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/fwd.ma".
19 include "basic_1/subst0/subst0.ma".
21 theorem subst1_subst1:
22 \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst1
23 j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst1 i
24 u u1 u2) \to (ex2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t:
25 T).(subst1 (S (plus i j)) u t t2)))))))))))
27 \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda
28 (H: (subst1 j u2 t1 t2)).(let TMP_5 \def (\lambda (t: T).(\forall (u1:
29 T).(\forall (u: T).(\forall (i: nat).((subst1 i u u1 u2) \to (let TMP_1 \def
30 (\lambda (t0: T).(subst1 j u1 t1 t0)) in (let TMP_4 \def (\lambda (t0:
31 T).(let TMP_2 \def (plus i j) in (let TMP_3 \def (S TMP_2) in (subst1 TMP_3 u
32 t0 t)))) in (ex2 T TMP_1 TMP_4)))))))) in (let TMP_14 \def (\lambda (u1:
33 T).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (subst1 i u u1 u2)).(let
34 TMP_6 \def (\lambda (t: T).(subst1 j u1 t1 t)) in (let TMP_9 \def (\lambda
35 (t: T).(let TMP_7 \def (plus i j) in (let TMP_8 \def (S TMP_7) in (subst1
36 TMP_8 u t t1)))) in (let TMP_10 \def (subst1_refl j u1 t1) in (let TMP_11
37 \def (plus i j) in (let TMP_12 \def (S TMP_11) in (let TMP_13 \def
38 (subst1_refl TMP_12 u t1) in (ex_intro2 T TMP_6 TMP_9 t1 TMP_10
39 TMP_13))))))))))) in (let TMP_63 \def (\lambda (t3: T).(\lambda (H0: (subst0
40 j u2 t1 t3)).(\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1:
41 (subst1 i u u1 u2)).(let TMP_15 \def (\lambda (t: T).(subst1 i u u1 t)) in
42 (let TMP_20 \def (\lambda (_: T).(let TMP_16 \def (\lambda (t0: T).(subst1 j
43 u1 t1 t0)) in (let TMP_19 \def (\lambda (t0: T).(let TMP_17 \def (plus i j)
44 in (let TMP_18 \def (S TMP_17) in (subst1 TMP_18 u t0 t3)))) in (ex2 T TMP_16
45 TMP_19)))) in (let TMP_62 \def (\lambda (y: T).(\lambda (H2: (subst1 i u u1
46 y)).(let TMP_25 \def (\lambda (t: T).((eq T t u2) \to (let TMP_21 \def
47 (\lambda (t0: T).(subst1 j u1 t1 t0)) in (let TMP_24 \def (\lambda (t0:
48 T).(let TMP_22 \def (plus i j) in (let TMP_23 \def (S TMP_22) in (subst1
49 TMP_23 u t0 t3)))) in (ex2 T TMP_21 TMP_24))))) in (let TMP_40 \def (\lambda
50 (H3: (eq T u1 u2)).(let TMP_30 \def (\lambda (t: T).(let TMP_26 \def (\lambda
51 (t0: T).(subst1 j t t1 t0)) in (let TMP_29 \def (\lambda (t0: T).(let TMP_27
52 \def (plus i j) in (let TMP_28 \def (S TMP_27) in (subst1 TMP_28 u t0 t3))))
53 in (ex2 T TMP_26 TMP_29)))) in (let TMP_31 \def (\lambda (t: T).(subst1 j u2
54 t1 t)) in (let TMP_34 \def (\lambda (t: T).(let TMP_32 \def (plus i j) in
55 (let TMP_33 \def (S TMP_32) in (subst1 TMP_33 u t t3)))) in (let TMP_35 \def
56 (subst1_single j u2 t1 t3 H0) in (let TMP_36 \def (plus i j) in (let TMP_37
57 \def (S TMP_36) in (let TMP_38 \def (subst1_refl TMP_37 u t3) in (let TMP_39
58 \def (ex_intro2 T TMP_31 TMP_34 t3 TMP_35 TMP_38) in (eq_ind_r T u2 TMP_30
59 TMP_39 u1 H3)))))))))) in (let TMP_61 \def (\lambda (t0: T).(\lambda (H3:
60 (subst0 i u u1 t0)).(\lambda (H4: (eq T t0 u2)).(let TMP_41 \def (\lambda (t:
61 T).(subst0 i u u1 t)) in (let H5 \def (eq_ind T t0 TMP_41 H3 u2 H4) in (let
62 TMP_42 \def (\lambda (t: T).(subst0 j u1 t1 t)) in (let TMP_45 \def (\lambda
63 (t: T).(let TMP_43 \def (plus i j) in (let TMP_44 \def (S TMP_43) in (subst0
64 TMP_44 u t t3)))) in (let TMP_46 \def (\lambda (t: T).(subst1 j u1 t1 t)) in
65 (let TMP_49 \def (\lambda (t: T).(let TMP_47 \def (plus i j) in (let TMP_48
66 \def (S TMP_47) in (subst1 TMP_48 u t t3)))) in (let TMP_50 \def (ex2 T
67 TMP_46 TMP_49) in (let TMP_59 \def (\lambda (x: T).(\lambda (H6: (subst0 j u1
68 t1 x)).(\lambda (H7: (subst0 (S (plus i j)) u x t3)).(let TMP_51 \def
69 (\lambda (t: T).(subst1 j u1 t1 t)) in (let TMP_54 \def (\lambda (t: T).(let
70 TMP_52 \def (plus i j) in (let TMP_53 \def (S TMP_52) in (subst1 TMP_53 u t
71 t3)))) in (let TMP_55 \def (subst1_single j u1 t1 x H6) in (let TMP_56 \def
72 (plus i j) in (let TMP_57 \def (S TMP_56) in (let TMP_58 \def (subst1_single
73 TMP_57 u x t3 H7) in (ex_intro2 T TMP_51 TMP_54 x TMP_55 TMP_58)))))))))) in
74 (let TMP_60 \def (subst0_subst0 t1 t3 u2 j H0 u1 u i H5) in (ex2_ind T TMP_42
75 TMP_45 TMP_50 TMP_59 TMP_60))))))))))))) in (subst1_ind i u u1 TMP_25 TMP_40
76 TMP_61 y H2)))))) in (insert_eq T u2 TMP_15 TMP_20 TMP_62 H1)))))))))) in
77 (subst1_ind j u2 t1 TMP_5 TMP_14 TMP_63 t2 H)))))))).
79 theorem subst1_subst1_back:
80 \forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst1
81 j u2 t1 t2) \to (\forall (u1: T).(\forall (u: T).(\forall (i: nat).((subst1 i
82 u u2 u1) \to (ex2 T (\lambda (t: T).(subst1 j u1 t1 t)) (\lambda (t:
83 T).(subst1 (S (plus i j)) u t2 t)))))))))))
85 \lambda (t1: T).(\lambda (t2: T).(\lambda (u2: T).(\lambda (j: nat).(\lambda
86 (H: (subst1 j u2 t1 t2)).(let TMP_5 \def (\lambda (t: T).(\forall (u1:
87 T).(\forall (u: T).(\forall (i: nat).((subst1 i u u2 u1) \to (let TMP_1 \def
88 (\lambda (t0: T).(subst1 j u1 t1 t0)) in (let TMP_4 \def (\lambda (t0:
89 T).(let TMP_2 \def (plus i j) in (let TMP_3 \def (S TMP_2) in (subst1 TMP_3 u
90 t t0)))) in (ex2 T TMP_1 TMP_4)))))))) in (let TMP_14 \def (\lambda (u1:
91 T).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (subst1 i u u2 u1)).(let
92 TMP_6 \def (\lambda (t: T).(subst1 j u1 t1 t)) in (let TMP_9 \def (\lambda
93 (t: T).(let TMP_7 \def (plus i j) in (let TMP_8 \def (S TMP_7) in (subst1
94 TMP_8 u t1 t)))) in (let TMP_10 \def (subst1_refl j u1 t1) in (let TMP_11
95 \def (plus i j) in (let TMP_12 \def (S TMP_11) in (let TMP_13 \def
96 (subst1_refl TMP_12 u t1) in (ex_intro2 T TMP_6 TMP_9 t1 TMP_10
97 TMP_13))))))))))) in (let TMP_49 \def (\lambda (t3: T).(\lambda (H0: (subst0
98 j u2 t1 t3)).(\lambda (u1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1:
99 (subst1 i u u2 u1)).(let TMP_19 \def (\lambda (t: T).(let TMP_15 \def
100 (\lambda (t0: T).(subst1 j t t1 t0)) in (let TMP_18 \def (\lambda (t0:
101 T).(let TMP_16 \def (plus i j) in (let TMP_17 \def (S TMP_16) in (subst1
102 TMP_17 u t3 t0)))) in (ex2 T TMP_15 TMP_18)))) in (let TMP_20 \def (\lambda
103 (t: T).(subst1 j u2 t1 t)) in (let TMP_23 \def (\lambda (t: T).(let TMP_21
104 \def (plus i j) in (let TMP_22 \def (S TMP_21) in (subst1 TMP_22 u t3 t))))
105 in (let TMP_24 \def (subst1_single j u2 t1 t3 H0) in (let TMP_25 \def (plus i
106 j) in (let TMP_26 \def (S TMP_25) in (let TMP_27 \def (subst1_refl TMP_26 u
107 t3) in (let TMP_28 \def (ex_intro2 T TMP_20 TMP_23 t3 TMP_24 TMP_27) in (let
108 TMP_48 \def (\lambda (t0: T).(\lambda (H2: (subst0 i u u2 t0)).(let TMP_29
109 \def (\lambda (t: T).(subst0 j t0 t1 t)) in (let TMP_32 \def (\lambda (t:
110 T).(let TMP_30 \def (plus i j) in (let TMP_31 \def (S TMP_30) in (subst0
111 TMP_31 u t3 t)))) in (let TMP_33 \def (\lambda (t: T).(subst1 j t0 t1 t)) in
112 (let TMP_36 \def (\lambda (t: T).(let TMP_34 \def (plus i j) in (let TMP_35
113 \def (S TMP_34) in (subst1 TMP_35 u t3 t)))) in (let TMP_37 \def (ex2 T
114 TMP_33 TMP_36) in (let TMP_46 \def (\lambda (x: T).(\lambda (H3: (subst0 j t0
115 t1 x)).(\lambda (H4: (subst0 (S (plus i j)) u t3 x)).(let TMP_38 \def
116 (\lambda (t: T).(subst1 j t0 t1 t)) in (let TMP_41 \def (\lambda (t: T).(let
117 TMP_39 \def (plus i j) in (let TMP_40 \def (S TMP_39) in (subst1 TMP_40 u t3
118 t)))) in (let TMP_42 \def (subst1_single j t0 t1 x H3) in (let TMP_43 \def
119 (plus i j) in (let TMP_44 \def (S TMP_43) in (let TMP_45 \def (subst1_single
120 TMP_44 u t3 x H4) in (ex_intro2 T TMP_38 TMP_41 x TMP_42 TMP_45)))))))))) in
121 (let TMP_47 \def (subst0_subst0_back t1 t3 u2 j H0 t0 u i H2) in (ex2_ind T
122 TMP_29 TMP_32 TMP_37 TMP_46 TMP_47)))))))))) in (subst1_ind i u u2 TMP_19
123 TMP_28 TMP_48 u1 H1)))))))))))))))) in (subst1_ind j u2 t1 TMP_5 TMP_14
126 theorem subst1_trans:
127 \forall (t2: T).(\forall (t1: T).(\forall (v: T).(\forall (i: nat).((subst1
128 i v t1 t2) \to (\forall (t3: T).((subst1 i v t2 t3) \to (subst1 i v t1
131 \lambda (t2: T).(\lambda (t1: T).(\lambda (v: T).(\lambda (i: nat).(\lambda
132 (H: (subst1 i v t1 t2)).(let TMP_1 \def (\lambda (t: T).(\forall (t3:
133 T).((subst1 i v t t3) \to (subst1 i v t1 t3)))) in (let TMP_2 \def (\lambda
134 (t3: T).(\lambda (H0: (subst1 i v t1 t3)).H0)) in (let TMP_7 \def (\lambda
135 (t3: T).(\lambda (H0: (subst0 i v t1 t3)).(\lambda (t4: T).(\lambda (H1:
136 (subst1 i v t3 t4)).(let TMP_3 \def (\lambda (t: T).(subst1 i v t1 t)) in
137 (let TMP_4 \def (subst1_single i v t1 t3 H0) in (let TMP_6 \def (\lambda (t0:
138 T).(\lambda (H2: (subst0 i v t3 t0)).(let TMP_5 \def (subst0_trans t3 t1 v i
139 H0 t0 H2) in (subst1_single i v t1 t0 TMP_5)))) in (subst1_ind i v t3 TMP_3
140 TMP_4 TMP_6 t4 H1)))))))) in (subst1_ind i v t1 TMP_1 TMP_2 TMP_7 t2
143 theorem subst1_confluence_neq:
144 \forall (t0: T).(\forall (t1: T).(\forall (u1: T).(\forall (i1:
145 nat).((subst1 i1 u1 t0 t1) \to (\forall (t2: T).(\forall (u2: T).(\forall
146 (i2: nat).((subst1 i2 u2 t0 t2) \to ((not (eq nat i1 i2)) \to (ex2 T (\lambda
147 (t: T).(subst1 i2 u2 t1 t)) (\lambda (t: T).(subst1 i1 u1 t2 t))))))))))))
149 \lambda (t0: T).(\lambda (t1: T).(\lambda (u1: T).(\lambda (i1:
150 nat).(\lambda (H: (subst1 i1 u1 t0 t1)).(let TMP_3 \def (\lambda (t:
151 T).(\forall (t2: T).(\forall (u2: T).(\forall (i2: nat).((subst1 i2 u2 t0 t2)
152 \to ((not (eq nat i1 i2)) \to (let TMP_1 \def (\lambda (t3: T).(subst1 i2 u2
153 t t3)) in (let TMP_2 \def (\lambda (t3: T).(subst1 i1 u1 t2 t3)) in (ex2 T
154 TMP_1 TMP_2))))))))) in (let TMP_7 \def (\lambda (t2: T).(\lambda (u2:
155 T).(\lambda (i2: nat).(\lambda (H0: (subst1 i2 u2 t0 t2)).(\lambda (_: (not
156 (eq nat i1 i2))).(let TMP_4 \def (\lambda (t: T).(subst1 i2 u2 t0 t)) in (let
157 TMP_5 \def (\lambda (t: T).(subst1 i1 u1 t2 t)) in (let TMP_6 \def
158 (subst1_refl i1 u1 t2) in (ex_intro2 T TMP_4 TMP_5 t2 H0 TMP_6))))))))) in
159 (let TMP_29 \def (\lambda (t2: T).(\lambda (H0: (subst0 i1 u1 t0
160 t2)).(\lambda (t3: T).(\lambda (u2: T).(\lambda (i2: nat).(\lambda (H1:
161 (subst1 i2 u2 t0 t3)).(\lambda (H2: (not (eq nat i1 i2))).(let TMP_10 \def
162 (\lambda (t: T).(let TMP_8 \def (\lambda (t4: T).(subst1 i2 u2 t2 t4)) in
163 (let TMP_9 \def (\lambda (t4: T).(subst1 i1 u1 t t4)) in (ex2 T TMP_8
164 TMP_9)))) in (let TMP_11 \def (\lambda (t: T).(subst1 i2 u2 t2 t)) in (let
165 TMP_12 \def (\lambda (t: T).(subst1 i1 u1 t0 t)) in (let TMP_13 \def
166 (subst1_refl i2 u2 t2) in (let TMP_14 \def (subst1_single i1 u1 t0 t2 H0) in
167 (let TMP_15 \def (ex_intro2 T TMP_11 TMP_12 t2 TMP_13 TMP_14) in (let TMP_28
168 \def (\lambda (t4: T).(\lambda (H3: (subst0 i2 u2 t0 t4)).(let TMP_16 \def
169 (\lambda (t: T).(subst0 i1 u1 t4 t)) in (let TMP_17 \def (\lambda (t:
170 T).(subst0 i2 u2 t2 t)) in (let TMP_18 \def (\lambda (t: T).(subst1 i2 u2 t2
171 t)) in (let TMP_19 \def (\lambda (t: T).(subst1 i1 u1 t4 t)) in (let TMP_20
172 \def (ex2 T TMP_18 TMP_19) in (let TMP_25 \def (\lambda (x: T).(\lambda (H4:
173 (subst0 i1 u1 t4 x)).(\lambda (H5: (subst0 i2 u2 t2 x)).(let TMP_21 \def
174 (\lambda (t: T).(subst1 i2 u2 t2 t)) in (let TMP_22 \def (\lambda (t:
175 T).(subst1 i1 u1 t4 t)) in (let TMP_23 \def (subst1_single i2 u2 t2 x H5) in
176 (let TMP_24 \def (subst1_single i1 u1 t4 x H4) in (ex_intro2 T TMP_21 TMP_22
177 x TMP_23 TMP_24)))))))) in (let TMP_26 \def (sym_not_eq nat i1 i2 H2) in (let
178 TMP_27 \def (subst0_confluence_neq t0 t4 u2 i2 H3 t2 u1 i1 H0 TMP_26) in
179 (ex2_ind T TMP_16 TMP_17 TMP_20 TMP_25 TMP_27))))))))))) in (subst1_ind i2 u2
180 t0 TMP_10 TMP_15 TMP_28 t3 H1))))))))))))))) in (subst1_ind i1 u1 t0 TMP_3
181 TMP_7 TMP_29 t1 H)))))))).
183 theorem subst1_confluence_eq:
184 \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst1
185 i u t0 t1) \to (\forall (t2: T).((subst1 i u t0 t2) \to (ex2 T (\lambda (t:
186 T).(subst1 i u t1 t)) (\lambda (t: T).(subst1 i u t2 t)))))))))
188 \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
189 (H: (subst1 i u t0 t1)).(let TMP_3 \def (\lambda (t: T).(\forall (t2:
190 T).((subst1 i u t0 t2) \to (let TMP_1 \def (\lambda (t3: T).(subst1 i u t
191 t3)) in (let TMP_2 \def (\lambda (t3: T).(subst1 i u t2 t3)) in (ex2 T TMP_1
192 TMP_2)))))) in (let TMP_7 \def (\lambda (t2: T).(\lambda (H0: (subst1 i u t0
193 t2)).(let TMP_4 \def (\lambda (t: T).(subst1 i u t0 t)) in (let TMP_5 \def
194 (\lambda (t: T).(subst1 i u t2 t)) in (let TMP_6 \def (subst1_refl i u t2) in
195 (ex_intro2 T TMP_4 TMP_5 t2 H0 TMP_6)))))) in (let TMP_57 \def (\lambda (t2:
196 T).(\lambda (H0: (subst0 i u t0 t2)).(\lambda (t3: T).(\lambda (H1: (subst1 i
197 u t0 t3)).(let TMP_10 \def (\lambda (t: T).(let TMP_8 \def (\lambda (t4:
198 T).(subst1 i u t2 t4)) in (let TMP_9 \def (\lambda (t4: T).(subst1 i u t t4))
199 in (ex2 T TMP_8 TMP_9)))) in (let TMP_11 \def (\lambda (t: T).(subst1 i u t2
200 t)) in (let TMP_12 \def (\lambda (t: T).(subst1 i u t0 t)) in (let TMP_13
201 \def (subst1_refl i u t2) in (let TMP_14 \def (subst1_single i u t0 t2 H0) in
202 (let TMP_15 \def (ex_intro2 T TMP_11 TMP_12 t2 TMP_13 TMP_14) in (let TMP_56
203 \def (\lambda (t4: T).(\lambda (H2: (subst0 i u t0 t4)).(let TMP_16 \def (eq
204 T t4 t2) in (let TMP_17 \def (\lambda (t: T).(subst0 i u t4 t)) in (let
205 TMP_18 \def (\lambda (t: T).(subst0 i u t2 t)) in (let TMP_19 \def (ex2 T
206 TMP_17 TMP_18) in (let TMP_20 \def (subst0 i u t4 t2) in (let TMP_21 \def
207 (subst0 i u t2 t4) in (let TMP_22 \def (\lambda (t: T).(subst1 i u t2 t)) in
208 (let TMP_23 \def (\lambda (t: T).(subst1 i u t4 t)) in (let TMP_24 \def (ex2
209 T TMP_22 TMP_23) in (let TMP_33 \def (\lambda (H3: (eq T t4 t2)).(let TMP_27
210 \def (\lambda (t: T).(let TMP_25 \def (\lambda (t5: T).(subst1 i u t2 t5)) in
211 (let TMP_26 \def (\lambda (t5: T).(subst1 i u t t5)) in (ex2 T TMP_25
212 TMP_26)))) in (let TMP_28 \def (\lambda (t: T).(subst1 i u t2 t)) in (let
213 TMP_29 \def (\lambda (t: T).(subst1 i u t2 t)) in (let TMP_30 \def
214 (subst1_refl i u t2) in (let TMP_31 \def (subst1_refl i u t2) in (let TMP_32
215 \def (ex_intro2 T TMP_28 TMP_29 t2 TMP_30 TMP_31) in (eq_ind_r T t2 TMP_27
216 TMP_32 t4 H3)))))))) in (let TMP_44 \def (\lambda (H3: (ex2 T (\lambda (t:
217 T).(subst0 i u t4 t)) (\lambda (t: T).(subst0 i u t2 t)))).(let TMP_34 \def
218 (\lambda (t: T).(subst0 i u t4 t)) in (let TMP_35 \def (\lambda (t:
219 T).(subst0 i u t2 t)) in (let TMP_36 \def (\lambda (t: T).(subst1 i u t2 t))
220 in (let TMP_37 \def (\lambda (t: T).(subst1 i u t4 t)) in (let TMP_38 \def
221 (ex2 T TMP_36 TMP_37) in (let TMP_43 \def (\lambda (x: T).(\lambda (H4:
222 (subst0 i u t4 x)).(\lambda (H5: (subst0 i u t2 x)).(let TMP_39 \def (\lambda
223 (t: T).(subst1 i u t2 t)) in (let TMP_40 \def (\lambda (t: T).(subst1 i u t4
224 t)) in (let TMP_41 \def (subst1_single i u t2 x H5) in (let TMP_42 \def
225 (subst1_single i u t4 x H4) in (ex_intro2 T TMP_39 TMP_40 x TMP_41
226 TMP_42)))))))) in (ex2_ind T TMP_34 TMP_35 TMP_38 TMP_43 H3)))))))) in (let
227 TMP_49 \def (\lambda (H3: (subst0 i u t4 t2)).(let TMP_45 \def (\lambda (t:
228 T).(subst1 i u t2 t)) in (let TMP_46 \def (\lambda (t: T).(subst1 i u t4 t))
229 in (let TMP_47 \def (subst1_refl i u t2) in (let TMP_48 \def (subst1_single i
230 u t4 t2 H3) in (ex_intro2 T TMP_45 TMP_46 t2 TMP_47 TMP_48)))))) in (let
231 TMP_54 \def (\lambda (H3: (subst0 i u t2 t4)).(let TMP_50 \def (\lambda (t:
232 T).(subst1 i u t2 t)) in (let TMP_51 \def (\lambda (t: T).(subst1 i u t4 t))
233 in (let TMP_52 \def (subst1_single i u t2 t4 H3) in (let TMP_53 \def
234 (subst1_refl i u t4) in (ex_intro2 T TMP_50 TMP_51 t4 TMP_52 TMP_53)))))) in
235 (let TMP_55 \def (subst0_confluence_eq t0 t4 u i H2 t2 H0) in (or4_ind TMP_16
236 TMP_19 TMP_20 TMP_21 TMP_24 TMP_33 TMP_44 TMP_49 TMP_54
237 TMP_55))))))))))))))))) in (subst1_ind i u t0 TMP_10 TMP_15 TMP_56 t3
238 H1)))))))))))) in (subst1_ind i u t0 TMP_3 TMP_7 TMP_57 t1 H)))))))).
240 theorem subst1_confluence_lift:
241 \forall (t0: T).(\forall (t1: T).(\forall (u: T).(\forall (i: nat).((subst1
242 i u t0 (lift (S O) i t1)) \to (\forall (t2: T).((subst1 i u t0 (lift (S O) i
243 t2)) \to (eq T t1 t2)))))))
245 \lambda (t0: T).(\lambda (t1: T).(\lambda (u: T).(\lambda (i: nat).(\lambda
246 (H: (subst1 i u t0 (lift (S O) i t1))).(let TMP_1 \def (S O) in (let TMP_2
247 \def (lift TMP_1 i t1) in (let TMP_3 \def (\lambda (t: T).(subst1 i u t0 t))
248 in (let TMP_4 \def (\lambda (_: T).(\forall (t2: T).((subst1 i u t0 (lift (S
249 O) i t2)) \to (eq T t1 t2)))) in (let TMP_70 \def (\lambda (y: T).(\lambda
250 (H0: (subst1 i u t0 y)).(let TMP_5 \def (\lambda (t: T).((eq T t (lift (S O)
251 i t1)) \to (\forall (t2: T).((subst1 i u t0 (lift (S O) i t2)) \to (eq T t1
252 t2))))) in (let TMP_32 \def (\lambda (H1: (eq T t0 (lift (S O) i
253 t1))).(\lambda (t2: T).(\lambda (H2: (subst1 i u t0 (lift (S O) i t2))).(let
254 TMP_8 \def (\lambda (t: T).(let TMP_6 \def (S O) in (let TMP_7 \def (lift
255 TMP_6 i t2) in (subst1 i u t TMP_7)))) in (let TMP_9 \def (S O) in (let
256 TMP_10 \def (lift TMP_9 i t1) in (let H3 \def (eq_ind T t0 TMP_8 H2 TMP_10
257 H1) in (let TMP_11 \def (S O) in (let TMP_12 \def (lift TMP_11 i t2) in (let
258 TMP_13 \def (S O) in (let TMP_14 \def (lift TMP_13 i t1) in (let TMP_15 \def
259 (S O) in (let TMP_16 \def (lift TMP_15 i t2) in (let TMP_17 \def (S O) in
260 (let TMP_18 \def (le_n i) in (let TMP_19 \def (S O) in (let TMP_20 \def (plus
261 TMP_19 i) in (let TMP_21 \def (\lambda (n: nat).(lt i n)) in (let TMP_22 \def
262 (S O) in (let TMP_23 \def (plus TMP_22 i) in (let TMP_24 \def (le_n TMP_23)
263 in (let TMP_25 \def (S O) in (let TMP_26 \def (plus i TMP_25) in (let TMP_27
264 \def (S O) in (let TMP_28 \def (plus_sym i TMP_27) in (let TMP_29 \def
265 (eq_ind_r nat TMP_20 TMP_21 TMP_24 TMP_26 TMP_28) in (let TMP_30 \def
266 (subst1_gen_lift_eq t1 u TMP_16 TMP_17 i i TMP_18 TMP_29 H3) in (let H4 \def
267 (sym_eq T TMP_12 TMP_14 TMP_30) in (let TMP_31 \def (S O) in (lift_inj t1 t2
268 TMP_31 i H4)))))))))))))))))))))))))))))) in (let TMP_69 \def (\lambda (t2:
269 T).(\lambda (H1: (subst0 i u t0 t2)).(\lambda (H2: (eq T t2 (lift (S O) i
270 t1))).(\lambda (t3: T).(\lambda (H3: (subst1 i u t0 (lift (S O) i t3))).(let
271 TMP_33 \def (\lambda (t: T).(subst0 i u t0 t)) in (let TMP_34 \def (S O) in
272 (let TMP_35 \def (lift TMP_34 i t1) in (let H4 \def (eq_ind T t2 TMP_33 H1
273 TMP_35 H2) in (let TMP_36 \def (S O) in (let TMP_37 \def (lift TMP_36 i t3)
274 in (let TMP_38 \def (\lambda (t: T).(subst1 i u t0 t)) in (let TMP_39 \def
275 (\lambda (_: T).(eq T t1 t3)) in (let TMP_68 \def (\lambda (y0: T).(\lambda
276 (H5: (subst1 i u t0 y0)).(let TMP_40 \def (\lambda (t: T).((eq T t (lift (S
277 O) i t3)) \to (eq T t1 t3))) in (let TMP_62 \def (\lambda (H6: (eq T t0 (lift
278 (S O) i t3))).(let TMP_43 \def (\lambda (t: T).(let TMP_41 \def (S O) in (let
279 TMP_42 \def (lift TMP_41 i t1) in (subst0 i u t TMP_42)))) in (let TMP_44
280 \def (S O) in (let TMP_45 \def (lift TMP_44 i t3) in (let H7 \def (eq_ind T
281 t0 TMP_43 H4 TMP_45 H6) in (let TMP_46 \def (S O) in (let TMP_47 \def (lift
282 TMP_46 i t1) in (let TMP_48 \def (S O) in (let TMP_49 \def (le_n i) in (let
283 TMP_50 \def (S O) in (let TMP_51 \def (plus TMP_50 i) in (let TMP_52 \def
284 (\lambda (n: nat).(lt i n)) in (let TMP_53 \def (S O) in (let TMP_54 \def
285 (plus TMP_53 i) in (let TMP_55 \def (le_n TMP_54) in (let TMP_56 \def (S O)
286 in (let TMP_57 \def (plus i TMP_56) in (let TMP_58 \def (S O) in (let TMP_59
287 \def (plus_sym i TMP_58) in (let TMP_60 \def (eq_ind_r nat TMP_51 TMP_52
288 TMP_55 TMP_57 TMP_59) in (let TMP_61 \def (eq T t1 t3) in
289 (subst0_gen_lift_false t3 u TMP_47 TMP_48 i i TMP_49 TMP_60 H7
290 TMP_61)))))))))))))))))))))) in (let TMP_67 \def (\lambda (t4: T).(\lambda
291 (H6: (subst0 i u t0 t4)).(\lambda (H7: (eq T t4 (lift (S O) i t3))).(let
292 TMP_63 \def (\lambda (t: T).(subst0 i u t0 t)) in (let TMP_64 \def (S O) in
293 (let TMP_65 \def (lift TMP_64 i t3) in (let H8 \def (eq_ind T t4 TMP_63 H6
294 TMP_65 H7) in (let TMP_66 \def (subst0_confluence_lift t0 t3 u i H8 t1 H4) in
295 (sym_eq T t3 t1 TMP_66))))))))) in (subst1_ind i u t0 TMP_40 TMP_62 TMP_67 y0
296 H5)))))) in (insert_eq T TMP_37 TMP_38 TMP_39 TMP_68 H3))))))))))))))) in
297 (subst1_ind i u t0 TMP_5 TMP_32 TMP_69 y H0)))))) in (insert_eq T TMP_2 TMP_3
298 TMP_4 TMP_70 H)))))))))).