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 set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/iso/fwd".
19 include "iso/defs.ma".
21 include "tlist/defs.ma".
23 theorem iso_flats_lref_bind_false:
24 \forall (f: F).(\forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall
25 (t: T).(\forall (vs: TList).((iso (THeads (Flat f) vs (TLRef i)) (THead (Bind
26 b) v t)) \to (\forall (P: Prop).P)))))))
28 \lambda (f: F).(\lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda
29 (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: TList).((iso (THeads
30 (Flat f) t0 (TLRef i)) (THead (Bind b) v t)) \to (\forall (P: Prop).P)))
31 (\lambda (H: (iso (TLRef i) (THead (Bind b) v t))).(\lambda (P: Prop).(let H0
32 \def (match H in iso return (\lambda (t0: T).(\lambda (t1: T).(\lambda (_:
33 (iso t0 t1)).((eq T t0 (TLRef i)) \to ((eq T t1 (THead (Bind b) v t)) \to
34 P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq T (TSort n1)
35 (TLRef i))).(\lambda (H1: (eq T (TSort n2) (THead (Bind b) v t))).((let H2
36 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e in T return (\lambda (_:
37 T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
38 (THead _ _ _) \Rightarrow False])) I (TLRef i) H0) in (False_ind ((eq T
39 (TSort n2) (THead (Bind b) v t)) \to P) H2)) H1))) | (iso_lref i1 i2)
40 \Rightarrow (\lambda (H0: (eq T (TLRef i1) (TLRef i))).(\lambda (H1: (eq T
41 (TLRef i2) (THead (Bind b) v t))).((let H2 \def (f_equal T nat (\lambda (e:
42 T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i1 |
43 (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i1])) (TLRef i1) (TLRef
44 i) H0) in (eq_ind nat i (\lambda (_: nat).((eq T (TLRef i2) (THead (Bind b) v
45 t)) \to P)) (\lambda (H3: (eq T (TLRef i2) (THead (Bind b) v t))).(let H4
46 \def (eq_ind T (TLRef i2) (\lambda (e: T).(match e in T return (\lambda (_:
47 T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
48 (THead _ _ _) \Rightarrow False])) I (THead (Bind b) v t) H3) in (False_ind P
49 H4))) i1 (sym_eq nat i1 i H2))) H1))) | (iso_head v1 v2 t1 t2 k) \Rightarrow
50 (\lambda (H0: (eq T (THead k v1 t1) (TLRef i))).(\lambda (H1: (eq T (THead k
51 v2 t2) (THead (Bind b) v t))).((let H2 \def (eq_ind T (THead k v1 t1)
52 (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _)
53 \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
54 True])) I (TLRef i) H0) in (False_ind ((eq T (THead k v2 t2) (THead (Bind b)
55 v t)) \to P) H2)) H1)))]) in (H0 (refl_equal T (TLRef i)) (refl_equal T
56 (THead (Bind b) v t)))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (_:
57 (((iso (THeads (Flat f) t1 (TLRef i)) (THead (Bind b) v t)) \to (\forall (P:
58 Prop).P)))).(\lambda (H0: (iso (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef
59 i))) (THead (Bind b) v t))).(\lambda (P: Prop).(let H1 \def (match H0 in iso
60 return (\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (iso t2 t3)).((eq T t2
61 (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i)))) \to ((eq T t3 (THead
62 (Bind b) v t)) \to P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H1:
63 (eq T (TSort n1) (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef
64 i))))).(\lambda (H2: (eq T (TSort n2) (THead (Bind b) v t))).((let H3 \def
65 (eq_ind T (TSort n1) (\lambda (e: T).(match e in T return (\lambda (_:
66 T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
67 (THead _ _ _) \Rightarrow False])) I (THead (Flat f) t0 (THeads (Flat f) t1
68 (TLRef i))) H1) in (False_ind ((eq T (TSort n2) (THead (Bind b) v t)) \to P)
69 H3)) H2))) | (iso_lref i1 i2) \Rightarrow (\lambda (H1: (eq T (TLRef i1)
70 (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))))).(\lambda (H2: (eq T
71 (TLRef i2) (THead (Bind b) v t))).((let H3 \def (eq_ind T (TLRef i1) (\lambda
72 (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
73 False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
74 (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))) H1) in (False_ind ((eq T
75 (TLRef i2) (THead (Bind b) v t)) \to P) H3)) H2))) | (iso_head v1 v2 t2 t3 k)
76 \Rightarrow (\lambda (H1: (eq T (THead k v1 t2) (THead (Flat f) t0 (THeads
77 (Flat f) t1 (TLRef i))))).(\lambda (H2: (eq T (THead k v2 t3) (THead (Bind b)
78 v t))).((let H3 \def (f_equal T T (\lambda (e: T).(match e in T return
79 (\lambda (_: T).T) with [(TSort _) \Rightarrow t2 | (TLRef _) \Rightarrow t2
80 | (THead _ _ t4) \Rightarrow t4])) (THead k v1 t2) (THead (Flat f) t0 (THeads
81 (Flat f) t1 (TLRef i))) H1) in ((let H4 \def (f_equal T T (\lambda (e:
82 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 |
83 (TLRef _) \Rightarrow v1 | (THead _ t4 _) \Rightarrow t4])) (THead k v1 t2)
84 (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))) H1) in ((let H5 \def
85 (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
86 [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _)
87 \Rightarrow k0])) (THead k v1 t2) (THead (Flat f) t0 (THeads (Flat f) t1
88 (TLRef i))) H1) in (eq_ind K (Flat f) (\lambda (k0: K).((eq T v1 t0) \to ((eq
89 T t2 (THeads (Flat f) t1 (TLRef i))) \to ((eq T (THead k0 v2 t3) (THead (Bind
90 b) v t)) \to P)))) (\lambda (H6: (eq T v1 t0)).(eq_ind T t0 (\lambda (_:
91 T).((eq T t2 (THeads (Flat f) t1 (TLRef i))) \to ((eq T (THead (Flat f) v2
92 t3) (THead (Bind b) v t)) \to P))) (\lambda (H7: (eq T t2 (THeads (Flat f) t1
93 (TLRef i)))).(eq_ind T (THeads (Flat f) t1 (TLRef i)) (\lambda (_: T).((eq T
94 (THead (Flat f) v2 t3) (THead (Bind b) v t)) \to P)) (\lambda (H8: (eq T
95 (THead (Flat f) v2 t3) (THead (Bind b) v t))).(let H9 \def (eq_ind T (THead
96 (Flat f) v2 t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop)
97 with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _
98 _) \Rightarrow (match k0 in K return (\lambda (_: K).Prop) with [(Bind _)
99 \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) v t) H8)
100 in (False_ind P H9))) t2 (sym_eq T t2 (THeads (Flat f) t1 (TLRef i)) H7))) v1
101 (sym_eq T v1 t0 H6))) k (sym_eq K k (Flat f) H5))) H4)) H3)) H2)))]) in (H1
102 (refl_equal T (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i)))) (refl_equal
103 T (THead (Bind b) v t))))))))) vs)))))).
105 theorem iso_flats_flat_bind_false:
106 \forall (f1: F).(\forall (f2: F).(\forall (b: B).(\forall (v: T).(\forall
107 (v2: T).(\forall (t: T).(\forall (t2: T).(\forall (vs: TList).((iso (THeads
108 (Flat f1) vs (THead (Flat f2) v2 t2)) (THead (Bind b) v t)) \to (\forall (P:
111 \lambda (f1: F).(\lambda (f2: F).(\lambda (b: B).(\lambda (v: T).(\lambda
112 (v2: T).(\lambda (t: T).(\lambda (t2: T).(\lambda (vs: TList).(TList_ind
113 (\lambda (t0: TList).((iso (THeads (Flat f1) t0 (THead (Flat f2) v2 t2))
114 (THead (Bind b) v t)) \to (\forall (P: Prop).P))) (\lambda (H: (iso (THead
115 (Flat f2) v2 t2) (THead (Bind b) v t))).(\lambda (P: Prop).(let H0 \def
116 (match H in iso return (\lambda (t0: T).(\lambda (t1: T).(\lambda (_: (iso t0
117 t1)).((eq T t0 (THead (Flat f2) v2 t2)) \to ((eq T t1 (THead (Bind b) v t))
118 \to P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq T (TSort n1)
119 (THead (Flat f2) v2 t2))).(\lambda (H1: (eq T (TSort n2) (THead (Bind b) v
120 t))).((let H2 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e in T return
121 (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
122 \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat f2) v2
123 t2) H0) in (False_ind ((eq T (TSort n2) (THead (Bind b) v t)) \to P) H2))
124 H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0: (eq T (TLRef i1) (THead
125 (Flat f2) v2 t2))).(\lambda (H1: (eq T (TLRef i2) (THead (Bind b) v
126 t))).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T return
127 (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
128 \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat f2) v2
129 t2) H0) in (False_ind ((eq T (TLRef i2) (THead (Bind b) v t)) \to P) H2))
130 H1))) | (iso_head v1 v0 t1 t0 k) \Rightarrow (\lambda (H0: (eq T (THead k v1
131 t1) (THead (Flat f2) v2 t2))).(\lambda (H1: (eq T (THead k v0 t0) (THead
132 (Bind b) v t))).((let H2 \def (f_equal T T (\lambda (e: T).(match e in T
133 return (\lambda (_: T).T) with [(TSort _) \Rightarrow t1 | (TLRef _)
134 \Rightarrow t1 | (THead _ _ t3) \Rightarrow t3])) (THead k v1 t1) (THead
135 (Flat f2) v2 t2) H0) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e
136 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _)
137 \Rightarrow v1 | (THead _ t3 _) \Rightarrow t3])) (THead k v1 t1) (THead
138 (Flat f2) v2 t2) H0) in ((let H4 \def (f_equal T K (\lambda (e: T).(match e
139 in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
140 \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k v1 t1) (THead (Flat
141 f2) v2 t2) H0) in (eq_ind K (Flat f2) (\lambda (k0: K).((eq T v1 v2) \to ((eq
142 T t1 t2) \to ((eq T (THead k0 v0 t0) (THead (Bind b) v t)) \to P)))) (\lambda
143 (H5: (eq T v1 v2)).(eq_ind T v2 (\lambda (_: T).((eq T t1 t2) \to ((eq T
144 (THead (Flat f2) v0 t0) (THead (Bind b) v t)) \to P))) (\lambda (H6: (eq T t1
145 t2)).(eq_ind T t2 (\lambda (_: T).((eq T (THead (Flat f2) v0 t0) (THead (Bind
146 b) v t)) \to P)) (\lambda (H7: (eq T (THead (Flat f2) v0 t0) (THead (Bind b)
147 v t))).(let H8 \def (eq_ind T (THead (Flat f2) v0 t0) (\lambda (e: T).(match
148 e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
149 (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K
150 return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
151 \Rightarrow True])])) I (THead (Bind b) v t) H7) in (False_ind P H8))) t1
152 (sym_eq T t1 t2 H6))) v1 (sym_eq T v1 v2 H5))) k (sym_eq K k (Flat f2) H4)))
153 H3)) H2)) H1)))]) in (H0 (refl_equal T (THead (Flat f2) v2 t2)) (refl_equal T
154 (THead (Bind b) v t)))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (_:
155 (((iso (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)) (THead (Bind b) v t))
156 \to (\forall (P: Prop).P)))).(\lambda (H0: (iso (THead (Flat f1) t0 (THeads
157 (Flat f1) t1 (THead (Flat f2) v2 t2))) (THead (Bind b) v t))).(\lambda (P:
158 Prop).(let H1 \def (match H0 in iso return (\lambda (t3: T).(\lambda (t4:
159 T).(\lambda (_: (iso t3 t4)).((eq T t3 (THead (Flat f1) t0 (THeads (Flat f1)
160 t1 (THead (Flat f2) v2 t2)))) \to ((eq T t4 (THead (Bind b) v t)) \to P)))))
161 with [(iso_sort n1 n2) \Rightarrow (\lambda (H1: (eq T (TSort n1) (THead
162 (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))))).(\lambda (H2:
163 (eq T (TSort n2) (THead (Bind b) v t))).((let H3 \def (eq_ind T (TSort n1)
164 (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _)
165 \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
166 False])) I (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)))
167 H1) in (False_ind ((eq T (TSort n2) (THead (Bind b) v t)) \to P) H3)) H2))) |
168 (iso_lref i1 i2) \Rightarrow (\lambda (H1: (eq T (TLRef i1) (THead (Flat f1)
169 t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))))).(\lambda (H2: (eq T
170 (TLRef i2) (THead (Bind b) v t))).((let H3 \def (eq_ind T (TLRef i1) (\lambda
171 (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
172 False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
173 (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) H1) in
174 (False_ind ((eq T (TLRef i2) (THead (Bind b) v t)) \to P) H3)) H2))) |
175 (iso_head v1 v0 t3 t4 k) \Rightarrow (\lambda (H1: (eq T (THead k v1 t3)
176 (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))))).(\lambda
177 (H2: (eq T (THead k v0 t4) (THead (Bind b) v t))).((let H3 \def (f_equal T T
178 (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
179 \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t5) \Rightarrow t5]))
180 (THead k v1 t3) (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2
181 t2))) H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e in T return
182 (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1
183 | (THead _ t5 _) \Rightarrow t5])) (THead k v1 t3) (THead (Flat f1) t0
184 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) H1) in ((let H5 \def (f_equal
185 T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
186 \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0]))
187 (THead k v1 t3) (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2
188 t2))) H1) in (eq_ind K (Flat f1) (\lambda (k0: K).((eq T v1 t0) \to ((eq T t3
189 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) \to ((eq T (THead k0 v0 t4)
190 (THead (Bind b) v t)) \to P)))) (\lambda (H6: (eq T v1 t0)).(eq_ind T t0
191 (\lambda (_: T).((eq T t3 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) \to
192 ((eq T (THead (Flat f1) v0 t4) (THead (Bind b) v t)) \to P))) (\lambda (H7:
193 (eq T t3 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)))).(eq_ind T (THeads
194 (Flat f1) t1 (THead (Flat f2) v2 t2)) (\lambda (_: T).((eq T (THead (Flat f1)
195 v0 t4) (THead (Bind b) v t)) \to P)) (\lambda (H8: (eq T (THead (Flat f1) v0
196 t4) (THead (Bind b) v t))).(let H9 \def (eq_ind T (THead (Flat f1) v0 t4)
197 (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _)
198 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow
199 (match k0 in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False
200 | (Flat _) \Rightarrow True])])) I (THead (Bind b) v t) H8) in (False_ind P
201 H9))) t3 (sym_eq T t3 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)) H7))) v1
202 (sym_eq T v1 t0 H6))) k (sym_eq K k (Flat f1) H5))) H4)) H3)) H2)))]) in (H1
203 (refl_equal T (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2
204 t2)))) (refl_equal T (THead (Bind b) v t))))))))) vs)))))))).
206 theorem iso_gen_sort:
207 \forall (u2: T).(\forall (n1: nat).((iso (TSort n1) u2) \to (ex nat (\lambda
208 (n2: nat).(eq T u2 (TSort n2))))))
210 \lambda (u2: T).(\lambda (n1: nat).(\lambda (H: (iso (TSort n1) u2)).(let H0
211 \def (match H in iso return (\lambda (t: T).(\lambda (t0: T).(\lambda (_:
212 (iso t t0)).((eq T t (TSort n1)) \to ((eq T t0 u2) \to (ex nat (\lambda (n2:
213 nat).(eq T u2 (TSort n2))))))))) with [(iso_sort n0 n2) \Rightarrow (\lambda
214 (H0: (eq T (TSort n0) (TSort n1))).(\lambda (H1: (eq T (TSort n2) u2)).((let
215 H2 \def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_:
216 T).nat) with [(TSort n) \Rightarrow n | (TLRef _) \Rightarrow n0 | (THead _ _
217 _) \Rightarrow n0])) (TSort n0) (TSort n1) H0) in (eq_ind nat n1 (\lambda (_:
218 nat).((eq T (TSort n2) u2) \to (ex nat (\lambda (n3: nat).(eq T u2 (TSort
219 n3)))))) (\lambda (H3: (eq T (TSort n2) u2)).(eq_ind T (TSort n2) (\lambda
220 (t: T).(ex nat (\lambda (n3: nat).(eq T t (TSort n3))))) (ex_intro nat
221 (\lambda (n3: nat).(eq T (TSort n2) (TSort n3))) n2 (refl_equal T (TSort
222 n2))) u2 H3)) n0 (sym_eq nat n0 n1 H2))) H1))) | (iso_lref i1 i2) \Rightarrow
223 (\lambda (H0: (eq T (TLRef i1) (TSort n1))).(\lambda (H1: (eq T (TLRef i2)
224 u2)).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T return
225 (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
226 \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n1) H0) in
227 (False_ind ((eq T (TLRef i2) u2) \to (ex nat (\lambda (n2: nat).(eq T u2
228 (TSort n2))))) H2)) H1))) | (iso_head v1 v2 t1 t2 k) \Rightarrow (\lambda
229 (H0: (eq T (THead k v1 t1) (TSort n1))).(\lambda (H1: (eq T (THead k v2 t2)
230 u2)).((let H2 \def (eq_ind T (THead k v1 t1) (\lambda (e: T).(match e in T
231 return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
232 \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n1) H0) in
233 (False_ind ((eq T (THead k v2 t2) u2) \to (ex nat (\lambda (n2: nat).(eq T u2
234 (TSort n2))))) H2)) H1)))]) in (H0 (refl_equal T (TSort n1)) (refl_equal T
237 theorem iso_gen_lref:
238 \forall (u2: T).(\forall (n1: nat).((iso (TLRef n1) u2) \to (ex nat (\lambda
239 (n2: nat).(eq T u2 (TLRef n2))))))
241 \lambda (u2: T).(\lambda (n1: nat).(\lambda (H: (iso (TLRef n1) u2)).(let H0
242 \def (match H in iso return (\lambda (t: T).(\lambda (t0: T).(\lambda (_:
243 (iso t t0)).((eq T t (TLRef n1)) \to ((eq T t0 u2) \to (ex nat (\lambda (n2:
244 nat).(eq T u2 (TLRef n2))))))))) with [(iso_sort n0 n2) \Rightarrow (\lambda
245 (H0: (eq T (TSort n0) (TLRef n1))).(\lambda (H1: (eq T (TSort n2) u2)).((let
246 H2 \def (eq_ind T (TSort n0) (\lambda (e: T).(match e in T return (\lambda
247 (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
248 (THead _ _ _) \Rightarrow False])) I (TLRef n1) H0) in (False_ind ((eq T
249 (TSort n2) u2) \to (ex nat (\lambda (n3: nat).(eq T u2 (TLRef n3))))) H2))
250 H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0: (eq T (TLRef i1) (TLRef
251 n1))).(\lambda (H1: (eq T (TLRef i2) u2)).((let H2 \def (f_equal T nat
252 (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _)
253 \Rightarrow i1 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i1]))
254 (TLRef i1) (TLRef n1) H0) in (eq_ind nat n1 (\lambda (_: nat).((eq T (TLRef
255 i2) u2) \to (ex nat (\lambda (n2: nat).(eq T u2 (TLRef n2)))))) (\lambda (H3:
256 (eq T (TLRef i2) u2)).(eq_ind T (TLRef i2) (\lambda (t: T).(ex nat (\lambda
257 (n2: nat).(eq T t (TLRef n2))))) (ex_intro nat (\lambda (n2: nat).(eq T
258 (TLRef i2) (TLRef n2))) i2 (refl_equal T (TLRef i2))) u2 H3)) i1 (sym_eq nat
259 i1 n1 H2))) H1))) | (iso_head v1 v2 t1 t2 k) \Rightarrow (\lambda (H0: (eq T
260 (THead k v1 t1) (TLRef n1))).(\lambda (H1: (eq T (THead k v2 t2) u2)).((let
261 H2 \def (eq_ind T (THead k v1 t1) (\lambda (e: T).(match e in T return
262 (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
263 \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n1) H0) in
264 (False_ind ((eq T (THead k v2 t2) u2) \to (ex nat (\lambda (n2: nat).(eq T u2
265 (TLRef n2))))) H2)) H1)))]) in (H0 (refl_equal T (TLRef n1)) (refl_equal T
268 theorem iso_gen_head:
269 \forall (k: K).(\forall (v1: T).(\forall (t1: T).(\forall (u2: T).((iso
270 (THead k v1 t1) u2) \to (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T u2
271 (THead k v2 t2)))))))))
273 \lambda (k: K).(\lambda (v1: T).(\lambda (t1: T).(\lambda (u2: T).(\lambda
274 (H: (iso (THead k v1 t1) u2)).(let H0 \def (match H in iso return (\lambda
275 (t: T).(\lambda (t0: T).(\lambda (_: (iso t t0)).((eq T t (THead k v1 t1))
276 \to ((eq T t0 u2) \to (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T u2
277 (THead k v2 t2)))))))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq
278 T (TSort n1) (THead k v1 t1))).(\lambda (H1: (eq T (TSort n2) u2)).((let H2
279 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e in T return (\lambda (_:
280 T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
281 (THead _ _ _) \Rightarrow False])) I (THead k v1 t1) H0) in (False_ind ((eq T
282 (TSort n2) u2) \to (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T u2
283 (THead k v2 t2)))))) H2)) H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0:
284 (eq T (TLRef i1) (THead k v1 t1))).(\lambda (H1: (eq T (TLRef i2) u2)).((let
285 H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T return (\lambda
286 (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
287 (THead _ _ _) \Rightarrow False])) I (THead k v1 t1) H0) in (False_ind ((eq T
288 (TLRef i2) u2) \to (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T u2
289 (THead k v2 t2)))))) H2)) H1))) | (iso_head v0 v2 t0 t2 k0) \Rightarrow
290 (\lambda (H0: (eq T (THead k0 v0 t0) (THead k v1 t1))).(\lambda (H1: (eq T
291 (THead k0 v2 t2) u2)).((let H2 \def (f_equal T T (\lambda (e: T).(match e in
292 T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
293 \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k0 v0 t0) (THead k v1
294 t1) H0) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e in T return
295 (\lambda (_: T).T) with [(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0
296 | (THead _ t _) \Rightarrow t])) (THead k0 v0 t0) (THead k v1 t1) H0) in
297 ((let H4 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_:
298 T).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _
299 _) \Rightarrow k1])) (THead k0 v0 t0) (THead k v1 t1) H0) in (eq_ind K k
300 (\lambda (k1: K).((eq T v0 v1) \to ((eq T t0 t1) \to ((eq T (THead k1 v2 t2)
301 u2) \to (ex_2 T T (\lambda (v3: T).(\lambda (t3: T).(eq T u2 (THead k v3
302 t3))))))))) (\lambda (H5: (eq T v0 v1)).(eq_ind T v1 (\lambda (_: T).((eq T
303 t0 t1) \to ((eq T (THead k v2 t2) u2) \to (ex_2 T T (\lambda (v3: T).(\lambda
304 (t3: T).(eq T u2 (THead k v3 t3)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T
305 t1 (\lambda (_: T).((eq T (THead k v2 t2) u2) \to (ex_2 T T (\lambda (v3:
306 T).(\lambda (t3: T).(eq T u2 (THead k v3 t3))))))) (\lambda (H7: (eq T (THead
307 k v2 t2) u2)).(eq_ind T (THead k v2 t2) (\lambda (t: T).(ex_2 T T (\lambda
308 (v3: T).(\lambda (t3: T).(eq T t (THead k v3 t3)))))) (ex_2_intro T T
309 (\lambda (v3: T).(\lambda (t3: T).(eq T (THead k v2 t2) (THead k v3 t3)))) v2
310 t2 (refl_equal T (THead k v2 t2))) u2 H7)) t0 (sym_eq T t0 t1 H6))) v0
311 (sym_eq T v0 v1 H5))) k0 (sym_eq K k0 k H4))) H3)) H2)) H1)))]) in (H0
312 (refl_equal T (THead k v1 t1)) (refl_equal T u2))))))).