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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/wf3/defs.ma".
19 theorem wf3_gen_sort1:
20 \forall (g: G).(\forall (x: C).(\forall (m: nat).((wf3 g (CSort m) x) \to
23 \lambda (g: G).(\lambda (x: C).(\lambda (m: nat).(\lambda (H: (wf3 g (CSort
24 m) x)).(insert_eq C (CSort m) (\lambda (c: C).(wf3 g c x)) (\lambda (c:
25 C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda
26 (c: C).(\lambda (c0: C).((eq C c (CSort m)) \to (eq C c0 c)))) (\lambda (m0:
27 nat).(\lambda (H1: (eq C (CSort m0) (CSort m))).(let H2 \def (f_equal C nat
28 (\lambda (e: C).(match e in C return (\lambda (_: C).nat) with [(CSort n)
29 \Rightarrow n | (CHead _ _ _) \Rightarrow m0])) (CSort m0) (CSort m) H1) in
30 (eq_ind_r nat m (\lambda (n: nat).(eq C (CSort n) (CSort n))) (refl_equal C
31 (CSort m)) m0 H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1
32 c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u:
33 T).(\lambda (t: T).(\lambda (_: (ty3 g c1 u t)).(\lambda (b: B).(\lambda (H4:
34 (eq C (CHead c1 (Bind b) u) (CSort m))).(let H5 \def (eq_ind C (CHead c1
35 (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with
36 [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort m)
37 H4) in (False_ind (eq C (CHead c2 (Bind b) u) (CHead c1 (Bind b) u))
38 H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (wf3 g c1
39 c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C c2 c1)))).(\lambda (u:
40 T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u t) \to False)))).(\lambda (b:
41 B).(\lambda (H4: (eq C (CHead c1 (Bind b) u) (CSort m))).(let H5 \def (eq_ind
42 C (CHead c1 (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_:
43 C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow
44 True])) I (CSort m) H4) in (False_ind (eq C (CHead c2 (Bind Void) (TSort O))
45 (CHead c1 (Bind b) u)) H5)))))))))) (\lambda (c1: C).(\lambda (c2:
46 C).(\lambda (_: (wf3 g c1 c2)).(\lambda (_: (((eq C c1 (CSort m)) \to (eq C
47 c2 c1)))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C (CHead c1 (Flat
48 f) u) (CSort m))).(let H4 \def (eq_ind C (CHead c1 (Flat f) u) (\lambda (ee:
49 C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
50 False | (CHead _ _ _) \Rightarrow True])) I (CSort m) H3) in (False_ind (eq C
51 c2 (CHead c1 (Flat f) u)) H4))))))))) y x H0))) H)))).
56 theorem wf3_gen_bind1:
57 \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (b:
58 B).((wf3 g (CHead c1 (Bind b) v) x) \to (or (ex3_2 C T (\lambda (c2:
59 C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2: C).(\lambda
60 (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3
61 C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O)))) (\lambda (c2:
62 C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to
65 \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (b:
66 B).(\lambda (H: (wf3 g (CHead c1 (Bind b) v) x)).(insert_eq C (CHead c1 (Bind
67 b) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(or (ex3_2 C T (\lambda
68 (c2: C).(\lambda (_: T).(eq C x (CHead c2 (Bind b) v)))) (\lambda (c2:
69 C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1
70 v w)))) (ex3 C (\lambda (c2: C).(eq C x (CHead c2 (Bind Void) (TSort O))))
71 (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v
72 w) \to False)))))) (\lambda (y: C).(\lambda (H0: (wf3 g y x)).(wf3_ind g
73 (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead c1 (Bind b) v)) \to (or
74 (ex3_2 C T (\lambda (c2: C).(\lambda (_: T).(eq C c0 (CHead c2 (Bind b) v))))
75 (\lambda (c2: C).(\lambda (_: T).(wf3 g c1 c2))) (\lambda (_: C).(\lambda (w:
76 T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2: C).(eq C c0 (CHead c2 (Bind Void)
77 (TSort O)))) (\lambda (c2: C).(wf3 g c1 c2)) (\lambda (_: C).(\forall (w:
78 T).((ty3 g c1 v w) \to False)))))))) (\lambda (m: nat).(\lambda (H1: (eq C
79 (CSort m) (CHead c1 (Bind b) v))).(let H2 \def (eq_ind C (CSort m) (\lambda
80 (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
81 \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead c1 (Bind b) v)
82 H1) in (False_ind (or (ex3_2 C T (\lambda (c2: C).(\lambda (_: T).(eq C
83 (CSort m) (CHead c2 (Bind b) v)))) (\lambda (c2: C).(\lambda (_: T).(wf3 g c1
84 c2))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c2:
85 C).(eq C (CSort m) (CHead c2 (Bind Void) (TSort O)))) (\lambda (c2: C).(wf3 g
86 c1 c2)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))))) H2))))
87 (\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2:
88 (((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda (c3:
89 C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3:
90 C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1
91 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O))))
92 (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v
93 w) \to False)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c0
94 u t)).(\lambda (b0: B).(\lambda (H4: (eq C (CHead c0 (Bind b0) u) (CHead c1
95 (Bind b) v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return
96 (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow
97 c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H6 \def
98 (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
99 [(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k in K return
100 (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow
101 b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in ((let H7 \def
102 (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
103 [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 (Bind
104 b0) u) (CHead c1 (Bind b) v) H4) in (\lambda (H8: (eq B b0 b)).(\lambda (H9:
105 (eq C c0 c1)).(eq_ind_r B b (\lambda (b1: B).(or (ex3_2 C T (\lambda (c3:
106 C).(\lambda (_: T).(eq C (CHead c2 (Bind b1) u) (CHead c3 (Bind b) v))))
107 (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w:
108 T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C (CHead c2 (Bind b1) u)
109 (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda
110 (_: C).(\forall (w: T).((ty3 g c1 v w) \to False)))))) (let H10 \def (eq_ind
111 T u (\lambda (t0: T).(ty3 g c0 t0 t)) H3 v H7) in (eq_ind_r T v (\lambda (t0:
112 T).(or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind b)
113 t0) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3)))
114 (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq
115 C (CHead c2 (Bind b) t0) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3:
116 C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to
117 False)))))) (let H11 \def (eq_ind C c0 (\lambda (c: C).(ty3 g c v t)) H10 c1
118 H9) in (let H12 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 (Bind b)
119 v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3
120 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_:
121 C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead
122 c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_:
123 C).(\forall (w: T).((ty3 g c1 v w) \to False))))))) H2 c1 H9) in (let H13
124 \def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H9) in (or_introl
125 (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind b) v)
126 (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3)))
127 (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq
128 C (CHead c2 (Bind b) v) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3:
129 C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to
130 False)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2
131 (Bind b) v) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g
132 c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w))) c2 t (refl_equal C
133 (CHead c2 (Bind b) v)) H13 H11))))) u H7)) b0 H8)))) H6)) H5)))))))))))
134 (\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2:
135 (((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T (\lambda (c3:
136 C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda (c3:
137 C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1
138 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort O))))
139 (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v
140 w) \to False)))))))).(\lambda (u: T).(\lambda (H3: ((\forall (t: T).((ty3 g
141 c0 u t) \to False)))).(\lambda (b0: B).(\lambda (H4: (eq C (CHead c0 (Bind
142 b0) u) (CHead c1 (Bind b) v))).(let H5 \def (f_equal C C (\lambda (e:
143 C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 |
144 (CHead c _ _) \Rightarrow c])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v)
145 H4) in ((let H6 \def (f_equal C B (\lambda (e: C).(match e in C return
146 (\lambda (_: C).B) with [(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow
147 (match k in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 |
148 (Flat _) \Rightarrow b0])])) (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4)
149 in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
150 (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t]))
151 (CHead c0 (Bind b0) u) (CHead c1 (Bind b) v) H4) in (\lambda (_: (eq B b0
152 b)).(\lambda (H9: (eq C c0 c1)).(let H10 \def (eq_ind T u (\lambda (t:
153 T).(\forall (t0: T).((ty3 g c0 t t0) \to False))) H3 v H7) in (let H11 \def
154 (eq_ind C c0 (\lambda (c: C).(\forall (t: T).((ty3 g c v t) \to False))) H10
155 c1 H9) in (let H12 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 (Bind
156 b) v)) \to (or (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3
157 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_:
158 C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead
159 c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_:
160 C).(\forall (w: T).((ty3 g c1 v w) \to False))))))) H2 c1 H9) in (let H13
161 \def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H9) in (or_intror
162 (ex3_2 C T (\lambda (c3: C).(\lambda (_: T).(eq C (CHead c2 (Bind Void)
163 (TSort O)) (CHead c3 (Bind b) v)))) (\lambda (c3: C).(\lambda (_: T).(wf3 g
164 c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3 g c1 v w)))) (ex3 C (\lambda
165 (c3: C).(eq C (CHead c2 (Bind Void) (TSort O)) (CHead c3 (Bind Void) (TSort
166 O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g
167 c1 v w) \to False)))) (ex3_intro C (\lambda (c3: C).(eq C (CHead c2 (Bind
168 Void) (TSort O)) (CHead c3 (Bind Void) (TSort O)))) (\lambda (c3: C).(wf3 g
169 c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g c1 v w) \to False))) c2
170 (refl_equal C (CHead c2 (Bind Void) (TSort O))) H13 H11))))))))) H6))
171 H5)))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0
172 c2)).(\lambda (_: (((eq C c0 (CHead c1 (Bind b) v)) \to (or (ex3_2 C T
173 (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda
174 (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3
175 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort
176 O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g
177 c1 v w) \to False)))))))).(\lambda (u: T).(\lambda (f: F).(\lambda (H3: (eq C
178 (CHead c0 (Flat f) u) (CHead c1 (Bind b) v))).(let H4 \def (eq_ind C (CHead
179 c0 (Flat f) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop)
180 with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K
181 return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
182 \Rightarrow True])])) I (CHead c1 (Bind b) v) H3) in (False_ind (or (ex3_2 C
183 T (\lambda (c3: C).(\lambda (_: T).(eq C c2 (CHead c3 (Bind b) v)))) (\lambda
184 (c3: C).(\lambda (_: T).(wf3 g c1 c3))) (\lambda (_: C).(\lambda (w: T).(ty3
185 g c1 v w)))) (ex3 C (\lambda (c3: C).(eq C c2 (CHead c3 (Bind Void) (TSort
186 O)))) (\lambda (c3: C).(wf3 g c1 c3)) (\lambda (_: C).(\forall (w: T).((ty3 g
187 c1 v w) \to False))))) H4))))))))) y x H0))) H)))))).
192 theorem wf3_gen_flat1:
193 \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (f:
194 F).((wf3 g (CHead c1 (Flat f) v) x) \to (wf3 g c1 x))))))
196 \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (f:
197 F).(\lambda (H: (wf3 g (CHead c1 (Flat f) v) x)).(insert_eq C (CHead c1 (Flat
198 f) v) (\lambda (c: C).(wf3 g c x)) (\lambda (_: C).(wf3 g c1 x)) (\lambda (y:
199 C).(\lambda (H0: (wf3 g y x)).(wf3_ind g (\lambda (c: C).(\lambda (c0:
200 C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c0)))) (\lambda (m:
201 nat).(\lambda (H1: (eq C (CSort m) (CHead c1 (Flat f) v))).(let H2 \def
202 (eq_ind C (CSort m) (\lambda (ee: C).(match ee in C return (\lambda (_:
203 C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
204 False])) I (CHead c1 (Flat f) v) H1) in (False_ind (wf3 g c1 (CSort m))
205 H2)))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (_: (wf3 g c0 c2)).(\lambda
206 (_: (((eq C c0 (CHead c1 (Flat f) v)) \to (wf3 g c1 c2)))).(\lambda (u:
207 T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (b: B).(\lambda (H4:
208 (eq C (CHead c0 (Bind b) u) (CHead c1 (Flat f) v))).(let H5 \def (eq_ind C
209 (CHead c0 (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_:
210 C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
211 k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat
212 _) \Rightarrow False])])) I (CHead c1 (Flat f) v) H4) in (False_ind (wf3 g c1
213 (CHead c2 (Bind b) u)) H5))))))))))) (\lambda (c0: C).(\lambda (c2:
214 C).(\lambda (_: (wf3 g c0 c2)).(\lambda (_: (((eq C c0 (CHead c1 (Flat f) v))
215 \to (wf3 g c1 c2)))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c0
216 u t) \to False)))).(\lambda (b: B).(\lambda (H4: (eq C (CHead c0 (Bind b) u)
217 (CHead c1 (Flat f) v))).(let H5 \def (eq_ind C (CHead c0 (Bind b) u) (\lambda
218 (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
219 \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
220 (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
221 False])])) I (CHead c1 (Flat f) v) H4) in (False_ind (wf3 g c1 (CHead c2
222 (Bind Void) (TSort O))) H5)))))))))) (\lambda (c0: C).(\lambda (c2:
223 C).(\lambda (H1: (wf3 g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 (Flat f)
224 v)) \to (wf3 g c1 c2)))).(\lambda (u: T).(\lambda (f0: F).(\lambda (H3: (eq C
225 (CHead c0 (Flat f0) u) (CHead c1 (Flat f) v))).(let H4 \def (f_equal C C
226 (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
227 \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Flat f0) u) (CHead
228 c1 (Flat f) v) H3) in ((let H5 \def (f_equal C F (\lambda (e: C).(match e in
229 C return (\lambda (_: C).F) with [(CSort _) \Rightarrow f0 | (CHead _ k _)
230 \Rightarrow (match k in K return (\lambda (_: K).F) with [(Bind _)
231 \Rightarrow f0 | (Flat f1) \Rightarrow f1])])) (CHead c0 (Flat f0) u) (CHead
232 c1 (Flat f) v) H3) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in
233 C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t)
234 \Rightarrow t])) (CHead c0 (Flat f0) u) (CHead c1 (Flat f) v) H3) in (\lambda
235 (_: (eq F f0 f)).(\lambda (H8: (eq C c0 c1)).(let H9 \def (eq_ind C c0
236 (\lambda (c: C).((eq C c (CHead c1 (Flat f) v)) \to (wf3 g c1 c2))) H2 c1 H8)
237 in (let H10 \def (eq_ind C c0 (\lambda (c: C).(wf3 g c c2)) H1 c1 H8) in
238 H10))))) H5)) H4))))))))) y x H0))) H)))))).
243 theorem wf3_gen_head2:
244 \forall (g: G).(\forall (x: C).(\forall (c: C).(\forall (v: T).(\forall (k:
245 K).((wf3 g x (CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b)))))))))
247 \lambda (g: G).(\lambda (x: C).(\lambda (c: C).(\lambda (v: T).(\lambda (k:
248 K).(\lambda (H: (wf3 g x (CHead c k v))).(insert_eq C (CHead c k v) (\lambda
249 (c0: C).(wf3 g x c0)) (\lambda (_: C).(ex B (\lambda (b: B).(eq K k (Bind
250 b))))) (\lambda (y: C).(\lambda (H0: (wf3 g x y)).(wf3_ind g (\lambda (_:
251 C).(\lambda (c1: C).((eq C c1 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K
252 k (Bind b))))))) (\lambda (m: nat).(\lambda (H1: (eq C (CSort m) (CHead c k
253 v))).(let H2 \def (eq_ind C (CSort m) (\lambda (ee: C).(match ee in C return
254 (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
255 \Rightarrow False])) I (CHead c k v) H1) in (False_ind (ex B (\lambda (b:
256 B).(eq K k (Bind b)))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1:
257 (wf3 g c1 c2)).(\lambda (H2: (((eq C c2 (CHead c k v)) \to (ex B (\lambda (b:
258 B).(eq K k (Bind b))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3
259 g c1 u t)).(\lambda (b: B).(\lambda (H4: (eq C (CHead c2 (Bind b) u) (CHead c
260 k v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return
261 (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _)
262 \Rightarrow c0])) (CHead c2 (Bind b) u) (CHead c k v) H4) in ((let H6 \def
263 (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
264 [(CSort _) \Rightarrow (Bind b) | (CHead _ k0 _) \Rightarrow k0])) (CHead c2
265 (Bind b) u) (CHead c k v) H4) in ((let H7 \def (f_equal C T (\lambda (e:
266 C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
267 (CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind b) u) (CHead c k v) H4) in
268 (\lambda (H8: (eq K (Bind b) k)).(\lambda (H9: (eq C c2 c)).(let H10 \def
269 (eq_ind T u (\lambda (t0: T).(ty3 g c1 t0 t)) H3 v H7) in (let H11 \def
270 (eq_ind C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex B (\lambda
271 (b0: B).(eq K k (Bind b0)))))) H2 c H9) in (let H12 \def (eq_ind C c2
272 (\lambda (c0: C).(wf3 g c1 c0)) H1 c H9) in (let H13 \def (eq_ind_r K k
273 (\lambda (k0: K).((eq C c (CHead c k0 v)) \to (ex B (\lambda (b0: B).(eq K k0
274 (Bind b0)))))) H11 (Bind b) H8) in (eq_ind K (Bind b) (\lambda (k0: K).(ex B
275 (\lambda (b0: B).(eq K k0 (Bind b0))))) (ex_intro B (\lambda (b0: B).(eq K
276 (Bind b) (Bind b0))) b (refl_equal K (Bind b))) k H8)))))))) H6))
277 H5))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1
278 c2)).(\lambda (H2: (((eq C c2 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K
279 k (Bind b))))))).(\lambda (u: T).(\lambda (_: ((\forall (t: T).((ty3 g c1 u
280 t) \to False)))).(\lambda (_: B).(\lambda (H4: (eq C (CHead c2 (Bind Void)
281 (TSort O)) (CHead c k v))).(let H5 \def (f_equal C C (\lambda (e: C).(match e
282 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _
283 _) \Rightarrow c0])) (CHead c2 (Bind Void) (TSort O)) (CHead c k v) H4) in
284 ((let H6 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_:
285 C).K) with [(CSort _) \Rightarrow (Bind Void) | (CHead _ k0 _) \Rightarrow
286 k0])) (CHead c2 (Bind Void) (TSort O)) (CHead c k v) H4) in ((let H7 \def
287 (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
288 [(CSort _) \Rightarrow (TSort O) | (CHead _ _ t) \Rightarrow t])) (CHead c2
289 (Bind Void) (TSort O)) (CHead c k v) H4) in (\lambda (H8: (eq K (Bind Void)
290 k)).(\lambda (H9: (eq C c2 c)).(let H10 \def (eq_ind C c2 (\lambda (c0:
291 C).((eq C c0 (CHead c k v)) \to (ex B (\lambda (b0: B).(eq K k (Bind b0))))))
292 H2 c H9) in (let H11 \def (eq_ind C c2 (\lambda (c0: C).(wf3 g c1 c0)) H1 c
293 H9) in (let H12 \def (eq_ind_r K k (\lambda (k0: K).((eq C c (CHead c k0 v))
294 \to (ex B (\lambda (b0: B).(eq K k0 (Bind b0)))))) H10 (Bind Void) H8) in
295 (eq_ind K (Bind Void) (\lambda (k0: K).(ex B (\lambda (b0: B).(eq K k0 (Bind
296 b0))))) (let H13 \def (eq_ind_r T v (\lambda (t: T).((eq C c (CHead c (Bind
297 Void) t)) \to (ex B (\lambda (b0: B).(eq K (Bind Void) (Bind b0)))))) H12
298 (TSort O) H7) in (ex_intro B (\lambda (b0: B).(eq K (Bind Void) (Bind b0)))
299 Void (refl_equal K (Bind Void)))) k H8))))))) H6)) H5)))))))))) (\lambda (c1:
300 C).(\lambda (c2: C).(\lambda (H1: (wf3 g c1 c2)).(\lambda (H2: (((eq C c2
301 (CHead c k v)) \to (ex B (\lambda (b: B).(eq K k (Bind b))))))).(\lambda (_:
302 T).(\lambda (_: F).(\lambda (H3: (eq C c2 (CHead c k v))).(let H4 \def
303 (f_equal C C (\lambda (e: C).e) c2 (CHead c k v) H3) in (let H5 \def (eq_ind
304 C c2 (\lambda (c0: C).((eq C c0 (CHead c k v)) \to (ex B (\lambda (b: B).(eq
305 K k (Bind b)))))) H2 (CHead c k v) H4) in (let H6 \def (eq_ind C c2 (\lambda
306 (c0: C).(wf3 g c1 c0)) H1 (CHead c k v) H4) in (H5 (refl_equal C (CHead c k
307 v))))))))))))) x y H0))) H)))))).