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 *********************)
19 include "pr2/defs.ma".
21 include "pr0/subst1.ma".
25 include "csubst1/getl.ma".
27 include "csubst1/fwd.ma".
29 include "subst1/subst1.ma".
31 include "getl/drop.ma".
34 \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c
35 (CHead d (Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2)
36 \to (\forall (t: T).((subst1 i u t2 t) \to (pr2 c t1 t))))))))))
38 \lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
39 (H: (getl i c (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2:
40 T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H1: (subst1 i u t2
41 t)).(subst1_ind i u t2 (\lambda (t0: T).(pr2 c t1 t0)) (pr2_free c t1 t2 H0)
42 (\lambda (t0: T).(\lambda (H2: (subst0 i u t2 t0)).(pr2_delta c d u i H t1 t2
43 H0 t0 H2))) t H1)))))))))).
46 \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c
47 (CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2)
48 \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c
49 w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))))))
51 \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
52 (H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2:
53 T).(\lambda (H0: (pr2 c t1 t2)).(let H1 \def (match H0 in pr2 return (\lambda
54 (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C
55 c0 c) \to ((eq T t t1) \to ((eq T t0 t2) \to (\forall (w1: T).((subst1 i v t1
56 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v
57 t2 w2)))))))))))) with [(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq
58 C c0 c)).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C c
59 (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (\forall
60 (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2))
61 (\lambda (w2: T).(subst1 i v t2 w2))))))))) (\lambda (H5: (eq T t0
62 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (\forall
63 (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2))
64 (\lambda (w2: T).(subst1 i v t2 w2)))))))) (\lambda (H6: (eq T t3
65 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (\forall (w1: T).((subst1 i
66 v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1
67 i v t2 w2))))))) (\lambda (H7: (pr0 t1 t2)).(\lambda (w1: T).(\lambda (H8:
68 (subst1 i v t1 w1)).(ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
69 T).(subst1 i v t2 w2)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2:
70 T).(subst1 i v t2 w2))) (\lambda (x: T).(\lambda (H9: (pr0 w1 x)).(\lambda
71 (H10: (subst1 i v t2 x)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2))
72 (\lambda (w2: T).(subst1 i v t2 w2)) x (pr2_free c w1 x H9) H10))))
73 (pr0_subst1 t1 t2 H7 v w1 i H8 v (pr0_refl v)))))) t3 (sym_eq T t3 t2 H6)))
74 t0 (sym_eq T t0 t1 H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d
75 u i0 H1 t0 t3 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5:
76 (eq T t0 t1)).(\lambda (H6: (eq T t t2)).(eq_ind C c (\lambda (c1: C).((eq T
77 t0 t1) \to ((eq T t t2) \to ((getl i0 c1 (CHead d (Bind Abbr) u)) \to ((pr0
78 t0 t3) \to ((subst0 i0 u t3 t) \to (\forall (w1: T).((subst1 i v t1 w1) \to
79 (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2
80 w2))))))))))) (\lambda (H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq
81 T t t2) \to ((getl i0 c (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to
82 ((subst0 i0 u t3 t) \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T
83 (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))))
84 (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i0 c (CHead d
85 (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i0 u t3 t4) \to (\forall (w1:
86 T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda
87 (w2: T).(subst1 i v t2 w2))))))))) (\lambda (H9: (getl i0 c (CHead d (Bind
88 Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda (H11: (subst0 i0 u t3
89 t2)).(\lambda (w1: T).(\lambda (H12: (subst1 i v t1 w1)).(ex2_ind T (\lambda
90 (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v t3 w2)) (ex2 T (\lambda
91 (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))) (\lambda (x:
92 T).(\lambda (H13: (pr0 w1 x)).(\lambda (H14: (subst1 i v t3 x)).(neq_eq_e i
93 i0 (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2
94 w2))) (\lambda (H15: (not (eq nat i i0))).(ex2_ind T (\lambda (t4: T).(subst1
95 i v t2 t4)) (\lambda (t4: T).(subst1 i0 u x t4)) (ex2 T (\lambda (w2: T).(pr2
96 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))) (\lambda (x0: T).(\lambda
97 (H16: (subst1 i v t2 x0)).(\lambda (H17: (subst1 i0 u x x0)).(ex_intro2 T
98 (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)) x0
99 (pr2_delta1 c d u i0 H9 w1 x H13 x0 H17) H16)))) (subst1_confluence_neq t3 t2
100 u i0 (subst1_single i0 u t3 t2 H11) x v i H14 (sym_not_eq nat i i0 H15))))
101 (\lambda (H15: (eq nat i i0)).(let H16 \def (eq_ind_r nat i0 (\lambda (n:
102 nat).(subst0 n u t3 t2)) H11 i H15) in (let H17 \def (eq_ind_r nat i0
103 (\lambda (n: nat).(getl n c (CHead d (Bind Abbr) u))) H9 i H15) in (let H18
104 \def (eq_ind C (CHead e (Bind Abbr) v) (\lambda (c1: C).(getl i c c1)) H
105 (CHead d (Bind Abbr) u) (getl_mono c (CHead e (Bind Abbr) v) i H (CHead d
106 (Bind Abbr) u) H17)) in (let H19 \def (f_equal C C (\lambda (e0: C).(match e0
107 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow e | (CHead c1 _ _)
108 \Rightarrow c1])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr) u) (getl_mono
109 c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H17)) in ((let H20 \def
110 (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with
111 [(CSort _) \Rightarrow v | (CHead _ _ t4) \Rightarrow t4])) (CHead e (Bind
112 Abbr) v) (CHead d (Bind Abbr) u) (getl_mono c (CHead e (Bind Abbr) v) i H
113 (CHead d (Bind Abbr) u) H17)) in (\lambda (H21: (eq C e d)).(let H22 \def
114 (eq_ind_r T u (\lambda (t4: T).(getl i c (CHead d (Bind Abbr) t4))) H18 v
115 H20) in (let H23 \def (eq_ind_r T u (\lambda (t4: T).(subst0 i t4 t3 t2)) H16
116 v H20) in (let H24 \def (eq_ind_r C d (\lambda (c1: C).(getl i c (CHead c1
117 (Bind Abbr) v))) H22 e H21) in (ex2_ind T (\lambda (t4: T).(subst1 i v t2
118 t4)) (\lambda (t4: T).(subst1 i v x t4)) (ex2 T (\lambda (w2: T).(pr2 c w1
119 w2)) (\lambda (w2: T).(subst1 i v t2 w2))) (\lambda (x0: T).(\lambda (H25:
120 (subst1 i v t2 x0)).(\lambda (H26: (subst1 i v x x0)).(ex_intro2 T (\lambda
121 (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)) x0 (pr2_delta1 c
122 e v i H24 w1 x H13 x0 H26) H25)))) (subst1_confluence_eq t3 t2 v i
123 (subst1_single i v t3 t2 H23) x H14))))))) H19)))))))))) (pr0_subst1 t1 t3
124 H10 v w1 i H12 v (pr0_refl v)))))))) t (sym_eq T t t2 H8))) t0 (sym_eq T t0
125 t1 H7))) c0 (sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C c)
126 (refl_equal T t1) (refl_equal T t2)))))))))).
128 theorem pr2_gen_cabbr:
129 \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
130 (e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u))
131 \to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d
132 a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T
133 (\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a
134 x1 x2))))))))))))))))
136 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
137 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (e:
138 C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to
139 (\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0
140 a) \to (\forall (x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda
141 (x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1
142 x2)))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4:
143 T).(\lambda (H0: (pr0 t3 t4)).(\lambda (e: C).(\lambda (u: T).(\lambda (d:
144 nat).(\lambda (_: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0:
145 C).(\lambda (_: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (_: (drop (S O)
146 d a0 a)).(\lambda (x1: T).(\lambda (H4: (subst1 d u t3 (lift (S O) d
147 x1))).(ex2_ind T (\lambda (w2: T).(pr0 (lift (S O) d x1) w2)) (\lambda (w2:
148 T).(subst1 d u t4 w2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d
149 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H5: (pr0
150 (lift (S O) d x1) x)).(\lambda (H6: (subst1 d u t4 x)).(ex2_ind T (\lambda
151 (t5: T).(eq T x (lift (S O) d t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T
152 (\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a
153 x1 x2))) (\lambda (x0: T).(\lambda (H7: (eq T x (lift (S O) d x0))).(\lambda
154 (H8: (pr0 x1 x0)).(let H9 \def (eq_ind T x (\lambda (t: T).(subst1 d u t4 t))
155 H6 (lift (S O) d x0) H7) in (ex_intro2 T (\lambda (x2: T).(subst1 d u t4
156 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 x2)) x0 H9 (pr2_free a x1 x0
157 H8)))))) (pr0_gen_lift x1 x (S O) d H5))))) (pr0_subst1 t3 t4 H0 u (lift (S
158 O) d x1) d H4 u (pr0_refl u))))))))))))))))) (\lambda (c0: C).(\lambda (d:
159 C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
160 Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3
161 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (e:
162 C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e
163 (Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0
164 a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 a)).(\lambda (x1:
165 T).(\lambda (H6: (subst1 d0 u0 t3 (lift (S O) d0 x1))).(ex2_ind T (\lambda
166 (w2: T).(pr0 (lift (S O) d0 x1) w2)) (\lambda (w2: T).(subst1 d0 u0 t4 w2))
167 (ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2:
168 T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H7: (pr0 (lift (S O) d0 x1)
169 x)).(\lambda (H8: (subst1 d0 u0 t4 x)).(ex2_ind T (\lambda (t5: T).(eq T x
170 (lift (S O) d0 t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T (\lambda (x2:
171 T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2)))
172 (\lambda (x0: T).(\lambda (H9: (eq T x (lift (S O) d0 x0))).(\lambda (H10:
173 (pr0 x1 x0)).(let H11 \def (eq_ind T x (\lambda (t0: T).(subst1 d0 u0 t4 t0))
174 H8 (lift (S O) d0 x0) H9) in (lt_eq_gt_e i d0 (ex2 T (\lambda (x2: T).(subst1
175 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (H12:
176 (lt i d0)).(ex2_ind T (\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0:
177 T).(subst1 i u (lift (S O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0
178 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2:
179 T).(\lambda (H13: (subst1 d0 u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O)
180 d0 x0) x2)).(ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 i) u0 (CHead d
181 (Bind Abbr) u) e2)) (\lambda (e2: C).(getl i a0 e2)) (ex2 T (\lambda (x3:
182 T).(subst1 d0 u0 t (lift (S O) d0 x3))) (\lambda (x3: T).(pr2 a x1 x3)))
183 (\lambda (x3: C).(\lambda (H15: (csubst1 (minus d0 i) u0 (CHead d (Bind Abbr)
184 u) x3)).(\lambda (H16: (getl i a0 x3)).(let H17 \def (eq_ind nat (minus d0 i)
185 (\lambda (n: nat).(csubst1 n u0 (CHead d (Bind Abbr) u) x3)) H15 (S (minus d0
186 (S i))) (minus_x_Sy d0 i H12)) in (let H18 \def (csubst1_gen_head (Bind Abbr)
187 d x3 u u0 (minus d0 (S i)) H17) in (ex3_2_ind T C (\lambda (u2: T).(\lambda
188 (c2: C).(eq C x3 (CHead c2 (Bind Abbr) u2)))) (\lambda (u2: T).(\lambda (_:
189 C).(subst1 (minus d0 (S i)) u0 u u2))) (\lambda (_: T).(\lambda (c2:
190 C).(csubst1 (minus d0 (S i)) u0 d c2))) (ex2 T (\lambda (x4: T).(subst1 d0 u0
191 t (lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4))) (\lambda (x4:
192 T).(\lambda (x5: C).(\lambda (H19: (eq C x3 (CHead x5 (Bind Abbr)
193 x4))).(\lambda (H20: (subst1 (minus d0 (S i)) u0 u x4)).(\lambda (_: (csubst1
194 (minus d0 (S i)) u0 d x5)).(let H22 \def (eq_ind C x3 (\lambda (c1: C).(getl
195 i a0 c1)) H16 (CHead x5 (Bind Abbr) x4) H19) in (let H23 \def (eq_ind nat d0
196 (\lambda (n: nat).(drop (S O) n a0 a)) H5 (S (plus i (minus d0 (S i))))
197 (lt_plus_minus i d0 H12)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
198 C).(eq T x4 (lift (S O) (minus d0 (S i)) v)))) (\lambda (v: T).(\lambda (e0:
199 C).(getl i a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0:
200 C).(drop (S O) (minus d0 (S i)) x5 e0))) (ex2 T (\lambda (x6: T).(subst1 d0
201 u0 t (lift (S O) d0 x6))) (\lambda (x6: T).(pr2 a x1 x6))) (\lambda (x6:
202 T).(\lambda (x7: C).(\lambda (H24: (eq T x4 (lift (S O) (minus d0 (S i))
203 x6))).(\lambda (H25: (getl i a (CHead x7 (Bind Abbr) x6))).(\lambda (_: (drop
204 (S O) (minus d0 (S i)) x5 x7)).(let H27 \def (eq_ind T x4 (\lambda (t0:
205 T).(subst1 (minus d0 (S i)) u0 u t0)) H20 (lift (S O) (minus d0 (S i)) x6)
206 H24) in (ex2_ind T (\lambda (t0: T).(subst1 i (lift (S O) (minus d0 (S i))
207 x6) (lift (S O) d0 x0) t0)) (\lambda (t0: T).(subst1 (S (plus (minus d0 (S
208 i)) i)) u0 x2 t0)) (ex2 T (\lambda (x8: T).(subst1 d0 u0 t (lift (S O) d0
209 x8))) (\lambda (x8: T).(pr2 a x1 x8))) (\lambda (x8: T).(\lambda (H28:
210 (subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S O) d0 x0) x8)).(\lambda
211 (H29: (subst1 (S (plus (minus d0 (S i)) i)) u0 x2 x8)).(let H30 \def (eq_ind
212 nat d0 (\lambda (n: nat).(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S
213 O) n x0) x8)) H28 (S (plus i (minus d0 (S i)))) (lt_plus_minus i d0 H12)) in
214 (ex2_ind T (\lambda (t5: T).(eq T x8 (lift (S O) (S (plus i (minus d0 (S
215 i)))) t5))) (\lambda (t5: T).(subst1 i x6 x0 t5)) (ex2 T (\lambda (x9:
216 T).(subst1 d0 u0 t (lift (S O) d0 x9))) (\lambda (x9: T).(pr2 a x1 x9)))
217 (\lambda (x9: T).(\lambda (H31: (eq T x8 (lift (S O) (S (plus i (minus d0 (S
218 i)))) x9))).(\lambda (H32: (subst1 i x6 x0 x9)).(let H33 \def (eq_ind T x8
219 (\lambda (t0: T).(subst1 (S (plus (minus d0 (S i)) i)) u0 x2 t0)) H29 (lift
220 (S O) (S (plus i (minus d0 (S i)))) x9) H31) in (let H34 \def (eq_ind_r nat
221 (S (plus i (minus d0 (S i)))) (\lambda (n: nat).(subst1 (S (plus (minus d0 (S
222 i)) i)) u0 x2 (lift (S O) n x9))) H33 d0 (lt_plus_minus i d0 H12)) in (let
223 H35 \def (eq_ind_r nat (S (plus (minus d0 (S i)) i)) (\lambda (n:
224 nat).(subst1 n u0 x2 (lift (S O) d0 x9))) H34 d0 (lt_plus_minus_r i d0 H12))
225 in (ex_intro2 T (\lambda (x10: T).(subst1 d0 u0 t (lift (S O) d0 x10)))
226 (\lambda (x10: T).(pr2 a x1 x10)) x9 (subst1_trans x2 t u0 d0 H13 (lift (S O)
227 d0 x9) H35) (pr2_delta1 a x7 x6 i H25 x1 x0 H10 x9 H32))))))))
228 (subst1_gen_lift_lt x6 x0 x8 i (S O) (minus d0 (S i)) H30))))))
229 (subst1_subst1_back (lift (S O) d0 x0) x2 u i H14 (lift (S O) (minus d0 (S
230 i)) x6) u0 (minus d0 (S i)) H27)))))))) (getl_drop_conf_lt Abbr a0 x5 x4 i
231 H22 a (S O) (minus d0 (S i)) H23))))))))) H18)))))) (csubst1_getl_lt d0 i H12
232 c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0))))) (subst1_confluence_neq t4 t u i
233 (subst1_single i u t4 t H2) (lift (S O) d0 x0) u0 d0 H11 (lt_neq i d0 H12))))
234 (\lambda (H12: (eq nat i d0)).(let H13 \def (eq_ind_r nat d0 (\lambda (n:
235 nat).(subst1 n u0 t4 (lift (S O) n x0))) H11 i H12) in (let H14 \def
236 (eq_ind_r nat d0 (\lambda (n: nat).(drop (S O) n a0 a)) H5 i H12) in (let H15
237 \def (eq_ind_r nat d0 (\lambda (n: nat).(csubst1 n u0 c0 a0)) H4 i H12) in
238 (let H16 \def (eq_ind_r nat d0 (\lambda (n: nat).(getl n c0 (CHead e (Bind
239 Abbr) u0))) H3 i H12) in (eq_ind nat i (\lambda (n: nat).(ex2 T (\lambda (x2:
240 T).(subst1 n u0 t (lift (S O) n x2))) (\lambda (x2: T).(pr2 a x1 x2)))) (let
241 H17 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1))
242 H0 (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead
243 e (Bind Abbr) u0) H16)) in (let H18 \def (f_equal C C (\lambda (e0: C).(match
244 e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _
245 _) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0)
246 (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in
247 ((let H19 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda
248 (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0]))
249 (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind
250 Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in (\lambda (H20: (eq C d
251 e)).(let H21 \def (eq_ind_r T u0 (\lambda (t0: T).(getl i c0 (CHead e (Bind
252 Abbr) t0))) H17 u H19) in (let H22 \def (eq_ind_r T u0 (\lambda (t0:
253 T).(subst1 i t0 t4 (lift (S O) i x0))) H13 u H19) in (let H23 \def (eq_ind_r
254 T u0 (\lambda (t0: T).(csubst1 i t0 c0 a0)) H15 u H19) in (eq_ind T u
255 (\lambda (t0: T).(ex2 T (\lambda (x2: T).(subst1 i t0 t (lift (S O) i x2)))
256 (\lambda (x2: T).(pr2 a x1 x2)))) (let H24 \def (eq_ind_r C e (\lambda (c1:
257 C).(getl i c0 (CHead c1 (Bind Abbr) u))) H21 d H20) in (ex2_ind T (\lambda
258 (t0: T).(subst1 i u t t0)) (\lambda (t0: T).(subst1 i u (lift (S O) i x0)
259 t0)) (ex2 T (\lambda (x2: T).(subst1 i u t (lift (S O) i x2))) (\lambda (x2:
260 T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H25: (subst1 i u t
261 x2)).(\lambda (H26: (subst1 i u (lift (S O) i x0) x2)).(let H27 \def (eq_ind
262 T x2 (\lambda (t0: T).(subst1 i u t t0)) H25 (lift (S O) i x0)
263 (subst1_gen_lift_eq x0 u x2 (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i)
264 (\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_comm i
265 (S O))) H26)) in (ex_intro2 T (\lambda (x3: T).(subst1 i u t (lift (S O) i
266 x3))) (\lambda (x3: T).(pr2 a x1 x3)) x0 H27 (pr2_free a x1 x0 H10))))))
267 (subst1_confluence_eq t4 t u i (subst1_single i u t4 t H2) (lift (S O) i x0)
268 H22))) u0 H19)))))) H18))) d0 H12)))))) (\lambda (H12: (lt d0 i)).(ex2_ind T
269 (\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0: T).(subst1 i u (lift (S
270 O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2)))
271 (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H13: (subst1 d0
272 u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O) d0 x0) x2)).(ex2_ind T
273 (\lambda (t5: T).(eq T x2 (lift (S O) d0 t5))) (\lambda (t5: T).(subst1
274 (minus i (S O)) u x0 t5)) (ex2 T (\lambda (x3: T).(subst1 d0 u0 t (lift (S O)
275 d0 x3))) (\lambda (x3: T).(pr2 a x1 x3))) (\lambda (x3: T).(\lambda (H15: (eq
276 T x2 (lift (S O) d0 x3))).(\lambda (H16: (subst1 (minus i (S O)) u x0
277 x3)).(let H17 \def (eq_ind T x2 (\lambda (t0: T).(subst1 d0 u0 t t0)) H13
278 (lift (S O) d0 x3) H15) in (ex_intro2 T (\lambda (x4: T).(subst1 d0 u0 t
279 (lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4)) x3 H17 (pr2_delta1 a d u
280 (minus i (S O)) (getl_drop_conf_ge i (CHead d (Bind Abbr) u) a0
281 (csubst1_getl_ge d0 i (le_S_n d0 i (le_S (S d0) i H12)) c0 a0 u0 H4 (CHead d
282 (Bind Abbr) u) H0) a (S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n:
283 nat).(le n i)) H12 (plus d0 (S O)) (plus_comm d0 (S O)))) x1 x0 H10 x3
284 H16)))))) (subst1_gen_lift_ge u x0 x2 i (S O) d0 H14 (eq_ind_r nat (plus (S
285 O) d0) (\lambda (n: nat).(le n i)) H12 (plus d0 (S O)) (plus_comm d0 (S
286 O)))))))) (subst1_confluence_neq t4 t u i (subst1_single i u t4 t H2) (lift
287 (S O) d0 x0) u0 d0 H11 (sym_not_eq nat d0 i (lt_neq d0 i H12))))))))))
288 (pr0_gen_lift x1 x (S O) d0 H7))))) (pr0_subst1 t3 t4 H1 u0 (lift (S O) d0
289 x1) d0 H6 u0 (pr0_refl u0))))))))))))))))))))))) c t1 t2 H)))).