP_4 \def (eq nat n i) in (let TMP_5 \def (S n) in (let TMP_6 \def
53 (lift TMP_5 O v) in (let TMP_7 \def (eq T t TMP_6) in (let TMP_8 \def (land
54 TMP_4 TMP_7) in (or TMP_3 TMP_8))))))))) in (let TMP_10 \def (TLRef n) in
55 (let TMP_11 \def (TLRef n) in (let TMP_12 \def (eq T TMP_10 TMP_11) in (let
56 TMP_13 \def (eq nat n i) in (let TMP_14 \def (TLRef n) in (let TMP_15 \def (S
57 n) in (let TMP_16 \def (lift TMP_15 O v) in (let TMP_17 \def (eq T TMP_14
58 TMP_16) in (let TMP_18 \def (land TMP_13 TMP_17) in (let TMP_19 \def (TLRef
59 n) in (let TMP_20 \def (refl_equal T TMP_19) in (let TMP_21 \def (or_introl
60 TMP_12 TMP_18 TMP_20) in (let TMP_48 \def (\lambda (t2: T).(\lambda (H0:
61 (subst0 i v (TLRef n) t2)).(let TMP_22 \def (eq nat n i) in (let TMP_23 \def
62 (S n) in (let TMP_24 \def (lift TMP_23 O v) in (let TMP_25 \def (eq T t2
63 TMP_24) in (let TMP_26 \def (TLRef n) in (let TMP_27 \def (eq T t2 TMP_26) in
64 (let TMP_28 \def (eq nat n i) in (let TMP_29 \def (S n) in (let TMP_30 \def
65 (lift TMP_29 O v) in (let TMP_31 \def (eq T t2 TMP_30) in (let TMP_32 \def
66 (land TMP_28 TMP_31) in (let TMP_33 \def (or TMP_27 TMP_32) in (let TMP_46
67 \def (\lambda (H1: (eq nat n i)).(\lambda (H2: (eq T t2 (lift (S n) O
68 v))).(let TMP_34 \def (TLRef n) in (let TMP_35 \def (eq T t2 TMP_34) in (let
69 TMP_36 \def (eq nat n i) in (let TMP_37 \def (S n) in (let TMP_38 \def (lift
70 TMP_37 O v) in (let TMP_39 \def (eq T t2 TMP_38) in (let TMP_40 \def (land
71 TMP_36 TMP_39) in (let TMP_41 \def (eq nat n i) in (let TMP_42 \def (S n) in
72 (let TMP_43 \def (lift TMP_42 O v) in (let TMP_44 \def (eq T t2 TMP_43) in
73 (let TMP_45 \def (conj TMP_41 TMP_44 H1 H2) in (or_intror TMP_35 TMP_40
74 TMP_45))))))))))))))) in (let TMP_47 \def (subst0_gen_lref v t2 i n H0) in
75 (land_ind TMP_22 TMP_25 TMP_33 TMP_46 TMP_47))))))))))))))))) in (subst1_ind
76 i v TMP_1 TMP_9 TMP_21 TMP_48 x H)))))))))))))))))))).
78 theorem subst1_gen_head:
79 \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall
80 (x: T).(\forall (i: nat).((subst1 i v (THead k u1 t1) x) \to (ex3_2 T T
81 (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2:
82 T).(\lambda (_: T).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (t2:
83 T).(subst1 (s k i) v t1 t2))))))))))
85 \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda
86 (x: T).(\lambda (i: nat).(\lambda (H: (subst1 i v (THead k u1 t1) x)).(let
87 TMP_1 \def (THead k u1 t1) in (let TMP_7 \def (\lambda (t: T).(let TMP_3 \def
88 (\lambda (u2: T).(\lambda (t2: T).(let TMP_2 \def (THead k u2 t2) in (eq T t
89 TMP_2)))) in (let TMP_4 \def (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1
90 u2))) in (let TMP_6 \def (\lambda (_: T).(\lambda (t2: T).(let TMP_5 \def (s
91 k i) in (subst1 TMP_5 v t1 t2)))) in (ex3_2 T T TMP_3 TMP_4 TMP_6))))) in
92 (let TMP_10 \def (\lambda (u2: T).(\lambda (t2: T).(let TMP_8 \def (THead k
93 u1 t1) in (let TMP_9 \def (THead k u2 t2) in (eq T TMP_8 TMP_9))))) in (let
94 TMP_11 \def (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) in (let
95 TMP_13 \def (\lambda (_: T).(\lambda (t2: T).(let TMP_12 \def (s k i) in
96 (subst1 TMP_12 v t1 t2)))) in (let TMP_14 \def (THead k u1 t1) in (let TMP_15
97 \def (refl_equal T TMP_14) in (let TMP_16 \def (subst1_refl i v u1) in (let
98 TMP_17 \def (s k i) in (let TMP_18 \def (subst1_refl TMP_17 v t1) in (let
99 TMP_19 \def (ex3_2_intro T T TMP_10 TMP_11 TMP_13 u1 t1 TMP_15 TMP_16 TMP_18)
100 in (let TMP_102 \def (\lambda (t2: T).(\lambda (H0: (subst0 i v (THead k u1
101 t1) t2)).(let TMP_21 \def (\lambda (u2: T).(let TMP_20 \def (THead k u2 t1)
102 in (eq T t2 TMP_20))) in (let TMP_22 \def (\lambda (u2: T).(subst0 i v u1
103 u2)) in (let TMP_23 \def (ex2 T TMP_21 TMP_22) in (let TMP_25 \def (\lambda
104 (t3: T).(let TMP_24 \def (THead k u1 t3) in (eq T t2 TMP_24))) in (let TMP_27
105 \def (\lambda (t3: T).(let TMP_26 \def (s k i) in (subst0 TMP_26 v t1 t3)))
106 in (let TMP_28 \def (ex2 T TMP_25 TMP_27) in (let TMP_30 \def (\lambda (u2:
107 T).(\lambda (t3: T).(let TMP_29 \def (THead k u2 t3) in (eq T t2 TMP_29))))
108 in (let TMP_31 \def (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) in
109 (let TMP_33 \def (\lambda (_: T).(\lambda (t3: T).(let TMP_32 \def (s k i) in
110 (subst0 TMP_32 v t1 t3)))) in (let TMP_34 \def (ex3_2 T T TMP_30 TMP_31
111 TMP_33) in (let TMP_36 \def (\lambda (u2: T).(\lambda (t3: T).(let TMP_35
112 \def (THead k u2 t3) in (eq T t2 TMP_35)))) in (let TMP_37 \def (\lambda (u2:
113 T).(\lambda (_: T).(subst1 i v u1 u2))) in (let TMP_39 \def (\lambda (_:
114 T).(\lambda (t3: T).(let TMP_38 \def (s k i) in (subst1 TMP_38 v t1 t3)))) in
115 (let TMP_40 \def (ex3_2 T T TMP_36 TMP_37 TMP_39) in (let TMP_59 \def
116 (\lambda (H1: (ex2 T (\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda
117 (u2: T).(subst0 i v u1 u2)))).(let TMP_42 \def (\lambda (u2: T).(let TMP_41
118 \def (THead k u2 t1) in (eq T t2 TMP_41))) in (let TMP_43 \def (\lambda (u2:
119 T).(subst0 i v u1 u2)) in (let TMP_45 \def (\lambda (u2: T).(\lambda (t3:
120 T).(let TMP_44 \def (THead k u2 t3) in (eq T t2 TMP_44)))) in (let TMP_46
121 \def (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) in (let TMP_48
122 \def (\lambda (_: T).(\lambda (t3: T).(let TMP_47 \def (s k i) in (subst1
123 TMP_47 v t1 t3)))) in (let TMP_49 \def (ex3_2 T T TMP_45 TMP_46 TMP_48) in
124 (let TMP_58 \def (\lambda (x0: T).(\lambda (H2: (eq T t2 (THead k x0
125 t1))).(\lambda (H3: (subst0 i v u1 x0)).(let TMP_51 \def (\lambda (u2:
126 T).(\lambda (t3: T).(let TMP_50 \def (THead k u2 t3) in (eq T t2 TMP_50))))
127 in (let TMP_52 \def (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) in
128 (let TMP_54 \def (\lambda (_: T).(\lambda (t3: T).(let TMP_53 \def (s k i) in
129 (subst1 TMP_53 v t1 t3)))) in (let TMP_55 \def (subst1_single i v u1 x0 H3)
130 in (let TMP_56 \def (s k i) in (let TMP_57 \def (subst1_refl TMP_56 v t1) in
131 (ex3_2_intro T T TMP_51 TMP_52 TMP_54 x0 t1 H2 TMP_55 TMP_57)))))))))) in
132 (ex2_ind T TMP_42 TMP_43 TMP_49 TMP_58 H1))))))))) in (let TMP_79 \def
133 (\lambda (H1: (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda
134 (t3: T).(subst0 (s k i) v t1 t3)))).(let TMP_61 \def (\lambda (t3: T).(let
135 TMP_60 \def (THead k u1 t3) in (eq T t2 TMP_60))) in (let TMP_63 \def
136 (\lambda (t3: T).(let TMP_62 \def (s k i) in (subst0 TMP_62 v t1 t3))) in
137 (let TMP_65 \def (\lambda (u2: T).(\lambda (t3: T).(let TMP_64 \def (THead k
138 u2 t3) in (eq T t2 TMP_64)))) in (let TMP_66 \def (\lambda (u2: T).(\lambda
139 (_: T).(subst1 i v u1 u2))) in (let TMP_68 \def (\lambda (_: T).(\lambda (t3:
140 T).(let TMP_67 \def (s k i) in (subst1 TMP_67 v t1 t3)))) in (let TMP_69 \def
141 (ex3_2 T T TMP_65 TMP_66 TMP_68) in (let TMP_78 \def (\lambda (x0:
142 T).(\lambda (H2: (eq T t2 (THead k u1 x0))).(\lambda (H3: (subst0 (s k i) v
143 t1 x0)).(let TMP_71 \def (\lambda (u2: T).(\lambda (t3: T).(let TMP_70 \def
144 (THead k u2 t3) in (eq T t2 TMP_70)))) in (let TMP_72 \def (\lambda (u2:
145 T).(\lambda (_: T).(subst1 i v u1 u2))) in (let TMP_74 \def (\lambda (_:
146 T).(\lambda (t3: T).(let TMP_73 \def (s k i) in (subst1 TMP_73 v t1 t3)))) in
147 (let TMP_75 \def (subst1_refl i v u1) in (let TMP_76 \def (s k i) in (let
148 TMP_77 \def (subst1_single TMP_76 v t1 x0 H3) in (ex3_2_intro T T TMP_71
149 TMP_72 TMP_74 u1 x0 H2 TMP_75 TMP_77)))))))))) in (ex2_ind T TMP_61 TMP_63
150 TMP_69 TMP_78 H1))))))))) in (let TMP_100 \def (\lambda (H1: (ex3_2 T T
151 (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2:
152 T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3:
153 T).(subst0 (s k i) v t1 t3))))).(let TMP_81 \def (\lambda (u2: T).(\lambda
154 (t3: T).(let TMP_80 \def (THead k u2 t3) in (eq T t2 TMP_80)))) in (let
155 TMP_82 \def (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) in (let
156 TMP_84 \def (\lambda (_: T).(\lambda (t3: T).(let TMP_83 \def (s k i) in
157 (subst0 TMP_83 v t1 t3)))) in (let TMP_86 \def (\lambda (u2: T).(\lambda (t3:
158 T).(let TMP_85 \def (THead k u2 t3) in (eq T t2 TMP_85)))) in (let TMP_87
159 \def (\lambda (u2: T).(\lambda (_: T).(subst1 i v u1 u2))) in (let TMP_89
160 \def (\lambda (_: T).(\lambda (t3: T).(let TMP_88 \def (s k i) in (subst1
161 TMP_88 v t1 t3)))) in (let TMP_90 \def (ex3_2 T T TMP_86 TMP_87 TMP_89) in
162 (let TMP_99 \def (\lambda (x0: T).(\lambda (x1: T).(\lambda (H2: (eq T t2
163 (THead k x0 x1))).(\lambda (H3: (subst0 i v u1 x0)).(\lambda (H4: (subst0 (s
164 k i) v t1 x1)).(let TMP_92 \def (\lambda (u2: T).(\lambda (t3: T).(let TMP_91
165 \def (THead k u2 t3) in (eq T t2 TMP_91)))) in (let TMP_93 \def (\lambda (u2:
166 T).(\lambda (_: T).(subst1 i v u1 u2))) in (let TMP_95 \def (\lambda (_:
167 T).(\lambda (t3: T).(let TMP_94 \def (s k i) in (subst1 TMP_94 v t1 t3)))) in
168 (let TMP_96 \def (subst1_single i v u1 x0 H3) in (let TMP_97 \def (s k i) in
169 (let TMP_98 \def (subst1_single TMP_97 v t1 x1 H4) in (ex3_2_intro T T TMP_92
170 TMP_93 TMP_95 x0 x1 H2 TMP_96 TMP_98)))))))))))) in (ex3_2_ind T T TMP_81
171 TMP_82 TMP_84 TMP_90 TMP_99 H1)))))))))) in (let TMP_101 \def
172 (subst0_gen_head k v u1 t1 t2 i H0) in (or3_ind TMP_23 TMP_28 TMP_34 TMP_40
173 TMP_59 TMP_79 TMP_100 TMP_101))))))))))))))))))))) in (subst1_ind i v TMP_1
174 TMP_7 TMP_19 TMP_102 x H))))))))))))))))))).
176 theorem subst1_gen_lift_lt:
177 \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
178 (h: nat).(\forall (d: nat).((subst1 i (lift h d u) (lift h (S (plus i d)) t1)
179 x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h (S (plus i d)) t2))) (\lambda
180 (t2: T).(subst1 i u t1 t2)))))))))
182 \lambda (u: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda
183 (h: nat).(\lambda (d: nat).(\lambda (H: (subst1 i (lift h d u) (lift h (S
184 (plus i d)) t1) x)).(let TMP_1 \def (lift h d u) in (let TMP_2 \def (plus i
185 d) in (let TMP_3 \def (S TMP_2) in (let TMP_4 \def (lift h TMP_3 t1) in (let
186 TMP_10 \def (\lambda (t: T).(let TMP_8 \def (\lambda (t2: T).(let TMP_5 \def
187 (plus i d) in (let TMP_6 \def (S TMP_5) in (let TMP_7 \def (lift h TMP_6 t2)
188 in (eq T t TMP_7))))) in (let TMP_9 \def (\lambda (t2: T).(subst1 i u t1 t2))
189 in (ex2 T TMP_8 TMP_9)))) in (let TMP_17 \def (\lambda (t2: T).(let TMP_11
190 \def (plus i d) in (let TMP_12 \def (S TMP_11) in (let TMP_13 \def (lift h
191 TMP_12 t1) in (let TMP_14 \def (plus i d) in (let TMP_15 \def (S TMP_14) in
192 (let TMP_16 \def (lift h TMP_15 t2) in (eq T TMP_13 TMP_16)))))))) in (let
193 TMP_18 \def (\lambda (t2: T).(subst1 i u t1 t2)) in (let TMP_19 \def (plus i
194 d) in (let TMP_20 \def (S TMP_19) in (let TMP_21 \def (lift h TMP_20 t1) in
195 (let TMP_22 \def (refl_equal T TMP_21) in (let TMP_23 \def (subst1_refl i u
196 t1) in (let TMP_24 \def (ex_intro2 T TMP_17 TMP_18 t1 TMP_22 TMP_23) in (let
197 TMP_44 \def (\lambda (t2: T).(\lambda (H0: (subst0 i (lift h d u) (lift h (S
198 (plus i d)) t1) t2)).(let TMP_28 \def (\lambda (t3: T).(let TMP_25 \def (plus
199 i d) in (let TMP_26 \def (S TMP_25) in (let TMP_27 \def (lift h TMP_26 t3) in
200 (eq T t2 TMP_27))))) in (let TMP_29 \def (\lambda (t3: T).(subst0 i u t1 t3))
201 in (let TMP_33 \def (\lambda (t3: T).(let TMP_30 \def (plus i d) in (let
202 TMP_31 \def (S TMP_30) in (let TMP_32 \def (lift h TMP_31 t3) in (eq T t2
203 TMP_32))))) in (let TMP_34 \def (\lambda (t3: T).(subst1 i u t1 t3)) in (let
204 TMP_35 \def (ex2 T TMP_33 TMP_34) in (let TMP_42 \def (\lambda (x0:
205 T).(\lambda (H1: (eq T t2 (lift h (S (plus i d)) x0))).(\lambda (H2: (subst0
206 i u t1 x0)).(let TMP_39 \def (\lambda (t3: T).(let TMP_36 \def (plus i d) in
207 (let TMP_37 \def (S TMP_36) in (let TMP_38 \def (lift h TMP_37 t3) in (eq T
208 t2 TMP_38))))) in (let TMP_40 \def (\lambda (t3: T).(subst1 i u t1 t3)) in
209 (let TMP_41 \def (subst1_single i u t1 x0 H2) in (ex_intro2 T TMP_39 TMP_40
210 x0 H1 TMP_41))))))) in (let TMP_43 \def (subst0_gen_lift_lt u t1 t2 i h d H0)
211 in (ex2_ind T TMP_28 TMP_29 TMP_35 TMP_42 TMP_43)))))))))) in (subst1_ind i
212 TMP_1 TMP_4 TMP_10 TMP_24 TMP_44 x H))))))))))))))))))))).
214 theorem subst1_gen_lift_eq:
215 \forall (t: T).(\forall (u: T).(\forall (x: T).(\forall (h: nat).(\forall
216 (d: nat).(\forall (i: nat).((le d i) \to ((lt i (plus d h)) \to ((subst1 i u
217 (lift h d t) x) \to (eq T x (lift h d t))))))))))
219 \lambda (t: T).(\lambda (u: T).(\lambda (x: T).(\lambda (h: nat).(\lambda
220 (d: nat).(\lambda (i: nat).(\lambda (H: (le d i)).(\lambda (H0: (lt i (plus d
221 h))).(\lambda (H1: (subst1 i u (lift h d t) x)).(let TMP_1 \def (lift h d t)
222 in (let TMP_3 \def (\lambda (t0: T).(let TMP_2 \def (lift h d t) in (eq T t0
223 TMP_2))) in (let TMP_4 \def (lift h d t) in (let TMP_5 \def (refl_equal T
224 TMP_4) in (let TMP_8 \def (\lambda (t2: T).(\lambda (H2: (subst0 i u (lift h
225 d t) t2)).(let TMP_6 \def (lift h d t) in (let TMP_7 \def (eq T t2 TMP_6) in
226 (subst0_gen_lift_false t u t2 h d i H H0 H2 TMP_7))))) in (subst1_ind i u
227 TMP_1 TMP_3 TMP_5 TMP_8 x H1)))))))))))))).
229 theorem subst1_gen_lift_ge:
230 \forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
231 (h: nat).(\forall (d: nat).((subst1 i u (lift h d t1) x) \to ((le (plus d h)
232 i) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2:
233 T).(subst1 (minus i h) u t1 t2))))))))))
235 \lambda (u: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda
236 (h: nat).(\lambda (d: nat).(\lambda (H: (subst1 i u (lift h d t1)
237 x)).(\lambda (H0: (le (plus d h) i)).(let TMP_1 \def (lift h d t1) in (let
238 TMP_6 \def (\lambda (t: T).(let TMP_3 \def (\lambda (t2: T).(let TMP_2 \def
239 (lift h d t2) in (eq T t TMP_2))) in (let TMP_5 \def (\lambda (t2: T).(let
240 TMP_4 \def (minus i h) in (subst1 TMP_4 u t1 t2))) in (ex2 T TMP_3 TMP_5))))
241 in (let TMP_9 \def (\lambda (t2: T).(let TMP_7 \def (lift h d t1) in (let
242 TMP_8 \def (lift h d t2) in (eq T TMP_7 TMP_8)))) in (let TMP_11 \def
243 (\lambda (t2: T).(let TMP_10 \def (minus i h) in (subst1 TMP_10 u t1 t2))) in
244 (let TMP_12 \def (lift h d t1) in (let TMP_13 \def (refl_equal T TMP_12) in
245 (let TMP_14 \def (minus i h) in (let TMP_15 \def (subst1_refl TMP_14 u t1) in
246 (let TMP_16 \def (ex_intro2 T TMP_9 TMP_11 t1 TMP_13 TMP_15) in (let TMP_34
247 \def (\lambda (t2: T).(\lambda (H1: (subst0 i u (lift h d t1) t2)).(let
248 TMP_18 \def (\lambda (t3: T).(let TMP_17 \def (lift h d t3) in (eq T t2
249 TMP_17))) in (let TMP_20 \def (\lambda (t3: T).(let TMP_19 \def (minus i h)
250 in (subst0 TMP_19 u t1 t3))) in (let TMP_22 \def (\lambda (t3: T).(let TMP_21
251 \def (lift h d t3) in (eq T t2 TMP_21))) in (let TMP_24 \def (\lambda (t3:
252 T).(let TMP_23 \def (minus i h) in (subst1 TMP_23 u t1 t3))) in (let TMP_25
253 \def (ex2 T TMP_22 TMP_24) in (let TMP_32 \def (\lambda (x0: T).(\lambda (H2:
254 (eq T t2 (lift h d x0))).(\lambda (H3: (subst0 (minus i h) u t1 x0)).(let
255 TMP_27 \def (\lambda (t3: T).(let TMP_26 \def (lift h d t3) in (eq T t2
256 TMP_26))) in (let TMP_29 \def (\lambda (t3: T).(let TMP_28 \def (minus i h)
257 in (subst1 TMP_28 u t1 t3))) in (let TMP_30 \def (minus i h) in (let TMP_31
258 \def (subst1_single TMP_30 u t1 x0 H3) in (ex_intro2 T TMP_27 TMP_29 x0 H2
259 TMP_31)))))))) in (let TMP_33 \def (subst0_gen_lift_ge u t1 t2 i h d H1 H0)
260 in (ex2_ind T TMP_18 TMP_20 TMP_25 TMP_32 TMP_33)))))))))) in (subst1_ind i u
261 TMP_1 TMP_6 TMP_16 TMP_34 x H)))))))))))))))))).