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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/arity/defs.ma".
19 include "basic_1/leq/asucc.ma".
21 include "basic_1/getl/drop.ma".
23 implied rec lemma arity_ind (g: G) (P: (C \to (T \to (A \to Prop)))) (f:
24 (\forall (c: C).(\forall (n: nat).(P c (TSort n) (ASort O n))))) (f0:
25 (\forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c
26 (CHead d (Bind Abbr) u)) \to (\forall (a: A).((arity g d u a) \to ((P d u a)
27 \to (P c (TLRef i) a)))))))))) (f1: (\forall (c: C).(\forall (d: C).(\forall
28 (u: T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) u)) \to (\forall (a:
29 A).((arity g d u (asucc g a)) \to ((P d u (asucc g a)) \to (P c (TLRef i)
30 a)))))))))) (f2: (\forall (b: B).((not (eq B b Abst)) \to (\forall (c:
31 C).(\forall (u: T).(\forall (a1: A).((arity g c u a1) \to ((P c u a1) \to
32 (\forall (t: T).(\forall (a2: A).((arity g (CHead c (Bind b) u) t a2) \to ((P
33 (CHead c (Bind b) u) t a2) \to (P c (THead (Bind b) u t) a2))))))))))))) (f3:
34 (\forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u (asucc g a1))
35 \to ((P c u (asucc g a1)) \to (\forall (t: T).(\forall (a2: A).((arity g
36 (CHead c (Bind Abst) u) t a2) \to ((P (CHead c (Bind Abst) u) t a2) \to (P c
37 (THead (Bind Abst) u t) (AHead a1 a2)))))))))))) (f4: (\forall (c:
38 C).(\forall (u: T).(\forall (a1: A).((arity g c u a1) \to ((P c u a1) \to
39 (\forall (t: T).(\forall (a2: A).((arity g c t (AHead a1 a2)) \to ((P c t
40 (AHead a1 a2)) \to (P c (THead (Flat Appl) u t) a2))))))))))) (f5: (\forall
41 (c: C).(\forall (u: T).(\forall (a: A).((arity g c u (asucc g a)) \to ((P c u
42 (asucc g a)) \to (\forall (t: T).((arity g c t a) \to ((P c t a) \to (P c
43 (THead (Flat Cast) u t) a)))))))))) (f6: (\forall (c: C).(\forall (t:
44 T).(\forall (a1: A).((arity g c t a1) \to ((P c t a1) \to (\forall (a2:
45 A).((leq g a1 a2) \to (P c t a2))))))))) (c: C) (t: T) (a: A) (a0: arity g c
46 t a) on a0: P c t a \def match a0 with [(arity_sort c0 n) \Rightarrow (f c0
47 n) | (arity_abbr c0 d u i g0 a1 a2) \Rightarrow (f0 c0 d u i g0 a1 a2
48 ((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) d u a1 a2)) | (arity_abst c0 d u i g0
49 a1 a2) \Rightarrow (f1 c0 d u i g0 a1 a2 ((arity_ind g P f f0 f1 f2 f3 f4 f5
50 f6) d u (asucc g a1) a2)) | (arity_bind b n c0 u a1 a2 t0 a3 a4) \Rightarrow
51 (f2 b n c0 u a1 a2 ((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) c0 u a1 a2) t0 a3
52 a4 ((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) (CHead c0 (Bind b) u) t0 a3 a4)) |
53 (arity_head c0 u a1 a2 t0 a3 a4) \Rightarrow (f3 c0 u a1 a2 ((arity_ind g P f
54 f0 f1 f2 f3 f4 f5 f6) c0 u (asucc g a1) a2) t0 a3 a4 ((arity_ind g P f f0 f1
55 f2 f3 f4 f5 f6) (CHead c0 (Bind Abst) u) t0 a3 a4)) | (arity_appl c0 u a1 a2
56 t0 a3 a4) \Rightarrow (f4 c0 u a1 a2 ((arity_ind g P f f0 f1 f2 f3 f4 f5 f6)
57 c0 u a1 a2) t0 a3 a4 ((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) c0 t0 (AHead a1
58 a3) a4)) | (arity_cast c0 u a1 a2 t0 a3) \Rightarrow (f5 c0 u a1 a2
59 ((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) c0 u (asucc g a1) a2) t0 a3
60 ((arity_ind g P f f0 f1 f2 f3 f4 f5 f6) c0 t0 a1 a3)) | (arity_repl c0 t0 a1
61 a2 a3 l) \Rightarrow (f6 c0 t0 a1 a2 ((arity_ind g P f f0 f1 f2 f3 f4 f5 f6)
65 \forall (g: G).(\forall (c: C).(\forall (n: nat).(\forall (a: A).((arity g c
66 (TSort n) a) \to (leq g a (ASort O n))))))
68 \lambda (g: G).(\lambda (c: C).(\lambda (n: nat).(\lambda (a: A).(\lambda
69 (H: (arity g c (TSort n) a)).(insert_eq T (TSort n) (\lambda (t: T).(arity g
70 c t a)) (\lambda (_: T).(leq g a (ASort O n))) (\lambda (y: T).(\lambda (H0:
71 (arity g c y a)).(arity_ind g (\lambda (_: C).(\lambda (t: T).(\lambda (a0:
72 A).((eq T t (TSort n)) \to (leq g a0 (ASort O n)))))) (\lambda (_:
73 C).(\lambda (n0: nat).(\lambda (H1: (eq T (TSort n0) (TSort n))).(let H2 \def
74 (f_equal T nat (\lambda (e: T).(match e with [(TSort n1) \Rightarrow n1 |
75 (TLRef _) \Rightarrow n0 | (THead _ _ _) \Rightarrow n0])) (TSort n0) (TSort
76 n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(leq g (ASort O n1) (ASort O
77 n))) (leq_refl g (ASort O n)) n0 H2))))) (\lambda (c0: C).(\lambda (d:
78 C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind
79 Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (_: (((eq
80 T u (TSort n)) \to (leq g a0 (ASort O n))))).(\lambda (H4: (eq T (TLRef i)
81 (TSort n))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with
82 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
83 \Rightarrow False])) I (TSort n) H4) in (False_ind (leq g a0 (ASort O n))
84 H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
85 nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u))).(\lambda (a0:
86 A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (_: (((eq T u (TSort n))
87 \to (leq g (asucc g a0) (ASort O n))))).(\lambda (H4: (eq T (TLRef i) (TSort
88 n))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with [(TSort
89 _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
90 False])) I (TSort n) H4) in (False_ind (leq g a0 (ASort O n)) H5)))))))))))
91 (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0: C).(\lambda
92 (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((eq T
93 u (TSort n)) \to (leq g a1 (ASort O n))))).(\lambda (t: T).(\lambda (a2:
94 A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t a2)).(\lambda (_: (((eq T t
95 (TSort n)) \to (leq g a2 (ASort O n))))).(\lambda (H6: (eq T (THead (Bind b)
96 u t) (TSort n))).(let H7 \def (eq_ind T (THead (Bind b) u t) (\lambda (ee:
97 T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
98 | (THead _ _ _) \Rightarrow True])) I (TSort n) H6) in (False_ind (leq g a2
99 (ASort O n)) H7)))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1:
100 A).(\lambda (_: (arity g c0 u (asucc g a1))).(\lambda (_: (((eq T u (TSort
101 n)) \to (leq g (asucc g a1) (ASort O n))))).(\lambda (t: T).(\lambda (a2:
102 A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t a2)).(\lambda (_: (((eq T
103 t (TSort n)) \to (leq g a2 (ASort O n))))).(\lambda (H5: (eq T (THead (Bind
104 Abst) u t) (TSort n))).(let H6 \def (eq_ind T (THead (Bind Abst) u t)
105 (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
106 \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in
107 (False_ind (leq g (AHead a1 a2) (ASort O n)) H6)))))))))))) (\lambda (c0:
108 C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda
109 (_: (((eq T u (TSort n)) \to (leq g a1 (ASort O n))))).(\lambda (t:
110 T).(\lambda (a2: A).(\lambda (_: (arity g c0 t (AHead a1 a2))).(\lambda (_:
111 (((eq T t (TSort n)) \to (leq g (AHead a1 a2) (ASort O n))))).(\lambda (H5:
112 (eq T (THead (Flat Appl) u t) (TSort n))).(let H6 \def (eq_ind T (THead (Flat
113 Appl) u t) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False |
114 (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n)
115 H5) in (False_ind (leq g a2 (ASort O n)) H6)))))))))))) (\lambda (c0:
116 C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g
117 a0))).(\lambda (_: (((eq T u (TSort n)) \to (leq g (asucc g a0) (ASort O
118 n))))).(\lambda (t: T).(\lambda (_: (arity g c0 t a0)).(\lambda (_: (((eq T t
119 (TSort n)) \to (leq g a0 (ASort O n))))).(\lambda (H5: (eq T (THead (Flat
120 Cast) u t) (TSort n))).(let H6 \def (eq_ind T (THead (Flat Cast) u t)
121 (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
122 \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in
123 (False_ind (leq g a0 (ASort O n)) H6))))))))))) (\lambda (c0: C).(\lambda (t:
124 T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t a1)).(\lambda (H2: (((eq T t
125 (TSort n)) \to (leq g a1 (ASort O n))))).(\lambda (a2: A).(\lambda (H3: (leq
126 g a1 a2)).(\lambda (H4: (eq T t (TSort n))).(let H5 \def (f_equal T T
127 (\lambda (e: T).e) t (TSort n) H4) in (let H6 \def (eq_ind T t (\lambda (t0:
128 T).((eq T t0 (TSort n)) \to (leq g a1 (ASort O n)))) H2 (TSort n) H5) in (let
129 H7 \def (eq_ind T t (\lambda (t0: T).(arity g c0 t0 a1)) H1 (TSort n) H5) in
130 (leq_trans g a2 a1 (leq_sym g a1 a2 H3) (ASort O n) (H6 (refl_equal T (TSort
131 n))))))))))))))) c y a H0))) H))))).
133 lemma arity_gen_lref:
134 \forall (g: G).(\forall (c: C).(\forall (i: nat).(\forall (a: A).((arity g c
135 (TLRef i) a) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c
136 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a))))
137 (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind Abst)
138 u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a))))))))))
140 \lambda (g: G).(\lambda (c: C).(\lambda (i: nat).(\lambda (a: A).(\lambda
141 (H: (arity g c (TLRef i) a)).(insert_eq T (TLRef i) (\lambda (t: T).(arity g
142 c t a)) (\lambda (_: T).(or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl
143 i c (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
144 a)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c (CHead d (Bind
145 Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a)))))))
146 (\lambda (y: T).(\lambda (H0: (arity g c y a)).(arity_ind g (\lambda (c0:
147 C).(\lambda (t: T).(\lambda (a0: A).((eq T t (TLRef i)) \to (or (ex2_2 C T
148 (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u))))
149 (\lambda (d: C).(\lambda (u: T).(arity g d u a0)))) (ex2_2 C T (\lambda (d:
150 C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
151 C).(\lambda (u: T).(arity g d u (asucc g a0)))))))))) (\lambda (c0:
152 C).(\lambda (n: nat).(\lambda (H1: (eq T (TSort n) (TLRef i))).(let H2 \def
153 (eq_ind T (TSort n) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow
154 True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
155 (TLRef i) H1) in (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u:
156 T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u:
157 T).(arity g d u (ASort O n))))) (ex2_2 C T (\lambda (d: C).(\lambda (u:
158 T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u:
159 T).(arity g d u (asucc g (ASort O n))))))) H2))))) (\lambda (c0: C).(\lambda
160 (d: C).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H1: (getl i0 c0 (CHead d
161 (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H2: (arity g d u a0)).(\lambda
162 (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d0: C).(\lambda (u0:
163 T).(getl i d (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0:
164 T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i
165 d (CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0
166 u0 (asucc g a0))))))))).(\lambda (H4: (eq T (TLRef i0) (TLRef i))).(let H5
167 \def (f_equal T nat (\lambda (e: T).(match e with [(TSort _) \Rightarrow i0 |
168 (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0])) (TLRef i0) (TLRef
169 i) H4) in (let H6 \def (eq_ind nat i0 (\lambda (n: nat).(getl n c0 (CHead d
170 (Bind Abbr) u))) H1 i H5) in (or_introl (ex2_2 C T (\lambda (d0: C).(\lambda
171 (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda
172 (u0: T).(arity g d0 u0 a0)))) (ex2_2 C T (\lambda (d0: C).(\lambda (u0:
173 T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0:
174 T).(arity g d0 u0 (asucc g a0))))) (ex2_2_intro C T (\lambda (d0: C).(\lambda
175 (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda
176 (u0: T).(arity g d0 u0 a0))) d u H6 H2))))))))))))) (\lambda (c0: C).(\lambda
177 (d: C).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H1: (getl i0 c0 (CHead d
178 (Bind Abst) u))).(\lambda (a0: A).(\lambda (H2: (arity g d u (asucc g
179 a0))).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d0:
180 C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abbr) u0)))) (\lambda (d0:
181 C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))) (ex2_2 C T (\lambda (d0:
182 C).(\lambda (u0: T).(getl i d (CHead d0 (Bind Abst) u0)))) (\lambda (d0:
183 C).(\lambda (u0: T).(arity g d0 u0 (asucc g (asucc g a0)))))))))).(\lambda
184 (H4: (eq T (TLRef i0) (TLRef i))).(let H5 \def (f_equal T nat (\lambda (e:
185 T).(match e with [(TSort _) \Rightarrow i0 | (TLRef n) \Rightarrow n | (THead
186 _ _ _) \Rightarrow i0])) (TLRef i0) (TLRef i) H4) in (let H6 \def (eq_ind nat
187 i0 (\lambda (n: nat).(getl n c0 (CHead d (Bind Abst) u))) H1 i H5) in
188 (or_intror (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0
189 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a0))))
190 (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst)
191 u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0)))))
192 (ex2_2_intro C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind
193 Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 (asucc g a0))))
194 d u H6 H2))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b
195 Abst))).(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity
196 g c0 u a1)).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d:
197 C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d:
198 C).(\lambda (u0: T).(arity g d u0 a1)))) (ex2_2 C T (\lambda (d: C).(\lambda
199 (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0:
200 T).(arity g d u0 (asucc g a1))))))))).(\lambda (t: T).(\lambda (a2:
201 A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t a2)).(\lambda (_: (((eq T t
202 (TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i (CHead
203 c0 (Bind b) u) (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0:
204 T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i
205 (CHead c0 (Bind b) u) (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda
206 (u0: T).(arity g d u0 (asucc g a2))))))))).(\lambda (H6: (eq T (THead (Bind
207 b) u t) (TLRef i))).(let H7 \def (eq_ind T (THead (Bind b) u t) (\lambda (ee:
208 T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
209 | (THead _ _ _) \Rightarrow True])) I (TLRef i) H6) in (False_ind (or (ex2_2
210 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0))))
211 (\lambda (d: C).(\lambda (u0: T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d:
212 C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d:
213 C).(\lambda (u0: T).(arity g d u0 (asucc g a2)))))) H7)))))))))))))) (\lambda
214 (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u (asucc g
215 a1))).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d:
216 C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d:
217 C).(\lambda (u0: T).(arity g d u0 (asucc g a1))))) (ex2_2 C T (\lambda (d:
218 C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d:
219 C).(\lambda (u0: T).(arity g d u0 (asucc g (asucc g a1)))))))))).(\lambda (t:
220 T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t
221 a2)).(\lambda (_: (((eq T t (TLRef i)) \to (or (ex2_2 C T (\lambda (d:
222 C).(\lambda (u0: T).(getl i (CHead c0 (Bind Abst) u) (CHead d (Bind Abbr)
223 u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 a2)))) (ex2_2 C T
224 (\lambda (d: C).(\lambda (u0: T).(getl i (CHead c0 (Bind Abst) u) (CHead d
225 (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g
226 a2))))))))).(\lambda (H5: (eq T (THead (Bind Abst) u t) (TLRef i))).(let H6
227 \def (eq_ind T (THead (Bind Abst) u t) (\lambda (ee: T).(match ee with
228 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
229 \Rightarrow True])) I (TLRef i) H5) in (False_ind (or (ex2_2 C T (\lambda (d:
230 C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d:
231 C).(\lambda (u0: T).(arity g d u0 (AHead a1 a2))))) (ex2_2 C T (\lambda (d:
232 C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d:
233 C).(\lambda (u0: T).(arity g d u0 (asucc g (AHead a1 a2))))))) H6))))))))))))
234 (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u
235 a1)).(\lambda (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d:
236 C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d:
237 C).(\lambda (u0: T).(arity g d u0 a1)))) (ex2_2 C T (\lambda (d: C).(\lambda
238 (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0:
239 T).(arity g d u0 (asucc g a1))))))))).(\lambda (t: T).(\lambda (a2:
240 A).(\lambda (_: (arity g c0 t (AHead a1 a2))).(\lambda (_: (((eq T t (TLRef
241 i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d
242 (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (AHead a1
243 a2))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind
244 Abst) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g (AHead
245 a1 a2)))))))))).(\lambda (H5: (eq T (THead (Flat Appl) u t) (TLRef i))).(let
246 H6 \def (eq_ind T (THead (Flat Appl) u t) (\lambda (ee: T).(match ee with
247 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
248 \Rightarrow True])) I (TLRef i) H5) in (False_ind (or (ex2_2 C T (\lambda (d:
249 C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d:
250 C).(\lambda (u0: T).(arity g d u0 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda
251 (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0:
252 T).(arity g d u0 (asucc g a2)))))) H6)))))))))))) (\lambda (c0: C).(\lambda
253 (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g a0))).(\lambda
254 (_: (((eq T u (TLRef i)) \to (or (ex2_2 C T (\lambda (d: C).(\lambda (u0:
255 T).(getl i c0 (CHead d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0:
256 T).(arity g d u0 (asucc g a0))))) (ex2_2 C T (\lambda (d: C).(\lambda (u0:
257 T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d: C).(\lambda (u0:
258 T).(arity g d u0 (asucc g (asucc g a0)))))))))).(\lambda (t: T).(\lambda (_:
259 (arity g c0 t a0)).(\lambda (_: (((eq T t (TLRef i)) \to (or (ex2_2 C T
260 (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abbr) u0))))
261 (\lambda (d: C).(\lambda (u0: T).(arity g d u0 a0)))) (ex2_2 C T (\lambda (d:
262 C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst) u0)))) (\lambda (d:
263 C).(\lambda (u0: T).(arity g d u0 (asucc g a0))))))))).(\lambda (H5: (eq T
264 (THead (Flat Cast) u t) (TLRef i))).(let H6 \def (eq_ind T (THead (Flat Cast)
265 u t) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
266 \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i) H5) in
267 (False_ind (or (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead
268 d (Bind Abbr) u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 a0))))
269 (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c0 (CHead d (Bind Abst)
270 u0)))) (\lambda (d: C).(\lambda (u0: T).(arity g d u0 (asucc g a0))))))
271 H6))))))))))) (\lambda (c0: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H1:
272 (arity g c0 t a1)).(\lambda (H2: (((eq T t (TLRef i)) \to (or (ex2_2 C T
273 (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u))))
274 (\lambda (d: C).(\lambda (u: T).(arity g d u a1)))) (ex2_2 C T (\lambda (d:
275 C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
276 C).(\lambda (u: T).(arity g d u (asucc g a1))))))))).(\lambda (a2:
277 A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t (TLRef i))).(let H5
278 \def (f_equal T T (\lambda (e: T).e) t (TLRef i) H4) in (let H6 \def (eq_ind
279 T t (\lambda (t0: T).((eq T t0 (TLRef i)) \to (or (ex2_2 C T (\lambda (d:
280 C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d:
281 C).(\lambda (u: T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: C).(\lambda
282 (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u:
283 T).(arity g d u (asucc g a1)))))))) H2 (TLRef i) H5) in (let H7 \def (eq_ind
284 T t (\lambda (t0: T).(arity g c0 t0 a1)) H1 (TLRef i) H5) in (let H8 \def (H6
285 (refl_equal T (TLRef i))) in (or_ind (ex2_2 C T (\lambda (d: C).(\lambda (u:
286 T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u:
287 T).(arity g d u a1)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
288 (CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
289 (asucc g a1))))) (or (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
290 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
291 a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind
292 Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a2))))))
293 (\lambda (H9: (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d
294 (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
295 a1))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d
296 (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a1))) (or
297 (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr)
298 u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a2)))) (ex2_2 C T (\lambda
299 (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
300 C).(\lambda (u: T).(arity g d u (asucc g a2)))))) (\lambda (x0: C).(\lambda
301 (x1: T).(\lambda (H10: (getl i c0 (CHead x0 (Bind Abbr) x1))).(\lambda (H11:
302 (arity g x0 x1 a1)).(or_introl (ex2_2 C T (\lambda (d: C).(\lambda (u:
303 T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u:
304 T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
305 (CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
306 (asucc g a2))))) (ex2_2_intro C T (\lambda (d: C).(\lambda (u: T).(getl i c0
307 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u a2)))
308 x0 x1 H10 (arity_repl g x0 x1 a1 H11 a2 H3))))))) H9)) (\lambda (H9: (ex2_2 C
309 T (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u))))
310 (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a1)))))).(ex2_2_ind C T
311 (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u))))
312 (\lambda (d: C).(\lambda (u: T).(arity g d u (asucc g a1)))) (or (ex2_2 C T
313 (\lambda (d: C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abbr) u))))
314 (\lambda (d: C).(\lambda (u: T).(arity g d u a2)))) (ex2_2 C T (\lambda (d:
315 C).(\lambda (u: T).(getl i c0 (CHead d (Bind Abst) u)))) (\lambda (d:
316 C).(\lambda (u: T).(arity g d u (asucc g a2)))))) (\lambda (x0: C).(\lambda
317 (x1: T).(\lambda (H10: (getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (H11:
318 (arity g x0 x1 (asucc g a1))).(or_intror (ex2_2 C T (\lambda (d: C).(\lambda
319 (u: T).(getl i c0 (CHead d (Bind Abbr) u)))) (\lambda (d: C).(\lambda (u:
320 T).(arity g d u a2)))) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl i c0
321 (CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
322 (asucc g a2))))) (ex2_2_intro C T (\lambda (d: C).(\lambda (u: T).(getl i c0
323 (CHead d (Bind Abst) u)))) (\lambda (d: C).(\lambda (u: T).(arity g d u
324 (asucc g a2)))) x0 x1 H10 (arity_repl g x0 x1 (asucc g a1) H11 (asucc g a2)
325 (asucc_repl g a1 a2 H3)))))))) H9)) H8))))))))))))) c y a H0))) H))))).
327 lemma arity_gen_bind:
328 \forall (b: B).((not (eq B b Abst)) \to (\forall (g: G).(\forall (c:
329 C).(\forall (u: T).(\forall (t: T).(\forall (a2: A).((arity g c (THead (Bind
330 b) u t) a2) \to (ex2 A (\lambda (a1: A).(arity g c u a1)) (\lambda (_:
331 A).(arity g (CHead c (Bind b) u) t a2))))))))))
333 \lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda (g: G).(\lambda
334 (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a2: A).(\lambda (H0: (arity
335 g c (THead (Bind b) u t) a2)).(insert_eq T (THead (Bind b) u t) (\lambda (t0:
336 T).(arity g c t0 a2)) (\lambda (_: T).(ex2 A (\lambda (a1: A).(arity g c u
337 a1)) (\lambda (_: A).(arity g (CHead c (Bind b) u) t a2)))) (\lambda (y:
338 T).(\lambda (H1: (arity g c y a2)).(arity_ind g (\lambda (c0: C).(\lambda
339 (t0: T).(\lambda (a: A).((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda
340 (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t
341 a))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H2: (eq T (TSort n)
342 (THead (Bind b) u t))).(let H3 \def (eq_ind T (TSort n) (\lambda (ee:
343 T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
344 (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t) H2) in (False_ind
345 (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0
346 (Bind b) u) t (ASort O n)))) H3))))) (\lambda (c0: C).(\lambda (d:
347 C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind
348 Abbr) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 a)).(\lambda (_: (((eq
349 T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g d u a1))
350 (\lambda (_: A).(arity g (CHead d (Bind b) u) t a)))))).(\lambda (H5: (eq T
351 (TLRef i) (THead (Bind b) u t))).(let H6 \def (eq_ind T (TLRef i) (\lambda
352 (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow
353 True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u t) H5) in
354 (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity
355 g (CHead c0 (Bind b) u) t a))) H6))))))))))) (\lambda (c0: C).(\lambda (d:
356 C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind
357 Abst) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 (asucc g a))).(\lambda
358 (_: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1: A).(arity g d u
359 a1)) (\lambda (_: A).(arity g (CHead d (Bind b) u) t (asucc g
360 a))))))).(\lambda (H5: (eq T (TLRef i) (THead (Bind b) u t))).(let H6 \def
361 (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow
362 False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
363 (THead (Bind b) u t) H5) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u
364 a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a))) H6)))))))))))
365 (\lambda (b0: B).(\lambda (H2: (not (eq B b0 Abst))).(\lambda (c0:
366 C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H3: (arity g c0 u0
367 a1)).(\lambda (H4: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3:
368 A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t
369 a1)))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (H5: (arity g (CHead c0
370 (Bind b0) u0) t0 a0)).(\lambda (H6: (((eq T t0 (THead (Bind b) u t)) \to (ex2
371 A (\lambda (a3: A).(arity g (CHead c0 (Bind b0) u0) u a3)) (\lambda (_:
372 A).(arity g (CHead (CHead c0 (Bind b0) u0) (Bind b) u) t a0)))))).(\lambda
373 (H7: (eq T (THead (Bind b0) u0 t0) (THead (Bind b) u t))).(let H8 \def
374 (f_equal T B (\lambda (e: T).(match e with [(TSort _) \Rightarrow b0 | (TLRef
375 _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k with [(Bind b1)
376 \Rightarrow b1 | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0 t0) (THead
377 (Bind b) u t) H7) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e with
378 [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _)
379 \Rightarrow t1])) (THead (Bind b0) u0 t0) (THead (Bind b) u t) H7) in ((let
380 H10 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0
381 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) (THead (Bind
382 b0) u0 t0) (THead (Bind b) u t) H7) in (\lambda (H11: (eq T u0 u)).(\lambda
383 (H12: (eq B b0 b)).(let H13 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1
384 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind
385 b0) u0) u a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind b0) u0) (Bind
386 b) u) t a0))))) H6 t H10) in (let H14 \def (eq_ind T t0 (\lambda (t1:
387 T).(arity g (CHead c0 (Bind b0) u0) t1 a0)) H5 t H10) in (let H15 \def
388 (eq_ind T u0 (\lambda (t1: T).((eq T t (THead (Bind b) u t)) \to (ex2 A
389 (\lambda (a3: A).(arity g (CHead c0 (Bind b0) t1) u a3)) (\lambda (_:
390 A).(arity g (CHead (CHead c0 (Bind b0) t1) (Bind b) u) t a0))))) H13 u H11)
391 in (let H16 \def (eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind b0)
392 t1) t a0)) H14 u H11) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).((eq T
393 t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3))
394 (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a1))))) H4 u H11) in (let
395 H18 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 a1)) H3 u H11) in (let
396 H19 \def (eq_ind B b0 (\lambda (b1: B).((eq T t (THead (Bind b) u t)) \to
397 (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind b1) u) u a3)) (\lambda (_:
398 A).(arity g (CHead (CHead c0 (Bind b1) u) (Bind b) u) t a0))))) H15 b H12) in
399 (let H20 \def (eq_ind B b0 (\lambda (b1: B).(arity g (CHead c0 (Bind b1) u) t
400 a0)) H16 b H12) in (let H21 \def (eq_ind B b0 (\lambda (b1: B).(not (eq B b1
401 Abst))) H2 b H12) in (ex_intro2 A (\lambda (a3: A).(arity g c0 u a3))
402 (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0)) a1 H18 H20)))))))))))))
403 H9)) H8)))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1:
404 A).(\lambda (H2: (arity g c0 u0 (asucc g a1))).(\lambda (H3: (((eq T u0
405 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda
406 (_: A).(arity g (CHead c0 (Bind b) u) t (asucc g a1))))))).(\lambda (t0:
407 T).(\lambda (a0: A).(\lambda (H4: (arity g (CHead c0 (Bind Abst) u0) t0
408 a0)).(\lambda (H5: (((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3:
409 A).(arity g (CHead c0 (Bind Abst) u0) u a3)) (\lambda (_: A).(arity g (CHead
410 (CHead c0 (Bind Abst) u0) (Bind b) u) t a0)))))).(\lambda (H6: (eq T (THead
411 (Bind Abst) u0 t0) (THead (Bind b) u t))).(let H7 \def (f_equal T B (\lambda
412 (e: T).(match e with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst
413 | (THead k _ _) \Rightarrow (match k with [(Bind b0) \Rightarrow b0 | (Flat
414 _) \Rightarrow Abst])])) (THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6)
415 in ((let H8 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _)
416 \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _) \Rightarrow t1]))
417 (THead (Bind Abst) u0 t0) (THead (Bind b) u t) H6) in ((let H9 \def (f_equal
418 T T (\lambda (e: T).(match e with [(TSort _) \Rightarrow t0 | (TLRef _)
419 \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1])) (THead (Bind Abst) u0 t0)
420 (THead (Bind b) u t) H6) in (\lambda (H10: (eq T u0 u)).(\lambda (H11: (eq B
421 Abst b)).(let H12 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Bind
422 b) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u0) u
423 a3)) (\lambda (_: A).(arity g (CHead (CHead c0 (Bind Abst) u0) (Bind b) u) t
424 a0))))) H5 t H9) in (let H13 \def (eq_ind T t0 (\lambda (t1: T).(arity g
425 (CHead c0 (Bind Abst) u0) t1 a0)) H4 t H9) in (let H14 \def (eq_ind T u0
426 (\lambda (t1: T).((eq T t (THead (Bind b) u t)) \to (ex2 A (\lambda (a3:
427 A).(arity g (CHead c0 (Bind Abst) t1) u a3)) (\lambda (_: A).(arity g (CHead
428 (CHead c0 (Bind Abst) t1) (Bind b) u) t a0))))) H12 u H10) in (let H15 \def
429 (eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind Abst) t1) t a0)) H13 u
430 H10) in (let H16 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Bind b)
431 u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g
432 (CHead c0 (Bind b) u) t (asucc g a1)))))) H3 u H10) in (let H17 \def (eq_ind
433 T u0 (\lambda (t1: T).(arity g c0 t1 (asucc g a1))) H2 u H10) in (let H18
434 \def (eq_ind_r B b (\lambda (b0: B).((eq T t (THead (Bind b0) u t)) \to (ex2
435 A (\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) u a3)) (\lambda (_:
436 A).(arity g (CHead (CHead c0 (Bind Abst) u) (Bind b0) u) t a0))))) H14 Abst
437 H11) in (let H19 \def (eq_ind_r B b (\lambda (b0: B).((eq T u (THead (Bind
438 b0) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_:
439 A).(arity g (CHead c0 (Bind b0) u) t (asucc g a1)))))) H16 Abst H11) in (let
440 H20 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H Abst H11) in
441 (eq_ind B Abst (\lambda (b0: B).(ex2 A (\lambda (a3: A).(arity g c0 u a3))
442 (\lambda (_: A).(arity g (CHead c0 (Bind b0) u) t (AHead a1 a0))))) (let H21
443 \def (match (H20 (refl_equal B Abst)) in False with []) in H21) b
444 H11))))))))))))) H8)) H7)))))))))))) (\lambda (c0: C).(\lambda (u0:
445 T).(\lambda (a1: A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0
446 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda
447 (_: A).(arity g (CHead c0 (Bind b) u) t a1)))))).(\lambda (t0: T).(\lambda
448 (a0: A).(\lambda (_: (arity g c0 t0 (AHead a1 a0))).(\lambda (_: (((eq T t0
449 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda
450 (_: A).(arity g (CHead c0 (Bind b) u) t (AHead a1 a0))))))).(\lambda (H6: (eq
451 T (THead (Flat Appl) u0 t0) (THead (Bind b) u t))).(let H7 \def (eq_ind T
452 (THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee with [(TSort _)
453 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
454 (match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I
455 (THead (Bind b) u t) H6) in (False_ind (ex2 A (\lambda (a3: A).(arity g c0 u
456 a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0))) H7))))))))))))
457 (\lambda (c0: C).(\lambda (u0: T).(\lambda (a: A).(\lambda (_: (arity g c0 u0
458 (asucc g a))).(\lambda (_: (((eq T u0 (THead (Bind b) u t)) \to (ex2 A
459 (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind
460 b) u) t (asucc g a))))))).(\lambda (t0: T).(\lambda (_: (arity g c0 t0
461 a)).(\lambda (_: (((eq T t0 (THead (Bind b) u t)) \to (ex2 A (\lambda (a1:
462 A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t
463 a)))))).(\lambda (H6: (eq T (THead (Flat Cast) u0 t0) (THead (Bind b) u
464 t))).(let H7 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee: T).(match
465 ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k
466 _ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _)
467 \Rightarrow True])])) I (THead (Bind b) u t) H6) in (False_ind (ex2 A
468 (\lambda (a1: A).(arity g c0 u a1)) (\lambda (_: A).(arity g (CHead c0 (Bind
469 b) u) t a))) H7))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1:
470 A).(\lambda (H2: (arity g c0 t0 a1)).(\lambda (H3: (((eq T t0 (THead (Bind b)
471 u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g
472 (CHead c0 (Bind b) u) t a1)))))).(\lambda (a0: A).(\lambda (H4: (leq g a1
473 a0)).(\lambda (H5: (eq T t0 (THead (Bind b) u t))).(let H6 \def (f_equal T T
474 (\lambda (e: T).e) t0 (THead (Bind b) u t) H5) in (let H7 \def (eq_ind T t0
475 (\lambda (t1: T).((eq T t1 (THead (Bind b) u t)) \to (ex2 A (\lambda (a3:
476 A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t
477 a1))))) H3 (THead (Bind b) u t) H6) in (let H8 \def (eq_ind T t0 (\lambda
478 (t1: T).(arity g c0 t1 a1)) H2 (THead (Bind b) u t) H6) in (let H9 \def (H7
479 (refl_equal T (THead (Bind b) u t))) in (ex2_ind A (\lambda (a3: A).(arity g
480 c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a1)) (ex2 A
481 (\lambda (a3: A).(arity g c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind
482 b) u) t a0))) (\lambda (x: A).(\lambda (H10: (arity g c0 u x)).(\lambda (H11:
483 (arity g (CHead c0 (Bind b) u) t a1)).(ex_intro2 A (\lambda (a3: A).(arity g
484 c0 u a3)) (\lambda (_: A).(arity g (CHead c0 (Bind b) u) t a0)) x H10
485 (arity_repl g (CHead c0 (Bind b) u) t a1 H11 a0 H4))))) H9))))))))))))) c y
488 lemma arity_gen_abst:
489 \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a:
490 A).((arity g c (THead (Bind Abst) u t) a) \to (ex3_2 A A (\lambda (a1:
491 A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
492 A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
493 (CHead c (Bind Abst) u) t a2)))))))))
495 \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a:
496 A).(\lambda (H: (arity g c (THead (Bind Abst) u t) a)).(insert_eq T (THead
497 (Bind Abst) u t) (\lambda (t0: T).(arity g c t0 a)) (\lambda (_: T).(ex3_2 A
498 A (\lambda (a1: A).(\lambda (a2: A).(eq A a (AHead a1 a2)))) (\lambda (a1:
499 A).(\lambda (_: A).(arity g c u (asucc g a1)))) (\lambda (_: A).(\lambda (a2:
500 A).(arity g (CHead c (Bind Abst) u) t a2))))) (\lambda (y: T).(\lambda (H0:
501 (arity g c y a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0:
502 A).((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1:
503 A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_:
504 A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
505 (CHead c0 (Bind Abst) u) t a2)))))))) (\lambda (c0: C).(\lambda (n:
506 nat).(\lambda (H1: (eq T (TSort n) (THead (Bind Abst) u t))).(let H2 \def
507 (eq_ind T (TSort n) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow
508 True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I
509 (THead (Bind Abst) u t) H1) in (False_ind (ex3_2 A A (\lambda (a1:
510 A).(\lambda (a2: A).(eq A (ASort O n) (AHead a1 a2)))) (\lambda (a1:
511 A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda
512 (a2: A).(arity g (CHead c0 (Bind Abst) u) t a2)))) H2))))) (\lambda (c0:
513 C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0
514 (CHead d (Bind Abbr) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0
515 a0)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda
516 (a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1: A).(\lambda
517 (_: A).(arity g d u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g
518 (CHead d (Bind Abst) u) t a2))))))).(\lambda (H4: (eq T (TLRef i) (THead
519 (Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match
520 ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _
521 _ _) \Rightarrow False])) I (THead (Bind Abst) u t) H4) in (False_ind (ex3_2
522 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1:
523 A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda
524 (a2: A).(arity g (CHead c0 (Bind Abst) u) t a2)))) H5))))))))))) (\lambda
525 (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl
526 i c0 (CHead d (Bind Abst) u0))).(\lambda (a0: A).(\lambda (_: (arity g d u0
527 (asucc g a0))).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A
528 A (\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g a0) (AHead a1 a2))))
529 (\lambda (a1: A).(\lambda (_: A).(arity g d u (asucc g a1)))) (\lambda (_:
530 A).(\lambda (a2: A).(arity g (CHead d (Bind Abst) u) t a2))))))).(\lambda
531 (H4: (eq T (TLRef i) (THead (Bind Abst) u t))).(let H5 \def (eq_ind T (TLRef
532 i) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
533 \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) u
534 t) H4) in (False_ind (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0
535 (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g
536 a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t
537 a2)))) H5))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b
538 Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H2:
539 (arity g c0 u0 a1)).(\lambda (H3: (((eq T u0 (THead (Bind Abst) u t)) \to
540 (ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead a2 a3))))
541 (\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) (\lambda (_:
542 A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda
543 (t0: T).(\lambda (a2: A).(\lambda (H4: (arity g (CHead c0 (Bind b) u0) t0
544 a2)).(\lambda (H5: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A
545 (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3:
546 A).(\lambda (_: A).(arity g (CHead c0 (Bind b) u0) u (asucc g a3)))) (\lambda
547 (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind b) u0) (Bind Abst) u)
548 t a4))))))).(\lambda (H6: (eq T (THead (Bind b) u0 t0) (THead (Bind Abst) u
549 t))).(let H7 \def (f_equal T B (\lambda (e: T).(match e with [(TSort _)
550 \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k
551 with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b)
552 u0 t0) (THead (Bind Abst) u t) H6) in ((let H8 \def (f_equal T T (\lambda (e:
553 T).(match e with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 |
554 (THead _ t1 _) \Rightarrow t1])) (THead (Bind b) u0 t0) (THead (Bind Abst) u
555 t) H6) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _)
556 \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1]))
557 (THead (Bind b) u0 t0) (THead (Bind Abst) u t) H6) in (\lambda (H10: (eq T u0
558 u)).(\lambda (H11: (eq B b Abst)).(let H12 \def (eq_ind T t0 (\lambda (t1:
559 T).((eq T t1 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3:
560 A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_:
561 A).(arity g (CHead c0 (Bind b) u0) u (asucc g a3)))) (\lambda (_: A).(\lambda
562 (a4: A).(arity g (CHead (CHead c0 (Bind b) u0) (Bind Abst) u) t a4)))))) H5 t
563 H9) in (let H13 \def (eq_ind T t0 (\lambda (t1: T).(arity g (CHead c0 (Bind
564 b) u0) t1 a2)) H4 t H9) in (let H14 \def (eq_ind T u0 (\lambda (t1: T).((eq T
565 t (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4:
566 A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead
567 c0 (Bind b) t1) u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g
568 (CHead (CHead c0 (Bind b) t1) (Bind Abst) u) t a4)))))) H12 u H10) in (let
569 H15 \def (eq_ind T u0 (\lambda (t1: T).(arity g (CHead c0 (Bind b) t1) t a2))
570 H13 u H10) in (let H16 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead
571 (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1
572 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g
573 a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t
574 a4)))))) H3 u H10) in (let H17 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0
575 t1 a1)) H2 u H10) in (let H18 \def (eq_ind B b (\lambda (b0: B).((eq T t
576 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq
577 A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0
578 (Bind b0) u) u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g
579 (CHead (CHead c0 (Bind b0) u) (Bind Abst) u) t a4)))))) H14 Abst H11) in (let
580 H19 \def (eq_ind B b (\lambda (b0: B).(arity g (CHead c0 (Bind b0) u) t a2))
581 H15 Abst H11) in (let H20 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0
582 Abst))) H1 Abst H11) in (let H21 \def (match (H20 (refl_equal B Abst)) in
583 False with []) in H21))))))))))))) H8)) H7)))))))))))))) (\lambda (c0:
584 C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 u0 (asucc g
585 a1))).(\lambda (H2: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A
586 (\lambda (a2: A).(\lambda (a3: A).(eq A (asucc g a1) (AHead a2 a3))))
587 (\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) (\lambda (_:
588 A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda
589 (t0: T).(\lambda (a2: A).(\lambda (H3: (arity g (CHead c0 (Bind Abst) u0) t0
590 a2)).(\lambda (H4: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A
591 (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3:
592 A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u0) u (asucc g a3))))
593 (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind Abst) u0)
594 (Bind Abst) u) t a4))))))).(\lambda (H5: (eq T (THead (Bind Abst) u0 t0)
595 (THead (Bind Abst) u t))).(let H6 \def (f_equal T T (\lambda (e: T).(match e
596 with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _)
597 \Rightarrow t1])) (THead (Bind Abst) u0 t0) (THead (Bind Abst) u t) H5) in
598 ((let H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _)
599 \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1]))
600 (THead (Bind Abst) u0 t0) (THead (Bind Abst) u t) H5) in (\lambda (H8: (eq T
601 u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Bind
602 Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead
603 a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) u0)
604 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0
605 (Bind Abst) u0) (Bind Abst) u) t a4)))))) H4 t H7) in (let H10 \def (eq_ind T
606 t0 (\lambda (t1: T).(arity g (CHead c0 (Bind Abst) u0) t1 a2)) H3 t H7) in
607 (let H11 \def (eq_ind T u0 (\lambda (t1: T).((eq T t (THead (Bind Abst) u t))
608 \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2 (AHead a3 a4))))
609 (\lambda (a3: A).(\lambda (_: A).(arity g (CHead c0 (Bind Abst) t1) u (asucc
610 g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead (CHead c0 (Bind
611 Abst) t1) (Bind Abst) u) t a4)))))) H9 u H8) in (let H12 \def (eq_ind T u0
612 (\lambda (t1: T).(arity g (CHead c0 (Bind Abst) t1) t a2)) H10 u H8) in (let
613 H13 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Bind Abst) u t)) \to
614 (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A (asucc g a1) (AHead a3
615 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3))))
616 (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))))))
617 H2 u H8) in (let H14 \def (eq_ind T u0 (\lambda (t1: T).(arity g c0 t1 (asucc
618 g a1))) H1 u H8) in (ex3_2_intro A A (\lambda (a3: A).(\lambda (a4: A).(eq A
619 (AHead a1 a2) (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u
620 (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind
621 Abst) u) t a4))) a1 a2 (refl_equal A (AHead a1 a2)) H14 H12)))))))))
622 H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda
623 (_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u t)) \to
624 (ex3_2 A A (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead a2 a3))))
625 (\lambda (a2: A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) (\lambda (_:
626 A).(\lambda (a3: A).(arity g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda
627 (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda
628 (_: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a3:
629 A).(\lambda (a4: A).(eq A (AHead a1 a2) (AHead a3 a4)))) (\lambda (a3:
630 A).(\lambda (_: A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda
631 (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))))))).(\lambda (H5: (eq T
632 (THead (Flat Appl) u0 t0) (THead (Bind Abst) u t))).(let H6 \def (eq_ind T
633 (THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee with [(TSort _)
634 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
635 (match k with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I
636 (THead (Bind Abst) u t) H5) in (False_ind (ex3_2 A A (\lambda (a3:
637 A).(\lambda (a4: A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_:
638 A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g
639 (CHead c0 (Bind Abst) u) t a4)))) H6)))))))))))) (\lambda (c0: C).(\lambda
640 (u0: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u0 (asucc g a0))).(\lambda
641 (_: (((eq T u0 (THead (Bind Abst) u t)) \to (ex3_2 A A (\lambda (a1:
642 A).(\lambda (a2: A).(eq A (asucc g a0) (AHead a1 a2)))) (\lambda (a1:
643 A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda
644 (a2: A).(arity g (CHead c0 (Bind Abst) u) t a2))))))).(\lambda (t0:
645 T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (_: (((eq T t0 (THead (Bind
646 Abst) u t)) \to (ex3_2 A A (\lambda (a1: A).(\lambda (a2: A).(eq A a0 (AHead
647 a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g a1))))
648 (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t
649 a2))))))).(\lambda (H5: (eq T (THead (Flat Cast) u0 t0) (THead (Bind Abst) u
650 t))).(let H6 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee: T).(match
651 ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k
652 _ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat _)
653 \Rightarrow True])])) I (THead (Bind Abst) u t) H5) in (False_ind (ex3_2 A A
654 (\lambda (a1: A).(\lambda (a2: A).(eq A a0 (AHead a1 a2)))) (\lambda (a1:
655 A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda
656 (a2: A).(arity g (CHead c0 (Bind Abst) u) t a2)))) H6))))))))))) (\lambda
657 (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t0
658 a1)).(\lambda (H2: (((eq T t0 (THead (Bind Abst) u t)) \to (ex3_2 A A
659 (\lambda (a2: A).(\lambda (a3: A).(eq A a1 (AHead a2 a3)))) (\lambda (a2:
660 A).(\lambda (_: A).(arity g c0 u (asucc g a2)))) (\lambda (_: A).(\lambda
661 (a3: A).(arity g (CHead c0 (Bind Abst) u) t a3))))))).(\lambda (a2:
662 A).(\lambda (H3: (leq g a1 a2)).(\lambda (H4: (eq T t0 (THead (Bind Abst) u
663 t))).(let H5 \def (f_equal T T (\lambda (e: T).e) t0 (THead (Bind Abst) u t)
664 H4) in (let H6 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Bind
665 Abst) u t)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a1 (AHead
666 a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g a3))))
667 (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t a4))))))
668 H2 (THead (Bind Abst) u t) H5) in (let H7 \def (eq_ind T t0 (\lambda (t1:
669 T).(arity g c0 t1 a1)) H1 (THead (Bind Abst) u t) H5) in (let H8 \def (H6
670 (refl_equal T (THead (Bind Abst) u t))) in (ex3_2_ind A A (\lambda (a3:
671 A).(\lambda (a4: A).(eq A a1 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_:
672 A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g
673 (CHead c0 (Bind Abst) u) t a4))) (ex3_2 A A (\lambda (a3: A).(\lambda (a4:
674 A).(eq A a2 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u
675 (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind
676 Abst) u) t a4)))) (\lambda (x0: A).(\lambda (x1: A).(\lambda (H9: (eq A a1
677 (AHead x0 x1))).(\lambda (H10: (arity g c0 u (asucc g x0))).(\lambda (H11:
678 (arity g (CHead c0 (Bind Abst) u) t x1)).(let H12 \def (eq_ind A a1 (\lambda
679 (a0: A).(leq g a0 a2)) H3 (AHead x0 x1) H9) in (let H13 \def (eq_ind A a1
680 (\lambda (a0: A).(arity g c0 (THead (Bind Abst) u t) a0)) H7 (AHead x0 x1)
681 H9) in (let H_x \def (leq_gen_head1 g x0 x1 a2 H12) in (let H14 \def H_x in
682 (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g x0 a3))) (\lambda (_:
683 A).(\lambda (a4: A).(leq g x1 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A
684 a2 (AHead a3 a4)))) (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a2
685 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(arity g c0 u (asucc g
686 a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g (CHead c0 (Bind Abst) u) t
687 a4)))) (\lambda (x2: A).(\lambda (x3: A).(\lambda (H15: (leq g x0
688 x2)).(\lambda (H16: (leq g x1 x3)).(\lambda (H17: (eq A a2 (AHead x2
689 x3))).(let H18 \def (f_equal A A (\lambda (e: A).e) a2 (AHead x2 x3) H17) in
690 (eq_ind_r A (AHead x2 x3) (\lambda (a0: A).(ex3_2 A A (\lambda (a3:
691 A).(\lambda (a4: A).(eq A a0 (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_:
692 A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g
693 (CHead c0 (Bind Abst) u) t a4))))) (ex3_2_intro A A (\lambda (a3: A).(\lambda
694 (a4: A).(eq A (AHead x2 x3) (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_:
695 A).(arity g c0 u (asucc g a3)))) (\lambda (_: A).(\lambda (a4: A).(arity g
696 (CHead c0 (Bind Abst) u) t a4))) x2 x3 (refl_equal A (AHead x2 x3))
697 (arity_repl g c0 u (asucc g x0) H10 (asucc g x2) (asucc_repl g x0 x2 H15))
698 (arity_repl g (CHead c0 (Bind Abst) u) t x1 H11 x3 H16)) a2 H18)))))))
699 H14)))))))))) H8))))))))))))) c y a H0))) H)))))).
701 lemma arity_gen_appl:
702 \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a2:
703 A).((arity g c (THead (Flat Appl) u t) a2) \to (ex2 A (\lambda (a1: A).(arity
704 g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1 a2)))))))))
706 \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a2:
707 A).(\lambda (H: (arity g c (THead (Flat Appl) u t) a2)).(insert_eq T (THead
708 (Flat Appl) u t) (\lambda (t0: T).(arity g c t0 a2)) (\lambda (_: T).(ex2 A
709 (\lambda (a1: A).(arity g c u a1)) (\lambda (a1: A).(arity g c t (AHead a1
710 a2))))) (\lambda (y: T).(\lambda (H0: (arity g c y a2)).(arity_ind g (\lambda
711 (c0: C).(\lambda (t0: T).(\lambda (a: A).((eq T t0 (THead (Flat Appl) u t))
712 \to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t
713 (AHead a1 a)))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T
714 (TSort n) (THead (Flat Appl) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda
715 (ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow
716 False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t) H1) in
717 (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity
718 g c0 t (AHead a1 (ASort O n))))) H2))))) (\lambda (c0: C).(\lambda (d:
719 C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind
720 Abbr) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 a)).(\lambda (_: (((eq
721 T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g d u a1))
722 (\lambda (a1: A).(arity g d t (AHead a1 a))))))).(\lambda (H4: (eq T (TLRef
723 i) (THead (Flat Appl) u t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee:
724 T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
725 (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u t) H4) in
726 (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity
727 g c0 t (AHead a1 a)))) H5))))))))))) (\lambda (c0: C).(\lambda (d:
728 C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (getl i c0 (CHead d (Bind
729 Abst) u0))).(\lambda (a: A).(\lambda (_: (arity g d u0 (asucc g a))).(\lambda
730 (_: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1: A).(arity g
731 d u a1)) (\lambda (a1: A).(arity g d t (AHead a1 (asucc g a)))))))).(\lambda
732 (H4: (eq T (TLRef i) (THead (Flat Appl) u t))).(let H5 \def (eq_ind T (TLRef
733 i) (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
734 \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) u
735 t) H4) in (False_ind (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1:
736 A).(arity g c0 t (AHead a1 a)))) H5))))))))))) (\lambda (b: B).(\lambda (_:
737 (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1:
738 A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Flat
739 Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3:
740 A).(arity g c0 t (AHead a3 a1))))))).(\lambda (t0: T).(\lambda (a0:
741 A).(\lambda (_: (arity g (CHead c0 (Bind b) u0) t0 a0)).(\lambda (_: (((eq T
742 t0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g (CHead c0
743 (Bind b) u0) u a3)) (\lambda (a3: A).(arity g (CHead c0 (Bind b) u0) t (AHead
744 a3 a0))))))).(\lambda (H6: (eq T (THead (Bind b) u0 t0) (THead (Flat Appl) u
745 t))).(let H7 \def (eq_ind T (THead (Bind b) u0 t0) (\lambda (ee: T).(match ee
746 with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _
747 _) \Rightarrow (match k with [(Bind _) \Rightarrow True | (Flat _)
748 \Rightarrow False])])) I (THead (Flat Appl) u t) H6) in (False_ind (ex2 A
749 (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3
750 a0)))) H7)))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1:
751 A).(\lambda (_: (arity g c0 u0 (asucc g a1))).(\lambda (_: (((eq T u0 (THead
752 (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda
753 (a3: A).(arity g c0 t (AHead a3 (asucc g a1)))))))).(\lambda (t0: T).(\lambda
754 (a0: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u0) t0 a0)).(\lambda (_:
755 (((eq T t0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g
756 (CHead c0 (Bind Abst) u0) u a3)) (\lambda (a3: A).(arity g (CHead c0 (Bind
757 Abst) u0) t (AHead a3 a0))))))).(\lambda (H5: (eq T (THead (Bind Abst) u0 t0)
758 (THead (Flat Appl) u t))).(let H6 \def (eq_ind T (THead (Bind Abst) u0 t0)
759 (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
760 \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _)
761 \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) u t)
762 H5) in (False_ind (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3:
763 A).(arity g c0 t (AHead a3 (AHead a1 a0))))) H6)))))))))))) (\lambda (c0:
764 C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 u0
765 a1)).(\lambda (H2: (((eq T u0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda
766 (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3
767 a1))))))).(\lambda (t0: T).(\lambda (a0: A).(\lambda (H3: (arity g c0 t0
768 (AHead a1 a0))).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u t)) \to (ex2 A
769 (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3
770 (AHead a1 a0)))))))).(\lambda (H5: (eq T (THead (Flat Appl) u0 t0) (THead
771 (Flat Appl) u t))).(let H6 \def (f_equal T T (\lambda (e: T).(match e with
772 [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t1 _)
773 \Rightarrow t1])) (THead (Flat Appl) u0 t0) (THead (Flat Appl) u t) H5) in
774 ((let H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort _)
775 \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow t1]))
776 (THead (Flat Appl) u0 t0) (THead (Flat Appl) u t) H5) in (\lambda (H8: (eq T
777 u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead (Flat
778 Appl) u t)) \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3:
779 A).(arity g c0 t (AHead a3 (AHead a1 a0))))))) H4 t H7) in (let H10 \def
780 (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a0))) H3 t H7) in (let
781 H11 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead (Flat Appl) u t)) \to
782 (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t
783 (AHead a3 a1)))))) H2 u H8) in (let H12 \def (eq_ind T u0 (\lambda (t1:
784 T).(arity g c0 t1 a1)) H1 u H8) in (ex_intro2 A (\lambda (a3: A).(arity g c0
785 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a0))) a1 H12 H10)))))))
786 H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a: A).(\lambda (_:
787 (arity g c0 u0 (asucc g a))).(\lambda (_: (((eq T u0 (THead (Flat Appl) u t))
788 \to (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t
789 (AHead a1 (asucc g a)))))))).(\lambda (t0: T).(\lambda (_: (arity g c0 t0
790 a)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a1:
791 A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1
792 a))))))).(\lambda (H5: (eq T (THead (Flat Cast) u0 t0) (THead (Flat Appl) u
793 t))).(let H6 \def (eq_ind T (THead (Flat Cast) u0 t0) (\lambda (ee: T).(match
794 ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k
795 _ _) \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f)
796 \Rightarrow (match f with [Appl \Rightarrow False | Cast \Rightarrow
797 True])])])) I (THead (Flat Appl) u t) H5) in (False_ind (ex2 A (\lambda (a1:
798 A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (AHead a1 a))))
799 H6))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda
800 (H1: (arity g c0 t0 a1)).(\lambda (H2: (((eq T t0 (THead (Flat Appl) u t))
801 \to (ex2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t
802 (AHead a3 a1))))))).(\lambda (a0: A).(\lambda (H3: (leq g a1 a0)).(\lambda
803 (H4: (eq T t0 (THead (Flat Appl) u t))).(let H5 \def (f_equal T T (\lambda
804 (e: T).e) t0 (THead (Flat Appl) u t) H4) in (let H6 \def (eq_ind T t0
805 (\lambda (t1: T).((eq T t1 (THead (Flat Appl) u t)) \to (ex2 A (\lambda (a3:
806 A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a1)))))) H2
807 (THead (Flat Appl) u t) H5) in (let H7 \def (eq_ind T t0 (\lambda (t1:
808 T).(arity g c0 t1 a1)) H1 (THead (Flat Appl) u t) H5) in (let H8 \def (H6
809 (refl_equal T (THead (Flat Appl) u t))) in (ex2_ind A (\lambda (a3: A).(arity
810 g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a1))) (ex2 A (\lambda
811 (a3: A).(arity g c0 u a3)) (\lambda (a3: A).(arity g c0 t (AHead a3 a0))))
812 (\lambda (x: A).(\lambda (H9: (arity g c0 u x)).(\lambda (H10: (arity g c0 t
813 (AHead x a1))).(ex_intro2 A (\lambda (a3: A).(arity g c0 u a3)) (\lambda (a3:
814 A).(arity g c0 t (AHead a3 a0))) x H9 (arity_repl g c0 t (AHead x a1) H10
815 (AHead x a0) (leq_head g x x (leq_refl g x) a1 a0 H3)))))) H8))))))))))))) c
818 lemma arity_gen_cast:
819 \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (a:
820 A).((arity g c (THead (Flat Cast) u t) a) \to (land (arity g c u (asucc g a))
821 (arity g c t a)))))))
823 \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (a:
824 A).(\lambda (H: (arity g c (THead (Flat Cast) u t) a)).(insert_eq T (THead
825 (Flat Cast) u t) (\lambda (t0: T).(arity g c t0 a)) (\lambda (_: T).(land
826 (arity g c u (asucc g a)) (arity g c t a))) (\lambda (y: T).(\lambda (H0:
827 (arity g c y a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a0:
828 A).((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g c0 u (asucc g a0))
829 (arity g c0 t a0)))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (H1: (eq T
830 (TSort n) (THead (Flat Cast) u t))).(let H2 \def (eq_ind T (TSort n) (\lambda
831 (ee: T).(match ee with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow
832 False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) u t) H1) in
833 (False_ind (land (arity g c0 u (asucc g (ASort O n))) (arity g c0 t (ASort O
834 n))) H2))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i:
835 nat).(\lambda (_: (getl i c0 (CHead d (Bind Abbr) u0))).(\lambda (a0:
836 A).(\lambda (_: (arity g d u0 a0)).(\lambda (_: (((eq T u0 (THead (Flat Cast)
837 u t)) \to (land (arity g d u (asucc g a0)) (arity g d t a0))))).(\lambda (H4:
838 (eq T (TLRef i) (THead (Flat Cast) u t))).(let H5 \def (eq_ind T (TLRef i)
839 (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
840 \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) u
841 t) H4) in (False_ind (land (arity g c0 u (asucc g a0)) (arity g c0 t a0))
842 H5))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (i:
843 nat).(\lambda (_: (getl i c0 (CHead d (Bind Abst) u0))).(\lambda (a0:
844 A).(\lambda (_: (arity g d u0 (asucc g a0))).(\lambda (_: (((eq T u0 (THead
845 (Flat Cast) u t)) \to (land (arity g d u (asucc g (asucc g a0))) (arity g d t
846 (asucc g a0)))))).(\lambda (H4: (eq T (TLRef i) (THead (Flat Cast) u
847 t))).(let H5 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee with [(TSort
848 _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
849 False])) I (THead (Flat Cast) u t) H4) in (False_ind (land (arity g c0 u
850 (asucc g a0)) (arity g c0 t a0)) H5))))))))))) (\lambda (b: B).(\lambda (_:
851 (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u0: T).(\lambda (a1:
852 A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Flat
853 Cast) u t)) \to (land (arity g c0 u (asucc g a1)) (arity g c0 t
854 a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0
855 (Bind b) u0) t0 a2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u t)) \to
856 (land (arity g (CHead c0 (Bind b) u0) u (asucc g a2)) (arity g (CHead c0
857 (Bind b) u0) t a2))))).(\lambda (H6: (eq T (THead (Bind b) u0 t0) (THead
858 (Flat Cast) u t))).(let H7 \def (eq_ind T (THead (Bind b) u0 t0) (\lambda
859 (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow
860 False | (THead k _ _) \Rightarrow (match k with [(Bind _) \Rightarrow True |
861 (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u t) H6) in (False_ind
862 (land (arity g c0 u (asucc g a2)) (arity g c0 t a2)) H7))))))))))))))
863 (\lambda (c0: C).(\lambda (u0: T).(\lambda (a1: A).(\lambda (_: (arity g c0
864 u0 (asucc g a1))).(\lambda (_: (((eq T u0 (THead (Flat Cast) u t)) \to (land
865 (arity g c0 u (asucc g (asucc g a1))) (arity g c0 t (asucc g
866 a1)))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0
867 (Bind Abst) u0) t0 a2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u t)) \to
868 (land (arity g (CHead c0 (Bind Abst) u0) u (asucc g a2)) (arity g (CHead c0
869 (Bind Abst) u0) t a2))))).(\lambda (H5: (eq T (THead (Bind Abst) u0 t0)
870 (THead (Flat Cast) u t))).(let H6 \def (eq_ind T (THead (Bind Abst) u0 t0)
871 (\lambda (ee: T).(match ee with [(TSort _) \Rightarrow False | (TLRef _)
872 \Rightarrow False | (THead k _ _) \Rightarrow (match k with [(Bind _)
873 \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u t)
874 H5) in (False_ind (land (arity g c0 u (asucc g (AHead a1 a2))) (arity g c0 t
875 (AHead a1 a2))) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda
876 (a1: A).(\lambda (_: (arity g c0 u0 a1)).(\lambda (_: (((eq T u0 (THead (Flat
877 Cast) u t)) \to (land (arity g c0 u (asucc g a1)) (arity g c0 t
878 a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead
879 a1 a2))).(\lambda (_: (((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g
880 c0 u (asucc g (AHead a1 a2))) (arity g c0 t (AHead a1 a2)))))).(\lambda (H5:
881 (eq T (THead (Flat Appl) u0 t0) (THead (Flat Cast) u t))).(let H6 \def
882 (eq_ind T (THead (Flat Appl) u0 t0) (\lambda (ee: T).(match ee with [(TSort
883 _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
884 \Rightarrow (match k with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow
885 (match f with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead
886 (Flat Cast) u t) H5) in (False_ind (land (arity g c0 u (asucc g a2)) (arity g
887 c0 t a2)) H6)))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (a0:
888 A).(\lambda (H1: (arity g c0 u0 (asucc g a0))).(\lambda (H2: (((eq T u0
889 (THead (Flat Cast) u t)) \to (land (arity g c0 u (asucc g (asucc g a0)))
890 (arity g c0 t (asucc g a0)))))).(\lambda (t0: T).(\lambda (H3: (arity g c0 t0
891 a0)).(\lambda (H4: (((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g c0
892 u (asucc g a0)) (arity g c0 t a0))))).(\lambda (H5: (eq T (THead (Flat Cast)
893 u0 t0) (THead (Flat Cast) u t))).(let H6 \def (f_equal T T (\lambda (e:
894 T).(match e with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 |
895 (THead _ t1 _) \Rightarrow t1])) (THead (Flat Cast) u0 t0) (THead (Flat Cast)
896 u t) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e with [(TSort
897 _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t1) \Rightarrow
898 t1])) (THead (Flat Cast) u0 t0) (THead (Flat Cast) u t) H5) in (\lambda (H8:
899 (eq T u0 u)).(let H9 \def (eq_ind T t0 (\lambda (t1: T).((eq T t1 (THead
900 (Flat Cast) u t)) \to (land (arity g c0 u (asucc g a0)) (arity g c0 t a0))))
901 H4 t H7) in (let H10 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 a0))
902 H3 t H7) in (let H11 \def (eq_ind T u0 (\lambda (t1: T).((eq T t1 (THead
903 (Flat Cast) u t)) \to (land (arity g c0 u (asucc g (asucc g a0))) (arity g c0
904 t (asucc g a0))))) H2 u H8) in (let H12 \def (eq_ind T u0 (\lambda (t1:
905 T).(arity g c0 t1 (asucc g a0))) H1 u H8) in (conj (arity g c0 u (asucc g
906 a0)) (arity g c0 t a0) H12 H10))))))) H6))))))))))) (\lambda (c0: C).(\lambda
907 (t0: T).(\lambda (a1: A).(\lambda (H1: (arity g c0 t0 a1)).(\lambda (H2:
908 (((eq T t0 (THead (Flat Cast) u t)) \to (land (arity g c0 u (asucc g a1))
909 (arity g c0 t a1))))).(\lambda (a2: A).(\lambda (H3: (leq g a1 a2)).(\lambda
910 (H4: (eq T t0 (THead (Flat Cast) u t))).(let H5 \def (f_equal T T (\lambda
911 (e: T).e) t0 (THead (Flat Cast) u t) H4) in (let H6 \def (eq_ind T t0
912 (\lambda (t1: T).((eq T t1 (THead (Flat Cast) u t)) \to (land (arity g c0 u
913 (asucc g a1)) (arity g c0 t a1)))) H2 (THead (Flat Cast) u t) H5) in (let H7
914 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 a1)) H1 (THead (Flat Cast)
915 u t) H5) in (let H8 \def (H6 (refl_equal T (THead (Flat Cast) u t))) in
916 (land_ind (arity g c0 u (asucc g a1)) (arity g c0 t a1) (land (arity g c0 u
917 (asucc g a2)) (arity g c0 t a2)) (\lambda (H9: (arity g c0 u (asucc g
918 a1))).(\lambda (H10: (arity g c0 t a1)).(conj (arity g c0 u (asucc g a2))
919 (arity g c0 t a2) (arity_repl g c0 u (asucc g a1) H9 (asucc g a2) (asucc_repl
920 g a1 a2 H3)) (arity_repl g c0 t a1 H10 a2 H3)))) H8))))))))))))) c y a H0)))
923 lemma arity_gen_appls:
924 \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (vs: TList).(\forall
925 (a2: A).((arity g c (THeads (Flat Appl) vs t) a2) \to (ex A (\lambda (a:
926 A).(arity g c t a))))))))
928 \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (vs:
929 TList).(TList_ind (\lambda (t0: TList).(\forall (a2: A).((arity g c (THeads
930 (Flat Appl) t0 t) a2) \to (ex A (\lambda (a: A).(arity g c t a)))))) (\lambda
931 (a2: A).(\lambda (H: (arity g c t a2)).(ex_intro A (\lambda (a: A).(arity g c
932 t a)) a2 H))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: ((\forall
933 (a2: A).((arity g c (THeads (Flat Appl) t1 t) a2) \to (ex A (\lambda (a:
934 A).(arity g c t a))))))).(\lambda (a2: A).(\lambda (H0: (arity g c (THead
935 (Flat Appl) t0 (THeads (Flat Appl) t1 t)) a2)).(let H1 \def (arity_gen_appl g
936 c t0 (THeads (Flat Appl) t1 t) a2 H0) in (ex2_ind A (\lambda (a1: A).(arity g
937 c t0 a1)) (\lambda (a1: A).(arity g c (THeads (Flat Appl) t1 t) (AHead a1
938 a2))) (ex A (\lambda (a: A).(arity g c t a))) (\lambda (x: A).(\lambda (_:
939 (arity g c t0 x)).(\lambda (H3: (arity g c (THeads (Flat Appl) t1 t) (AHead x
940 a2))).(let H_x \def (H (AHead x a2) H3) in (let H4 \def H_x in (ex_ind A
941 (\lambda (a: A).(arity g c t a)) (ex A (\lambda (a: A).(arity g c t a)))
942 (\lambda (x0: A).(\lambda (H5: (arity g c t x0)).(ex_intro A (\lambda (a:
943 A).(arity g c t a)) x0 H5))) H4)))))) H1))))))) vs)))).
945 lemma arity_gen_lift:
946 \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).(\forall (h:
947 nat).(\forall (d: nat).((arity g c1 (lift h d t) a) \to (\forall (c2:
948 C).((drop h d c1 c2) \to (arity g c2 t a)))))))))
950 \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (h:
951 nat).(\lambda (d: nat).(\lambda (H: (arity g c1 (lift h d t) a)).(insert_eq T
952 (lift h d t) (\lambda (t0: T).(arity g c1 t0 a)) (\lambda (_: T).(\forall
953 (c2: C).((drop h d c1 c2) \to (arity g c2 t a)))) (\lambda (y: T).(\lambda
954 (H0: (arity g c1 y a)).(unintro T t (\lambda (t0: T).((eq T y (lift h d t0))
955 \to (\forall (c2: C).((drop h d c1 c2) \to (arity g c2 t0 a))))) (unintro nat
956 d (\lambda (n: nat).(\forall (x: T).((eq T y (lift h n x)) \to (\forall (c2:
957 C).((drop h n c1 c2) \to (arity g c2 x a)))))) (arity_ind g (\lambda (c:
958 C).(\lambda (t0: T).(\lambda (a0: A).(\forall (x: nat).(\forall (x0: T).((eq
959 T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0
960 a0))))))))) (\lambda (c: C).(\lambda (n: nat).(\lambda (x: nat).(\lambda (x0:
961 T).(\lambda (H1: (eq T (TSort n) (lift h x x0))).(\lambda (c2: C).(\lambda
962 (_: (drop h x c c2)).(eq_ind_r T (TSort n) (\lambda (t0: T).(arity g c2 t0
963 (ASort O n))) (arity_sort g c2 n) x0 (lift_gen_sort h x n x0 H1)))))))))
964 (\lambda (c: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
965 (H1: (getl i c (CHead d0 (Bind Abbr) u))).(\lambda (a0: A).(\lambda (H2:
966 (arity g d0 u a0)).(\lambda (H3: ((\forall (x: nat).(\forall (x0: T).((eq T u
967 (lift h x x0)) \to (\forall (c2: C).((drop h x d0 c2) \to (arity g c2 x0
968 a0)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H4: (eq T (TLRef i)
969 (lift h x x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(let H_x \def
970 (lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq
971 T x0 (TLRef i))) (land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))
972 (arity g c2 x0 a0) (\lambda (H7: (land (lt i x) (eq T x0 (TLRef
973 i)))).(land_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8:
974 (lt i x)).(\lambda (H9: (eq T x0 (TLRef i))).(eq_ind_r T (TLRef i) (\lambda
975 (t0: T).(arity g c2 t0 a0)) (let H10 \def (eq_ind nat x (\lambda (n:
976 nat).(drop h n c c2)) H5 (S (plus i (minus x (S i)))) (lt_plus_minus i x H8))
977 in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x (S
978 i)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl i c2 (CHead e0 (Bind Abbr)
979 v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x (S i)) d0 e0)))
980 (arity g c2 (TLRef i) a0) (\lambda (x1: T).(\lambda (x2: C).(\lambda (H11:
981 (eq T u (lift h (minus x (S i)) x1))).(\lambda (H12: (getl i c2 (CHead x2
982 (Bind Abbr) x1))).(\lambda (H13: (drop h (minus x (S i)) d0 x2)).(let H14
983 \def (eq_ind T u (\lambda (t0: T).(\forall (x3: nat).(\forall (x4: T).((eq T
984 t0 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 d0 c3) \to (arity g c3 x4
985 a0))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def (eq_ind T u
986 (\lambda (t0: T).(arity g d0 t0 a0)) H2 (lift h (minus x (S i)) x1) H11) in
987 (arity_abbr g c2 x2 x1 i H12 a0 (H14 (minus x (S i)) x1 (refl_equal T (lift h
988 (minus x (S i)) x1)) x2 H13))))))))) (getl_drop_conf_lt Abbr c d0 u i H1 c2 h
989 (minus x (S i)) H10))) x0 H9))) H7)) (\lambda (H7: (land (le (plus x h) i)
990 (eq T x0 (TLRef (minus i h))))).(land_ind (le (plus x h) i) (eq T x0 (TLRef
991 (minus i h))) (arity g c2 x0 a0) (\lambda (H8: (le (plus x h) i)).(\lambda
992 (H9: (eq T x0 (TLRef (minus i h)))).(eq_ind_r T (TLRef (minus i h)) (\lambda
993 (t0: T).(arity g c2 t0 a0)) (arity_abbr g c2 d0 u (minus i h)
994 (getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c H1 c2 h x H5 H8) a0 H2) x0
995 H9))) H7)) H6)))))))))))))))) (\lambda (c: C).(\lambda (d0: C).(\lambda (u:
996 T).(\lambda (i: nat).(\lambda (H1: (getl i c (CHead d0 (Bind Abst)
997 u))).(\lambda (a0: A).(\lambda (H2: (arity g d0 u (asucc g a0))).(\lambda
998 (H3: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall
999 (c2: C).((drop h x d0 c2) \to (arity g c2 x0 (asucc g a0))))))))).(\lambda
1000 (x: nat).(\lambda (x0: T).(\lambda (H4: (eq T (TLRef i) (lift h x
1001 x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(let H_x \def
1002 (lift_gen_lref x0 x h i H4) in (let H6 \def H_x in (or_ind (land (lt i x) (eq
1003 T x0 (TLRef i))) (land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))
1004 (arity g c2 x0 a0) (\lambda (H7: (land (lt i x) (eq T x0 (TLRef
1005 i)))).(land_ind (lt i x) (eq T x0 (TLRef i)) (arity g c2 x0 a0) (\lambda (H8:
1006 (lt i x)).(\lambda (H9: (eq T x0 (TLRef i))).(eq_ind_r T (TLRef i) (\lambda
1007 (t0: T).(arity g c2 t0 a0)) (let H10 \def (eq_ind nat x (\lambda (n:
1008 nat).(drop h n c c2)) H5 (S (plus i (minus x (S i)))) (lt_plus_minus i x H8))
1009 in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus x (S
1010 i)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl i c2 (CHead e0 (Bind Abst)
1011 v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus x (S i)) d0 e0)))
1012 (arity g c2 (TLRef i) a0) (\lambda (x1: T).(\lambda (x2: C).(\lambda (H11:
1013 (eq T u (lift h (minus x (S i)) x1))).(\lambda (H12: (getl i c2 (CHead x2
1014 (Bind Abst) x1))).(\lambda (H13: (drop h (minus x (S i)) d0 x2)).(let H14
1015 \def (eq_ind T u (\lambda (t0: T).(\forall (x3: nat).(\forall (x4: T).((eq T
1016 t0 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 d0 c3) \to (arity g c3 x4
1017 (asucc g a0)))))))) H3 (lift h (minus x (S i)) x1) H11) in (let H15 \def
1018 (eq_ind T u (\lambda (t0: T).(arity g d0 t0 (asucc g a0))) H2 (lift h (minus
1019 x (S i)) x1) H11) in (arity_abst g c2 x2 x1 i H12 a0 (H14 (minus x (S i)) x1
1020 (refl_equal T (lift h (minus x (S i)) x1)) x2 H13))))))))) (getl_drop_conf_lt
1021 Abst c d0 u i H1 c2 h (minus x (S i)) H10))) x0 H9))) H7)) (\lambda (H7:
1022 (land (le (plus x h) i) (eq T x0 (TLRef (minus i h))))).(land_ind (le (plus x
1023 h) i) (eq T x0 (TLRef (minus i h))) (arity g c2 x0 a0) (\lambda (H8: (le
1024 (plus x h) i)).(\lambda (H9: (eq T x0 (TLRef (minus i h)))).(eq_ind_r T
1025 (TLRef (minus i h)) (\lambda (t0: T).(arity g c2 t0 a0)) (arity_abst g c2 d0
1026 u (minus i h) (getl_drop_conf_ge i (CHead d0 (Bind Abst) u) c H1 c2 h x H5
1027 H8) a0 H2) x0 H9))) H7)) H6)))))))))))))))) (\lambda (b: B).(\lambda (H1:
1028 (not (eq B b Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1:
1029 A).(\lambda (H2: (arity g c u a1)).(\lambda (H3: ((\forall (x: nat).(\forall
1030 (x0: T).((eq T u (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to
1031 (arity g c2 x0 a1)))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H4:
1032 (arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H5: ((\forall (x:
1033 nat).(\forall (x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h
1034 x (CHead c (Bind b) u) c2) \to (arity g c2 x0 a2)))))))).(\lambda (x:
1035 nat).(\lambda (x0: T).(\lambda (H6: (eq T (THead (Bind b) u t0) (lift h x
1036 x0))).(\lambda (c2: C).(\lambda (H7: (drop h x c c2)).(ex3_2_ind T T (\lambda
1037 (y0: T).(\lambda (z: T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0:
1038 T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: T).(\lambda (z:
1039 T).(eq T t0 (lift h (S x) z)))) (arity g c2 x0 a2) (\lambda (x1: T).(\lambda
1040 (x2: T).(\lambda (H8: (eq T x0 (THead (Bind b) x1 x2))).(\lambda (H9: (eq T u
1041 (lift h x x1))).(\lambda (H10: (eq T t0 (lift h (S x) x2))).(eq_ind_r T
1042 (THead (Bind b) x1 x2) (\lambda (t1: T).(arity g c2 t1 a2)) (let H11 \def
1043 (eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T t1
1044 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 (CHead c (Bind b) u) c3) \to
1045 (arity g c3 x4 a2))))))) H5 (lift h (S x) x2) H10) in (let H12 \def (eq_ind T
1046 t0 (\lambda (t1: T).(arity g (CHead c (Bind b) u) t1 a2)) H4 (lift h (S x)
1047 x2) H10) in (let H13 \def (eq_ind T u (\lambda (t1: T).(arity g (CHead c
1048 (Bind b) t1) (lift h (S x) x2) a2)) H12 (lift h x x1) H9) in (let H14 \def
1049 (eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T (lift
1050 h (S x) x2) (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 (CHead c (Bind
1051 b) t1) c3) \to (arity g c3 x4 a2))))))) H11 (lift h x x1) H9) in (let H15
1052 \def (eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T
1053 t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to (arity g c3 x4
1054 a1))))))) H3 (lift h x x1) H9) in (let H16 \def (eq_ind T u (\lambda (t1:
1055 T).(arity g c t1 a1)) H2 (lift h x x1) H9) in (arity_bind g b H1 c2 x1 a1
1056 (H15 x x1 (refl_equal T (lift h x x1)) c2 H7) x2 a2 (H14 (S x) x2 (refl_equal
1057 T (lift h (S x) x2)) (CHead c2 (Bind b) x1) (drop_skip_bind h x c c2 H7 b
1058 x1))))))))) x0 H8)))))) (lift_gen_bind b u t0 x0 h x H6))))))))))))))))))
1059 (\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H1: (arity g c u
1060 (asucc g a1))).(\lambda (H2: ((\forall (x: nat).(\forall (x0: T).((eq T u
1061 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0
1062 (asucc g a1))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H3: (arity g
1063 (CHead c (Bind Abst) u) t0 a2)).(\lambda (H4: ((\forall (x: nat).(\forall
1064 (x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x (CHead c
1065 (Bind Abst) u) c2) \to (arity g c2 x0 a2)))))))).(\lambda (x: nat).(\lambda
1066 (x0: T).(\lambda (H5: (eq T (THead (Bind Abst) u t0) (lift h x x0))).(\lambda
1067 (c2: C).(\lambda (H6: (drop h x c c2)).(ex3_2_ind T T (\lambda (y0:
1068 T).(\lambda (z: T).(eq T x0 (THead (Bind Abst) y0 z)))) (\lambda (y0:
1069 T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_: T).(\lambda (z:
1070 T).(eq T t0 (lift h (S x) z)))) (arity g c2 x0 (AHead a1 a2)) (\lambda (x1:
1071 T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Bind Abst) x1
1072 x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h (S
1073 x) x2))).(eq_ind_r T (THead (Bind Abst) x1 x2) (\lambda (t1: T).(arity g c2
1074 t1 (AHead a1 a2))) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3:
1075 nat).(\forall (x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h
1076 x3 (CHead c (Bind Abst) u) c3) \to (arity g c3 x4 a2))))))) H4 (lift h (S x)
1077 x2) H9) in (let H11 \def (eq_ind T t0 (\lambda (t1: T).(arity g (CHead c
1078 (Bind Abst) u) t1 a2)) H3 (lift h (S x) x2) H9) in (let H12 \def (eq_ind T u
1079 (\lambda (t1: T).(arity g (CHead c (Bind Abst) t1) (lift h (S x) x2) a2)) H11
1080 (lift h x x1) H8) in (let H13 \def (eq_ind T u (\lambda (t1: T).(\forall (x3:
1081 nat).(\forall (x4: T).((eq T (lift h (S x) x2) (lift h x3 x4)) \to (\forall
1082 (c3: C).((drop h x3 (CHead c (Bind Abst) t1) c3) \to (arity g c3 x4 a2)))))))
1083 H10 (lift h x x1) H8) in (let H14 \def (eq_ind T u (\lambda (t1: T).(\forall
1084 (x3: nat).(\forall (x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3:
1085 C).((drop h x3 c c3) \to (arity g c3 x4 (asucc g a1)))))))) H2 (lift h x x1)
1086 H8) in (let H15 \def (eq_ind T u (\lambda (t1: T).(arity g c t1 (asucc g
1087 a1))) H1 (lift h x x1) H8) in (arity_head g c2 x1 a1 (H14 x x1 (refl_equal T
1088 (lift h x x1)) c2 H6) x2 a2 (H13 (S x) x2 (refl_equal T (lift h (S x) x2))
1089 (CHead c2 (Bind Abst) x1) (drop_skip_bind h x c c2 H6 Abst x1))))))))) x0
1090 H7)))))) (lift_gen_bind Abst u t0 x0 h x H5)))))))))))))))) (\lambda (c:
1091 C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H1: (arity g c u a1)).(\lambda
1092 (H2: ((\forall (x: nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall
1093 (c2: C).((drop h x c c2) \to (arity g c2 x0 a1)))))))).(\lambda (t0:
1094 T).(\lambda (a2: A).(\lambda (H3: (arity g c t0 (AHead a1 a2))).(\lambda (H4:
1095 ((\forall (x: nat).(\forall (x0: T).((eq T t0 (lift h x x0)) \to (\forall
1096 (c2: C).((drop h x c c2) \to (arity g c2 x0 (AHead a1 a2))))))))).(\lambda
1097 (x: nat).(\lambda (x0: T).(\lambda (H5: (eq T (THead (Flat Appl) u t0) (lift
1098 h x x0))).(\lambda (c2: C).(\lambda (H6: (drop h x c c2)).(ex3_2_ind T T
1099 (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat Appl) y0 z))))
1100 (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x y0)))) (\lambda (_:
1101 T).(\lambda (z: T).(eq T t0 (lift h x z)))) (arity g c2 x0 a2) (\lambda (x1:
1102 T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Flat Appl) x1
1103 x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h x
1104 x2))).(eq_ind_r T (THead (Flat Appl) x1 x2) (\lambda (t1: T).(arity g c2 t1
1105 a2)) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall
1106 (x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to
1107 (arity g c3 x4 (AHead a1 a2)))))))) H4 (lift h x x2) H9) in (let H11 \def
1108 (eq_ind T t0 (\lambda (t1: T).(arity g c t1 (AHead a1 a2))) H3 (lift h x x2)
1109 H9) in (let H12 \def (eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall
1110 (x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to
1111 (arity g c3 x4 a1))))))) H2 (lift h x x1) H8) in (let H13 \def (eq_ind T u
1112 (\lambda (t1: T).(arity g c t1 a1)) H1 (lift h x x1) H8) in (arity_appl g c2
1113 x1 a1 (H12 x x1 (refl_equal T (lift h x x1)) c2 H6) x2 a2 (H10 x x2
1114 (refl_equal T (lift h x x2)) c2 H6)))))) x0 H7)))))) (lift_gen_flat Appl u t0
1115 x0 h x H5)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0:
1116 A).(\lambda (H1: (arity g c u (asucc g a0))).(\lambda (H2: ((\forall (x:
1117 nat).(\forall (x0: T).((eq T u (lift h x x0)) \to (\forall (c2: C).((drop h x
1118 c c2) \to (arity g c2 x0 (asucc g a0))))))))).(\lambda (t0: T).(\lambda (H3:
1119 (arity g c t0 a0)).(\lambda (H4: ((\forall (x: nat).(\forall (x0: T).((eq T
1120 t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to (arity g c2 x0
1121 a0)))))))).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H5: (eq T (THead
1122 (Flat Cast) u t0) (lift h x x0))).(\lambda (c2: C).(\lambda (H6: (drop h x c
1123 c2)).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Flat
1124 Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x y0))))
1125 (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h x z)))) (arity g c2 x0 a0)
1126 (\lambda (x1: T).(\lambda (x2: T).(\lambda (H7: (eq T x0 (THead (Flat Cast)
1127 x1 x2))).(\lambda (H8: (eq T u (lift h x x1))).(\lambda (H9: (eq T t0 (lift h
1128 x x2))).(eq_ind_r T (THead (Flat Cast) x1 x2) (\lambda (t1: T).(arity g c2 t1
1129 a0)) (let H10 \def (eq_ind T t0 (\lambda (t1: T).(\forall (x3: nat).(\forall
1130 (x4: T).((eq T t1 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to
1131 (arity g c3 x4 a0))))))) H4 (lift h x x2) H9) in (let H11 \def (eq_ind T t0
1132 (\lambda (t1: T).(arity g c t1 a0)) H3 (lift h x x2) H9) in (let H12 \def
1133 (eq_ind T u (\lambda (t1: T).(\forall (x3: nat).(\forall (x4: T).((eq T t1
1134 (lift h x3 x4)) \to (\forall (c3: C).((drop h x3 c c3) \to (arity g c3 x4
1135 (asucc g a0)))))))) H2 (lift h x x1) H8) in (let H13 \def (eq_ind T u
1136 (\lambda (t1: T).(arity g c t1 (asucc g a0))) H1 (lift h x x1) H8) in
1137 (arity_cast g c2 x1 a0 (H12 x x1 (refl_equal T (lift h x x1)) c2 H6) x2 (H10
1138 x x2 (refl_equal T (lift h x x2)) c2 H6)))))) x0 H7)))))) (lift_gen_flat Cast
1139 u t0 x0 h x H5))))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1:
1140 A).(\lambda (_: (arity g c t0 a1)).(\lambda (H2: ((\forall (x: nat).(\forall
1141 (x0: T).((eq T t0 (lift h x x0)) \to (\forall (c2: C).((drop h x c c2) \to
1142 (arity g c2 x0 a1)))))))).(\lambda (a2: A).(\lambda (H3: (leq g a1
1143 a2)).(\lambda (x: nat).(\lambda (x0: T).(\lambda (H4: (eq T t0 (lift h x
1144 x0))).(\lambda (c2: C).(\lambda (H5: (drop h x c c2)).(arity_repl g c2 x0 a1
1145 (H2 x x0 H4 c2 H5) a2 H3))))))))))))) c1 y a H0))))) H))))))).
1148 \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c
1149 t a1) \to (\forall (a2: A).((arity g c t a2) \to (leq g a1 a2)))))))
1151 \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a1: A).(\lambda (H:
1152 (arity g c t a1)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (a:
1153 A).(\forall (a2: A).((arity g c0 t0 a2) \to (leq g a a2)))))) (\lambda (c0:
1154 C).(\lambda (n: nat).(\lambda (a2: A).(\lambda (H0: (arity g c0 (TSort n)
1155 a2)).(leq_sym g a2 (ASort O n) (arity_gen_sort g c0 n a2 H0)))))) (\lambda
1156 (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl
1157 i c0 (CHead d (Bind Abbr) u))).(\lambda (a: A).(\lambda (_: (arity g d u
1158 a)).(\lambda (H2: ((\forall (a2: A).((arity g d u a2) \to (leq g a
1159 a2))))).(\lambda (a2: A).(\lambda (H3: (arity g c0 (TLRef i) a2)).(let H4
1160 \def (arity_gen_lref g c0 i a2 H3) in (or_ind (ex2_2 C T (\lambda (d0:
1161 C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0:
1162 C).(\lambda (u0: T).(arity g d0 u0 a2)))) (ex2_2 C T (\lambda (d0:
1163 C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0:
1164 C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2))))) (leq g a a2) (\lambda
1165 (H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind
1166 Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0
1167 a2))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0
1168 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2)))
1169 (leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0
1170 (CHead x0 (Bind Abbr) x1))).(\lambda (H7: (arity g x0 x1 a2)).(let H8 \def
1171 (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead
1172 x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind
1173 Abbr) x1) H6)) in (let H9 \def (f_equal C C (\lambda (e: C).(match e with
1174 [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind
1175 Abbr) u) (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i H0
1176 (CHead x0 (Bind Abbr) x1) H6)) in ((let H10 \def (f_equal C T (\lambda (e:
1177 C).(match e with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0]))
1178 (CHead d (Bind Abbr) u) (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d
1179 (Bind Abbr) u) i H0 (CHead x0 (Bind Abbr) x1) H6)) in (\lambda (H11: (eq C d
1180 x0)).(let H12 \def (eq_ind_r T x1 (\lambda (t0: T).(getl i c0 (CHead x0 (Bind
1181 Abbr) t0))) H8 u H10) in (let H13 \def (eq_ind_r T x1 (\lambda (t0: T).(arity
1182 g x0 t0 a2)) H7 u H10) in (let H14 \def (eq_ind_r C x0 (\lambda (c1: C).(getl
1183 i c0 (CHead c1 (Bind Abbr) u))) H12 d H11) in (let H15 \def (eq_ind_r C x0
1184 (\lambda (c1: C).(arity g c1 u a2)) H13 d H11) in (H2 a2 H15))))))) H9)))))))
1185 H5)) (\lambda (H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0
1186 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0
1187 (asucc g a2)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0
1188 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0
1189 (asucc g a2)))) (leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6:
1190 (getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (_: (arity g x0 x1 (asucc g
1191 a2))).(let H8 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i
1192 c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abbr) u) i
1193 H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind
1194 Abbr) u) (\lambda (ee: C).(match ee with [(CSort _) \Rightarrow False |
1195 (CHead _ k _) \Rightarrow (match k with [(Bind b) \Rightarrow (match b with
1196 [Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) |
1197 (Flat _) \Rightarrow False])])) I (CHead x0 (Bind Abst) x1) (getl_mono c0
1198 (CHead d (Bind Abbr) u) i H0 (CHead x0 (Bind Abst) x1) H6)) in (False_ind
1199 (leq g a a2) H9))))))) H5)) H4)))))))))))) (\lambda (c0: C).(\lambda (d:
1200 C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
1201 Abst) u))).(\lambda (a: A).(\lambda (_: (arity g d u (asucc g a))).(\lambda
1202 (H2: ((\forall (a2: A).((arity g d u a2) \to (leq g (asucc g a)
1203 a2))))).(\lambda (a2: A).(\lambda (H3: (arity g c0 (TLRef i) a2)).(let H4
1204 \def (arity_gen_lref g c0 i a2 H3) in (or_ind (ex2_2 C T (\lambda (d0:
1205 C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (d0:
1206 C).(\lambda (u0: T).(arity g d0 u0 a2)))) (ex2_2 C T (\lambda (d0:
1207 C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0:
1208 C).(\lambda (u0: T).(arity g d0 u0 (asucc g a2))))) (leq g a a2) (\lambda
1209 (H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0 (Bind
1210 Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0
1211 a2))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda (u0: T).(getl i c0 (CHead d0
1212 (Bind Abbr) u0)))) (\lambda (d0: C).(\lambda (u0: T).(arity g d0 u0 a2)))
1213 (leq g a a2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl i c0
1214 (CHead x0 (Bind Abbr) x1))).(\lambda (_: (arity g x0 x1 a2)).(let H8 \def
1215 (eq_ind C (CHead d (Bind Abst) u) (\lambda (c1: C).(getl i c0 c1)) H0 (CHead
1216 x0 (Bind Abbr) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind
1217 Abbr) x1) H6)) in (let H9 \def (eq_ind C (CHead d (Bind Abst) u) (\lambda
1218 (ee: C).(match ee with [(CSort _) \Rightarrow False | (CHead _ k _)
1219 \Rightarrow (match k with [(Bind b) \Rightarrow (match b with [Abbr
1220 \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat
1221 _) \Rightarrow False])])) I (CHead x0 (Bind Abbr) x1) (getl_mono c0 (CHead d
1222 (Bind Abst) u) i H0 (CHead x0 (Bind Abbr) x1) H6)) in (False_ind (leq g a a2)
1223 H9))))))) H5)) (\lambda (H5: (ex2_2 C T (\lambda (d0: C).(\lambda (u0:
1224 T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda (u0:
1225 T).(arity g d0 u0 (asucc g a2)))))).(ex2_2_ind C T (\lambda (d0: C).(\lambda
1226 (u0: T).(getl i c0 (CHead d0 (Bind Abst) u0)))) (\lambda (d0: C).(\lambda
1227 (u0: T).(arity g d0 u0 (asucc g a2)))) (leq g a a2) (\lambda (x0: C).(\lambda
1228 (x1: T).(\lambda (H6: (getl i c0 (CHead x0 (Bind Abst) x1))).(\lambda (H7:
1229 (arity g x0 x1 (asucc g a2))).(let H8 \def (eq_ind C (CHead d (Bind Abst) u)
1230 (\lambda (c1: C).(getl i c0 c1)) H0 (CHead x0 (Bind Abst) x1) (getl_mono c0
1231 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind Abst) x1) H6)) in (let H9 \def
1232 (f_equal C C (\lambda (e: C).(match e with [(CSort _) \Rightarrow d | (CHead
1233 c1 _ _) \Rightarrow c1])) (CHead d (Bind Abst) u) (CHead x0 (Bind Abst) x1)
1234 (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead x0 (Bind Abst) x1) H6)) in
1235 ((let H10 \def (f_equal C T (\lambda (e: C).(match e with [(CSort _)
1236 \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abst) u)
1237 (CHead x0 (Bind Abst) x1) (getl_mono c0 (CHead d (Bind Abst) u) i H0 (CHead
1238 x0 (Bind Abst) x1) H6)) in (\lambda (H11: (eq C d x0)).(let H12 \def
1239 (eq_ind_r T x1 (\lambda (t0: T).(getl i c0 (CHead x0 (Bind Abst) t0))) H8 u
1240 H10) in (let H13 \def (eq_ind_r T x1 (\lambda (t0: T).(arity g x0 t0 (asucc g
1241 a2))) H7 u H10) in (let H14 \def (eq_ind_r C x0 (\lambda (c1: C).(getl i c0
1242 (CHead c1 (Bind Abst) u))) H12 d H11) in (let H15 \def (eq_ind_r C x0
1243 (\lambda (c1: C).(arity g c1 u (asucc g a2))) H13 d H11) in (asucc_inj g a a2
1244 (H2 (asucc g a2) H15)))))))) H9))))))) H5)) H4)))))))))))) (\lambda (b:
1245 B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u:
1246 T).(\lambda (a2: A).(\lambda (_: (arity g c0 u a2)).(\lambda (_: ((\forall
1247 (a3: A).((arity g c0 u a3) \to (leq g a2 a3))))).(\lambda (t0: T).(\lambda
1248 (a3: A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t0 a3)).(\lambda (H4:
1249 ((\forall (a4: A).((arity g (CHead c0 (Bind b) u) t0 a4) \to (leq g a3
1250 a4))))).(\lambda (a0: A).(\lambda (H5: (arity g c0 (THead (Bind b) u t0)
1251 a0)).(let H6 \def (arity_gen_bind b H0 g c0 u t0 a0 H5) in (ex2_ind A
1252 (\lambda (a4: A).(arity g c0 u a4)) (\lambda (_: A).(arity g (CHead c0 (Bind
1253 b) u) t0 a0)) (leq g a3 a0) (\lambda (x: A).(\lambda (_: (arity g c0 u
1254 x)).(\lambda (H8: (arity g (CHead c0 (Bind b) u) t0 a0)).(H4 a0 H8))))
1255 H6))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a2: A).(\lambda
1256 (_: (arity g c0 u (asucc g a2))).(\lambda (H1: ((\forall (a3: A).((arity g c0
1257 u a3) \to (leq g (asucc g a2) a3))))).(\lambda (t0: T).(\lambda (a3:
1258 A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 a3)).(\lambda (H3:
1259 ((\forall (a4: A).((arity g (CHead c0 (Bind Abst) u) t0 a4) \to (leq g a3
1260 a4))))).(\lambda (a0: A).(\lambda (H4: (arity g c0 (THead (Bind Abst) u t0)
1261 a0)).(let H5 \def (arity_gen_abst g c0 u t0 a0 H4) in (ex3_2_ind A A (\lambda
1262 (a4: A).(\lambda (a5: A).(eq A a0 (AHead a4 a5)))) (\lambda (a4: A).(\lambda
1263 (_: A).(arity g c0 u (asucc g a4)))) (\lambda (_: A).(\lambda (a5: A).(arity
1264 g (CHead c0 (Bind Abst) u) t0 a5))) (leq g (AHead a2 a3) a0) (\lambda (x0:
1265 A).(\lambda (x1: A).(\lambda (H6: (eq A a0 (AHead x0 x1))).(\lambda (H7:
1266 (arity g c0 u (asucc g x0))).(\lambda (H8: (arity g (CHead c0 (Bind Abst) u)
1267 t0 x1)).(eq_ind_r A (AHead x0 x1) (\lambda (a: A).(leq g (AHead a2 a3) a))
1268 (leq_head g a2 x0 (asucc_inj g a2 x0 (H1 (asucc g x0) H7)) a3 x1 (H3 x1 H8))
1269 a0 H6)))))) H5))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a2:
1270 A).(\lambda (_: (arity g c0 u a2)).(\lambda (_: ((\forall (a3: A).((arity g
1271 c0 u a3) \to (leq g a2 a3))))).(\lambda (t0: T).(\lambda (a3: A).(\lambda (_:
1272 (arity g c0 t0 (AHead a2 a3))).(\lambda (H3: ((\forall (a4: A).((arity g c0
1273 t0 a4) \to (leq g (AHead a2 a3) a4))))).(\lambda (a0: A).(\lambda (H4: (arity
1274 g c0 (THead (Flat Appl) u t0) a0)).(let H5 \def (arity_gen_appl g c0 u t0 a0
1275 H4) in (ex2_ind A (\lambda (a4: A).(arity g c0 u a4)) (\lambda (a4: A).(arity
1276 g c0 t0 (AHead a4 a0))) (leq g a3 a0) (\lambda (x: A).(\lambda (_: (arity g
1277 c0 u x)).(\lambda (H7: (arity g c0 t0 (AHead x a0))).(ahead_inj_snd g a2 a3 x
1278 a0 (H3 (AHead x a0) H7))))) H5))))))))))))) (\lambda (c0: C).(\lambda (u:
1279 T).(\lambda (a: A).(\lambda (_: (arity g c0 u (asucc g a))).(\lambda (_:
1280 ((\forall (a2: A).((arity g c0 u a2) \to (leq g (asucc g a) a2))))).(\lambda
1281 (t0: T).(\lambda (_: (arity g c0 t0 a)).(\lambda (H3: ((\forall (a2:
1282 A).((arity g c0 t0 a2) \to (leq g a a2))))).(\lambda (a2: A).(\lambda (H4:
1283 (arity g c0 (THead (Flat Cast) u t0) a2)).(let H5 \def (arity_gen_cast g c0 u
1284 t0 a2 H4) in (land_ind (arity g c0 u (asucc g a2)) (arity g c0 t0 a2) (leq g
1285 a a2) (\lambda (_: (arity g c0 u (asucc g a2))).(\lambda (H7: (arity g c0 t0
1286 a2)).(H3 a2 H7))) H5)))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda
1287 (a2: A).(\lambda (_: (arity g c0 t0 a2)).(\lambda (H1: ((\forall (a3:
1288 A).((arity g c0 t0 a3) \to (leq g a2 a3))))).(\lambda (a3: A).(\lambda (H2:
1289 (leq g a2 a3)).(\lambda (a0: A).(\lambda (H3: (arity g c0 t0 a0)).(leq_trans
1290 g a3 a2 (leq_sym g a2 a3 H2) a0 (H1 a0 H3))))))))))) c t a1 H))))).
projects / helm.git / blob