1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 include "LambdaDelta-1/nf2/fwd.ma".
19 include "LambdaDelta-1/arity/subst0.ma".
21 theorem 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 return (\lambda
124 (_: False).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead
125 (Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_:
126 T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst)
127 w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) u t0) (TSort
128 n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead
129 (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
130 TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
131 nat).(nf2 c0 (TLRef i))))))) with []) in H9)))) (\lambda (_: (not (eq B Void
132 Abst))).(\lambda (H7: (arity g (CHead c0 (Bind Void) u) t0 a2)).(\lambda (H8:
133 (nf2 c0 (THead (Bind Void) u t0))).(let H9 \def (arity_gen_cvoid g (CHead c0
134 (Bind Void) u) t0 a2 H7 c0 u O (getl_refl Void c0 u)) in (ex_ind T (\lambda
135 (v: T).(eq T t0 (lift (S O) O v))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda
136 (u0: T).(eq T (THead (Bind Void) u t0) (THead (Bind Abst) w u0)))) (\lambda
137 (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2
138 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind
139 Void) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
140 nat).(eq T (THead (Bind Void) u t0) (THeads (Flat Appl) ws (TLRef i)))))
141 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
142 TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x: T).(\lambda
143 (H10: (eq T t0 (lift (S O) O x))).(let H11 \def (eq_ind T t0 (\lambda (t1:
144 T).(nf2 c0 (THead (Bind Void) u t1))) H8 (lift (S O) O x) H10) in (eq_ind_r T
145 (lift (S O) O x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda
146 (u0: T).(eq T (THead (Bind Void) u t1) (THead (Bind Abst) w u0)))) (\lambda
147 (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2
148 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind
149 Void) u t1) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
150 nat).(eq T (THead (Bind Void) u t1) (THeads (Flat Appl) ws (TLRef i)))))
151 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
152 TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (nf2_gen_void c0 u x H11
153 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Void) u
154 (lift (S O) O x)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_:
155 T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst)
156 w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Void) u (lift (S O) O
157 x)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq
158 T (THead (Bind Void) u (lift (S O) O x)) (THeads (Flat Appl) ws (TLRef i)))))
159 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
160 TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) t0 H10)))) H9))))) b H0 H3
161 H5))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda
162 (_: (arity g c0 u (asucc g a1))).(\lambda (_: (((nf2 c0 u) \to (or3 (ex3_2 T
163 T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind Abst) w u0))))
164 (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0:
165 T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T u
166 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T u
167 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
168 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
169 i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0
170 (Bind Abst) u) t0 a2)).(\lambda (_: (((nf2 (CHead c0 (Bind Abst) u) t0) \to
171 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst)
172 w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 (CHead c0 (Bind Abst) u) w)))
173 (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind Abst) u) (Bind
174 Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList
175 nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws
176 (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 (CHead c0 (Bind
177 Abst) u) ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 (CHead c0 (Bind
178 Abst) u) (TLRef i))))))))).(\lambda (H4: (nf2 c0 (THead (Bind Abst) u
179 t0))).(let H5 \def (nf2_gen_abst c0 u t0 H4) in (and_ind (nf2 c0 u) (nf2
180 (CHead c0 (Bind Abst) u) t0) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0:
181 T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda (w:
182 T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead
183 c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) u
184 t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq
185 T (THead (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
186 TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
187 nat).(nf2 c0 (TLRef i)))))) (\lambda (H6: (nf2 c0 u)).(\lambda (H7: (nf2
188 (CHead c0 (Bind Abst) u) t0)).(or3_intro0 (ex3_2 T T (\lambda (w: T).(\lambda
189 (u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w u0)))) (\lambda
190 (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2
191 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind
192 Abst) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
193 nat).(eq T (THead (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i)))))
194 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
195 TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro T T (\lambda (w:
196 T).(\lambda (u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w
197 u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
198 (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))) u t0 (refl_equal T (THead (Bind
199 Abst) u t0)) H6 H7)))) H5)))))))))))) (\lambda (c0: C).(\lambda (u:
200 T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((nf2 c0 u)
201 \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind
202 Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
203 T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n:
204 nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
205 nat).(eq T u (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
206 TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
207 nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda
208 (H2: (arity g c0 t0 (AHead a1 a2))).(\lambda (H3: (((nf2 c0 t0) \to (or3
209 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w
210 u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
211 (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
212 t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
213 t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
214 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
215 i))))))))).(\lambda (H4: (nf2 c0 (THead (Flat Appl) u t0))).(let H5 \def
216 (nf2_gen_flat Appl c0 u t0 H4) in (and_ind (nf2 c0 u) (nf2 c0 t0) (or3 (ex3_2
217 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead
218 (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda
219 (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda
220 (n: nat).(eq T (THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat
221 (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads
222 (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0
223 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda
224 (H6: (nf2 c0 u)).(\lambda (H7: (nf2 c0 t0)).(let H_x \def (H3 H7) in (let H8
225 \def H_x in (or3_ind (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0
226 (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
227 (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat
228 (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws:
229 TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
230 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
231 TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (or3 (ex3_2 T T (\lambda (w:
232 T).(\lambda (u0: T).(eq T (THead (Flat Appl) u 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)))) (ex nat (\lambda (n: nat).(eq T
235 (THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws:
236 TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl)
237 ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
238 (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (H9:
239 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w
240 u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
241 (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))).(ex3_2_ind T T (\lambda (w:
242 T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w:
243 T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead
244 c0 (Bind Abst) w) u0))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq
245 T (THead (Flat Appl) u t0) (THead (Bind Abst) w u0)))) (\lambda (w:
246 T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead
247 c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u
248 t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq
249 T (THead (Flat Appl) u t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
250 TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
251 nat).(nf2 c0 (TLRef i)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H10:
252 (eq T t0 (THead (Bind Abst) x0 x1))).(\lambda (_: (nf2 c0 x0)).(\lambda (_:
253 (nf2 (CHead c0 (Bind Abst) x0) x1)).(let H13 \def (eq_ind T t0 (\lambda (t1:
254 T).(nf2 c0 (THead (Flat Appl) u t1))) H4 (THead (Bind Abst) x0 x1) H10) in
255 (let H14 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2
256 (THead (Bind Abst) x0 x1) H10) in (eq_ind_r T (THead (Bind Abst) x0 x1)
257 (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T
258 (THead (Flat Appl) u t1) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda
259 (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind
260 Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u t1)
261 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
262 (THead (Flat Appl) u t1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
263 TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
264 nat).(nf2 c0 (TLRef i))))))) (nf2_gen_beta c0 u x0 x1 H13 (or3 (ex3_2 T T
265 (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (THead (Bind
266 Abst) x0 x1)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_:
267 T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst)
268 w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (THead (Bind
269 Abst) x0 x1)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
270 nat).(eq T (THead (Flat Appl) u (THead (Bind Abst) x0 x1)) (THeads (Flat
271 Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
272 (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) t0 H10))))))))
273 H9)) (\lambda (H9: (ex nat (\lambda (n: nat).(eq T t0 (TSort n))))).(ex_ind
274 nat (\lambda (n: nat).(eq T t0 (TSort n))) (or3 (ex3_2 T T (\lambda (w:
275 T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w
276 u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
277 (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
278 (THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws:
279 TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl)
280 ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
281 (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x:
282 nat).(\lambda (H10: (eq T t0 (TSort x))).(let H11 \def (eq_ind T t0 (\lambda
283 (t1: T).(nf2 c0 (THead (Flat Appl) u t1))) H4 (TSort x) H10) in (let H12 \def
284 (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 (TSort x)
285 H10) in (eq_ind_r T (TSort x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w:
286 T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t1) (THead (Bind Abst) w
287 u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
288 (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
289 (THead (Flat Appl) u t1) (TSort n)))) (ex3_2 TList nat (\lambda (ws:
290 TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t1) (THeads (Flat Appl)
291 ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
292 (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (let H13 \def
293 (match (arity_gen_sort g c0 x (AHead a1 a2) H12) in leq return (\lambda (a0:
294 A).(\lambda (a3: A).(\lambda (_: (leq ? a0 a3)).((eq A a0 (AHead a1 a2)) \to
295 ((eq A a3 (ASort O x)) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0:
296 T).(eq T (THead (Flat Appl) u (TSort x)) (THead (Bind Abst) w u0)))) (\lambda
297 (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2
298 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat
299 Appl) u (TSort x)) (TSort n)))) (ex3_2 TList nat (\lambda (ws:
300 TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (TSort x)) (THeads (Flat
301 Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
302 (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))))))) with
303 [(leq_sort h1 h2 n1 n2 k H13) \Rightarrow (\lambda (H14: (eq A (ASort h1 n1)
304 (AHead a1 a2))).(\lambda (H15: (eq A (ASort h2 n2) (ASort O x))).((let H16
305 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e in A return (\lambda
306 (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow
307 False])) I (AHead a1 a2) H14) in (False_ind ((eq A (ASort h2 n2) (ASort O x))
308 \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (or3
309 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (TSort
310 x)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
311 (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat
312 (\lambda (n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n)))) (ex3_2
313 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u
314 (TSort x)) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda
315 (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
316 i)))))))) H16)) H15 H13))) | (leq_head a0 a3 H13 a4 a5 H14) \Rightarrow
317 (\lambda (H15: (eq A (AHead a0 a4) (AHead a1 a2))).(\lambda (H16: (eq A
318 (AHead a3 a5) (ASort O x))).((let H17 \def (f_equal A A (\lambda (e:
319 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 |
320 (AHead _ a6) \Rightarrow a6])) (AHead a0 a4) (AHead a1 a2) H15) in ((let H18
321 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
322 with [(ASort _ _) \Rightarrow a0 | (AHead a6 _) \Rightarrow a6])) (AHead a0
323 a4) (AHead a1 a2) H15) in (eq_ind A a1 (\lambda (a6: A).((eq A a4 a2) \to
324 ((eq A (AHead a3 a5) (ASort O x)) \to ((leq g a6 a3) \to ((leq g a4 a5) \to
325 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u
326 (TSort x)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2
327 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))
328 (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n))))
329 (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat
330 Appl) u (TSort x)) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
331 TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
332 nat).(nf2 c0 (TLRef i))))))))))) (\lambda (H19: (eq A a4 a2)).(eq_ind A a2
333 (\lambda (a6: A).((eq A (AHead a3 a5) (ASort O x)) \to ((leq g a1 a3) \to
334 ((leq g a6 a5) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T
335 (THead (Flat Appl) u (TSort x)) (THead (Bind Abst) w u0)))) (\lambda (w:
336 T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead
337 c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u
338 (TSort x)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
339 nat).(eq T (THead (Flat Appl) u (TSort x)) (THeads (Flat Appl) ws (TLRef
340 i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
341 TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))))))) (\lambda (H20: (eq A
342 (AHead a3 a5) (ASort O x))).(let H21 \def (eq_ind A (AHead a3 a5) (\lambda
343 (e: A).(match e in A return (\lambda (_: A).Prop) with [(ASort _ _)
344 \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O x) H20) in
345 (False_ind ((leq g a1 a3) \to ((leq g a2 a5) \to (or3 (ex3_2 T T (\lambda (w:
346 T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (TSort x)) (THead (Bind Abst)
347 w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
348 T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n:
349 nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n)))) (ex3_2 TList nat
350 (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (TSort x))
351 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
352 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
353 i)))))))) H21))) a4 (sym_eq A a4 a2 H19))) a0 (sym_eq A a0 a1 H18))) H17))
354 H16 H13 H14)))]) in (H13 (refl_equal A (AHead a1 a2)) (refl_equal A (ASort O
355 x)))) t0 H10))))) H9)) (\lambda (H9: (ex3_2 TList nat (\lambda (ws:
356 TList).(\lambda (i: nat).(eq T t0 (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)))))).(ex3_2_ind TList nat (\lambda
359 (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
360 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
361 TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))) (or3 (ex3_2 T T (\lambda (w:
362 T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w
363 u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
364 (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
365 (THead (Flat Appl) u t0) (TSort n)))) (ex3_2 TList nat (\lambda (ws:
366 TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl)
367 ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws)))
368 (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda (x0:
369 TList).(\lambda (x1: nat).(\lambda (H10: (eq T t0 (THeads (Flat Appl) x0
370 (TLRef x1)))).(\lambda (H11: (nfs2 c0 x0)).(\lambda (H12: (nf2 c0 (TLRef
371 x1))).(let H13 \def (eq_ind T t0 (\lambda (t1: T).(nf2 c0 (THead (Flat Appl)
372 u t1))) H4 (THeads (Flat Appl) x0 (TLRef x1)) H10) in (let H14 \def (eq_ind T
373 t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 (THeads (Flat Appl) x0
374 (TLRef x1)) H10) in (eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) (\lambda
375 (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat
376 Appl) u t1) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2
377 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))
378 (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u t1) (TSort n)))) (ex3_2
379 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u
380 t1) (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))))))) (or3_intro2 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead
383 (Flat Appl) u (THeads (Flat Appl) x0 (TLRef x1))) (THead (Bind Abst) w u0))))
384 (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0:
385 T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
386 (THead (Flat Appl) u (THeads (Flat Appl) x0 (TLRef x1))) (TSort n)))) (ex3_2
387 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u
388 (THeads (Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws (TLRef i)))))
389 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
390 TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro TList nat
391 (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Appl) u (THeads
392 (Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws (TLRef i))))) (\lambda
393 (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
394 nat).(nf2 c0 (TLRef i)))) (TCons u x0) x1 (refl_equal T (THead (Flat Appl) u
395 (THeads (Flat Appl) x0 (TLRef x1)))) (conj (nf2 c0 u) (nfs2 c0 x0) H6 H11)
396 H12)) t0 H10)))))))) H9)) H8))))) H5)))))))))))) (\lambda (c0: C).(\lambda
397 (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g a0))).(\lambda
398 (_: (((nf2 c0 u) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u
399 (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
400 (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat
401 (\lambda (n: nat).(eq T u (TSort n)))) (ex3_2 TList nat (\lambda (ws:
402 TList).(\lambda (i: nat).(eq T u (THeads (Flat Appl) ws (TLRef i)))))
403 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
404 TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))))).(\lambda (t0: T).(\lambda
405 (_: (arity g c0 t0 a0)).(\lambda (_: (((nf2 c0 t0) \to (or3 (ex3_2 T T
406 (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0))))
407 (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0:
408 T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0
409 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0
410 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
411 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
412 i))))))))).(\lambda (H4: (nf2 c0 (THead (Flat Cast) u t0))).(nf2_gen_cast c0
413 u t0 H4 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat
414 Cast) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2
415 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))
416 (ex nat (\lambda (n: nat).(eq T (THead (Flat Cast) u t0) (TSort n)))) (ex3_2
417 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Flat Cast) u
418 t0) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
419 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
420 i)))))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda
421 (_: (arity g c0 t0 a1)).(\lambda (H1: (((nf2 c0 t0) \to (or3 (ex3_2 T T
422 (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda
423 (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2
424 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort
425 n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0
426 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
427 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
428 i))))))))).(\lambda (a2: A).(\lambda (_: (leq g a1 a2)).(\lambda (H3: (nf2 c0
429 t0)).(let H_x \def (H1 H3) in (let H4 \def H_x in (or3_ind (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))))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind
437 Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
438 T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n:
439 nat).(eq T t0 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i:
440 nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
441 TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
442 nat).(nf2 c0 (TLRef i)))))) (\lambda (H5: (ex3_2 T T (\lambda (w: T).(\lambda
443 (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_:
444 T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w)
445 u))))).(ex3_2_ind T T (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind
446 Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
447 T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u))) (or3 (ex3_2 T T
448 (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda
449 (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2
450 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort
451 n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0
452 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
453 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
454 i)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t0 (THead (Bind
455 Abst) x0 x1))).(\lambda (H7: (nf2 c0 x0)).(\lambda (H8: (nf2 (CHead c0 (Bind
456 Abst) x0) x1)).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t1: T).(or3
457 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind Abst) w
458 u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
459 (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t1
460 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t1
461 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
462 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
463 i))))))) (or3_intro0 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T (THead
464 (Bind Abst) x0 x1) (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_:
465 T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w)
466 u)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Abst) x0 x1) (TSort n))))
467 (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T (THead (Bind
468 Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws:
469 TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i:
470 nat).(nf2 c0 (TLRef i))))) (ex3_2_intro T T (\lambda (w: T).(\lambda (u:
471 T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w u)))) (\lambda (w:
472 T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead
473 c0 (Bind Abst) w) u))) x0 x1 (refl_equal T (THead (Bind Abst) x0 x1)) H7 H8))
474 t0 H6)))))) H5)) (\lambda (H5: (ex nat (\lambda (n: nat).(eq T t0 (TSort
475 n))))).(ex_ind nat (\lambda (n: nat).(eq T t0 (TSort n))) (or3 (ex3_2 T T
476 (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda
477 (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2
478 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort
479 n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0
480 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
481 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
482 i)))))) (\lambda (x: nat).(\lambda (H6: (eq T t0 (TSort x))).(eq_ind_r T
483 (TSort x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u:
484 T).(eq T t1 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2
485 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u))))
486 (ex nat (\lambda (n: nat).(eq T t1 (TSort n)))) (ex3_2 TList nat (\lambda
487 (ws: TList).(\lambda (i: nat).(eq T t1 (THeads (Flat Appl) ws (TLRef i)))))
488 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
489 TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))))) (or3_intro1 (ex3_2 T T
490 (\lambda (w: T).(\lambda (u: T).(eq T (TSort x) (THead (Bind Abst) w u))))
491 (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u:
492 T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T (TSort
493 x) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T
494 (TSort x) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda
495 (_: nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
496 i))))) (ex_intro nat (\lambda (n: nat).(eq T (TSort x) (TSort n))) x
497 (refl_equal T (TSort x)))) t0 H6))) H5)) (\lambda (H5: (ex3_2 TList nat
498 (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef
499 i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
500 TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))).(ex3_2_ind TList nat (\lambda
501 (ws: TList).(\lambda (i: nat).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))
502 (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda (_:
503 TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))) (or3 (ex3_2 T T (\lambda (w:
504 T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda (w:
505 T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead
506 c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n))))
507 (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t0 (THeads
508 (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0
509 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i)))))) (\lambda
510 (x0: TList).(\lambda (x1: nat).(\lambda (H6: (eq T t0 (THeads (Flat Appl) x0
511 (TLRef x1)))).(\lambda (H7: (nfs2 c0 x0)).(\lambda (H8: (nf2 c0 (TLRef
512 x1))).(eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) (\lambda (t1: T).(or3
513 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind Abst) w
514 u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
515 (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t1
516 (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t1
517 (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
518 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
519 i))))))) (or3_intro2 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T (THeads
520 (Flat Appl) x0 (TLRef x1)) (THead (Bind Abst) w u)))) (\lambda (w:
521 T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead
522 c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T (THeads (Flat Appl)
523 x0 (TLRef x1)) (TSort n)))) (ex3_2 TList nat (\lambda (ws: TList).(\lambda
524 (i: nat).(eq T (THeads (Flat Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws
525 (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c0 ws))) (\lambda
526 (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef i))))) (ex3_2_intro TList nat
527 (\lambda (ws: TList).(\lambda (i: nat).(eq T (THeads (Flat Appl) x0 (TLRef
528 x1)) (THeads (Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_:
529 nat).(nfs2 c0 ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c0 (TLRef
530 i)))) x0 x1 (refl_equal T (THeads (Flat Appl) x0 (TLRef x1))) H7 H8)) t0
531 H6)))))) H5)) H4))))))))))) c t a H))))).
projects / helm.git / blob