1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/ty3/pr3_props".
22 \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr2 c v1
23 v2) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c
24 (Bind b) v1) t1 t2) \to (ty3 g (CHead c (Bind b) v2) t1 t2)))))))))
26 \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda
27 (H: (pr2 c v1 v2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
28 T).(\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c0 (Bind
29 b) t) t1 t2) \to (ty3 g (CHead c0 (Bind b) t0) t1 t2)))))))) (\lambda (c0:
30 C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (b:
31 B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (ty3 g (CHead c0 (Bind b)
32 t1) t0 t3)).(ty3_sred_wcpr0_pr0 g (CHead c0 (Bind b) t1) t0 t3 H1 (CHead c0
33 (Bind b) t2) (wcpr0_comp c0 c0 (wcpr0_refl c0) t1 t2 H0 (Bind b)) t0
34 (pr0_refl t0)))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u:
35 T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr)
36 u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda
37 (t: T).(\lambda (H2: (subst0 i u t2 t)).(\lambda (b: B).(\lambda (t0:
38 T).(\lambda (t3: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b) t1) t0
39 t3)).(ty3_csubst0 g (CHead c0 (Bind b) t2) t0 t3 (ty3_sred_wcpr0_pr0 g (CHead
40 c0 (Bind b) t1) t0 t3 H3 (CHead c0 (Bind b) t2) (wcpr0_comp c0 c0 (wcpr0_refl
41 c0) t1 t2 H1 (Bind b)) t0 (pr0_refl t0)) d u (S i) (getl_clear_bind b (CHead
42 c0 (Bind b) t2) c0 t2 (clear_bind b c0 t2) (CHead d (Bind Abbr) u) i H0)
43 (CHead c0 (Bind b) t) (csubst0_snd_bind b i u t2 t H2 c0)))))))))))))))) c v1
47 \forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr3 c v1
48 v2) \to (\forall (b: B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c
49 (Bind b) v1) t1 t2) \to (ty3 g (CHead c (Bind b) v2) t1 t2)))))))))
51 \lambda (g: G).(\lambda (c: C).(\lambda (v1: T).(\lambda (v2: T).(\lambda
52 (H: (pr3 c v1 v2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (b:
53 B).(\forall (t1: T).(\forall (t2: T).((ty3 g (CHead c (Bind b) t) t1 t2) \to
54 (ty3 g (CHead c (Bind b) t0) t1 t2))))))) (\lambda (t: T).(\lambda (b:
55 B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (ty3 g (CHead c (Bind b)
56 t) t1 t2)).H0))))) (\lambda (t2: T).(\lambda (t1: T).(\lambda (H0: (pr2 c t1
57 t2)).(\lambda (t3: T).(\lambda (_: (pr3 c t2 t3)).(\lambda (H2: ((\forall (b:
58 B).(\forall (t4: T).(\forall (t5: T).((ty3 g (CHead c (Bind b) t2) t4 t5) \to
59 (ty3 g (CHead c (Bind b) t3) t4 t5))))))).(\lambda (b: B).(\lambda (t0:
60 T).(\lambda (t4: T).(\lambda (H3: (ty3 g (CHead c (Bind b) t1) t0 t4)).(H2 b
61 t0 t4 (ty3_cred_pr2 g c t1 t2 H0 b t0 t4 H3)))))))))))) v1 v2 H))))).
64 \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h:
65 nat).(\forall (d: nat).((ty3 g c (lift h d t1) x) \to (\forall (e: C).((drop
66 h d c e) \to (ex2 T (\lambda (t2: T).(pc3 c (lift h d t2) x)) (\lambda (t2:
67 T).(ty3 g e t1 t2)))))))))))
69 \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h:
70 nat).(\lambda (d: nat).(\lambda (H: (ty3 g c (lift h d t1) x)).(insert_eq T
71 (lift h d t1) (\lambda (t: T).(ty3 g c t x)) (\forall (e: C).((drop h d c e)
72 \to (ex2 T (\lambda (t2: T).(pc3 c (lift h d t2) x)) (\lambda (t2: T).(ty3 g
73 e t1 t2))))) (\lambda (y: T).(\lambda (H0: (ty3 g c y x)).(unintro nat d
74 (\lambda (n: nat).((eq T y (lift h n t1)) \to (\forall (e: C).((drop h n c e)
75 \to (ex2 T (\lambda (t2: T).(pc3 c (lift h n t2) x)) (\lambda (t2: T).(ty3 g
76 e t1 t2))))))) (unintro T t1 (\lambda (t: T).(\forall (x0: nat).((eq T y
77 (lift h x0 t)) \to (\forall (e: C).((drop h x0 c e) \to (ex2 T (\lambda (t2:
78 T).(pc3 c (lift h x0 t2) x)) (\lambda (t2: T).(ty3 g e t t2)))))))) (ty3_ind
79 g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (x0: T).(\forall
80 (x1: nat).((eq T t (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to
81 (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) t0)) (\lambda (t2: T).(ty3 g e
82 x0 t2))))))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t: T).(\lambda
83 (_: (ty3 g c0 t2 t)).(\lambda (_: ((\forall (x0: T).(\forall (x1: nat).((eq T
84 t2 (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda
85 (t3: T).(pc3 c0 (lift h x1 t3) t)) (\lambda (t3: T).(ty3 g e x0
86 t3)))))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (H3: (ty3 g c0 u
87 t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1
88 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t4: T).(pc3
89 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)))))))))).(\lambda
90 (H5: (pc3 c0 t3 t2)).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H6: (eq T
91 u (lift h x1 x0))).(\lambda (e: C).(\lambda (H7: (drop h x1 c0 e)).(let H8
92 \def (eq_ind T u (\lambda (t0: T).(\forall (x2: T).(\forall (x3: nat).((eq T
93 t0 (lift h x3 x2)) \to (\forall (e0: C).((drop h x3 c0 e0) \to (ex2 T
94 (\lambda (t4: T).(pc3 c0 (lift h x3 t4) t3)) (\lambda (t4: T).(ty3 g e0 x2
95 t4))))))))) H4 (lift h x1 x0) H6) in (let H9 \def (eq_ind T u (\lambda (t0:
96 T).(ty3 g c0 t0 t3)) H3 (lift h x1 x0) H6) in (let H10 \def (H8 x0 x1
97 (refl_equal T (lift h x1 x0)) e H7) in (ex2_ind T (\lambda (t4: T).(pc3 c0
98 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)) (ex2 T (\lambda (t4:
99 T).(pc3 c0 (lift h x1 t4) t2)) (\lambda (t4: T).(ty3 g e x0 t4))) (\lambda
100 (x2: T).(\lambda (H11: (pc3 c0 (lift h x1 x2) t3)).(\lambda (H12: (ty3 g e x0
101 x2)).(ex_intro2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t2)) (\lambda (t4:
102 T).(ty3 g e x0 t4)) x2 (pc3_t t3 c0 (lift h x1 x2) H11 t2 H5) H12))))
103 H10))))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (x0:
104 T).(\lambda (x1: nat).(\lambda (H1: (eq T (TSort m) (lift h x1 x0))).(\lambda
105 (e: C).(\lambda (_: (drop h x1 c0 e)).(eq_ind_r T (TSort m) (\lambda (t:
106 T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (TSort (next g m))))
107 (\lambda (t2: T).(ty3 g e t t2)))) (ex_intro2 T (\lambda (t2: T).(pc3 c0
108 (lift h x1 t2) (TSort (next g m)))) (\lambda (t2: T).(ty3 g e (TSort m) t2))
109 (TSort (next g m)) (eq_ind_r T (TSort (next g m)) (\lambda (t: T).(pc3 c0 t
110 (TSort (next g m)))) (pc3_refl c0 (TSort (next g m))) (lift h x1 (TSort (next
111 g m))) (lift_sort (next g m) h x1)) (ty3_sort g e m)) x0 (lift_gen_sort h x1
112 m x0 H1))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d0: C).(\lambda
113 (u: T).(\lambda (H1: (getl n c0 (CHead d0 (Bind Abbr) u))).(\lambda (t:
114 T).(\lambda (H2: (ty3 g d0 u t)).(\lambda (H3: ((\forall (x0: T).(\forall
115 (x1: nat).((eq T u (lift h x1 x0)) \to (\forall (e: C).((drop h x1 d0 e) \to
116 (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x1 t2) t)) (\lambda (t2: T).(ty3 g e
117 x0 t2)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H4: (eq T
118 (TLRef n) (lift h x1 x0))).(\lambda (e: C).(\lambda (H5: (drop h x1 c0
119 e)).(let H_x \def (lift_gen_lref x0 x1 h n H4) in (let H6 \def H_x in (or_ind
120 (land (lt n x1) (eq T x0 (TLRef n))) (land (le (plus x1 h) n) (eq T x0 (TLRef
121 (minus n h)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O
122 t))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H7: (land (lt n x1) (eq T
123 x0 (TLRef n)))).(and_ind (lt n x1) (eq T x0 (TLRef n)) (ex2 T (\lambda (t2:
124 T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e x0
125 t2))) (\lambda (H8: (lt n x1)).(\lambda (H9: (eq T x0 (TLRef n))).(eq_ind_r T
126 (TLRef n) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2)
127 (lift (S n) O t))) (\lambda (t2: T).(ty3 g e t0 t2)))) (let H10 \def (eq_ind
128 nat x1 (\lambda (n0: nat).(drop h n0 c0 e)) H5 (S (plus n (minus x1 (S n))))
129 (lt_plus_minus n x1 H8)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
130 C).(eq T u (lift h (minus x1 (S n)) v)))) (\lambda (v: T).(\lambda (e0:
131 C).(getl n e (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0:
132 C).(drop h (minus x1 (S n)) d0 e0))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h
133 x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e (TLRef n) t2))) (\lambda
134 (x2: T).(\lambda (x3: C).(\lambda (H11: (eq T u (lift h (minus x1 (S n))
135 x2))).(\lambda (H12: (getl n e (CHead x3 (Bind Abbr) x2))).(\lambda (H13:
136 (drop h (minus x1 (S n)) d0 x3)).(let H14 \def (eq_ind T u (\lambda (t0:
137 T).(\forall (x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to (\forall
138 (e0: C).((drop h x5 d0 e0) \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x5 t2)
139 t)) (\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H3 (lift h (minus x1 (S n)) x2)
140 H11) in (let H15 \def (eq_ind T u (\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift
141 h (minus x1 (S n)) x2) H11) in (let H16 \def (H14 x2 (minus x1 (S n))
142 (refl_equal T (lift h (minus x1 (S n)) x2)) x3 H13) in (ex2_ind T (\lambda
143 (t2: T).(pc3 d0 (lift h (minus x1 (S n)) t2) t)) (\lambda (t2: T).(ty3 g x3
144 x2 t2)) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t)))
145 (\lambda (t2: T).(ty3 g e (TLRef n) t2))) (\lambda (x4: T).(\lambda (H17:
146 (pc3 d0 (lift h (minus x1 (S n)) x4) t)).(\lambda (H18: (ty3 g x3 x2
147 x4)).(eq_ind_r nat (plus (S n) (minus x1 (S n))) (\lambda (n0: nat).(ex2 T
148 (\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift (S n) O t))) (\lambda (t2:
149 T).(ty3 g e (TLRef n) t2)))) (ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h
150 (plus (S n) (minus x1 (S n))) t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g
151 e (TLRef n) t2)) (lift (S n) O x4) (eq_ind_r T (lift (S n) O (lift h (minus
152 x1 (S n)) x4)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O t))) (pc3_lift c0 d0
153 (S n) O (getl_drop Abbr c0 d0 u n H1) (lift h (minus x1 (S n)) x4) t H17)
154 (lift h (plus (S n) (minus x1 (S n))) (lift (S n) O x4)) (lift_d x4 h (S n)
155 (minus x1 (S n)) O (le_O_n (minus x1 (S n))))) (ty3_abbr g n e x3 x2 H12 x4
156 H18)) x1 (le_plus_minus (S n) x1 H8))))) H16))))))))) (getl_drop_conf_lt Abbr
157 c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9))) H7)) (\lambda (H7: (land
158 (le (plus x1 h) n) (eq T x0 (TLRef (minus n h))))).(and_ind (le (plus x1 h)
159 n) (eq T x0 (TLRef (minus n h))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1
160 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H8: (le
161 (plus x1 h) n)).(\lambda (H9: (eq T x0 (TLRef (minus n h)))).(eq_ind_r T
162 (TLRef (minus n h)) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h
163 x1 t2) (lift (S n) O t))) (\lambda (t2: T).(ty3 g e t0 t2)))) (ex_intro2 T
164 (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t))) (\lambda (t2:
165 T).(ty3 g e (TLRef (minus n h)) t2)) (lift (S (minus n h)) O t) (eq_ind_r T
166 (lift (plus h (S (minus n h))) O t) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O
167 t))) (eq_ind nat (S (plus h (minus n h))) (\lambda (n0: nat).(pc3 c0 (lift n0
168 O t) (lift (S n) O t))) (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0)
169 O t) (lift (S n) O t))) (pc3_refl c0 (lift (S n) O t)) (plus h (minus n h))
170 (le_plus_minus h n (le_trans_plus_r x1 h n H8))) (plus h (S (minus n h)))
171 (plus_n_Sm h (minus n h))) (lift h x1 (lift (S (minus n h)) O t)) (lift_free
172 t (S (minus n h)) h O x1 (le_trans x1 (S (minus n h)) (plus O (S (minus n
173 h))) (le_S_minus x1 h n H8) (le_n (plus O (S (minus n h))))) (le_O_n x1)))
174 (ty3_abbr g (minus n h) e d0 u (getl_drop_conf_ge n (CHead d0 (Bind Abbr) u)
175 c0 H1 e h x1 H5 H8) t H2)) x0 H9))) H7)) H6)))))))))))))))) (\lambda (n:
176 nat).(\lambda (c0: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (H1: (getl n
177 c0 (CHead d0 (Bind Abst) u))).(\lambda (t: T).(\lambda (H2: (ty3 g d0 u
178 t)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1
179 x0)) \to (\forall (e: C).((drop h x1 d0 e) \to (ex2 T (\lambda (t2: T).(pc3
180 d0 (lift h x1 t2) t)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (x0:
181 T).(\lambda (x1: nat).(\lambda (H4: (eq T (TLRef n) (lift h x1 x0))).(\lambda
182 (e: C).(\lambda (H5: (drop h x1 c0 e)).(let H_x \def (lift_gen_lref x0 x1 h n
183 H4) in (let H6 \def H_x in (or_ind (land (lt n x1) (eq T x0 (TLRef n))) (land
184 (le (plus x1 h) n) (eq T x0 (TLRef (minus n h)))) (ex2 T (\lambda (t2:
185 T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0
186 t2))) (\lambda (H7: (land (lt n x1) (eq T x0 (TLRef n)))).(and_ind (lt n x1)
187 (eq T x0 (TLRef n)) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S
188 n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H8: (lt n
189 x1)).(\lambda (H9: (eq T x0 (TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0:
190 T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda
191 (t2: T).(ty3 g e t0 t2)))) (let H10 \def (eq_ind nat x1 (\lambda (n0:
192 nat).(drop h n0 c0 e)) H5 (S (plus n (minus x1 (S n)))) (lt_plus_minus n x1
193 H8)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus
194 x1 (S n)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind
195 Abst) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x1 (S n)) d0
196 e0))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u)))
197 (\lambda (t2: T).(ty3 g e (TLRef n) t2))) (\lambda (x2: T).(\lambda (x3:
198 C).(\lambda (H11: (eq T u (lift h (minus x1 (S n)) x2))).(\lambda (H12: (getl
199 n e (CHead x3 (Bind Abst) x2))).(\lambda (H13: (drop h (minus x1 (S n)) d0
200 x3)).(let H14 \def (eq_ind T u (\lambda (t0: T).(\forall (x4: T).(\forall
201 (x5: nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 d0 e0)
202 \to (ex2 T (\lambda (t2: T).(pc3 d0 (lift h x5 t2) t)) (\lambda (t2: T).(ty3
203 g e0 x4 t2))))))))) H3 (lift h (minus x1 (S n)) x2) H11) in (let H15 \def
204 (eq_ind T u (\lambda (t0: T).(ty3 g d0 t0 t)) H2 (lift h (minus x1 (S n)) x2)
205 H11) in (eq_ind_r T (lift h (minus x1 (S n)) x2) (\lambda (t0: T).(ex2 T
206 (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O t0))) (\lambda (t2:
207 T).(ty3 g e (TLRef n) t2)))) (let H16 \def (H14 x2 (minus x1 (S n))
208 (refl_equal T (lift h (minus x1 (S n)) x2)) x3 H13) in (ex2_ind T (\lambda
209 (t2: T).(pc3 d0 (lift h (minus x1 (S n)) t2) t)) (\lambda (t2: T).(ty3 g x3
210 x2 t2)) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O (lift h
211 (minus x1 (S n)) x2)))) (\lambda (t2: T).(ty3 g e (TLRef n) t2))) (\lambda
212 (x4: T).(\lambda (_: (pc3 d0 (lift h (minus x1 (S n)) x4) t)).(\lambda (H18:
213 (ty3 g x3 x2 x4)).(eq_ind_r nat (plus (S n) (minus x1 (S n))) (\lambda (n0:
214 nat).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h n0 t2) (lift (S n) O (lift h
215 (minus n0 (S n)) x2)))) (\lambda (t2: T).(ty3 g e (TLRef n) t2)))) (ex_intro2
216 T (\lambda (t2: T).(pc3 c0 (lift h (plus (S n) (minus x1 (S n))) t2) (lift (S
217 n) O (lift h (minus (plus (S n) (minus x1 (S n))) (S n)) x2)))) (\lambda (t2:
218 T).(ty3 g e (TLRef n) t2)) (lift (S n) O x2) (eq_ind_r T (lift (S n) O (lift
219 h (minus x1 (S n)) x2)) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O (lift h
220 (minus (plus (S n) (minus x1 (S n))) (S n)) x2)))) (eq_ind nat x1 (\lambda
221 (n0: nat).(pc3 c0 (lift (S n) O (lift h (minus x1 (S n)) x2)) (lift (S n) O
222 (lift h (minus n0 (S n)) x2)))) (pc3_refl c0 (lift (S n) O (lift h (minus x1
223 (S n)) x2))) (plus (S n) (minus x1 (S n))) (le_plus_minus (S n) x1 H8)) (lift
224 h (plus (S n) (minus x1 (S n))) (lift (S n) O x2)) (lift_d x2 h (S n) (minus
225 x1 (S n)) O (le_O_n (minus x1 (S n))))) (ty3_abst g n e x3 x2 H12 x4 H18)) x1
226 (le_plus_minus (S n) x1 H8))))) H16)) u H11)))))))) (getl_drop_conf_lt Abst
227 c0 d0 u n H1 e h (minus x1 (S n)) H10))) x0 H9))) H7)) (\lambda (H7: (land
228 (le (plus x1 h) n) (eq T x0 (TLRef (minus n h))))).(and_ind (le (plus x1 h)
229 n) (eq T x0 (TLRef (minus n h))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1
230 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (H8: (le
231 (plus x1 h) n)).(\lambda (H9: (eq T x0 (TLRef (minus n h)))).(eq_ind_r T
232 (TLRef (minus n h)) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0 (lift h
233 x1 t2) (lift (S n) O u))) (\lambda (t2: T).(ty3 g e t0 t2)))) (ex_intro2 T
234 (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (lift (S n) O u))) (\lambda (t2:
235 T).(ty3 g e (TLRef (minus n h)) t2)) (lift (S (minus n h)) O u) (eq_ind_r T
236 (lift (plus h (S (minus n h))) O u) (\lambda (t0: T).(pc3 c0 t0 (lift (S n) O
237 u))) (eq_ind nat (S (plus h (minus n h))) (\lambda (n0: nat).(pc3 c0 (lift n0
238 O u) (lift (S n) O u))) (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0)
239 O u) (lift (S n) O u))) (pc3_refl c0 (lift (S n) O u)) (plus h (minus n h))
240 (le_plus_minus h n (le_trans_plus_r x1 h n H8))) (plus h (S (minus n h)))
241 (plus_n_Sm h (minus n h))) (lift h x1 (lift (S (minus n h)) O u)) (lift_free
242 u (S (minus n h)) h O x1 (le_trans x1 (S (minus n h)) (plus O (S (minus n
243 h))) (le_S_minus x1 h n H8) (le_n (plus O (S (minus n h))))) (le_O_n x1)))
244 (ty3_abst g (minus n h) e d0 u (getl_drop_conf_ge n (CHead d0 (Bind Abst) u)
245 c0 H1 e h x1 H5 H8) t H2)) x0 H9))) H7)) H6)))))))))))))))) (\lambda (c0:
246 C).(\lambda (u: T).(\lambda (t: T).(\lambda (H1: (ty3 g c0 u t)).(\lambda
247 (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T u (lift h x1 x0)) \to
248 (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h
249 x1 t2) t)) (\lambda (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (b: B).(\lambda
250 (t2: T).(\lambda (t3: T).(\lambda (H3: (ty3 g (CHead c0 (Bind b) u) t2
251 t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1
252 x0)) \to (\forall (e: C).((drop h x1 (CHead c0 (Bind b) u) e) \to (ex2 T
253 (\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (lift h x1 t4) t3)) (\lambda (t4:
254 T).(ty3 g e x0 t4)))))))))).(\lambda (t0: T).(\lambda (H5: (ty3 g (CHead c0
255 (Bind b) u) t3 t0)).(\lambda (H6: ((\forall (x0: T).(\forall (x1: nat).((eq T
256 t3 (lift h x1 x0)) \to (\forall (e: C).((drop h x1 (CHead c0 (Bind b) u) e)
257 \to (ex2 T (\lambda (t4: T).(pc3 (CHead c0 (Bind b) u) (lift h x1 t4) t0))
258 (\lambda (t4: T).(ty3 g e x0 t4)))))))))).(\lambda (x0: T).(\lambda (x1:
259 nat).(\lambda (H7: (eq T (THead (Bind b) u t2) (lift h x1 x0))).(\lambda (e:
260 C).(\lambda (H8: (drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda
261 (z: T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq
262 T u (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1)
263 z)))) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) u t3)))
264 (\lambda (t4: T).(ty3 g e x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda
265 (H9: (eq T x0 (THead (Bind b) x2 x3))).(\lambda (H10: (eq T u (lift h x1
266 x2))).(\lambda (H11: (eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind
267 b) x2 x3) (\lambda (t4: T).(ex2 T (\lambda (t5: T).(pc3 c0 (lift h x1 t5)
268 (THead (Bind b) u t3))) (\lambda (t5: T).(ty3 g e t4 t5)))) (let H12 \def
269 (eq_ind T t2 (\lambda (t4: T).(\forall (x4: T).(\forall (x5: nat).((eq T t4
270 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 (CHead c0 (Bind b) u) e0)
271 \to (ex2 T (\lambda (t5: T).(pc3 (CHead c0 (Bind b) u) (lift h x5 t5) t3))
272 (\lambda (t5: T).(ty3 g e0 x4 t5))))))))) H4 (lift h (S x1) x3) H11) in (let
273 H13 \def (eq_ind T t2 (\lambda (t4: T).(ty3 g (CHead c0 (Bind b) u) t4 t3))
274 H3 (lift h (S x1) x3) H11) in (let H14 \def (eq_ind T u (\lambda (t4: T).(ty3
275 g (CHead c0 (Bind b) t4) (lift h (S x1) x3) t3)) H13 (lift h x1 x2) H10) in
276 (let H15 \def (eq_ind T u (\lambda (t4: T).(\forall (x4: T).(\forall (x5:
277 nat).((eq T (lift h (S x1) x3) (lift h x5 x4)) \to (\forall (e0: C).((drop h
278 x5 (CHead c0 (Bind b) t4) e0) \to (ex2 T (\lambda (t5: T).(pc3 (CHead c0
279 (Bind b) t4) (lift h x5 t5) t3)) (\lambda (t5: T).(ty3 g e0 x4 t5)))))))))
280 H12 (lift h x1 x2) H10) in (let H16 \def (eq_ind T u (\lambda (t4:
281 T).(\forall (x4: T).(\forall (x5: nat).((eq T t3 (lift h x5 x4)) \to (\forall
282 (e0: C).((drop h x5 (CHead c0 (Bind b) t4) e0) \to (ex2 T (\lambda (t5:
283 T).(pc3 (CHead c0 (Bind b) t4) (lift h x5 t5) t0)) (\lambda (t5: T).(ty3 g e0
284 x4 t5))))))))) H6 (lift h x1 x2) H10) in (let H17 \def (eq_ind T u (\lambda
285 (t4: T).(ty3 g (CHead c0 (Bind b) t4) t3 t0)) H5 (lift h x1 x2) H10) in (let
286 H18 \def (eq_ind T u (\lambda (t4: T).(\forall (x4: T).(\forall (x5:
287 nat).((eq T t4 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to
288 (ex2 T (\lambda (t5: T).(pc3 c0 (lift h x5 t5) t)) (\lambda (t5: T).(ty3 g e0
289 x4 t5))))))))) H2 (lift h x1 x2) H10) in (let H19 \def (eq_ind T u (\lambda
290 (t4: T).(ty3 g c0 t4 t)) H1 (lift h x1 x2) H10) in (eq_ind_r T (lift h x1 x2)
291 (\lambda (t4: T).(ex2 T (\lambda (t5: T).(pc3 c0 (lift h x1 t5) (THead (Bind
292 b) t4 t3))) (\lambda (t5: T).(ty3 g e (THead (Bind b) x2 x3) t5)))) (let H20
293 \def (H18 x2 x1 (refl_equal T (lift h x1 x2)) e H8) in (ex2_ind T (\lambda
294 (t4: T).(pc3 c0 (lift h x1 t4) t)) (\lambda (t4: T).(ty3 g e x2 t4)) (ex2 T
295 (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b) (lift h x1 x2) t3)))
296 (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4))) (\lambda (x4:
297 T).(\lambda (_: (pc3 c0 (lift h x1 x4) t)).(\lambda (H22: (ty3 g e x2
298 x4)).(let H23 \def (H15 x3 (S x1) (refl_equal T (lift h (S x1) x3)) (CHead e
299 (Bind b) x2) (drop_skip_bind h x1 c0 e H8 b x2)) in (ex2_ind T (\lambda (t4:
300 T).(pc3 (CHead c0 (Bind b) (lift h x1 x2)) (lift h (S x1) t4) t3)) (\lambda
301 (t4: T).(ty3 g (CHead e (Bind b) x2) x3 t4)) (ex2 T (\lambda (t4: T).(pc3 c0
302 (lift h x1 t4) (THead (Bind b) (lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e
303 (THead (Bind b) x2 x3) t4))) (\lambda (x5: T).(\lambda (H24: (pc3 (CHead c0
304 (Bind b) (lift h x1 x2)) (lift h (S x1) x5) t3)).(\lambda (H25: (ty3 g (CHead
305 e (Bind b) x2) x3 x5)).(ex_ind T (\lambda (t4: T).(ty3 g (CHead e (Bind b)
306 x2) x5 t4)) (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b)
307 (lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4)))
308 (\lambda (x6: T).(\lambda (H26: (ty3 g (CHead e (Bind b) x2) x5
309 x6)).(ex_intro2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) (THead (Bind b)
310 (lift h x1 x2) t3))) (\lambda (t4: T).(ty3 g e (THead (Bind b) x2 x3) t4))
311 (THead (Bind b) x2 x5) (eq_ind_r T (THead (Bind b) (lift h x1 x2) (lift h (S
312 x1) x5)) (\lambda (t4: T).(pc3 c0 t4 (THead (Bind b) (lift h x1 x2) t3)))
313 (pc3_head_2 c0 (lift h x1 x2) (lift h (S x1) x5) t3 (Bind b) H24) (lift h x1
314 (THead (Bind b) x2 x5)) (lift_bind b x2 x5 h x1)) (ty3_bind g e x2 x4 H22 b
315 x3 x5 H25 x6 H26)))) (ty3_correct g (CHead e (Bind b) x2) x3 x5 H25)))))
316 H23))))) H20)) u H10))))))))) x0 H9)))))) (lift_gen_bind b u t2 x0 h x1
317 H7)))))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u:
318 T).(\lambda (H1: (ty3 g c0 w u)).(\lambda (H2: ((\forall (x0: T).(\forall
319 (x1: nat).((eq T w (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to
320 (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e
321 x0 t2)))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H3: (ty3 g c0 v
322 (THead (Bind Abst) u t))).(\lambda (H4: ((\forall (x0: T).(\forall (x1:
323 nat).((eq T v (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to (ex2
324 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Bind Abst) u t))) (\lambda
325 (t2: T).(ty3 g e x0 t2)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda
326 (H5: (eq T (THead (Flat Appl) w v) (lift h x1 x0))).(\lambda (e: C).(\lambda
327 (H6: (drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T
328 x0 (THead (Flat Appl) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T w (lift
329 h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T v (lift h x1 z)))) (ex2 T
330 (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) w (THead (Bind
331 Abst) u t)))) (\lambda (t2: T).(ty3 g e x0 t2))) (\lambda (x2: T).(\lambda
332 (x3: T).(\lambda (H7: (eq T x0 (THead (Flat Appl) x2 x3))).(\lambda (H8: (eq
333 T w (lift h x1 x2))).(\lambda (H9: (eq T v (lift h x1 x3))).(eq_ind_r T
334 (THead (Flat Appl) x2 x3) (\lambda (t0: T).(ex2 T (\lambda (t2: T).(pc3 c0
335 (lift h x1 t2) (THead (Flat Appl) w (THead (Bind Abst) u t)))) (\lambda (t2:
336 T).(ty3 g e t0 t2)))) (let H10 \def (eq_ind T v (\lambda (t0: T).(\forall
337 (x4: T).(\forall (x5: nat).((eq T t0 (lift h x5 x4)) \to (\forall (e0:
338 C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x5 t2)
339 (THead (Bind Abst) u t))) (\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H4 (lift
340 h x1 x3) H9) in (let H11 \def (eq_ind T v (\lambda (t0: T).(ty3 g c0 t0
341 (THead (Bind Abst) u t))) H3 (lift h x1 x3) H9) in (let H12 \def (eq_ind T w
342 (\lambda (t0: T).(\forall (x4: T).(\forall (x5: nat).((eq T t0 (lift h x5
343 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda (t2: T).(pc3
344 c0 (lift h x5 t2) u)) (\lambda (t2: T).(ty3 g e0 x4 t2))))))))) H2 (lift h x1
345 x2) H8) in (let H13 \def (eq_ind T w (\lambda (t0: T).(ty3 g c0 t0 u)) H1
346 (lift h x1 x2) H8) in (eq_ind_r T (lift h x1 x2) (\lambda (t0: T).(ex2 T
347 (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) t0 (THead (Bind
348 Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))) (let
349 H14 \def (H12 x2 x1 (refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T
350 (\lambda (t2: T).(pc3 c0 (lift h x1 t2) u)) (\lambda (t2: T).(ty3 g e x2 t2))
351 (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1
352 x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl)
353 x2 x3) t2))) (\lambda (x4: T).(\lambda (H15: (pc3 c0 (lift h x1 x4)
354 u)).(\lambda (H16: (ty3 g e x2 x4)).(let H17 \def (H10 x3 x1 (refl_equal T
355 (lift h x1 x3)) e H6) in (ex2_ind T (\lambda (t2: T).(pc3 c0 (lift h x1 t2)
356 (THead (Bind Abst) u t))) (\lambda (t2: T).(ty3 g e x3 t2)) (ex2 T (\lambda
357 (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind
358 Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2)))
359 (\lambda (x5: T).(\lambda (H18: (pc3 c0 (lift h x1 x5) (THead (Bind Abst) u
360 t))).(\lambda (H19: (ty3 g e x3 x5)).(ex3_2_ind T T (\lambda (u1: T).(\lambda
361 (t2: T).(pr3 e x5 (THead (Bind Abst) u1 t2)))) (\lambda (u1: T).(\lambda (_:
362 T).(pr3 c0 u (lift h x1 u1)))) (\lambda (_: T).(\lambda (t2: T).(\forall (b:
363 B).(\forall (u0: T).(pr3 (CHead c0 (Bind b) u0) t (lift h (S x1) t2))))))
364 (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1
365 x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl)
366 x2 x3) t2))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (pr3 e x5
367 (THead (Bind Abst) x6 x7))).(\lambda (H21: (pr3 c0 u (lift h x1
368 x6))).(\lambda (H22: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c0 (Bind
369 b) u0) t (lift h (S x1) x7)))))).(ex_ind T (\lambda (t0: T).(ty3 g e x5 t0))
370 (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl) (lift h x1
371 x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead (Flat Appl)
372 x2 x3) t2))) (\lambda (x8: T).(\lambda (H23: (ty3 g e x5 x8)).(ex4_3_ind T T
373 T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 e (THead (Bind Abst)
374 x6 t2) x8)))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g e x6
375 t0)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead e (Bind
376 Abst) x6) x7 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g
377 (CHead e (Bind Abst) x6) t2 t3)))) (ex2 T (\lambda (t2: T).(pc3 c0 (lift h x1
378 t2) (THead (Flat Appl) (lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda
379 (t2: T).(ty3 g e (THead (Flat Appl) x2 x3) t2))) (\lambda (x9: T).(\lambda
380 (x10: T).(\lambda (x11: T).(\lambda (_: (pc3 e (THead (Bind Abst) x6 x9)
381 x8)).(\lambda (H25: (ty3 g e x6 x10)).(\lambda (H26: (ty3 g (CHead e (Bind
382 Abst) x6) x7 x9)).(\lambda (H27: (ty3 g (CHead e (Bind Abst) x6) x9
383 x11)).(ex_intro2 T (\lambda (t2: T).(pc3 c0 (lift h x1 t2) (THead (Flat Appl)
384 (lift h x1 x2) (THead (Bind Abst) u t)))) (\lambda (t2: T).(ty3 g e (THead
385 (Flat Appl) x2 x3) t2)) (THead (Flat Appl) x2 (THead (Bind Abst) x6 x7))
386 (eq_ind_r T (THead (Flat Appl) (lift h x1 x2) (lift h x1 (THead (Bind Abst)
387 x6 x7))) (\lambda (t0: T).(pc3 c0 t0 (THead (Flat Appl) (lift h x1 x2) (THead
388 (Bind Abst) u t)))) (pc3_thin_dx c0 (lift h x1 (THead (Bind Abst) x6 x7))
389 (THead (Bind Abst) u t) (eq_ind_r T (THead (Bind Abst) (lift h x1 x6) (lift h
390 (S x1) x7)) (\lambda (t0: T).(pc3 c0 t0 (THead (Bind Abst) u t)))
391 (pc3_head_21 c0 (lift h x1 x6) u (pc3_pr3_x c0 (lift h x1 x6) u H21) (Bind
392 Abst) (lift h (S x1) x7) t (pc3_pr3_x (CHead c0 (Bind Abst) (lift h x1 x6))
393 (lift h (S x1) x7) t (H22 Abst (lift h x1 x6)))) (lift h x1 (THead (Bind
394 Abst) x6 x7)) (lift_bind Abst x6 x7 h x1)) (lift h x1 x2) Appl) (lift h x1
395 (THead (Flat Appl) x2 (THead (Bind Abst) x6 x7))) (lift_flat Appl x2 (THead
396 (Bind Abst) x6 x7) h x1)) (ty3_appl g e x2 x6 (ty3_conv g e x6 x10 H25 x2 x4
397 H16 (pc3_gen_lift c0 x4 x6 h x1 (pc3_t u c0 (lift h x1 x4) H15 (lift h x1 x6)
398 (pc3_pr3_r c0 u (lift h x1 x6) H21)) e H6)) x3 x7 (ty3_conv g e (THead (Bind
399 Abst) x6 x7) (THead (Bind Abst) x6 x9) (ty3_bind g e x6 x10 H25 Abst x7 x9
400 H26 x11 H27) x3 x5 H19 (pc3_pr3_r e x5 (THead (Bind Abst) x6 x7)
401 H20))))))))))) (ty3_gen_bind g Abst e x6 x7 x8 (ty3_sred_pr3 e x5 (THead
402 (Bind Abst) x6 x7) H20 g x8 H23))))) (ty3_correct g e x3 x5 H19)))))))
403 (pc3_gen_lift_abst c0 x5 t u h x1 H18 e H6))))) H17))))) H14)) w H8))))) x0
404 H7)))))) (lift_gen_flat Appl w v x0 h x1 H5)))))))))))))))) (\lambda (c0:
405 C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H1: (ty3 g c0 t2 t3)).(\lambda
406 (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to
407 (\forall (e: C).((drop h x1 c0 e) \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h
408 x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0 t4)))))))))).(\lambda (t0:
409 T).(\lambda (H3: (ty3 g c0 t3 t0)).(\lambda (H4: ((\forall (x0: T).(\forall
410 (x1: nat).((eq T t3 (lift h x1 x0)) \to (\forall (e: C).((drop h x1 c0 e) \to
411 (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t0)) (\lambda (t4: T).(ty3 g e
412 x0 t4)))))))))).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T
413 (THead (Flat Cast) t3 t2) (lift h x1 x0))).(\lambda (e: C).(\lambda (H6:
414 (drop h x1 c0 e)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0
415 (THead (Flat Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T t3 (lift h
416 x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T
417 (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e x0
418 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T x0 (THead (Flat
419 Cast) x2 x3))).(\lambda (H8: (eq T t3 (lift h x1 x2))).(\lambda (H9: (eq T t2
420 (lift h x1 x3))).(eq_ind_r T (THead (Flat Cast) x2 x3) (\lambda (t: T).(ex2 T
421 (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t3)) (\lambda (t4: T).(ty3 g e t
422 t4)))) (let H10 \def (eq_ind T t3 (\lambda (t: T).(\forall (x4: T).(\forall
423 (x5: nat).((eq T t (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0)
424 \to (ex2 T (\lambda (t4: T).(pc3 c0 (lift h x5 t4) t0)) (\lambda (t4: T).(ty3
425 g e0 x4 t4))))))))) H4 (lift h x1 x2) H8) in (let H11 \def (eq_ind T t3
426 (\lambda (t: T).(ty3 g c0 t t0)) H3 (lift h x1 x2) H8) in (let H12 \def
427 (eq_ind T t3 (\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t2
428 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda
429 (t4: T).(pc3 c0 (lift h x5 t4) t)) (\lambda (t4: T).(ty3 g e0 x4 t4)))))))))
430 H2 (lift h x1 x2) H8) in (let H13 \def (eq_ind T t3 (\lambda (t: T).(ty3 g c0
431 t2 t)) H1 (lift h x1 x2) H8) in (eq_ind_r T (lift h x1 x2) (\lambda (t:
432 T).(ex2 T (\lambda (t4: T).(pc3 c0 (lift h x1 t4) t)) (\lambda (t4: T).(ty3 g
433 e (THead (Flat Cast) x2 x3) t4)))) (let H14 \def (eq_ind T t2 (\lambda (t:
434 T).(ty3 g c0 t (lift h x1 x2))) H13 (lift h x1 x3) H9) in (let H15 \def
435 (eq_ind T t2 (\lambda (t: T).(\forall (x4: T).(\forall (x5: nat).((eq T t
436 (lift h x5 x4)) \to (\forall (e0: C).((drop h x5 c0 e0) \to (ex2 T (\lambda
437 (t4: T).(pc3 c0 (lift h x5 t4) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e0 x4
438 t4))))))))) H12 (lift h x1 x3) H9) in (let H16 \def (H15 x3 x1 (refl_equal T
439 (lift h x1 x3)) e H6) in (ex2_ind T (\lambda (t4: T).(pc3 c0 (lift h x1 t4)
440 (lift h x1 x2))) (\lambda (t4: T).(ty3 g e x3 t4)) (ex2 T (\lambda (t4:
441 T).(pc3 c0 (lift h x1 t4) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e (THead
442 (Flat Cast) x2 x3) t4))) (\lambda (x4: T).(\lambda (H17: (pc3 c0 (lift h x1
443 x4) (lift h x1 x2))).(\lambda (H18: (ty3 g e x3 x4)).(let H19 \def (H10 x2 x1
444 (refl_equal T (lift h x1 x2)) e H6) in (ex2_ind T (\lambda (t4: T).(pc3 c0
445 (lift h x1 t4) t0)) (\lambda (t4: T).(ty3 g e x2 t4)) (ex2 T (\lambda (t4:
446 T).(pc3 c0 (lift h x1 t4) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e (THead
447 (Flat Cast) x2 x3) t4))) (\lambda (x5: T).(\lambda (_: (pc3 c0 (lift h x1 x5)
448 t0)).(\lambda (H21: (ty3 g e x2 x5)).(ex_intro2 T (\lambda (t4: T).(pc3 c0
449 (lift h x1 t4) (lift h x1 x2))) (\lambda (t4: T).(ty3 g e (THead (Flat Cast)
450 x2 x3) t4)) x2 (pc3_refl c0 (lift h x1 x2)) (ty3_cast g e x3 x2 (ty3_conv g e
451 x2 x5 H21 x3 x4 H18 (pc3_gen_lift c0 x4 x2 h x1 H17 e H6)) x5 H21)))))
452 H19))))) H16)))) t3 H8))))) x0 H7)))))) (lift_gen_flat Cast t3 t2 x0 h x1
453 H5))))))))))))))) c y x H0))))) H))))))).
456 \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u
457 t1) \to (\forall (t2: T).((pr3 c t1 t2) \to (ty3 g c u t2)))))))
459 \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H:
460 (ty3 g c u t1)).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(ex_ind T
461 (\lambda (t: T).(ty3 g c t1 t)) (ty3 g c u t2) (\lambda (x: T).(\lambda (H1:
462 (ty3 g c t1 x)).(ty3_conv g c t2 x (ty3_sred_pr3 c t1 t2 H0 g x H1) u t1 H
463 (pc3_pr3_r c t1 t2 H0)))) (ty3_correct g c u t1 H)))))))).
465 theorem ty3_sconv_pc3:
466 \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c
467 u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to ((pc3 c u1
468 u2) \to (pc3 c t1 t2)))))))))
470 \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda
471 (H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c
472 u2 t2)).(\lambda (H1: (pc3 c u1 u2)).(let H2 \def H1 in (ex2_ind T (\lambda
473 (t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) (pc3 c t1 t2) (\lambda (x:
474 T).(\lambda (H3: (pr3 c u1 x)).(\lambda (H4: (pr3 c u2 x)).(ty3_unique g c x
475 t1 (ty3_sred_pr3 c u1 x H3 g t1 H) t2 (ty3_sred_pr3 c u2 x H4 g t2 H0)))))
478 theorem ty3_sred_back:
479 \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t0: T).((ty3 g c
480 t1 t0) \to (\forall (t2: T).((pr3 c t1 t2) \to (\forall (t: T).((ty3 g c t2
481 t) \to (ty3 g c t1 t)))))))))
483 \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t0: T).(\lambda
484 (H: (ty3 g c t1 t0)).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).(\lambda
485 (t: T).(\lambda (H1: (ty3 g c t2 t)).(ex_ind T (\lambda (t3: T).(ty3 g c t
486 t3)) (ty3 g c t1 t) (\lambda (x: T).(\lambda (H2: (ty3 g c t x)).(ty3_conv g
487 c t x H2 t1 t0 H (ty3_unique g c t2 t0 (ty3_sred_pr3 c t1 t2 H0 g t0 H) t
488 H1)))) (ty3_correct g c t2 t H1)))))))))).
491 \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c
492 u1 t1) \to (\forall (u2: T).(\forall (t2: T).((ty3 g c u2 t2) \to ((pc3 c u1
493 u2) \to (ty3 g c u1 t2)))))))))
495 \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda
496 (H: (ty3 g c u1 t1)).(\lambda (u2: T).(\lambda (t2: T).(\lambda (H0: (ty3 g c
497 u2 t2)).(\lambda (H1: (pc3 c u1 u2)).(let H2 \def H1 in (ex2_ind T (\lambda
498 (t: T).(pr3 c u1 t)) (\lambda (t: T).(pr3 c u2 t)) (ty3 g c u1 t2) (\lambda
499 (x: T).(\lambda (H3: (pr3 c u1 x)).(\lambda (H4: (pr3 c u2 x)).(ty3_sred_back
500 g c u1 t1 H x H3 t2 (ty3_sred_pr3 c u2 x H4 g t2 H0))))) H2)))))))))).
projects / helm.git / blob