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_1/nf2/fwd.ma".
19 include "basic_1/arity/subst0.ma".
21 lemma arity_nf2_inv_all:
22 \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t
23 a) \to ((nf2 c t) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t
24 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w)))
25 (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) u)))) (ex nat
26 (\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws:
27 TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i)))))
28 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
29 TList).(\lambda (i: nat).(nf2 c (TLRef i)))))))))))
31 \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
32 (arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_:
33 A).((nf2 c0 t0) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0
34 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
35 (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat
36 (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws:
37 TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
38 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
39 TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))))))) (\lambda (c0: C).(\lambda
40 (n: nat).(\lambda (_: (nf2 c0 (TSort n))).(or3_intro1 (ex3_2 T T (\lambda (w:
41 T).(\lambda (u: T).(eq T (TSort n) (THead (Bind Abst) w u)))) (\lambda (w:
42 T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead
43 c0 (Bind Abst) w) u)))) (ex nat (\lambda (n0: nat).(eq T (TSort n) (TSort
44 n0)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (TSort
45 n) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
46 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
47 i))))) (ex_intro nat (\lambda (n0: nat).(eq T (TSort n) (TSort n0))) n
48 (refl_equal T (TSort n))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u:
49 T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr)
50 u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (_: (((nf2 d u)
51 \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind
52 Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 d w))) (\lambda (w:
53 T).(\lambda (u0: T).(nf2 (CHead d (Bind Abst) w) u0)))) (ex nat (\lambda (n:
54 nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0:
55 nat).(eq T u (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws:
56 TList).(\lambda (_: nat).(nfs2 d ws))) (\lambda (_: TList).(\lambda (i0:
57 nat).(nf2 d (TLRef i0))))))))).(\lambda (H3: (nf2 c0 (TLRef
58 i))).(nf2_gen_lref c0 d u i H0 H3 (or3 (ex3_2 T T (\lambda (w: T).(\lambda
59 (u0: T).(eq T (TLRef i) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda
60 (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind
61 Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (TLRef i) (TSort n)))) (ex3_2
62 TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T (TLRef i) (THeads
63 (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0
64 ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef
65 i0)))))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
66 (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u))).(\lambda (a0:
67 A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (_: (((nf2 d u) \to (or3
68 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind Abst) w
69 u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 d w))) (\lambda (w: T).(\lambda
70 (u0: T).(nf2 (CHead d (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T u
71 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T u
72 (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda (_:
73 nat).(nfs2 d ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 d (TLRef
74 i0))))))))).(\lambda (H3: (nf2 c0 (TLRef i))).(or3_intro2 (ex3_2 T T (\lambda
75 (w: T).(\lambda (u0: T).(eq T (TLRef i) (THead (Bind Abst) w u0)))) (\lambda
76 (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2
77 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (TLRef i)
78 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T
79 (TLRef i) (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda
80 (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef
81 i0))))) (ex3_2_intro TList nat (\lambda (ws: TList).(\lambda (i0: nat).(eq T
82 (TLRef i) (THeads (Flat Appl) ws (TLRef i0))))) (\lambda (ws: TList).(\lambda
83 (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i0: nat).(nf2 c0 (TLRef
84 i0)))) TNil i (refl_equal T (TLRef i)) I H3))))))))))) (\lambda (b:
85 B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u:
86 T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((nf2 c0 u)
87 \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind
88 Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
89 T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n:
90 nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
91 nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
92 TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
93 nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda
94 (H3: (arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (_: (((nf2 (CHead c0
95 (Bind b) u) t0) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0
96 (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 (CHead c0
97 (Bind b) u) w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind
98 b) u) (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n))))
99 (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads
100 (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2
101 (CHead c0 (Bind b) u) ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 (CHead
102 c0 (Bind b) u) (TLRef i))))))))).(\lambda (H5: (nf2 c0 (THead (Bind b) u
103 t0))).(B_ind (\lambda (b0: B).((not (eq B b0 Abst)) \to ((arity g (CHead c0
104 (Bind b0) u) t0 a2) \to ((nf2 c0 (THead (Bind b0) u t0)) \to (or3 (ex3_2 T T
105 (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind b0) u t0) (THead (Bind
106 Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
107 T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n:
108 nat).(eq T (THead (Bind b0) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws:
109 TList).(\lambda (i: nat).(eq T (THead (Bind b0) u t0) (THeads (Flat Appl) ws
110 (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda
111 (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))))))) (\lambda (_: (not (eq
112 B Abbr Abst))).(\lambda (_: (arity g (CHead c0 (Bind Abbr) u) t0
113 a2)).(\lambda (H8: (nf2 c0 (THead (Bind Abbr) u t0))).(nf2_gen_abbr c0 u t0
114 H8 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Abbr)
115 u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0
116 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))
117 (ex nat (\lambda (n: nat).(eq T (THead (Bind Abbr) u t0) (TSort n)))) (ex3_2
118 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind Abbr) u
119 t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
120 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
121 i)))))))))) (\lambda (H6: (not (eq B Abst Abst))).(\lambda (_: (arity g
122 (CHead c0 (Bind Abst) u) t0 a2)).(\lambda (_: (nf2 c0 (THead (Bind Abst) u
123 t0))).(let H9 \def (match (H6 (refl_equal B Abst)) in False with []) in
124 H9)))) (\lambda (_: (not (eq B Void Abst))).(\lambda (H7: (arity g (CHead c0
125 (Bind Void) u) t0 a2)).(\lambda (H8: (nf2 c0 (THead (Bind Void) u t0))).(let
126 H9 \def (arity_gen_cvoid g (CHead c0 (Bind Void) u) t0 a2 H7 c0 u O
127 (getl_refl Void c0 u)) in (ex_ind T (\lambda (v: T).(eq T t0 (lift (S O) O
128 v))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind
129 Void) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2
130 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))
131 (ex nat (\lambda (n: nat).(eq T (THead (Bind Void) u t0) (TSort n)))) (ex3_2
132 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind Void) u
133 t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
134 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
135 i)))))) (\lambda (x: T).(\lambda (H10: (eq T t0 (lift (S O) O x))).(let H11
136 \def (eq_ind T t0 (\lambda (t1: T).(nf2 c0 (THead (Bind Void) u t1))) H8
137 (lift (S O) O x) H10) in (eq_ind_r T (lift (S O) O x) (\lambda (t1: T).(or3
138 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Void) u t1)
139 (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
140 (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat
141 (\lambda (n: nat).(eq T (THead (Bind Void) u t1) (TSort n)))) (ex3_2 TList
142 nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind Void) u t1)
143 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
144 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
145 i))))))) (nf2_gen_void c0 u x H11 (or3 (ex3_2 T T (\lambda (w: T).(\lambda
146 (u0: T).(eq T (THead (Bind Void) u (lift (S O) O x)) (THead (Bind Abst) w
147 u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
148 (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
149 (THead (Bind Void) u (lift (S O) O x)) (TSort n)))) (ex3_2 TList nat (\lambda
150 (ws: TList).(\lambda (i: nat).(eq T (THead (Bind Void) u (lift (S O) O x))
151 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
152 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
153 i))))))) t0 H10)))) H9))))) b H0 H3 H5))))))))))))) (\lambda (c0: C).(\lambda
154 (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u (asucc g a1))).(\lambda
155 (_: (((nf2 c0 u) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u
156 (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
157 (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat
158 (\lambda (n: nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws:
159 TList).(\lambda (i: nat).(eq T u (THeads (Flat Appl) ws (TLRef i)))))
160 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
161 TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda
162 (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 a2)).(\lambda (_:
163 (((nf2 (CHead c0 (Bind Abst) u) t0) \to (or3 (ex3_2 T T (\lambda (w:
164 T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w:
165 T).(\lambda (_: T).(nf2 (CHead c0 (Bind Abst) u) w))) (\lambda (w:
166 T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind Abst) u) (Bind Abst) w)
167 u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList nat
168 (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef
169 i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 (CHead c0 (Bind Abst) u)
170 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 (CHead c0 (Bind Abst) u)
171 (TLRef i))))))))).(\lambda (H4: (nf2 c0 (THead (Bind Abst) u t0))).(let H5
172 \def (nf2_gen_abst c0 u t0 H4) in (land_ind (nf2 c0 u) (nf2 (CHead c0 (Bind
173 Abst) u) t0) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead
174 (Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_:
175 T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst)
176 w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) u t0) (TSort
177 n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead
178 (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
179 TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
180 nat).(nf2 c0 (TLRef i)))))) (\lambda (H6: (nf2 c0 u)).(\lambda (H7: (nf2
181 (CHead c0 (Bind Abst) u) t0)).(or3_intro0 (ex3_2 T T (\lambda (w: T).(\lambda
182 (u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda
183 (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2
184 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind
185 Abst) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
186 nat).(eq T (THead (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i)))))
187 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
188 TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro T T (\lambda (w:
189 T).(\lambda (u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w
190 u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
191 (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))) u t0 (refl_equal T (THead (Bind
192 Abst) u t0)) H6 H7)))) H5)))))))))))) (\lambda (c0: C).(\lambda (u:
193 T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((nf2 c0 u)
194 \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind
195 Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
196 T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n:
197 nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
198 nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
199 TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
200 nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda
201 (H2: (arity g c0 t0 (AHead a1 a2))).(\lambda (H3: (((nf2 c0 t0) \to (or3
202 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w
203 u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
204 (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
205 t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
206 t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
207 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
208 i))))))))).(\lambda (H4: (nf2 c0 (THead (Flat Appl) u t0))).(let H5 \def
209 (nf2_gen_flat Appl c0 u t0 H4) in (land_ind (nf2 c0 u) (nf2 c0 t0) (or3
210 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0)
211 (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
212 (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat
213 (\lambda (n: nat).(eq T (THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList
214 nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0)
215 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
216 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
217 i)))))) (\lambda (H6: (nf2 c0 u)).(\lambda (H7: (nf2 c0 t0)).(let H_x \def
218 (H3 H7) in (let H8 \def H_x in (or3_ind (ex3_2 T T (\lambda (w: T).(\lambda
219 (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_:
220 T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst)
221 w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList nat
222 (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef
223 i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
224 TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (or3 (ex3_2 T T (\lambda (w:
225 T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w
226 u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
227 (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
228 (THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws:
229 TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl)
230 ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
231 (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (H9:
232 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w
233 u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
234 (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))).(ex3_2_ind T T (\lambda (w:
235 T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w:
236 T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead
237 c0 (Bind Abst) w) u0))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq
238 T (THead (Flat Appl) u t0) (THead (Bind Abst) w u0)))) (\lambda (w:
239 T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead
240 c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u
241 t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq
242 T (THead (Flat Appl) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
243 TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
244 nat).(nf2 c0 (TLRef i)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H10:
245 (eq T t0 (THead (Bind Abst) x0 x1))).(\lambda (_: (nf2 c0 x0)).(\lambda (_:
246 (nf2 (CHead c0 (Bind Abst) x0) x1)).(let H13 \def (eq_ind T t0 (\lambda (t1:
247 T).(nf2 c0 (THead (Flat Appl) u t1))) H4 (THead (Bind Abst) x0 x1) H10) in
248 (let H14 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2
249 (THead (Bind Abst) x0 x1) H10) in (eq_ind_r T (THead (Bind Abst) x0 x1)
250 (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T
251 (THead (Flat Appl) u t1) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda
252 (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind
253 Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u t1)
254 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
255 (THead (Flat Appl) u t1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
256 TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
257 nat).(nf2 c0 (TLRef i))))))) (nf2_gen_beta c0 u x0 x1 H13 (or3 (ex3_2 T T
258 (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (THead (Bind
259 Abst) x0 x1)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_:
260 T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst)
261 w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (THead (Bind
262 Abst) x0 x1)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
263 nat).(eq T (THead (Flat Appl) u (THead (Bind Abst) x0 x1)) (THeads (Flat
264 Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
265 (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) t0 H10))))))))
266 H9)) (\lambda (H9: (ex nat (\lambda (n: nat).(eq T t0 (TSort n))))).(ex_ind
267 nat (\lambda (n: nat).(eq T t0 (TSort n))) (or3 (ex3_2 T T (\lambda (w:
268 T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w
269 u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
270 (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
271 (THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws:
272 TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl)
273 ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
274 (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x:
275 nat).(\lambda (H10: (eq T t0 (TSort x))).(let H11 \def (eq_ind T t0 (\lambda
276 (t1: T).(nf2 c0 (THead (Flat Appl) u t1))) H4 (TSort x) H10) in (let H12 \def
277 (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 (TSort x)
278 H10) in (eq_ind_r T (TSort x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w:
279 T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t1) (THead (Bind Abst) w
280 u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
281 (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
282 (THead (Flat Appl) u t1) (TSort n)))) (ex3_2 TList nat (\lambda (ws:
283 TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t1) (THeads (Flat Appl)
284 ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
285 (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (let H_x0 \def
286 (leq_gen_head1 g a1 a2 (ASort O x) (arity_gen_sort g c0 x (AHead a1 a2) H12))
287 in (let H13 \def H_x0 in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq
288 g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3:
289 A).(\lambda (a4: A).(eq A (ASort O x) (AHead a3 a4)))) (or3 (ex3_2 T T
290 (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (TSort x)) (THead
291 (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda
292 (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda
293 (n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n)))) (ex3_2 TList nat
294 (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (TSort x))
295 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
296 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
297 i)))))) (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1
298 x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H16: (eq A (ASort O x) (AHead x0
299 x1))).(let H17 \def (eq_ind A (ASort O x) (\lambda (ee: A).(match ee with
300 [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0
301 x1) H16) in (False_ind (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T
302 (THead (Flat Appl) u (TSort x)) (THead (Bind Abst) w u0)))) (\lambda (w:
303 T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead
304 c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u
305 (TSort x)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
306 nat).(eq T (THead (Flat Appl) u (TSort x)) (THeads (Flat Appl) ws (TLRef
307 i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
308 TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) H17))))))) H13))) t0 H10)))))
309 H9)) (\lambda (H9: (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
310 nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
311 TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
312 nat).(nf2 c0 (TLRef i)))))).(ex3_2_ind TList nat (\lambda (ws:
313 TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
314 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
315 TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))) (or3 (ex3_2 T T (\lambda (w:
316 T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w
317 u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
318 (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
319 (THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws:
320 TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl)
321 ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
322 (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x0:
323 TList).(\lambda (x1: nat).(\lambda (H10: (eq T t0 (THeads (Flat Appl) x0
324 (TLRef x1)))).(\lambda (H11: (nfs2 c0 x0)).(\lambda (H12: (nf2 c0 (TLRef
325 x1))).(let H13 \def (eq_ind T t0 (\lambda (t1: T).(nf2 c0 (THead (Flat Appl)
326 u t1))) H4 (THeads (Flat Appl) x0 (TLRef x1)) H10) in (let H14 \def (eq_ind T
327 t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 (THeads (Flat Appl) x0
328 (TLRef x1)) H10) in (eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) (\lambda
329 (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat
330 Appl) u t1) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2
331 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))
332 (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u t1) (TSort n)))) (ex3_2
333 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u
334 t1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
335 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
336 i))))))) (or3_intro2 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead
337 (Flat Appl) u (THeads (Flat Appl) x0 (TLRef x1))) (THead (Bind Abst) w u0))))
338 (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0:
339 T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
340 (THead (Flat Appl) u (THeads (Flat Appl) x0 (TLRef x1))) (TSort n)))) (ex3_2
341 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u
342 (THeads (Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws (TLRef i)))))
343 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
344 TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro TList nat
345 (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (THeads
346 (Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws (TLRef i))))) (\lambda
347 (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
348 nat).(nf2 c0 (TLRef i)))) (TCons u x0) x1 (refl_equal T (THead (Flat Appl) u
349 (THeads (Flat Appl) x0 (TLRef x1)))) (conj (nf2 c0 u) (nfs2 c0 x0) H6 H11)
350 H12)) t0 H10)))))))) H9)) H8))))) H5)))))))))))) (\lambda (c0: C).(\lambda
351 (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g a0))).(\lambda
352 (_: (((nf2 c0 u) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u
353 (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
354 (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat
355 (\lambda (n: nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws:
356 TList).(\lambda (i: nat).(eq T u (THeads (Flat Appl) ws (TLRef i)))))
357 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
358 TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda
359 (_: (arity g c0 t0 a0)).(\lambda (_: (((nf2 c0 t0) \to (or3 (ex3_2 T T
360 (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0))))
361 (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0:
362 T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0
363 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0
364 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
365 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
366 i))))))))).(\lambda (H4: (nf2 c0 (THead (Flat Cast) u t0))).(nf2_gen_cast c0
367 u t0 H4 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat
368 Cast) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2
369 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))
370 (ex nat (\lambda (n: nat).(eq T (THead (Flat Cast) u t0) (TSort n)))) (ex3_2
371 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Cast) u
372 t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
373 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
374 i)))))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda
375 (_: (arity g c0 t0 a1)).(\lambda (H1: (((nf2 c0 t0) \to (or3 (ex3_2 T T
376 (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda
377 (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2
378 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort
379 n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0
380 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
381 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
382 i))))))))).(\lambda (a2: A).(\lambda (_: (leq g a1 a2)).(\lambda (H3: (nf2 c0
383 t0)).(let H_x \def (H1 H3) in (let H4 \def H_x in (or3_ind (ex3_2 T T
384 (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda
385 (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2
386 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort
387 n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0
388 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
389 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
390 i))))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind
391 Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
392 T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n:
393 nat).(eq T t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
394 nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
395 TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
396 nat).(nf2 c0 (TLRef i)))))) (\lambda (H5: (ex3_2 T T (\lambda (w: T).(\lambda
397 (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_:
398 T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w)
399 u))))).(ex3_2_ind T T (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind
400 Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
401 T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u))) (or3 (ex3_2 T T
402 (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda
403 (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2
404 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort
405 n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0
406 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
407 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
408 i)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t0 (THead (Bind
409 Abst) x0 x1))).(\lambda (H7: (nf2 c0 x0)).(\lambda (H8: (nf2 (CHead c0 (Bind
410 Abst) x0) x1)).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t1: T).(or3
411 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind Abst) w
412 u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
413 (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t1
414 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t1
415 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
416 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
417 i))))))) (or3_intro0 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T (THead
418 (Bind Abst) x0 x1) (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_:
419 T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w)
420 u)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) x0 x1) (TSort n))))
421 (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind
422 Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
423 TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
424 nat).(nf2 c0 (TLRef i))))) (ex3_2_intro T T (\lambda (w: T).(\lambda (u:
425 T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w u)))) (\lambda (w:
426 T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead
427 c0 (Bind Abst) w) u))) x0 x1 (refl_equal T (THead (Bind Abst) x0 x1)) H7 H8))
428 t0 H6)))))) H5)) (\lambda (H5: (ex nat (\lambda (n: nat).(eq T t0 (TSort
429 n))))).(ex_ind nat (\lambda (n: nat).(eq T t0 (TSort n))) (or3 (ex3_2 T T
430 (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda
431 (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2
432 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort
433 n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0
434 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
435 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
436 i)))))) (\lambda (x: nat).(\lambda (H6: (eq T t0 (TSort x))).(eq_ind_r T
437 (TSort x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u:
438 T).(eq T t1 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2
439 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u))))
440 (ex nat (\lambda (n: nat).(eq T t1 (TSort n)))) (ex3_2 TList nat (\lambda
441 (ws: TList).(\lambda (i: nat).(eq T t1 (THeads (Flat Appl) ws (TLRef i)))))
442 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
443 TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (or3_intro1 (ex3_2 T T
444 (\lambda (w: T).(\lambda (u: T).(eq T (TSort x) (THead (Bind Abst) w u))))
445 (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u:
446 T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T (TSort
447 x) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
448 (TSort x) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda
449 (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
450 i))))) (ex_intro nat (\lambda (n: nat).(eq T (TSort x) (TSort n))) x
451 (refl_equal T (TSort x)))) t0 H6))) H5)) (\lambda (H5: (ex3_2 TList nat
452 (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef
453 i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
454 TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))).(ex3_2_ind TList nat (\lambda
455 (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
456 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
457 TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))) (or3 (ex3_2 T T (\lambda (w:
458 T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda (w:
459 T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead
460 c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n))))
461 (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads
462 (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0
463 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda
464 (x0: TList).(\lambda (x1: nat).(\lambda (H6: (eq T t0 (THeads (Flat Appl) x0
465 (TLRef x1)))).(\lambda (H7: (nfs2 c0 x0)).(\lambda (H8: (nf2 c0 (TLRef
466 x1))).(eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) (\lambda (t1: T).(or3
467 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind Abst) w
468 u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
469 (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t1
470 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t1
471 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
472 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
473 i))))))) (or3_intro2 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T (THeads
474 (Flat Appl) x0 (TLRef x1)) (THead (Bind Abst) w u)))) (\lambda (w:
475 T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead
476 c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T (THeads (Flat Appl)
477 x0 (TLRef x1)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda
478 (i: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws
479 (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda
480 (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro TList nat
481 (\lambda (ws: TList).(\lambda (i: nat).(eq T (THeads (Flat Appl) x0 (TLRef
482 x1)) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
483 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
484 i)))) x0 x1 (refl_equal T (THeads (Flat Appl) x0 (TLRef x1))) H7 H8)) t0
485 H6)))))) H5)) H4))))))))))) c t a H))))).
projects / helm.git / blob