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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 (* This file was automatically generated: do not edit *********************)
17 include "Basic-1/ty3/arity_props.ma".
19 include "Basic-1/pc3/nf2.ma".
21 include "Basic-1/nf2/fwd.ma".
23 theorem ty3_gen_appl_nf2:
24 \forall (g: G).(\forall (c: C).(\forall (w: T).(\forall (v: T).(\forall (x:
25 T).((ty3 g c (THead (Flat Appl) w v) x) \to (ex4_2 T T (\lambda (u:
26 T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x)))
27 (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
28 (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t:
29 T).(nf2 c (THead (Bind Abst) u t))))))))))
31 \lambda (g: G).(\lambda (c: C).(\lambda (w: T).(\lambda (v: T).(\lambda (x:
32 T).(\lambda (H: (ty3 g c (THead (Flat Appl) w v) x)).(ex3_2_ind T T (\lambda
33 (u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t))
34 x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
35 (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (ex4_2 T T (\lambda (u:
36 T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x)))
37 (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
38 (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t:
39 T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x0: T).(\lambda (x1:
40 T).(\lambda (H0: (pc3 c (THead (Flat Appl) w (THead (Bind Abst) x0 x1))
41 x)).(\lambda (H1: (ty3 g c v (THead (Bind Abst) x0 x1))).(\lambda (H2: (ty3 g
42 c w x0)).(let H_x \def (ty3_correct g c v (THead (Bind Abst) x0 x1) H1) in
43 (let H3 \def H_x in (ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) x0
44 x1) t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl)
45 w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v
46 (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c w u)))
47 (\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t))))) (\lambda
48 (x2: T).(\lambda (H4: (ty3 g c (THead (Bind Abst) x0 x1) x2)).(let H_x0 \def
49 (ty3_correct g c w x0 H2) in (let H5 \def H_x0 in (ex_ind T (\lambda (t:
50 T).(ty3 g c x0 t)) (ex4_2 T T (\lambda (u: T).(\lambda (t: T).(pc3 c (THead
51 (Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u: T).(\lambda (t:
52 T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u: T).(\lambda (_: T).(ty3
53 g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c (THead (Bind Abst) u t)))))
54 (\lambda (x3: T).(\lambda (H6: (ty3 g c x0 x3)).(let H7 \def (ty3_sn3 g c
55 (THead (Bind Abst) x0 x1) x2 H4) in (let H_x1 \def (nf2_sn3 c (THead (Bind
56 Abst) x0 x1) H7) in (let H8 \def H_x1 in (ex2_ind T (\lambda (u: T).(pr3 c
57 (THead (Bind Abst) x0 x1) u)) (\lambda (u: T).(nf2 c u)) (ex4_2 T T (\lambda
58 (u: T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t))
59 x))) (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
60 (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t:
61 T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x4: T).(\lambda (H9: (pr3 c
62 (THead (Bind Abst) x0 x1) x4)).(\lambda (H10: (nf2 c x4)).(let H11 \def
63 (pr3_gen_abst c x0 x1 x4 H9) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2:
64 T).(eq T x4 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_:
65 T).(pr3 c x0 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
66 (u: T).(pr3 (CHead c (Bind b) u) x1 t2))))) (ex4_2 T T (\lambda (u:
67 T).(\lambda (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x)))
68 (\lambda (u: T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t))))
69 (\lambda (u: T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t:
70 T).(nf2 c (THead (Bind Abst) u t))))) (\lambda (x5: T).(\lambda (x6:
71 T).(\lambda (H12: (eq T x4 (THead (Bind Abst) x5 x6))).(\lambda (H13: (pr3 c
72 x0 x5)).(\lambda (H14: ((\forall (b: B).(\forall (u: T).(pr3 (CHead c (Bind
73 b) u) x1 x6))))).(let H15 \def (eq_ind T x4 (\lambda (t: T).(nf2 c t)) H10
74 (THead (Bind Abst) x5 x6) H12) in (let H16 \def (pr3_head_12 c x0 x5 H13
75 (Bind Abst) x1 x6 (H14 Abst x5)) in (ex4_2_intro T T (\lambda (u: T).(\lambda
76 (t: T).(pc3 c (THead (Flat Appl) w (THead (Bind Abst) u t)) x))) (\lambda (u:
77 T).(\lambda (t: T).(ty3 g c v (THead (Bind Abst) u t)))) (\lambda (u:
78 T).(\lambda (_: T).(ty3 g c w u))) (\lambda (u: T).(\lambda (t: T).(nf2 c
79 (THead (Bind Abst) u t)))) x5 x6 (pc3_pr3_conf c (THead (Flat Appl) w (THead
80 (Bind Abst) x0 x1)) x H0 (THead (Flat Appl) w (THead (Bind Abst) x5 x6))
81 (pr3_thin_dx c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 w
82 Appl)) (ty3_conv g c (THead (Bind Abst) x5 x6) x2 (ty3_sred_pr3 c (THead
83 (Bind Abst) x0 x1) (THead (Bind Abst) x5 x6) H16 g x2 H4) v (THead (Bind
84 Abst) x0 x1) H1 (pc3_pr3_r c (THead (Bind Abst) x0 x1) (THead (Bind Abst) x5
85 x6) H16)) (ty3_conv g c x5 x3 (ty3_sred_pr3 c x0 x5 H13 g x3 H6) w x0 H2
86 (pc3_pr3_r c x0 x5 H13)) H15)))))))) H11))))) H8)))))) H5))))) H3))))))))
87 (ty3_gen_appl g c w v x H))))))).
92 theorem ty3_inv_lref_nf2_pc3:
93 \forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (i: nat).((ty3 g c
94 (TLRef i) u1) \to ((nf2 c (TLRef i)) \to (\forall (u2: T).((nf2 c u2) \to
95 ((pc3 c u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))))
97 \lambda (g: G).(\lambda (c: C).(\lambda (u1: T).(\lambda (i: nat).(\lambda
98 (H: (ty3 g c (TLRef i) u1)).(insert_eq T (TLRef i) (\lambda (t: T).(ty3 g c t
99 u1)) (\lambda (t: T).((nf2 c t) \to (\forall (u2: T).((nf2 c u2) \to ((pc3 c
100 u1 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))) (\lambda
101 (y: T).(\lambda (H0: (ty3 g c y u1)).(ty3_ind g (\lambda (c0: C).(\lambda (t:
102 T).(\lambda (t0: T).((eq T t (TLRef i)) \to ((nf2 c0 t) \to (\forall (u2:
103 T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift
104 (S i) O u)))))))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t:
105 T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: (((eq T t2 (TLRef i)) \to ((nf2
106 c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to (ex T
107 (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (u: T).(\lambda
108 (t1: T).(\lambda (H3: (ty3 g c0 u t1)).(\lambda (H4: (((eq T u (TLRef i)) \to
109 ((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t1 u2) \to (ex T
110 (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (pc3 c0
111 t1 t2)).(\lambda (H6: (eq T u (TLRef i))).(\lambda (H7: (nf2 c0 u)).(\lambda
112 (u2: T).(\lambda (H8: (nf2 c0 u2)).(\lambda (H9: (pc3 c0 t2 u2)).(let H10
113 \def (eq_ind T u (\lambda (t0: T).(nf2 c0 t0)) H7 (TLRef i) H6) in (let H11
114 \def (eq_ind T u (\lambda (t0: T).((eq T t0 (TLRef i)) \to ((nf2 c0 t0) \to
115 (\forall (u3: T).((nf2 c0 u3) \to ((pc3 c0 t1 u3) \to (ex T (\lambda (u0:
116 T).(eq T u3 (lift (S i) O u0)))))))))) H4 (TLRef i) H6) in (let H12 \def
117 (eq_ind T u (\lambda (t0: T).(ty3 g c0 t0 t1)) H3 (TLRef i) H6) in (let H_y
118 \def (H11 (refl_equal T (TLRef i)) H10 u2 H8) in (H_y (pc3_t t2 c0 t1 H5 u2
119 H9))))))))))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (H1: (eq
120 T (TSort m) (TLRef i))).(\lambda (_: (nf2 c0 (TSort m))).(\lambda (u2:
121 T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (TSort (next g m))
122 u2)).(let H5 \def (eq_ind T (TSort m) (\lambda (ee: T).(match ee in T return
123 (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
124 \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef i) H1) in
125 (False_ind (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u)))) H5)))))))))
126 (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
127 (H1: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g
128 d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2:
129 T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S
130 i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5:
131 (nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (H7:
132 (pc3 c0 (lift (S n) O t) u2)).(let H8 \def (f_equal T nat (\lambda (e:
133 T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n |
134 (TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef
135 i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0)
136 O t) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0
137 (TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl
138 n0 c0 (CHead d (Bind Abbr) u))) H1 i H8) in (nf2_gen_lref c0 d u i H11 H10
139 (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))))))))))))))
140 (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
141 (H1: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g
142 d u t)).(\lambda (_: (((eq T u (TLRef i)) \to ((nf2 d u) \to (\forall (u2:
143 T).((nf2 d u2) \to ((pc3 d t u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S
144 i) O u0))))))))))).(\lambda (H4: (eq T (TLRef n) (TLRef i))).(\lambda (H5:
145 (nf2 c0 (TLRef n))).(\lambda (u2: T).(\lambda (H6: (nf2 c0 u2)).(\lambda (H7:
146 (pc3 c0 (lift (S n) O u) u2)).(let H8 \def (f_equal T nat (\lambda (e:
147 T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n |
148 (TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef
149 i) H4) in (let H9 \def (eq_ind nat n (\lambda (n0: nat).(pc3 c0 (lift (S n0)
150 O u) u2)) H7 i H8) in (let H10 \def (eq_ind nat n (\lambda (n0: nat).(nf2 c0
151 (TLRef n0))) H5 i H8) in (let H11 \def (eq_ind nat n (\lambda (n0: nat).(getl
152 n0 c0 (CHead d (Bind Abst) u))) H1 i H8) in (let H_y \def (pc3_nf2_unfold c0
153 (lift (S i) O u) u2 H9 H6) in (let H12 \def (pr3_gen_lift c0 u u2 (S i) O H_y
154 d (getl_drop Abst c0 d u i H11)) in (ex2_ind T (\lambda (t2: T).(eq T u2
155 (lift (S i) O t2))) (\lambda (t2: T).(pr3 d u t2)) (ex T (\lambda (u0: T).(eq
156 T u2 (lift (S i) O u0)))) (\lambda (x: T).(\lambda (H13: (eq T u2 (lift (S i)
157 O x))).(\lambda (_: (pr3 d u x)).(eq_ind_r T (lift (S i) O x) (\lambda (t0:
158 T).(ex T (\lambda (u0: T).(eq T t0 (lift (S i) O u0))))) (ex_intro T (\lambda
159 (u0: T).(eq T (lift (S i) O x) (lift (S i) O u0))) x (refl_equal T (lift (S
160 i) O x))) u2 H13)))) H12)))))))))))))))))))) (\lambda (c0: C).(\lambda (u:
161 T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (_: (((eq T u (TLRef
162 i)) \to ((nf2 c0 u) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t u2) \to
163 (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (b:
164 B).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g (CHead c0 (Bind b)
165 u) t1 t2)).(\lambda (_: (((eq T t1 (TLRef i)) \to ((nf2 (CHead c0 (Bind b) u)
166 t1) \to (\forall (u2: T).((nf2 (CHead c0 (Bind b) u) u2) \to ((pc3 (CHead c0
167 (Bind b) u) t2 u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O
168 u0))))))))))).(\lambda (H5: (eq T (THead (Bind b) u t1) (TLRef i))).(\lambda
169 (_: (nf2 c0 (THead (Bind b) u t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0
170 u2)).(\lambda (_: (pc3 c0 (THead (Bind b) u t2) u2)).(let H9 \def (eq_ind T
171 (THead (Bind b) u t1) (\lambda (ee: T).(match ee in T return (\lambda (_:
172 T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
173 (THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T
174 (\lambda (u0: T).(eq T u2 (lift (S i) O u0)))) H9))))))))))))))))) (\lambda
175 (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda
176 (_: (((eq T w (TLRef i)) \to ((nf2 c0 w) \to (\forall (u2: T).((nf2 c0 u2)
177 \to ((pc3 c0 u u2) \to (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O
178 u0))))))))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead
179 (Bind Abst) u t))).(\lambda (_: (((eq T v (TLRef i)) \to ((nf2 c0 v) \to
180 (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 (THead (Bind Abst) u t) u2) \to
181 (ex T (\lambda (u0: T).(eq T u2 (lift (S i) O u0))))))))))).(\lambda (H5: (eq
182 T (THead (Flat Appl) w v) (TLRef i))).(\lambda (_: (nf2 c0 (THead (Flat Appl)
183 w v))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0 (THead
184 (Flat Appl) w (THead (Bind Abst) u t)) u2)).(let H9 \def (eq_ind T (THead
185 (Flat Appl) w v) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
186 with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
187 _) \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u0:
188 T).(eq T u2 (lift (S i) O u0)))) H9)))))))))))))))) (\lambda (c0: C).(\lambda
189 (t1: T).(\lambda (t2: T).(\lambda (_: (ty3 g c0 t1 t2)).(\lambda (_: (((eq T
190 t1 (TLRef i)) \to ((nf2 c0 t1) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0
191 t2 u2) \to (ex T (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda
192 (t0: T).(\lambda (_: (ty3 g c0 t2 t0)).(\lambda (_: (((eq T t2 (TLRef i)) \to
193 ((nf2 c0 t2) \to (\forall (u2: T).((nf2 c0 u2) \to ((pc3 c0 t0 u2) \to (ex T
194 (\lambda (u: T).(eq T u2 (lift (S i) O u))))))))))).(\lambda (H5: (eq T
195 (THead (Flat Cast) t2 t1) (TLRef i))).(\lambda (_: (nf2 c0 (THead (Flat Cast)
196 t2 t1))).(\lambda (u2: T).(\lambda (_: (nf2 c0 u2)).(\lambda (_: (pc3 c0
197 (THead (Flat Cast) t0 t2) u2)).(let H9 \def (eq_ind T (THead (Flat Cast) t2
198 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
199 _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
200 \Rightarrow True])) I (TLRef i) H5) in (False_ind (ex T (\lambda (u: T).(eq T
201 u2 (lift (S i) O u)))) H9))))))))))))))) c y u1 H0))) H))))).
206 theorem ty3_inv_lref_nf2:
207 \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (i: nat).((ty3 g c
208 (TLRef i) u) \to ((nf2 c (TLRef i)) \to ((nf2 c u) \to (ex T (\lambda (u0:
209 T).(eq T u (lift (S i) O u0))))))))))
211 \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
212 (H: (ty3 g c (TLRef i) u)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1:
213 (nf2 c u)).(ty3_inv_lref_nf2_pc3 g c u i H H0 u H1 (pc3_refl c u)))))))).
218 theorem ty3_inv_appls_lref_nf2:
219 \forall (g: G).(\forall (c: C).(\forall (vs: TList).(\forall (u1:
220 T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) vs (TLRef i)) u1) \to
221 ((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S
222 i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) vs (lift (S i) O u))
225 \lambda (g: G).(\lambda (c: C).(\lambda (vs: TList).(TList_ind (\lambda (t:
226 TList).(\forall (u1: T).(\forall (i: nat).((ty3 g c (THeads (Flat Appl) t
227 (TLRef i)) u1) \to ((nf2 c (TLRef i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u:
228 T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t
229 (lift (S i) O u)) u1))))))))) (\lambda (u1: T).(\lambda (i: nat).(\lambda (H:
230 (ty3 g c (TLRef i) u1)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (nf2 c
231 u1)).(let H_x \def (ty3_inv_lref_nf2 g c u1 i H H0 H1) in (let H2 \def H_x in
232 (ex_ind T (\lambda (u0: T).(eq T u1 (lift (S i) O u0))) (ex2 T (\lambda (u:
233 T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) u1)))
234 (\lambda (x: T).(\lambda (H3: (eq T u1 (lift (S i) O x))).(let H4 \def
235 (eq_ind T u1 (\lambda (t: T).(nf2 c t)) H1 (lift (S i) O x) H3) in (eq_ind_r
236 T (lift (S i) O x) (\lambda (t: T).(ex2 T (\lambda (u: T).(nf2 c (lift (S i)
237 O u))) (\lambda (u: T).(pc3 c (lift (S i) O u) t)))) (ex_intro2 T (\lambda
238 (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3 c (lift (S i) O u)
239 (lift (S i) O x))) x H4 (pc3_refl c (lift (S i) O x))) u1 H3)))) H2))))))))
240 (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: ((\forall (u1: T).(\forall
241 (i: nat).((ty3 g c (THeads (Flat Appl) t0 (TLRef i)) u1) \to ((nf2 c (TLRef
242 i)) \to ((nf2 c u1) \to (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u)))
243 (\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i) O u))
244 u1)))))))))).(\lambda (u1: T).(\lambda (i: nat).(\lambda (H0: (ty3 g c (THead
245 (Flat Appl) t (THeads (Flat Appl) t0 (TLRef i))) u1)).(\lambda (H1: (nf2 c
246 (TLRef i))).(\lambda (_: (nf2 c u1)).(let H_x \def (ty3_gen_appl_nf2 g c t
247 (THeads (Flat Appl) t0 (TLRef i)) u1 H0) in (let H3 \def H_x in (ex4_2_ind T
248 T (\lambda (u: T).(\lambda (t1: T).(pc3 c (THead (Flat Appl) t (THead (Bind
249 Abst) u t1)) u1))) (\lambda (u: T).(\lambda (t1: T).(ty3 g c (THeads (Flat
250 Appl) t0 (TLRef i)) (THead (Bind Abst) u t1)))) (\lambda (u: T).(\lambda (_:
251 T).(ty3 g c t u))) (\lambda (u: T).(\lambda (t1: T).(nf2 c (THead (Bind Abst)
252 u t1)))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u:
253 T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u)))
254 u1))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (pc3 c (THead (Flat
255 Appl) t (THead (Bind Abst) x0 x1)) u1)).(\lambda (H5: (ty3 g c (THeads (Flat
256 Appl) t0 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c t
257 x0)).(\lambda (H7: (nf2 c (THead (Bind Abst) x0 x1))).(let H8 \def
258 (nf2_gen_abst c x0 x1 H7) in (land_ind (nf2 c x0) (nf2 (CHead c (Bind Abst)
259 x0) x1) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O u))) (\lambda (u: T).(pc3
260 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O u))) u1)))
261 (\lambda (H9: (nf2 c x0)).(\lambda (H10: (nf2 (CHead c (Bind Abst) x0)
262 x1)).(let H_y \def (H (THead (Bind Abst) x0 x1) i H5 H1) in (let H11 \def
263 (H_y (nf2_abst_shift c x0 H9 x1 H10)) in (ex2_ind T (\lambda (u: T).(nf2 c
264 (lift (S i) O u))) (\lambda (u: T).(pc3 c (THeads (Flat Appl) t0 (lift (S i)
265 O u)) (THead (Bind Abst) x0 x1))) (ex2 T (\lambda (u: T).(nf2 c (lift (S i) O
266 u))) (\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift
267 (S i) O u))) u1))) (\lambda (x: T).(\lambda (H12: (nf2 c (lift (S i) O
268 x))).(\lambda (H13: (pc3 c (THeads (Flat Appl) t0 (lift (S i) O x)) (THead
269 (Bind Abst) x0 x1))).(ex_intro2 T (\lambda (u: T).(nf2 c (lift (S i) O u)))
270 (\lambda (u: T).(pc3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S
271 i) O u))) u1)) x H12 (pc3_t (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) c
272 (THead (Flat Appl) t (THeads (Flat Appl) t0 (lift (S i) O x))) (pc3_thin_dx c
273 (THeads (Flat Appl) t0 (lift (S i) O x)) (THead (Bind Abst) x0 x1) H13 t
274 Appl) u1 H4))))) H11))))) H8)))))))) H3))))))))))) vs))).
279 theorem ty3_inv_lref_lref_nf2:
280 \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (j: nat).((ty3 g c
281 (TLRef i) (TLRef j)) \to ((nf2 c (TLRef i)) \to ((nf2 c (TLRef j)) \to (lt i
284 \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (j: nat).(\lambda
285 (H: (ty3 g c (TLRef i) (TLRef j))).(\lambda (H0: (nf2 c (TLRef i))).(\lambda
286 (H1: (nf2 c (TLRef j))).(let H_x \def (ty3_inv_lref_nf2 g c (TLRef j) i H H0
287 H1) in (let H2 \def H_x in (ex_ind T (\lambda (u0: T).(eq T (TLRef j) (lift
288 (S i) O u0))) (lt i j) (\lambda (x: T).(\lambda (H3: (eq T (TLRef j) (lift (S
289 i) O x))).(let H_x0 \def (lift_gen_lref x O (S i) j H3) in (let H4 \def H_x0
290 in (or_ind (land (lt j O) (eq T x (TLRef j))) (land (le (S i) j) (eq T x
291 (TLRef (minus j (S i))))) (lt i j) (\lambda (H5: (land (lt j O) (eq T x
292 (TLRef j)))).(land_ind (lt j O) (eq T x (TLRef j)) (lt i j) (\lambda (H6: (lt
293 j O)).(\lambda (_: (eq T x (TLRef j))).(lt_x_O j H6 (lt i j)))) H5)) (\lambda
294 (H5: (land (le (S i) j) (eq T x (TLRef (minus j (S i)))))).(land_ind (le (S
295 i) j) (eq T x (TLRef (minus j (S i)))) (lt i j) (\lambda (H6: (le (S i)
296 j)).(\lambda (_: (eq T x (TLRef (minus j (S i))))).H6)) H5)) H4)))))