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