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_1A/csubt/ty3.ma".
19 include "basic_1A/ty3/subst1.ma".
21 include "basic_1A/ty3/fsubst0.ma".
23 include "basic_1A/pc3/pc1.ma".
25 include "basic_1A/pc3/wcpr0.ma".
27 include "basic_1A/pc1/props.ma".
29 lemma ty3_sred_wcpr0_pr0:
30 \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1
31 t1 t) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1 t2)
32 \to (ty3 g c2 t2 t)))))))))
34 \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda
35 (H: (ty3 g c1 t1 t)).(ty3_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda
36 (t2: T).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t3: T).((pr0 t0 t3) \to
37 (ty3 g c2 t3 t2)))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t0:
38 T).(\lambda (_: (ty3 g c t2 t0)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c
39 c2) \to (\forall (t3: T).((pr0 t2 t3) \to (ty3 g c2 t3 t0))))))).(\lambda (u:
40 T).(\lambda (t3: T).(\lambda (_: (ty3 g c u t3)).(\lambda (H3: ((\forall (c2:
41 C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 u t4) \to (ty3 g c2 t4
42 t3))))))).(\lambda (H4: (pc3 c t3 t2)).(\lambda (c2: C).(\lambda (H5: (wcpr0
43 c c2)).(\lambda (t4: T).(\lambda (H6: (pr0 u t4)).(ty3_conv g c2 t2 t0 (H1 c2
44 H5 t2 (pr0_refl t2)) t4 t3 (H3 c2 H5 t4 H6) (pc3_wcpr0 c c2 H5 t3 t2
45 H4)))))))))))))))) (\lambda (c: C).(\lambda (m: nat).(\lambda (c2:
46 C).(\lambda (_: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H1: (pr0 (TSort m)
47 t2)).(eq_ind_r T (TSort m) (\lambda (t0: T).(ty3 g c2 t0 (TSort (next g m))))
48 (ty3_sort g c2 m) t2 (pr0_gen_sort t2 m H1)))))))) (\lambda (n: nat).(\lambda
49 (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind
50 Abbr) u))).(\lambda (t0: T).(\lambda (_: (ty3 g d u t0)).(\lambda (H2:
51 ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: T).((pr0 u t2) \to (ty3 g
52 c2 t2 t0))))))).(\lambda (c2: C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2:
53 T).(\lambda (H4: (pr0 (TLRef n) t2)).(eq_ind_r T (TLRef n) (\lambda (t3:
54 T).(ty3 g c2 t3 (lift (S n) O t0))) (ex3_2_ind C T (\lambda (e2: C).(\lambda
55 (u2: T).(getl n c2 (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_:
56 T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u u2))) (ty3 g c2
57 (TLRef n) (lift (S n) O t0)) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5:
58 (getl n c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda
59 (H7: (pr0 u x1)).(ty3_abbr g n c2 x0 x1 H5 t0 (H2 x0 H6 x1 H7)))))))
60 (wcpr0_getl c c2 H3 n d u (Bind Abbr) H0)) t2 (pr0_gen_lref t2 n
61 H4)))))))))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda
62 (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abst) u))).(\lambda (t0:
63 T).(\lambda (_: (ty3 g d u t0)).(\lambda (H2: ((\forall (c2: C).((wcpr0 d c2)
64 \to (\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (c2:
65 C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef n)
66 t2)).(eq_ind_r T (TLRef n) (\lambda (t3: T).(ty3 g c2 t3 (lift (S n) O u)))
67 (ex3_2_ind C T (\lambda (e2: C).(\lambda (u2: T).(getl n c2 (CHead e2 (Bind
68 Abst) u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_:
69 C).(\lambda (u2: T).(pr0 u u2))) (ty3 g c2 (TLRef n) (lift (S n) O u))
70 (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (getl n c2 (CHead x0 (Bind
71 Abst) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda (H7: (pr0 u x1)).(ty3_conv g
72 c2 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g x0 u t0 (H2 x0 H6 u
73 (pr0_refl u)) c2 O (S n) (getl_drop Abst c2 x0 x1 n H5)) (TLRef n) (lift (S
74 n) O x1) (ty3_abst g n c2 x0 x1 H5 t0 (H2 x0 H6 x1 H7)) (pc3_lift c2 x0 (S n)
75 O (getl_drop Abst c2 x0 x1 n H5) x1 u (pc3_pr2_x x0 x1 u (pr2_free x0 u x1
76 H7))))))))) (wcpr0_getl c c2 H3 n d u (Bind Abst) H0)) t2 (pr0_gen_lref t2 n
77 H4)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t0: T).(\lambda
78 (_: (ty3 g c u t0)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to
79 (\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (b:
80 B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2: (ty3 g (CHead c (Bind b)
81 u) t2 t3)).(\lambda (H3: ((\forall (c2: C).((wcpr0 (CHead c (Bind b) u) c2)
82 \to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda (c2:
83 C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t4: T).(\lambda (H5: (pr0 (THead
84 (Bind b) u t2) t4)).(let H6 \def (match H5 with [(pr0_refl t5) \Rightarrow
85 (\lambda (H6: (eq T t5 (THead (Bind b) u t2))).(\lambda (H7: (eq T t5
86 t4)).(eq_ind T (THead (Bind b) u t2) (\lambda (t6: T).((eq T t6 t4) \to (ty3
87 g c2 t4 (THead (Bind b) u t3)))) (\lambda (H8: (eq T (THead (Bind b) u t2)
88 t4)).(eq_ind T (THead (Bind b) u t2) (\lambda (t6: T).(ty3 g c2 t6 (THead
89 (Bind b) u t3))) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) b t2 t3 (H3
90 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) t2
91 (pr0_refl t2))) t4 H8)) t5 (sym_eq T t5 (THead (Bind b) u t2) H6) H7))) |
92 (pr0_comp u1 u2 H6 t5 t6 H7 k) \Rightarrow (\lambda (H8: (eq T (THead k u1
93 t5) (THead (Bind b) u t2))).(\lambda (H9: (eq T (THead k u2 t6) t4)).((let
94 H10 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t5
95 | (TLRef _) \Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t5)
96 (THead (Bind b) u t2) H8) in ((let H11 \def (f_equal T T (\lambda (e:
97 T).(match e with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 |
98 (THead _ t7 _) \Rightarrow t7])) (THead k u1 t5) (THead (Bind b) u t2) H8) in
99 ((let H12 \def (f_equal T K (\lambda (e: T).(match e with [(TSort _)
100 \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0]))
101 (THead k u1 t5) (THead (Bind b) u t2) H8) in (eq_ind K (Bind b) (\lambda (k0:
102 K).((eq T u1 u) \to ((eq T t5 t2) \to ((eq T (THead k0 u2 t6) t4) \to ((pr0
103 u1 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))))
104 (\lambda (H13: (eq T u1 u)).(eq_ind T u (\lambda (t7: T).((eq T t5 t2) \to
105 ((eq T (THead (Bind b) u2 t6) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3
106 g c2 t4 (THead (Bind b) u t3))))))) (\lambda (H14: (eq T t5 t2)).(eq_ind T t2
107 (\lambda (t7: T).((eq T (THead (Bind b) u2 t6) t4) \to ((pr0 u u2) \to ((pr0
108 t7 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))))) (\lambda (H15: (eq T
109 (THead (Bind b) u2 t6) t4)).(eq_ind T (THead (Bind b) u2 t6) (\lambda (t7:
110 T).((pr0 u u2) \to ((pr0 t2 t6) \to (ty3 g c2 t7 (THead (Bind b) u t3)))))
111 (\lambda (H16: (pr0 u u2)).(\lambda (H17: (pr0 t2 t6)).(ex_ind T (\lambda
112 (t7: T).(ty3 g (CHead c2 (Bind b) u) t3 t7)) (ty3 g c2 (THead (Bind b) u2 t6)
113 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H18: (ty3 g (CHead c2 (Bind
114 b) u) t3 x)).(ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind b) u2) t3 t7))
115 (ty3 g c2 (THead (Bind b) u2 t6) (THead (Bind b) u t3)) (\lambda (x0:
116 T).(\lambda (_: (ty3 g (CHead c2 (Bind b) u2) t3 x0)).(ty3_conv g c2 (THead
117 (Bind b) u t3) (THead (Bind b) u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl
118 u)) b t3 x H18) (THead (Bind b) u2 t6) (THead (Bind b) u2 t3) (ty3_bind g c2
119 u2 t0 (H1 c2 H4 u2 H16) b t6 t3 (H3 (CHead c2 (Bind b) u2) (wcpr0_comp c c2
120 H4 u u2 H16 (Bind b)) t6 H17)) (pc3_pr2_x c2 (THead (Bind b) u2 t3) (THead
121 (Bind b) u t3) (pr2_head_1 c2 u u2 (pr2_free c2 u u2 H16) (Bind b) t3)))))
122 (ty3_correct g (CHead c2 (Bind b) u2) t6 t3 (H3 (CHead c2 (Bind b) u2)
123 (wcpr0_comp c c2 H4 u u2 H16 (Bind b)) t6 H17))))) (ty3_correct g (CHead c2
124 (Bind b) u) t2 t3 (H3 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u (pr0_refl
125 u) (Bind b)) t2 (pr0_refl t2)))))) t4 H15)) t5 (sym_eq T t5 t2 H14))) u1
126 (sym_eq T u1 u H13))) k (sym_eq K k (Bind b) H12))) H11)) H10)) H9 H6 H7))) |
127 (pr0_beta u0 v1 v2 H6 t5 t6 H7) \Rightarrow (\lambda (H8: (eq T (THead (Flat
128 Appl) v1 (THead (Bind Abst) u0 t5)) (THead (Bind b) u t2))).(\lambda (H9: (eq
129 T (THead (Bind Abbr) v2 t6) t4)).((let H10 \def (eq_ind T (THead (Flat Appl)
130 v1 (THead (Bind Abst) u0 t5)) (\lambda (e: T).(match e with [(TSort _)
131 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
132 (match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I
133 (THead (Bind b) u t2) H8) in (False_ind ((eq T (THead (Bind Abbr) v2 t6) t4)
134 \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))))
135 H10)) H9 H6 H7))) | (pr0_upsilon b0 H6 v1 v2 H7 u1 u2 H8 t5 t6 H9)
136 \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1 (THead (Bind b0) u1
137 t5)) (THead (Bind b) u t2))).(\lambda (H11: (eq T (THead (Bind b0) u2 (THead
138 (Flat Appl) (lift (S O) O v2) t6)) t4)).((let H12 \def (eq_ind T (THead (Flat
139 Appl) v1 (THead (Bind b0) u1 t5)) (\lambda (e: T).(match e with [(TSort _)
140 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
141 (match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I
142 (THead (Bind b) u t2) H10) in (False_ind ((eq T (THead (Bind b0) u2 (THead
143 (Flat Appl) (lift (S O) O v2) t6)) t4) \to ((not (eq B b0 Abst)) \to ((pr0 v1
144 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u
145 t3))))))) H12)) H11 H6 H7 H8 H9))) | (pr0_delta u1 u2 H6 t5 t6 H7 w H8)
146 \Rightarrow (\lambda (H9: (eq T (THead (Bind Abbr) u1 t5) (THead (Bind b) u
147 t2))).(\lambda (H10: (eq T (THead (Bind Abbr) u2 w) t4)).((let H11 \def
148 (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t5 | (TLRef
149 _) \Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1
150 t5) (THead (Bind b) u t2) H9) in ((let H12 \def (f_equal T T (\lambda (e:
151 T).(match e with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 |
152 (THead _ t7 _) \Rightarrow t7])) (THead (Bind Abbr) u1 t5) (THead (Bind b) u
153 t2) H9) in ((let H13 \def (f_equal T B (\lambda (e: T).(match e with [(TSort
154 _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow
155 (match k with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])]))
156 (THead (Bind Abbr) u1 t5) (THead (Bind b) u t2) H9) in (eq_ind B Abbr
157 (\lambda (b0: B).((eq T u1 u) \to ((eq T t5 t2) \to ((eq T (THead (Bind Abbr)
158 u2 w) t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ty3
159 g c2 t4 (THead (Bind b0) u t3))))))))) (\lambda (H14: (eq T u1 u)).(eq_ind T
160 u (\lambda (t7: T).((eq T t5 t2) \to ((eq T (THead (Bind Abbr) u2 w) t4) \to
161 ((pr0 t7 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2 t4 (THead
162 (Bind Abbr) u t3)))))))) (\lambda (H15: (eq T t5 t2)).(eq_ind T t2 (\lambda
163 (t7: T).((eq T (THead (Bind Abbr) u2 w) t4) \to ((pr0 u u2) \to ((pr0 t7 t6)
164 \to ((subst0 O u2 t6 w) \to (ty3 g c2 t4 (THead (Bind Abbr) u t3)))))))
165 (\lambda (H16: (eq T (THead (Bind Abbr) u2 w) t4)).(eq_ind T (THead (Bind
166 Abbr) u2 w) (\lambda (t7: T).((pr0 u u2) \to ((pr0 t2 t6) \to ((subst0 O u2
167 t6 w) \to (ty3 g c2 t7 (THead (Bind Abbr) u t3)))))) (\lambda (H17: (pr0 u
168 u2)).(\lambda (H18: (pr0 t2 t6)).(\lambda (H19: (subst0 O u2 t6 w)).(let H20
169 \def (eq_ind_r B b (\lambda (b0: B).(\forall (c3: C).((wcpr0 (CHead c (Bind
170 b0) u) c3) \to (\forall (t7: T).((pr0 t2 t7) \to (ty3 g c3 t7 t3)))))) H3
171 Abbr H13) in (let H21 \def (eq_ind_r B b (\lambda (b0: B).(ty3 g (CHead c
172 (Bind b0) u) t2 t3)) H2 Abbr H13) in (ex_ind T (\lambda (t7: T).(ty3 g (CHead
173 c2 (Bind Abbr) u) t3 t7)) (ty3 g c2 (THead (Bind Abbr) u2 w) (THead (Bind
174 Abbr) u t3)) (\lambda (x: T).(\lambda (H22: (ty3 g (CHead c2 (Bind Abbr) u)
175 t3 x)).(ex_ind T (\lambda (t7: T).(ty3 g (CHead c2 (Bind Abbr) u2) t3 t7))
176 (ty3 g c2 (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u t3)) (\lambda (x0:
177 T).(\lambda (_: (ty3 g (CHead c2 (Bind Abbr) u2) t3 x0)).(ty3_conv g c2
178 (THead (Bind Abbr) u t3) (THead (Bind Abbr) u x) (ty3_bind g c2 u t0 (H1 c2
179 H4 u (pr0_refl u)) Abbr t3 x H22) (THead (Bind Abbr) u2 w) (THead (Bind Abbr)
180 u2 t3) (ty3_bind g c2 u2 t0 (H1 c2 H4 u2 H17) Abbr w t3 (ty3_subst0 g (CHead
181 c2 (Bind Abbr) u2) t6 t3 (H20 (CHead c2 (Bind Abbr) u2) (wcpr0_comp c c2 H4 u
182 u2 H17 (Bind Abbr)) t6 H18) c2 u2 O (getl_refl Abbr c2 u2) w H19)) (pc3_pr2_x
183 c2 (THead (Bind Abbr) u2 t3) (THead (Bind Abbr) u t3) (pr2_head_1 c2 u u2
184 (pr2_free c2 u u2 H17) (Bind Abbr) t3))))) (ty3_correct g (CHead c2 (Bind
185 Abbr) u2) t6 t3 (H20 (CHead c2 (Bind Abbr) u2) (wcpr0_comp c c2 H4 u u2 H17
186 (Bind Abbr)) t6 H18))))) (ty3_correct g (CHead c2 (Bind Abbr) u) t2 t3 (H20
187 (CHead c2 (Bind Abbr) u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind Abbr)) t2
188 (pr0_refl t2))))))))) t4 H16)) t5 (sym_eq T t5 t2 H15))) u1 (sym_eq T u1 u
189 H14))) b H13)) H12)) H11)) H10 H6 H7 H8))) | (pr0_zeta b0 H6 t5 t6 H7 u0)
190 \Rightarrow (\lambda (H8: (eq T (THead (Bind b0) u0 (lift (S O) O t5)) (THead
191 (Bind b) u t2))).(\lambda (H9: (eq T t6 t4)).((let H10 \def (f_equal T T
192 (\lambda (e: T).(match e with [(TSort _) \Rightarrow (lref_map (\lambda (x:
193 nat).(plus x (S O))) O t5) | (TLRef _) \Rightarrow (lref_map (\lambda (x:
194 nat).(plus x (S O))) O t5) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind
195 b0) u0 (lift (S O) O t5)) (THead (Bind b) u t2) H8) in ((let H11 \def
196 (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u0 | (TLRef
197 _) \Rightarrow u0 | (THead _ t7 _) \Rightarrow t7])) (THead (Bind b0) u0
198 (lift (S O) O t5)) (THead (Bind b) u t2) H8) in ((let H12 \def (f_equal T B
199 (\lambda (e: T).(match e with [(TSort _) \Rightarrow b0 | (TLRef _)
200 \Rightarrow b0 | (THead k _ _) \Rightarrow (match k with [(Bind b1)
201 \Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0 (lift (S O)
202 O t5)) (THead (Bind b) u t2) H8) in (eq_ind B b (\lambda (b1: B).((eq T u0 u)
203 \to ((eq T (lift (S O) O t5) t2) \to ((eq T t6 t4) \to ((not (eq B b1 Abst))
204 \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3)))))))) (\lambda (H13:
205 (eq T u0 u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t5) t2) \to
206 ((eq T t6 t4) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4
207 (THead (Bind b) u t3))))))) (\lambda (H14: (eq T (lift (S O) O t5)
208 t2)).(eq_ind T (lift (S O) O t5) (\lambda (_: T).((eq T t6 t4) \to ((not (eq
209 B b Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Bind b) u t3))))))
210 (\lambda (H15: (eq T t6 t4)).(eq_ind T t4 (\lambda (t7: T).((not (eq B b
211 Abst)) \to ((pr0 t5 t7) \to (ty3 g c2 t4 (THead (Bind b) u t3))))) (\lambda
212 (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 t5 t4)).(let H18 \def
213 (eq_ind_r T t2 (\lambda (t7: T).(\forall (c3: C).((wcpr0 (CHead c (Bind b) u)
214 c3) \to (\forall (t8: T).((pr0 t7 t8) \to (ty3 g c3 t8 t3)))))) H3 (lift (S
215 O) O t5) H14) in (let H19 \def (eq_ind_r T t2 (\lambda (t7: T).(ty3 g (CHead
216 c (Bind b) u) t7 t3)) H2 (lift (S O) O t5) H14) in (ex_ind T (\lambda (t7:
217 T).(ty3 g (CHead c2 (Bind b) u) t3 t7)) (ty3 g c2 t4 (THead (Bind b) u t3))
218 (\lambda (x: T).(\lambda (H20: (ty3 g (CHead c2 (Bind b) u) t3 x)).(B_ind
219 (\lambda (b1: B).((not (eq B b1 Abst)) \to ((ty3 g (CHead c2 (Bind b1) u) t3
220 x) \to ((ty3 g (CHead c2 (Bind b1) u) (lift (S O) O t4) t3) \to (ty3 g c2 t4
221 (THead (Bind b1) u t3)))))) (\lambda (H21: (not (eq B Abbr Abst))).(\lambda
222 (H22: (ty3 g (CHead c2 (Bind Abbr) u) t3 x)).(\lambda (H23: (ty3 g (CHead c2
223 (Bind Abbr) u) (lift (S O) O t4) t3)).(let H24 \def (ty3_gen_cabbr g (CHead
224 c2 (Bind Abbr) u) (lift (S O) O t4) t3 H23 c2 u O (getl_refl Abbr c2 u)
225 (CHead c2 (Bind Abbr) u) (csubst1_refl O u (CHead c2 (Bind Abbr) u)) c2
226 (drop_drop (Bind Abbr) O c2 c2 (drop_refl c2) u)) in (ex3_2_ind T T (\lambda
227 (y1: T).(\lambda (_: T).(subst1 O u (lift (S O) O t4) (lift (S O) O y1))))
228 (\lambda (_: T).(\lambda (y2: T).(subst1 O u t3 (lift (S O) O y2)))) (\lambda
229 (y1: T).(\lambda (y2: T).(ty3 g c2 y1 y2))) (ty3 g c2 t4 (THead (Bind Abbr) u
230 t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H25: (subst1 O u (lift (S O)
231 O t4) (lift (S O) O x0))).(\lambda (H26: (subst1 O u t3 (lift (S O) O
232 x1))).(\lambda (H27: (ty3 g c2 x0 x1)).(let H28 \def (eq_ind T x0 (\lambda
233 (t7: T).(ty3 g c2 t7 x1)) H27 t4 (lift_inj x0 t4 (S O) O (subst1_gen_lift_eq
234 t4 u (lift (S O) O x0) (S O) O O (le_O_n O) (eq_ind_r nat (plus (S O) O)
235 (\lambda (n: nat).(lt O n)) (le_n (plus (S O) O)) (plus O (S O)) (plus_sym O
236 (S O))) H25))) in (ty3_conv g c2 (THead (Bind Abbr) u t3) (THead (Bind Abbr)
237 u x) (ty3_bind g c2 u t0 (H1 c2 H4 u (pr0_refl u)) Abbr t3 x H22) t4 x1 H28
238 (pc3_pr3_x c2 x1 (THead (Bind Abbr) u t3) (pr3_t (THead (Bind Abbr) u (lift
239 (S O) O x1)) (THead (Bind Abbr) u t3) c2 (pr3_pr2 c2 (THead (Bind Abbr) u t3)
240 (THead (Bind Abbr) u (lift (S O) O x1)) (pr2_free c2 (THead (Bind Abbr) u t3)
241 (THead (Bind Abbr) u (lift (S O) O x1)) (pr0_delta1 u u (pr0_refl u) t3 t3
242 (pr0_refl t3) (lift (S O) O x1) H26))) x1 (pr3_pr2 c2 (THead (Bind Abbr) u
243 (lift (S O) O x1)) x1 (pr2_free c2 (THead (Bind Abbr) u (lift (S O) O x1)) x1
244 (pr0_zeta Abbr H21 x1 x1 (pr0_refl x1) u)))))))))))) H24))))) (\lambda (H21:
245 (not (eq B Abst Abst))).(\lambda (_: (ty3 g (CHead c2 (Bind Abst) u) t3
246 x)).(\lambda (_: (ty3 g (CHead c2 (Bind Abst) u) (lift (S O) O t4) t3)).(let
247 H24 \def (match (H21 (refl_equal B Abst)) in False with []) in H24))))
248 (\lambda (H21: (not (eq B Void Abst))).(\lambda (H22: (ty3 g (CHead c2 (Bind
249 Void) u) t3 x)).(\lambda (H23: (ty3 g (CHead c2 (Bind Void) u) (lift (S O) O
250 t4) t3)).(let H24 \def (ty3_gen_cvoid g (CHead c2 (Bind Void) u) (lift (S O)
251 O t4) t3 H23 c2 u O (getl_refl Void c2 u) c2 (drop_drop (Bind Void) O c2 c2
252 (drop_refl c2) u)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T
253 (lift (S O) O t4) (lift (S O) O y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T
254 t3 (lift (S O) O y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g c2 y1 y2)))
255 (ty3 g c2 t4 (THead (Bind Void) u t3)) (\lambda (x0: T).(\lambda (x1:
256 T).(\lambda (H25: (eq T (lift (S O) O t4) (lift (S O) O x0))).(\lambda (H26:
257 (eq T t3 (lift (S O) O x1))).(\lambda (H27: (ty3 g c2 x0 x1)).(let H28 \def
258 (eq_ind T t3 (\lambda (t7: T).(ty3 g (CHead c2 (Bind Void) u) t7 x)) H22
259 (lift (S O) O x1) H26) in (eq_ind_r T (lift (S O) O x1) (\lambda (t7: T).(ty3
260 g c2 t4 (THead (Bind Void) u t7))) (let H29 \def (eq_ind_r T x0 (\lambda (t7:
261 T).(ty3 g c2 t7 x1)) H27 t4 (lift_inj t4 x0 (S O) O H25)) in (ty3_conv g c2
262 (THead (Bind Void) u (lift (S O) O x1)) (THead (Bind Void) u x) (ty3_bind g
263 c2 u t0 (H1 c2 H4 u (pr0_refl u)) Void (lift (S O) O x1) x H28) t4 x1 H29
264 (pc3_s c2 x1 (THead (Bind Void) u (lift (S O) O x1)) (pc3_pr2_r c2 (THead
265 (Bind Void) u (lift (S O) O x1)) x1 (pr2_free c2 (THead (Bind Void) u (lift
266 (S O) O x1)) x1 (pr0_zeta Void H21 x1 x1 (pr0_refl x1) u)))))) t3 H26)))))))
267 H24))))) b H16 H20 (H18 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H4 u u
268 (pr0_refl u) (Bind b)) (lift (S O) O t4) (pr0_lift t5 t4 H17 (S O) O)))))
269 (ty3_correct g (CHead c2 (Bind b) u) (lift (S O) O t4) t3 (H18 (CHead c2
270 (Bind b) u) (wcpr0_comp c c2 H4 u u (pr0_refl u) (Bind b)) (lift (S O) O t4)
271 (pr0_lift t5 t4 H17 (S O) O)))))))) t6 (sym_eq T t6 t4 H15))) t2 H14)) u0
272 (sym_eq T u0 u H13))) b0 (sym_eq B b0 b H12))) H11)) H10)) H9 H6 H7))) |
273 (pr0_tau t5 t6 H6 u0) \Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0
274 t5) (THead (Bind b) u t2))).(\lambda (H8: (eq T t6 t4)).((let H9 \def (eq_ind
275 T (THead (Flat Cast) u0 t5) (\lambda (e: T).(match e with [(TSort _)
276 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
277 (match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I
278 (THead (Bind b) u t2) H7) in (False_ind ((eq T t6 t4) \to ((pr0 t5 t6) \to
279 (ty3 g c2 t4 (THead (Bind b) u t3)))) H9)) H8 H6)))]) in (H6 (refl_equal T
280 (THead (Bind b) u t2)) (refl_equal T t4))))))))))))))))) (\lambda (c:
281 C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c w u)).(\lambda (H1:
282 ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 w t2) \to (ty3 g
283 c2 t2 u))))))).(\lambda (v: T).(\lambda (t0: T).(\lambda (H2: (ty3 g c v
284 (THead (Bind Abst) u t0))).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) \to
285 (\forall (t2: T).((pr0 v t2) \to (ty3 g c2 t2 (THead (Bind Abst) u
286 t0)))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t2:
287 T).(\lambda (H5: (pr0 (THead (Flat Appl) w v) t2)).(let H6 \def (match H5
288 with [(pr0_refl t3) \Rightarrow (\lambda (H6: (eq T t3 (THead (Flat Appl) w
289 v))).(\lambda (H7: (eq T t3 t2)).(eq_ind T (THead (Flat Appl) w v) (\lambda
290 (t4: T).((eq T t4 t2) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind
291 Abst) u t0))))) (\lambda (H8: (eq T (THead (Flat Appl) w v) t2)).(eq_ind T
292 (THead (Flat Appl) w v) (\lambda (t4: T).(ty3 g c2 t4 (THead (Flat Appl) w
293 (THead (Bind Abst) u t0)))) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) v t0
294 (H3 c2 H4 v (pr0_refl v))) t2 H8)) t3 (sym_eq T t3 (THead (Flat Appl) w v)
295 H6) H7))) | (pr0_comp u1 u2 H6 t3 t4 H7 k) \Rightarrow (\lambda (H8: (eq T
296 (THead k u1 t3) (THead (Flat Appl) w v))).(\lambda (H9: (eq T (THead k u2 t4)
297 t2)).((let H10 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _)
298 \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t5) \Rightarrow t5]))
299 (THead k u1 t3) (THead (Flat Appl) w v) H8) in ((let H11 \def (f_equal T T
300 (\lambda (e: T).(match e with [(TSort _) \Rightarrow u1 | (TLRef _)
301 \Rightarrow u1 | (THead _ t5 _) \Rightarrow t5])) (THead k u1 t3) (THead
302 (Flat Appl) w v) H8) in ((let H12 \def (f_equal T K (\lambda (e: T).(match e
303 with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _)
304 \Rightarrow k0])) (THead k u1 t3) (THead (Flat Appl) w v) H8) in (eq_ind K
305 (Flat Appl) (\lambda (k0: K).((eq T u1 w) \to ((eq T t3 v) \to ((eq T (THead
306 k0 u2 t4) t2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat
307 Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H13: (eq T u1 w)).(eq_ind
308 T w (\lambda (t5: T).((eq T t3 v) \to ((eq T (THead (Flat Appl) u2 t4) t2)
309 \to ((pr0 t5 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w
310 (THead (Bind Abst) u t0)))))))) (\lambda (H14: (eq T t3 v)).(eq_ind T v
311 (\lambda (t5: T).((eq T (THead (Flat Appl) u2 t4) t2) \to ((pr0 w u2) \to
312 ((pr0 t5 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u
313 t0))))))) (\lambda (H15: (eq T (THead (Flat Appl) u2 t4) t2)).(eq_ind T
314 (THead (Flat Appl) u2 t4) (\lambda (t5: T).((pr0 w u2) \to ((pr0 v t4) \to
315 (ty3 g c2 t5 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))) (\lambda
316 (H16: (pr0 w u2)).(\lambda (H17: (pr0 v t4)).(ex_ind T (\lambda (t5: T).(ty3
317 g c2 (THead (Bind Abst) u t0) t5)) (ty3 g c2 (THead (Flat Appl) u2 t4) (THead
318 (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H18: (ty3
319 g c2 (THead (Bind Abst) u t0) x)).(ex3_2_ind T T (\lambda (t5: T).(\lambda
320 (_: T).(pc3 c2 (THead (Bind Abst) u t5) x))) (\lambda (_: T).(\lambda (t6:
321 T).(ty3 g c2 u t6))) (\lambda (t5: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind
322 Abst) u) t0 t5))) (ty3 g c2 (THead (Flat Appl) u2 t4) (THead (Flat Appl) w
323 (THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_:
324 (pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H20: (ty3 g c2 u x1)).(\lambda
325 (H21: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(ty3_conv g c2 (THead (Flat
326 Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u
327 x0)) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) (THead (Bind Abst) u t0) x0
328 (ty3_bind g c2 u x1 H20 Abst t0 x0 H21)) (THead (Flat Appl) u2 t4) (THead
329 (Flat Appl) u2 (THead (Bind Abst) u t0)) (ty3_appl g c2 u2 u (H1 c2 H4 u2
330 H16) t4 t0 (H3 c2 H4 t4 H17)) (pc3_pr2_x c2 (THead (Flat Appl) u2 (THead
331 (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pr2_head_1
332 c2 w u2 (pr2_free c2 w u2 H16) (Flat Appl) (THead (Bind Abst) u t0)))))))))
333 (ty3_gen_bind g Abst c2 u t0 x H18)))) (ty3_correct g c2 v (THead (Bind Abst)
334 u t0) (H3 c2 H4 v (pr0_refl v)))))) t2 H15)) t3 (sym_eq T t3 v H14))) u1
335 (sym_eq T u1 w H13))) k (sym_eq K k (Flat Appl) H12))) H11)) H10)) H9 H6
336 H7))) | (pr0_beta u0 v1 v2 H6 t3 t4 H7) \Rightarrow (\lambda (H8: (eq T
337 (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3)) (THead (Flat Appl) w
338 v))).(\lambda (H9: (eq T (THead (Bind Abbr) v2 t4) t2)).((let H10 \def
339 (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow (THead
340 (Bind Abst) u0 t3) | (TLRef _) \Rightarrow (THead (Bind Abst) u0 t3) | (THead
341 _ _ t5) \Rightarrow t5])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t3))
342 (THead (Flat Appl) w v) H8) in ((let H11 \def (f_equal T T (\lambda (e:
343 T).(match e with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 |
344 (THead _ t5 _) \Rightarrow t5])) (THead (Flat Appl) v1 (THead (Bind Abst) u0
345 t3)) (THead (Flat Appl) w v) H8) in (eq_ind T w (\lambda (t5: T).((eq T
346 (THead (Bind Abst) u0 t3) v) \to ((eq T (THead (Bind Abbr) v2 t4) t2) \to
347 ((pr0 t5 v2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead
348 (Bind Abst) u t0)))))))) (\lambda (H12: (eq T (THead (Bind Abst) u0 t3)
349 v)).(eq_ind T (THead (Bind Abst) u0 t3) (\lambda (_: T).((eq T (THead (Bind
350 Abbr) v2 t4) t2) \to ((pr0 w v2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead
351 (Flat Appl) w (THead (Bind Abst) u t0))))))) (\lambda (H13: (eq T (THead
352 (Bind Abbr) v2 t4) t2)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda (t5:
353 T).((pr0 w v2) \to ((pr0 t3 t4) \to (ty3 g c2 t5 (THead (Flat Appl) w (THead
354 (Bind Abst) u t0)))))) (\lambda (H14: (pr0 w v2)).(\lambda (H15: (pr0 t3
355 t4)).(let H16 \def (eq_ind_r T v (\lambda (t5: T).(\forall (c3: C).((wcpr0 c
356 c3) \to (\forall (t6: T).((pr0 t5 t6) \to (ty3 g c3 t6 (THead (Bind Abst) u
357 t0))))))) H3 (THead (Bind Abst) u0 t3) H12) in (let H17 \def (eq_ind_r T v
358 (\lambda (t5: T).(ty3 g c t5 (THead (Bind Abst) u t0))) H2 (THead (Bind Abst)
359 u0 t3) H12) in (ex_ind T (\lambda (t5: T).(ty3 g c2 (THead (Bind Abst) u t0)
360 t5)) (ty3 g c2 (THead (Bind Abbr) v2 t4) (THead (Flat Appl) w (THead (Bind
361 Abst) u t0))) (\lambda (x: T).(\lambda (H18: (ty3 g c2 (THead (Bind Abst) u
362 t0) x)).(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 (THead (Bind
363 Abst) u t5) x))) (\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u t6))) (\lambda
364 (t5: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t5))) (ty3 g c2
365 (THead (Bind Abbr) v2 t4) (THead (Flat Appl) w (THead (Bind Abst) u t0)))
366 (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Bind Abst) u
367 x0) x)).(\lambda (H20: (ty3 g c2 u x1)).(\lambda (H21: (ty3 g (CHead c2 (Bind
368 Abst) u) t0 x0)).(ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2
369 (THead (Bind Abst) u0 t5) (THead (Bind Abst) u t0)))) (\lambda (_:
370 T).(\lambda (t6: T).(ty3 g c2 u0 t6))) (\lambda (t5: T).(\lambda (_: T).(ty3
371 g (CHead c2 (Bind Abst) u0) t4 t5))) (ty3 g c2 (THead (Bind Abbr) v2 t4)
372 (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x2: T).(\lambda
373 (x3: T).(\lambda (H22: (pc3 c2 (THead (Bind Abst) u0 x2) (THead (Bind Abst) u
374 t0))).(\lambda (H23: (ty3 g c2 u0 x3)).(\lambda (H24: (ty3 g (CHead c2 (Bind
375 Abst) u0) t4 x2)).(land_ind (pc3 c2 u0 u) (\forall (b: B).(\forall (u1:
376 T).(pc3 (CHead c2 (Bind b) u1) x2 t0))) (ty3 g c2 (THead (Bind Abbr) v2 t4)
377 (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H25: (pc3 c2 u0
378 u)).(\lambda (H26: ((\forall (b: B).(\forall (u1: T).(pc3 (CHead c2 (Bind b)
379 u1) x2 t0))))).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0))
380 (THead (Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w
381 (pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H20 Abst t0 x0
382 H21)) (THead (Bind Abbr) v2 t4) (THead (Bind Abbr) v2 x2) (ty3_bind g c2 v2 u
383 (H1 c2 H4 v2 H14) Abbr t4 x2 (csubt_ty3_ld g c2 v2 u0 (ty3_conv g c2 u0 x3
384 H23 v2 u (H1 c2 H4 v2 H14) (pc3_s c2 u u0 H25)) t4 x2 H24)) (pc3_t (THead
385 (Bind Abbr) v2 t0) c2 (THead (Bind Abbr) v2 x2) (pc3_head_2 c2 v2 x2 t0 (Bind
386 Abbr) (H26 Abbr v2)) (THead (Flat Appl) w (THead (Bind Abst) u t0))
387 (pc3_pr2_x c2 (THead (Bind Abbr) v2 t0) (THead (Flat Appl) w (THead (Bind
388 Abst) u t0)) (pr2_free c2 (THead (Flat Appl) w (THead (Bind Abst) u t0))
389 (THead (Bind Abbr) v2 t0) (pr0_beta u w v2 H14 t0 t0 (pr0_refl t0))))))))
390 (pc3_gen_abst c2 u0 u x2 t0 H22))))))) (ty3_gen_bind g Abst c2 u0 t4 (THead
391 (Bind Abst) u t0) (H16 c2 H4 (THead (Bind Abst) u0 t4) (pr0_comp u0 u0
392 (pr0_refl u0) t3 t4 H15 (Bind Abst)))))))))) (ty3_gen_bind g Abst c2 u t0 x
393 H18)))) (ty3_correct g c2 (THead (Bind Abst) u0 t3) (THead (Bind Abst) u t0)
394 (H16 c2 H4 (THead (Bind Abst) u0 t3) (pr0_refl (THead (Bind Abst) u0
395 t3))))))))) t2 H13)) v H12)) v1 (sym_eq T v1 w H11))) H10)) H9 H6 H7))) |
396 (pr0_upsilon b H6 v1 v2 H7 u1 u2 H8 t3 t4 H9) \Rightarrow (\lambda (H10: (eq
397 T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) w
398 v))).(\lambda (H11: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
399 v2) t4)) t2)).((let H12 \def (f_equal T T (\lambda (e: T).(match e with
400 [(TSort _) \Rightarrow (THead (Bind b) u1 t3) | (TLRef _) \Rightarrow (THead
401 (Bind b) u1 t3) | (THead _ _ t5) \Rightarrow t5])) (THead (Flat Appl) v1
402 (THead (Bind b) u1 t3)) (THead (Flat Appl) w v) H10) in ((let H13 \def
403 (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow v1 | (TLRef
404 _) \Rightarrow v1 | (THead _ t5 _) \Rightarrow t5])) (THead (Flat Appl) v1
405 (THead (Bind b) u1 t3)) (THead (Flat Appl) w v) H10) in (eq_ind T w (\lambda
406 (t5: T).((eq T (THead (Bind b) u1 t3) v) \to ((eq T (THead (Bind b) u2 (THead
407 (Flat Appl) (lift (S O) O v2) t4)) t2) \to ((not (eq B b Abst)) \to ((pr0 t5
408 v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w
409 (THead (Bind Abst) u t0)))))))))) (\lambda (H14: (eq T (THead (Bind b) u1 t3)
410 v)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (_: T).((eq T (THead (Bind b)
411 u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2) \to ((not (eq B b Abst)) \to
412 ((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat
413 Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H15: (eq T (THead (Bind b)
414 u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2)).(eq_ind T (THead (Bind b)
415 u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (\lambda (t5: T).((not (eq B b
416 Abst)) \to ((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ty3 g c2 t5
417 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) (\lambda (H16: (not (eq
418 B b Abst))).(\lambda (H17: (pr0 w v2)).(\lambda (H18: (pr0 u1 u2)).(\lambda
419 (H19: (pr0 t3 t4)).(let H20 \def (eq_ind_r T v (\lambda (t5: T).(\forall (c3:
420 C).((wcpr0 c c3) \to (\forall (t6: T).((pr0 t5 t6) \to (ty3 g c3 t6 (THead
421 (Bind Abst) u t0))))))) H3 (THead (Bind b) u1 t3) H14) in (let H21 \def
422 (eq_ind_r T v (\lambda (t5: T).(ty3 g c t5 (THead (Bind Abst) u t0))) H2
423 (THead (Bind b) u1 t3) H14) in (ex_ind T (\lambda (t5: T).(ty3 g c2 (THead
424 (Bind Abst) u t0) t5)) (ty3 g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift
425 (S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x:
426 T).(\lambda (H22: (ty3 g c2 (THead (Bind Abst) u t0) x)).(let H23 \def H22 in
427 (ex3_2_ind T T (\lambda (t5: T).(\lambda (_: T).(pc3 c2 (THead (Bind Abst) u
428 t5) x))) (\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u t6))) (\lambda (t5:
429 T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t5))) (ty3 g c2 (THead
430 (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w
431 (THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (_:
432 (pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H25: (ty3 g c2 u x1)).(\lambda
433 (H26: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(ex3_2_ind T T (\lambda (t5:
434 T).(\lambda (_: T).(pc3 c2 (THead (Bind b) u2 t5) (THead (Bind Abst) u t0))))
435 (\lambda (_: T).(\lambda (t6: T).(ty3 g c2 u2 t6))) (\lambda (t5: T).(\lambda
436 (_: T).(ty3 g (CHead c2 (Bind b) u2) t4 t5))) (ty3 g c2 (THead (Bind b) u2
437 (THead (Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind
438 Abst) u t0))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H27: (pc3 c2 (THead
439 (Bind b) u2 x2) (THead (Bind Abst) u t0))).(\lambda (H28: (ty3 g c2 u2
440 x3)).(\lambda (H29: (ty3 g (CHead c2 (Bind b) u2) t4 x2)).(let H30 \def
441 (eq_ind T (lift (S O) O (THead (Bind Abst) u t0)) (\lambda (t5: T).(pc3
442 (CHead c2 (Bind b) u2) x2 t5)) (pc3_gen_not_abst b H16 c2 x2 t0 u2 u H27)
443 (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u
444 t0 (S O) O)) in (let H31 \def (eq_ind T (lift (S O) O (THead (Bind Abst) u
445 t0)) (\lambda (t5: T).(ty3 g (CHead c2 (Bind b) u2) t5 (lift (S O) O x)))
446 (ty3_lift g c2 (THead (Bind Abst) u t0) x H22 (CHead c2 (Bind b) u2) O (S O)
447 (drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) (THead (Bind Abst) (lift (S
448 O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u t0 (S O) O)) in (ex3_2_ind T
449 T (\lambda (t5: T).(\lambda (_: T).(pc3 (CHead c2 (Bind b) u2) (THead (Bind
450 Abst) (lift (S O) O u) t5) (lift (S O) O x)))) (\lambda (_: T).(\lambda (t6:
451 T).(ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) t6))) (\lambda (t5:
452 T).(\lambda (_: T).(ty3 g (CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S
453 O) O u)) (lift (S O) (S O) t0) t5))) (ty3 g c2 (THead (Bind b) u2 (THead
454 (Flat Appl) (lift (S O) O v2) t4)) (THead (Flat Appl) w (THead (Bind Abst) u
455 t0))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 (CHead c2 (Bind b)
456 u2) (THead (Bind Abst) (lift (S O) O u) x4) (lift (S O) O x))).(\lambda (H33:
457 (ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) x5)).(\lambda (H34: (ty3 g
458 (CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S O) O u)) (lift (S O) (S O)
459 t0) x4)).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead
460 (Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w
461 (pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H25 Abst t0 x0
462 H26)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (THead
463 (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (THead (Bind Abst) (lift (S
464 O) O u) (lift (S O) (S O) t0)))) (ty3_bind g c2 u2 x3 H28 b (THead (Flat
465 Appl) (lift (S O) O v2) t4) (THead (Flat Appl) (lift (S O) O v2) (THead (Bind
466 Abst) (lift (S O) O u) (lift (S O) (S O) t0))) (ty3_appl g (CHead c2 (Bind b)
467 u2) (lift (S O) O v2) (lift (S O) O u) (ty3_lift g c2 v2 u (H1 c2 H4 v2 H17)
468 (CHead c2 (Bind b) u2) O (S O) (drop_drop (Bind b) O c2 c2 (drop_refl c2)
469 u2)) t4 (lift (S O) (S O) t0) (ty3_conv g (CHead c2 (Bind b) u2) (THead (Bind
470 Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (THead (Bind Abst) (lift (S O)
471 O u) x4) (ty3_bind g (CHead c2 (Bind b) u2) (lift (S O) O u) x5 H33 Abst
472 (lift (S O) (S O) t0) x4 H34) t4 x2 H29 H30))) (eq_ind T (lift (S O) O (THead
473 (Bind Abst) u t0)) (\lambda (t5: T).(pc3 c2 (THead (Bind b) u2 (THead (Flat
474 Appl) (lift (S O) O v2) t5)) (THead (Flat Appl) w (THead (Bind Abst) u t0))))
475 (pc3_pc1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O)
476 O (THead (Bind Abst) u t0)))) (THead (Flat Appl) w (THead (Bind Abst) u t0))
477 (pc1_pr0_u2 (THead (Flat Appl) v2 (THead (Bind b) u2 (lift (S O) O (THead
478 (Bind Abst) u t0)))) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
479 (lift (S O) O (THead (Bind Abst) u t0)))) (pr0_upsilon b H16 v2 v2 (pr0_refl
480 v2) u2 u2 (pr0_refl u2) (lift (S O) O (THead (Bind Abst) u t0)) (lift (S O) O
481 (THead (Bind Abst) u t0)) (pr0_refl (lift (S O) O (THead (Bind Abst) u t0))))
482 (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc1_head v2 w (pc1_pr0_x v2 w
483 H17) (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u t0))) (THead (Bind
484 Abst) u t0) (pc1_pr0_r (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u
485 t0))) (THead (Bind Abst) u t0) (pr0_zeta b H16 (THead (Bind Abst) u t0)
486 (THead (Bind Abst) u t0) (pr0_refl (THead (Bind Abst) u t0)) u2)) (Flat
487 Appl))) c2) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0))
488 (lift_bind Abst u t0 (S O) O)))))))) (ty3_gen_bind g Abst (CHead c2 (Bind b)
489 u2) (lift (S O) O u) (lift (S O) (S O) t0) (lift (S O) O x) H31)))))))))
490 (ty3_gen_bind g b c2 u2 t4 (THead (Bind Abst) u t0) (H20 c2 H4 (THead (Bind
491 b) u2 t4) (pr0_comp u1 u2 H18 t3 t4 H19 (Bind b)))))))))) (ty3_gen_bind g
492 Abst c2 u t0 x H23))))) (ty3_correct g c2 (THead (Bind b) u2 t4) (THead (Bind
493 Abst) u t0) (H20 c2 H4 (THead (Bind b) u2 t4) (pr0_comp u1 u2 H18 t3 t4 H19
494 (Bind b))))))))))) t2 H15)) v H14)) v1 (sym_eq T v1 w H13))) H12)) H11 H6 H7
495 H8 H9))) | (pr0_delta u1 u2 H6 t3 t4 H7 w0 H8) \Rightarrow (\lambda (H9: (eq
496 T (THead (Bind Abbr) u1 t3) (THead (Flat Appl) w v))).(\lambda (H10: (eq T
497 (THead (Bind Abbr) u2 w0) t2)).((let H11 \def (eq_ind T (THead (Bind Abbr) u1
498 t3) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False | (TLRef _)
499 \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _)
500 \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) w v)
501 H9) in (False_ind ((eq T (THead (Bind Abbr) u2 w0) t2) \to ((pr0 u1 u2) \to
502 ((pr0 t3 t4) \to ((subst0 O u2 t4 w0) \to (ty3 g c2 t2 (THead (Flat Appl) w
503 (THead (Bind Abst) u t0))))))) H11)) H10 H6 H7 H8))) | (pr0_zeta b H6 t3 t4
504 H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Bind b) u0 (lift (S O) O t3))
505 (THead (Flat Appl) w v))).(\lambda (H9: (eq T t4 t2)).((let H10 \def (eq_ind
506 T (THead (Bind b) u0 (lift (S O) O t3)) (\lambda (e: T).(match e with [(TSort
507 _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
508 \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
509 False])])) I (THead (Flat Appl) w v) H8) in (False_ind ((eq T t4 t2) \to
510 ((not (eq B b Abst)) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl) w
511 (THead (Bind Abst) u t0)))))) H10)) H9 H6 H7))) | (pr0_tau t3 t4 H6 u0)
512 \Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0 t3) (THead (Flat Appl)
513 w v))).(\lambda (H8: (eq T t4 t2)).((let H9 \def (eq_ind T (THead (Flat Cast)
514 u0 t3) (\lambda (e: T).(match e with [(TSort _) \Rightarrow False | (TLRef _)
515 \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _)
516 \Rightarrow False | (Flat f) \Rightarrow (match f with [Appl \Rightarrow
517 False | Cast \Rightarrow True])])])) I (THead (Flat Appl) w v) H7) in
518 (False_ind ((eq T t4 t2) \to ((pr0 t3 t4) \to (ty3 g c2 t2 (THead (Flat Appl)
519 w (THead (Bind Abst) u t0))))) H9)) H8 H6)))]) in (H6 (refl_equal T (THead
520 (Flat Appl) w v)) (refl_equal T t2)))))))))))))))) (\lambda (c: C).(\lambda
521 (t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t2 t3)).(\lambda (H1:
522 ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g
523 c2 t4 t3))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c t3 t0)).(\lambda (H3:
524 ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 t3 t4) \to (ty3 g
525 c2 t4 t0))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t4:
526 T).(\lambda (H5: (pr0 (THead (Flat Cast) t3 t2) t4)).(let H6 \def (match H5
527 with [(pr0_refl t5) \Rightarrow (\lambda (H6: (eq T t5 (THead (Flat Cast) t3
528 t2))).(\lambda (H7: (eq T t5 t4)).(eq_ind T (THead (Flat Cast) t3 t2)
529 (\lambda (t6: T).((eq T t6 t4) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))
530 (\lambda (H8: (eq T (THead (Flat Cast) t3 t2) t4)).(eq_ind T (THead (Flat
531 Cast) t3 t2) (\lambda (t6: T).(ty3 g c2 t6 (THead (Flat Cast) t0 t3)))
532 (ty3_cast g c2 t2 t3 (H1 c2 H4 t2 (pr0_refl t2)) t0 (H3 c2 H4 t3 (pr0_refl
533 t3))) t4 H8)) t5 (sym_eq T t5 (THead (Flat Cast) t3 t2) H6) H7))) | (pr0_comp
534 u1 u2 H6 t5 t6 H7 k) \Rightarrow (\lambda (H8: (eq T (THead k u1 t5) (THead
535 (Flat Cast) t3 t2))).(\lambda (H9: (eq T (THead k u2 t6) t4)).((let H10 \def
536 (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t5 | (TLRef
537 _) \Rightarrow t5 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t5) (THead
538 (Flat Cast) t3 t2) H8) in ((let H11 \def (f_equal T T (\lambda (e: T).(match
539 e with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _)
540 \Rightarrow t7])) (THead k u1 t5) (THead (Flat Cast) t3 t2) H8) in ((let H12
541 \def (f_equal T K (\lambda (e: T).(match e with [(TSort _) \Rightarrow k |
542 (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t5)
543 (THead (Flat Cast) t3 t2) H8) in (eq_ind K (Flat Cast) (\lambda (k0: K).((eq
544 T u1 t3) \to ((eq T t5 t2) \to ((eq T (THead k0 u2 t6) t4) \to ((pr0 u1 u2)
545 \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))))) (\lambda
546 (H13: (eq T u1 t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t5 t2) \to ((eq T
547 (THead (Flat Cast) u2 t6) t4) \to ((pr0 t7 u2) \to ((pr0 t5 t6) \to (ty3 g c2
548 t4 (THead (Flat Cast) t0 t3))))))) (\lambda (H14: (eq T t5 t2)).(eq_ind T t2
549 (\lambda (t7: T).((eq T (THead (Flat Cast) u2 t6) t4) \to ((pr0 t3 u2) \to
550 ((pr0 t7 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))) (\lambda (H15:
551 (eq T (THead (Flat Cast) u2 t6) t4)).(eq_ind T (THead (Flat Cast) u2 t6)
552 (\lambda (t7: T).((pr0 t3 u2) \to ((pr0 t2 t6) \to (ty3 g c2 t7 (THead (Flat
553 Cast) t0 t3))))) (\lambda (H16: (pr0 t3 u2)).(\lambda (H17: (pr0 t2
554 t6)).(ex_ind T (\lambda (t7: T).(ty3 g c2 t0 t7)) (ty3 g c2 (THead (Flat
555 Cast) u2 t6) (THead (Flat Cast) t0 t3)) (\lambda (x: T).(\lambda (H18: (ty3 g
556 c2 t0 x)).(ty3_conv g c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) x t0)
557 (ty3_cast g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3)) x H18) (THead (Flat Cast) u2
558 t6) (THead (Flat Cast) t0 u2) (ty3_cast g c2 t6 u2 (ty3_conv g c2 u2 t0 (H3
559 c2 H4 u2 H16) t6 t3 (H1 c2 H4 t6 H17) (pc3_pr2_r c2 t3 u2 (pr2_free c2 t3 u2
560 H16))) t0 (H3 c2 H4 u2 H16)) (pc3_s c2 (THead (Flat Cast) t0 u2) (THead (Flat
561 Cast) t0 t3) (pc3_pr2_r c2 (THead (Flat Cast) t0 t3) (THead (Flat Cast) t0
562 u2) (pr2_thin_dx c2 t3 u2 (pr2_free c2 t3 u2 H16) t0 Cast)))))) (ty3_correct
563 g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3)))))) t4 H15)) t5 (sym_eq T t5 t2 H14)))
564 u1 (sym_eq T u1 t3 H13))) k (sym_eq K k (Flat Cast) H12))) H11)) H10)) H9 H6
565 H7))) | (pr0_beta u v1 v2 H6 t5 t6 H7) \Rightarrow (\lambda (H8: (eq T (THead
566 (Flat Appl) v1 (THead (Bind Abst) u t5)) (THead (Flat Cast) t3 t2))).(\lambda
567 (H9: (eq T (THead (Bind Abbr) v2 t6) t4)).((let H10 \def (eq_ind T (THead
568 (Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (e: T).(match e with
569 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
570 \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow
571 (match f with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead
572 (Flat Cast) t3 t2) H8) in (False_ind ((eq T (THead (Bind Abbr) v2 t6) t4) \to
573 ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3)))))
574 H10)) H9 H6 H7))) | (pr0_upsilon b H6 v1 v2 H7 u1 u2 H8 t5 t6 H9) \Rightarrow
575 (\lambda (H10: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) (THead
576 (Flat Cast) t3 t2))).(\lambda (H11: (eq T (THead (Bind b) u2 (THead (Flat
577 Appl) (lift (S O) O v2) t6)) t4)).((let H12 \def (eq_ind T (THead (Flat Appl)
578 v1 (THead (Bind b) u1 t5)) (\lambda (e: T).(match e with [(TSort _)
579 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
580 (match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f
581 with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat
582 Cast) t3 t2) H10) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl)
583 (lift (S O) O v2) t6)) t4) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to
584 ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0
585 t3))))))) H12)) H11 H6 H7 H8 H9))) | (pr0_delta u1 u2 H6 t5 t6 H7 w H8)
586 \Rightarrow (\lambda (H9: (eq T (THead (Bind Abbr) u1 t5) (THead (Flat Cast)
587 t3 t2))).(\lambda (H10: (eq T (THead (Bind Abbr) u2 w) t4)).((let H11 \def
588 (eq_ind T (THead (Bind Abbr) u1 t5) (\lambda (e: T).(match e with [(TSort _)
589 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
590 (match k with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I
591 (THead (Flat Cast) t3 t2) H9) in (False_ind ((eq T (THead (Bind Abbr) u2 w)
592 t4) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ty3 g c2
593 t4 (THead (Flat Cast) t0 t3)))))) H11)) H10 H6 H7 H8))) | (pr0_zeta b H6 t5
594 t6 H7 u) \Rightarrow (\lambda (H8: (eq T (THead (Bind b) u (lift (S O) O t5))
595 (THead (Flat Cast) t3 t2))).(\lambda (H9: (eq T t6 t4)).((let H10 \def
596 (eq_ind T (THead (Bind b) u (lift (S O) O t5)) (\lambda (e: T).(match e with
597 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
598 \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
599 False])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T t6 t4) \to
600 ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0
601 t3))))) H10)) H9 H6 H7))) | (pr0_tau t5 t6 H6 u) \Rightarrow (\lambda (H7:
602 (eq T (THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq
603 T t6 t4)).((let H9 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _)
604 \Rightarrow t5 | (TLRef _) \Rightarrow t5 | (THead _ _ t7) \Rightarrow t7]))
605 (THead (Flat Cast) u t5) (THead (Flat Cast) t3 t2) H7) in ((let H10 \def
606 (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow u | (TLRef
607 _) \Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Cast) u t5)
608 (THead (Flat Cast) t3 t2) H7) in (eq_ind T t3 (\lambda (_: T).((eq T t5 t2)
609 \to ((eq T t6 t4) \to ((pr0 t5 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0
610 t3)))))) (\lambda (H11: (eq T t5 t2)).(eq_ind T t2 (\lambda (t7: T).((eq T t6
611 t4) \to ((pr0 t7 t6) \to (ty3 g c2 t4 (THead (Flat Cast) t0 t3))))) (\lambda
612 (H12: (eq T t6 t4)).(eq_ind T t4 (\lambda (t7: T).((pr0 t2 t7) \to (ty3 g c2
613 t4 (THead (Flat Cast) t0 t3)))) (\lambda (H13: (pr0 t2 t4)).(ex_ind T
614 (\lambda (t7: T).(ty3 g c2 t0 t7)) (ty3 g c2 t4 (THead (Flat Cast) t0 t3))
615 (\lambda (x: T).(\lambda (H14: (ty3 g c2 t0 x)).(ty3_conv g c2 (THead (Flat
616 Cast) t0 t3) (THead (Flat Cast) x t0) (ty3_cast g c2 t3 t0 (H3 c2 H4 t3
617 (pr0_refl t3)) x H14) t4 t3 (H1 c2 H4 t4 H13) (pc3_pr2_x c2 t3 (THead (Flat
618 Cast) t0 t3) (pr2_free c2 (THead (Flat Cast) t0 t3) t3 (pr0_tau t3 t3
619 (pr0_refl t3) t0)))))) (ty3_correct g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3)))))
620 t6 (sym_eq T t6 t4 H12))) t5 (sym_eq T t5 t2 H11))) u (sym_eq T u t3 H10)))
621 H9)) H8 H6)))]) in (H6 (refl_equal T (THead (Flat Cast) t3 t2)) (refl_equal T
622 t4))))))))))))))) c1 t1 t H))))).
625 \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (g: G).(\forall
626 (c: C).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
628 \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(\lambda (g:
629 G).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (ty3 g c t1
630 t)).(ty3_sred_wcpr0_pr0 g c t1 t H0 c (wcpr0_refl c) t2 H))))))).
633 \forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall
634 (c: C).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
636 \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr1 t1 t2)).(pr1_ind (\lambda
637 (t: T).(\lambda (t0: T).(\forall (g: G).(\forall (c: C).(\forall (t3:
638 T).((ty3 g c t t3) \to (ty3 g c t0 t3))))))) (\lambda (t: T).(\lambda (g:
639 G).(\lambda (c: C).(\lambda (t0: T).(\lambda (H0: (ty3 g c t t0)).H0)))))
640 (\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t4 t3)).(\lambda (t5:
641 T).(\lambda (_: (pr1 t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (c:
642 C).(\forall (t: T).((ty3 g c t3 t) \to (ty3 g c t5 t))))))).(\lambda (g:
643 G).(\lambda (c: C).(\lambda (t: T).(\lambda (H3: (ty3 g c t4 t)).(H2 g c t
644 (ty3_sred_pr0 t4 t3 H0 g c t H3)))))))))))) t1 t2 H))).
647 \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
648 (g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
650 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
651 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (g:
652 G).(\forall (t3: T).((ty3 g c0 t t3) \to (ty3 g c0 t0 t3))))))) (\lambda (c0:
653 C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (g:
654 G).(\lambda (t: T).(\lambda (H1: (ty3 g c0 t3 t)).(ty3_sred_wcpr0_pr0 g c0 t3
655 t H1 c0 (wcpr0_refl c0) t4 H0)))))))) (\lambda (c0: C).(\lambda (d:
656 C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
657 Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3
658 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (g:
659 G).(\lambda (t0: T).(\lambda (H3: (ty3 g c0 t3 t0)).(ty3_subst0 g c0 t4 t0
660 (ty3_sred_wcpr0_pr0 g c0 t3 t0 H3 c0 (wcpr0_refl c0) t4 H1) d u i H0 t
661 H2)))))))))))))) c t1 t2 H)))).
664 \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to (\forall
665 (g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
667 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
668 t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (g: G).(\forall
669 (t3: T).((ty3 g c t t3) \to (ty3 g c t0 t3)))))) (\lambda (t: T).(\lambda (g:
670 G).(\lambda (t0: T).(\lambda (H0: (ty3 g c t t0)).H0)))) (\lambda (t3:
671 T).(\lambda (t4: T).(\lambda (H0: (pr2 c t4 t3)).(\lambda (t5: T).(\lambda
672 (_: (pr3 c t3 t5)).(\lambda (H2: ((\forall (g: G).(\forall (t: T).((ty3 g c
673 t3 t) \to (ty3 g c t5 t)))))).(\lambda (g: G).(\lambda (t: T).(\lambda (H3:
674 (ty3 g c t4 t)).(H2 g t (ty3_sred_pr2 c t4 t3 H0 g t H3))))))))))) t1 t2
projects / helm.git / blob