1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/spare".
22 C \to (TList \to Prop)
24 let rec nfs2 (c: C) (ts: TList) on ts: Prop \def (match ts with [TNil
25 \Rightarrow True | (TCons t ts0) \Rightarrow (land (nf2 c t) (nfs2 c ts0))])
29 \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((nf2 c
30 (THead (Flat Appl) u (THead (Bind Abst) v t))) \to (\forall (P: Prop).P)))))
32 \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H:
33 ((\forall (t2: T).((pr2 c (THead (Flat Appl) u (THead (Bind Abst) v t)) t2)
34 \to (eq T (THead (Flat Appl) u (THead (Bind Abst) v t)) t2))))).(\lambda (P:
35 Prop).(let H0 \def (eq_ind T (THead (Flat Appl) u (THead (Bind Abst) v t))
36 (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
37 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
38 (match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
39 (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u t) (H (THead (Bind
40 Abbr) u t) (pr2_free c (THead (Flat Appl) u (THead (Bind Abst) v t)) (THead
41 (Bind Abbr) u t) (pr0_beta v u u (pr0_refl u) t t (pr0_refl t))))) in
42 (False_ind P H0))))))).
45 \forall (b: B).(\forall (x: T).(\forall (u: T).(\forall (d: nat).((eq T
46 (THead (Bind b) u (lift (S O) d x)) x) \to (\forall (P: Prop).P)))))
48 \lambda (b: B).(\lambda (x: T).(T_ind (\lambda (t: T).(\forall (u:
49 T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to
50 (\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (d:
51 nat).(\lambda (H: (eq T (THead (Bind b) u (lift (S O) d (TSort n))) (TSort
52 n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead (Bind b) u (lift (S O)
53 d (TSort n))) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
54 with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
55 _) \Rightarrow True])) I (TSort n) H) in (False_ind P H0))))))) (\lambda (n:
56 nat).(\lambda (u: T).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u
57 (lift (S O) d (TLRef n))) (TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind
58 T (THead (Bind b) u (lift (S O) d (TLRef n))) (\lambda (ee: T).(match ee in T
59 return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
60 \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H) in
61 (False_ind P H0))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: ((\forall
62 (u: T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to
63 (\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: ((\forall (u:
64 T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t0)) t0) \to
65 (\forall (P: Prop).P)))))).(\lambda (u: T).(\lambda (d: nat).(\lambda (H1:
66 (eq T (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t
67 t0))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: T).(match e
68 in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef
69 _) \Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u
70 (lift (S O) d (THead k t t0))) (THead k t t0) H1) in ((let H3 \def (f_equal T
71 T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
72 \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t1 _) \Rightarrow t1]))
73 (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t t0) H1) in ((let
74 H4 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
75 with [(TSort _) \Rightarrow (THead k ((let rec lref_map (f: ((nat \to nat)))
76 (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort n) \Rightarrow (TSort
77 n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i
78 | false \Rightarrow (f i)])) | (THead k0 u0 t2) \Rightarrow (THead k0
79 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in lref_map) (\lambda (x0:
80 nat).(plus x0 (S O))) d t) ((let rec lref_map (f: ((nat \to nat))) (d0: nat)
81 (t1: T) on t1: T \def (match t1 with [(TSort n) \Rightarrow (TSort n) |
82 (TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i |
83 false \Rightarrow (f i)])) | (THead k0 u0 t2) \Rightarrow (THead k0 (lref_map
84 f d0 u0) (lref_map f (s k0 d0) t2))]) in lref_map) (\lambda (x0: nat).(plus
85 x0 (S O))) (s k d) t0)) | (TLRef _) \Rightarrow (THead k ((let rec lref_map
86 (f: ((nat \to nat))) (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort
87 n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0)
88 with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u0 t2)
89 \Rightarrow (THead k0 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in
90 lref_map) (\lambda (x0: nat).(plus x0 (S O))) d t) ((let rec lref_map (f:
91 ((nat \to nat))) (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort n)
92 \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with
93 [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u0 t2)
94 \Rightarrow (THead k0 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in
95 lref_map) (\lambda (x0: nat).(plus x0 (S O))) (s k d) t0)) | (THead _ _ t1)
96 \Rightarrow t1])) (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t
97 t0) H1) in (\lambda (_: (eq T u t)).(\lambda (H6: (eq K (Bind b) k)).(let H7
98 \def (eq_ind_r K k (\lambda (k0: K).(eq T (lift (S O) d (THead k0 t t0)) t0))
99 H4 (Bind b) H6) in (let H8 \def (eq_ind T (lift (S O) d (THead (Bind b) t
100 t0)) (\lambda (t1: T).(eq T t1 t0)) H7 (THead (Bind b) (lift (S O) d t) (lift
101 (S O) (S d) t0)) (lift_bind b t t0 (S O) d)) in (H0 (lift (S O) d t) (S d) H8
102 P)))))) H3)) H2))))))))))) x)).
104 theorem nf2_gen_abbr:
105 \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abbr) u
106 t)) \to (\forall (P: Prop).P))))
108 \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2:
109 T).((pr2 c (THead (Bind Abbr) u t) t2) \to (eq T (THead (Bind Abbr) u t)
110 t2))))).(\lambda (P: Prop).(let H_x \def (dnf_dec u t O) in (let H0 \def H_x
111 in (ex_ind T (\lambda (v: T).(or (subst0 O u t (lift (S O) O v)) (eq T t
112 (lift (S O) O v)))) P (\lambda (x: T).(\lambda (H1: (or (subst0 O u t (lift
113 (S O) O x)) (eq T t (lift (S O) O x)))).(or_ind (subst0 O u t (lift (S O) O
114 x)) (eq T t (lift (S O) O x)) P (\lambda (H2: (subst0 O u t (lift (S O) O
115 x))).(let H3 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
116 (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _
117 _ t0) \Rightarrow t0])) (THead (Bind Abbr) u t) (THead (Bind Abbr) u (lift (S
118 O) O x)) (H (THead (Bind Abbr) u (lift (S O) O x)) (pr2_free c (THead (Bind
119 Abbr) u t) (THead (Bind Abbr) u (lift (S O) O x)) (pr0_delta u u (pr0_refl u)
120 t t (pr0_refl t) (lift (S O) O x) H2)))) in (let H4 \def (eq_ind T t (\lambda
121 (t0: T).(subst0 O u t0 (lift (S O) O x))) H2 (lift (S O) O x) H3) in
122 (subst0_refl u (lift (S O) O x) O H4 P)))) (\lambda (H2: (eq T t (lift (S O)
123 O x))).(let H3 \def (eq_ind T t (\lambda (t0: T).(\forall (t2: T).((pr2 c
124 (THead (Bind Abbr) u t0) t2) \to (eq T (THead (Bind Abbr) u t0) t2)))) H
125 (lift (S O) O x) H2) in (nf2_gen__aux Abbr x u O (H3 x (pr2_free c (THead
126 (Bind Abbr) u (lift (S O) O x)) x (pr0_zeta Abbr not_abbr_abst x x (pr0_refl
127 x) u))) P))) H1))) H0))))))).
129 theorem nf2_gen_void:
130 \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Void) u
131 (lift (S O) O t))) \to (\forall (P: Prop).P))))
133 \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2:
134 T).((pr2 c (THead (Bind Void) u (lift (S O) O t)) t2) \to (eq T (THead (Bind
135 Void) u (lift (S O) O t)) t2))))).(\lambda (P: Prop).(nf2_gen__aux Void t u O
136 (H t (pr2_free c (THead (Bind Void) u (lift (S O) O t)) t (pr0_zeta Void
137 not_void_abst t t (pr0_refl t) u))) P))))).
139 theorem arity_nf2_inv_all:
140 \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t
141 a) \to ((nf2 c t) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t
142 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c w)))
143 (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c (Bind Abst) w) u)))) (ex nat
144 (\lambda (n: nat).(eq T t (TSort n)))) (ex3_4 TList nat C T (\lambda (ws:
145 TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t (THeads
146 (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda
147 (d: C).(\lambda (v: T).(getl i c (CHead d (Bind Abst) v)))))) (\lambda (ws:
148 TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c
151 \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
152 (arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (t0: T).(\lambda (_:
153 A).((nf2 c0 t0) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0
154 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
155 (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat
156 (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_4 TList nat C T (\lambda (ws:
157 TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads
158 (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda
159 (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
160 TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0
161 ws))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (_: (nf2 c0 (TSort
162 n))).(or3_intro1 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T (TSort n)
163 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
164 (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat
165 (\lambda (n0: nat).(eq T (TSort n) (TSort n0)))) (ex3_4 TList nat C T
166 (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T
167 (TSort n) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda
168 (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst)
169 v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_:
170 T).(nfs2 c0 ws)))))) (ex_intro nat (\lambda (n0: nat).(eq T (TSort n) (TSort
171 n0))) n (refl_equal T (TSort n))))))) (\lambda (c0: C).(\lambda (d:
172 C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind
173 Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (_:
174 (((nf2 d u) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u
175 (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 d w)))
176 (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead d (Bind Abst) w) u0)))) (ex nat
177 (\lambda (n: nat).(eq T u (TSort n)))) (ex3_4 TList nat C T (\lambda (ws:
178 TList).(\lambda (i0: nat).(\lambda (_: C).(\lambda (_: T).(eq T u (THeads
179 (Flat Appl) ws (TLRef i0))))))) (\lambda (_: TList).(\lambda (i0:
180 nat).(\lambda (d0: C).(\lambda (v: T).(getl i0 d (CHead d0 (Bind Abst)
181 v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_:
182 T).(nfs2 d ws)))))))))).(\lambda (H3: (nf2 c0 (TLRef i))).(nf2_gen_lref c0 d
183 u i H0 H3 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (TLRef i)
184 (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
185 (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat
186 (\lambda (n: nat).(eq T (TLRef i) (TSort n)))) (ex3_4 TList nat C T (\lambda
187 (ws: TList).(\lambda (i0: nat).(\lambda (_: C).(\lambda (_: T).(eq T (TLRef
188 i) (THeads (Flat Appl) ws (TLRef i0))))))) (\lambda (_: TList).(\lambda (i0:
189 nat).(\lambda (d0: C).(\lambda (v: T).(getl i0 c0 (CHead d0 (Bind Abst)
190 v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_:
191 T).(nfs2 c0 ws))))))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u:
192 T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst)
193 u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (_:
194 (((nf2 d u) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u
195 (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 d w)))
196 (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead d (Bind Abst) w) u0)))) (ex nat
197 (\lambda (n: nat).(eq T u (TSort n)))) (ex3_4 TList nat C T (\lambda (ws:
198 TList).(\lambda (i0: nat).(\lambda (_: C).(\lambda (_: T).(eq T u (THeads
199 (Flat Appl) ws (TLRef i0))))))) (\lambda (_: TList).(\lambda (i0:
200 nat).(\lambda (d0: C).(\lambda (v: T).(getl i0 d (CHead d0 (Bind Abst)
201 v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_:
202 T).(nfs2 d ws)))))))))).(\lambda (_: (nf2 c0 (TLRef i))).(or3_intro2 (ex3_2 T
203 T (\lambda (w: T).(\lambda (u0: T).(eq T (TLRef i) (THead (Bind Abst) w
204 u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
205 (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
206 (TLRef i) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda
207 (i0: nat).(\lambda (_: C).(\lambda (_: T).(eq T (TLRef i) (THeads (Flat Appl)
208 ws (TLRef i0))))))) (\lambda (_: TList).(\lambda (i0: nat).(\lambda (d0:
209 C).(\lambda (v: T).(getl i0 c0 (CHead d0 (Bind Abst) v)))))) (\lambda (ws:
210 TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))
211 (ex3_4_intro TList nat C T (\lambda (ws: TList).(\lambda (i0: nat).(\lambda
212 (_: C).(\lambda (_: T).(eq T (TLRef i) (THeads (Flat Appl) ws (TLRef
213 i0))))))) (\lambda (_: TList).(\lambda (i0: nat).(\lambda (d0: C).(\lambda
214 (v: T).(getl i0 c0 (CHead d0 (Bind Abst) v)))))) (\lambda (ws:
215 TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))
216 TNil i d u (refl_equal T (TLRef i)) H0 I))))))))))) (\lambda (b: B).(\lambda
217 (H0: (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: T).(\lambda (a1:
218 A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((nf2 c0 u) \to (or3 (ex3_2
219 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind Abst) w u0))))
220 (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0:
221 T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T u
222 (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i:
223 nat).(\lambda (_: C).(\lambda (_: T).(eq T u (THeads (Flat Appl) ws (TLRef
224 i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v:
225 T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
226 nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))))).(\lambda (t0:
227 T).(\lambda (a2: A).(\lambda (H3: (arity g (CHead c0 (Bind b) u) t0
228 a2)).(\lambda (_: (((nf2 (CHead c0 (Bind b) u) t0) \to (or3 (ex3_2 T T
229 (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0))))
230 (\lambda (w: T).(\lambda (_: T).(nf2 (CHead c0 (Bind b) u) w))) (\lambda (w:
231 T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind b) u) (Bind Abst) w) u0))))
232 (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_4 TList nat C T (\lambda
233 (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t0
234 (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i:
235 nat).(\lambda (d: C).(\lambda (v: T).(getl i (CHead c0 (Bind b) u) (CHead d
236 (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
237 C).(\lambda (_: T).(nfs2 (CHead c0 (Bind b) u) ws)))))))))).(\lambda (H5:
238 (nf2 c0 (THead (Bind b) u t0))).(B_ind (\lambda (b0: B).((not (eq B b0 Abst))
239 \to ((arity g (CHead c0 (Bind b0) u) t0 a2) \to ((nf2 c0 (THead (Bind b0) u
240 t0)) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind
241 b0) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0
242 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))
243 (ex nat (\lambda (n: nat).(eq T (THead (Bind b0) u t0) (TSort n)))) (ex3_4
244 TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda
245 (_: T).(eq T (THead (Bind b0) u t0) (THeads (Flat Appl) ws (TLRef i)))))))
246 (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i
247 c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
248 nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))))))) (\lambda (_: (not
249 (eq B Abbr Abst))).(\lambda (_: (arity g (CHead c0 (Bind Abbr) u) t0
250 a2)).(\lambda (H8: (nf2 c0 (THead (Bind Abbr) u t0))).(nf2_gen_abbr c0 u t0
251 H8 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Abbr)
252 u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0
253 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))
254 (ex nat (\lambda (n: nat).(eq T (THead (Bind Abbr) u t0) (TSort n)))) (ex3_4
255 TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda
256 (_: T).(eq T (THead (Bind Abbr) u t0) (THeads (Flat Appl) ws (TLRef i)))))))
257 (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i
258 c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
259 nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))))))) (\lambda (H6:
260 (not (eq B Abst Abst))).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0
261 a2)).(\lambda (_: (nf2 c0 (THead (Bind Abst) u t0))).(let H9 \def (match (H6
262 (refl_equal B Abst)) in False return (\lambda (_: False).(or3 (ex3_2 T T
263 (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind
264 Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
265 T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n:
266 nat).(eq T (THead (Bind Abst) u t0) (TSort n)))) (ex3_4 TList nat C T
267 (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T
268 (THead (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
269 TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
270 (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
271 C).(\lambda (_: T).(nfs2 c0 ws)))))))) with []) in H9)))) (\lambda (_: (not
272 (eq B Void Abst))).(\lambda (H7: (arity g (CHead c0 (Bind Void) u) t0
273 a2)).(\lambda (H8: (nf2 c0 (THead (Bind Void) u t0))).(let H9 \def
274 (arity_gen_cvoid g (CHead c0 (Bind Void) u) t0 a2 H7 c0 u O (getl_refl Void
275 c0 u)) in (ex_ind T (\lambda (v: T).(eq T t0 (lift (S O) O v))) (or3 (ex3_2 T
276 T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Void) u t0) (THead
277 (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda
278 (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda
279 (n: nat).(eq T (THead (Bind Void) u t0) (TSort n)))) (ex3_4 TList nat C T
280 (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T
281 (THead (Bind Void) u t0) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
282 TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
283 (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
284 C).(\lambda (_: T).(nfs2 c0 ws))))))) (\lambda (x: T).(\lambda (H10: (eq T t0
285 (lift (S O) O x))).(let H11 \def (eq_ind T t0 (\lambda (t1: T).(nf2 c0 (THead
286 (Bind Void) u t1))) H8 (lift (S O) O x) H10) in (eq_ind_r T (lift (S O) O x)
287 (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T
288 (THead (Bind Void) u t1) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda
289 (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind
290 Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind Void) u t1)
291 (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i:
292 nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Bind Void) u t1) (THeads
293 (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda
294 (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
295 TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))))
296 (nf2_gen_void c0 u x H11 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq
297 T (THead (Bind Void) u (lift (S O) O x)) (THead (Bind Abst) w u0)))) (\lambda
298 (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2
299 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Bind
300 Void) u (lift (S O) O x)) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws:
301 TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Bind
302 Void) u (lift (S O) O x)) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
303 TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
304 (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
305 C).(\lambda (_: T).(nfs2 c0 ws)))))))) t0 H10)))) H9))))) b H0 H3
306 H5))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda
307 (_: (arity g c0 u (asucc g a1))).(\lambda (_: (((nf2 c0 u) \to (or3 (ex3_2 T
308 T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind Abst) w u0))))
309 (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0:
310 T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T u
311 (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i:
312 nat).(\lambda (_: C).(\lambda (_: T).(eq T u (THeads (Flat Appl) ws (TLRef
313 i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v:
314 T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
315 nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))))).(\lambda (t0:
316 T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0
317 a2)).(\lambda (_: (((nf2 (CHead c0 (Bind Abst) u) t0) \to (or3 (ex3_2 T T
318 (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0))))
319 (\lambda (w: T).(\lambda (_: T).(nf2 (CHead c0 (Bind Abst) u) w))) (\lambda
320 (w: T).(\lambda (u0: T).(nf2 (CHead (CHead c0 (Bind Abst) u) (Bind Abst) w)
321 u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_4 TList nat C T
322 (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T
323 t0 (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i:
324 nat).(\lambda (d: C).(\lambda (v: T).(getl i (CHead c0 (Bind Abst) u) (CHead
325 d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
326 C).(\lambda (_: T).(nfs2 (CHead c0 (Bind Abst) u) ws)))))))))).(\lambda (H4:
327 (nf2 c0 (THead (Bind Abst) u t0))).(let H5 \def (nf2_gen_abst c0 u t0 H4) in
328 (and_ind (nf2 c0 u) (nf2 (CHead c0 (Bind Abst) u) t0) (or3 (ex3_2 T T
329 (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind
330 Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
331 T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n:
332 nat).(eq T (THead (Bind Abst) u t0) (TSort n)))) (ex3_4 TList nat C T
333 (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T
334 (THead (Bind Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
335 TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
336 (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
337 C).(\lambda (_: T).(nfs2 c0 ws))))))) (\lambda (H6: (nf2 c0 u)).(\lambda (H7:
338 (nf2 (CHead c0 (Bind Abst) u) t0)).(or3_intro0 (ex3_2 T T (\lambda (w:
339 T).(\lambda (u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w
340 u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
341 (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
342 (THead (Bind Abst) u t0) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws:
343 TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Bind
344 Abst) u t0) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
345 TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
346 (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
347 C).(\lambda (_: T).(nfs2 c0 ws)))))) (ex3_2_intro T T (\lambda (w:
348 T).(\lambda (u0: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) w
349 u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
350 (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))) u t0 (refl_equal T (THead (Bind
351 Abst) u t0)) H6 H7)))) H5)))))))))))) (\lambda (c0: C).(\lambda (u:
352 T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: (((nf2 c0 u)
353 \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind
354 Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
355 T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n:
356 nat).(eq T u (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda
357 (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T u (THeads (Flat Appl) ws
358 (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d:
359 C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
360 TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0
361 ws)))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H2: (arity g c0 t0
362 (AHead a1 a2))).(\lambda (H3: (((nf2 c0 t0) \to (or3 (ex3_2 T T (\lambda (w:
363 T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w:
364 T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead
365 c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort n))))
366 (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_:
367 C).(\lambda (_: T).(eq T t0 (THeads (Flat Appl) ws (TLRef i))))))) (\lambda
368 (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0
369 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda
370 (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))))).(\lambda (H4: (nf2 c0 (THead
371 (Flat Appl) u t0))).(let H5 \def (nf2_gen_flat Appl c0 u t0 H4) in (and_ind
372 (nf2 c0 u) (nf2 c0 t0) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T
373 (THead (Flat Appl) u t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda
374 (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind
375 Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u t0)
376 (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i:
377 nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Flat Appl) u t0) (THeads
378 (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda
379 (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
380 TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))
381 (\lambda (H6: (nf2 c0 u)).(\lambda (H7: (nf2 c0 t0)).(let H_x \def (H3 H7) in
382 (let H8 \def H_x in (or3_ind (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq
383 T t0 (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
384 (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat
385 (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_4 TList nat C T (\lambda (ws:
386 TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads
387 (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda
388 (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
389 TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))
390 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u
391 t0) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
392 (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat
393 (\lambda (n: nat).(eq T (THead (Flat Appl) u t0) (TSort n)))) (ex3_4 TList
394 nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_:
395 T).(eq T (THead (Flat Appl) u t0) (THeads (Flat Appl) ws (TLRef i)))))))
396 (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i
397 c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
398 nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))) (\lambda (H9: (ex3_2
399 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0))))
400 (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0:
401 T).(nf2 (CHead c0 (Bind Abst) w) u0))))).(ex3_2_ind T T (\lambda (w:
402 T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w u0)))) (\lambda (w:
403 T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead
404 c0 (Bind Abst) w) u0))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq
405 T (THead (Flat Appl) u t0) (THead (Bind Abst) w u0)))) (\lambda (w:
406 T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead
407 c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u
408 t0) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i:
409 nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Flat Appl) u t0) (THeads
410 (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda
411 (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
412 TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))
413 (\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: (eq T t0 (THead (Bind Abst)
414 x0 x1))).(\lambda (_: (nf2 c0 x0)).(\lambda (_: (nf2 (CHead c0 (Bind Abst)
415 x0) x1)).(let H13 \def (eq_ind T t0 (\lambda (t1: T).(nf2 c0 (THead (Flat
416 Appl) u t1))) H4 (THead (Bind Abst) x0 x1) H10) in (let H14 \def (eq_ind T t0
417 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2 (THead (Bind Abst) x0 x1)
418 H10) in (eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t1: T).(or3 (ex3_2 T
419 T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t1) (THead
420 (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda
421 (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda
422 (n: nat).(eq T (THead (Flat Appl) u t1) (TSort n)))) (ex3_4 TList nat C T
423 (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T
424 (THead (Flat Appl) u t1) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
425 TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
426 (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
427 C).(\lambda (_: T).(nfs2 c0 ws)))))))) (nf2_gen_beta c0 u x0 x1 H13 (or3
428 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (THead
429 (Bind Abst) x0 x1)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_:
430 T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst)
431 w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (THead (Bind
432 Abst) x0 x1)) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda
433 (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Flat Appl) u (THead
434 (Bind Abst) x0 x1)) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
435 TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
436 (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
437 C).(\lambda (_: T).(nfs2 c0 ws)))))))) t0 H10)))))))) H9)) (\lambda (H9: (ex
438 nat (\lambda (n: nat).(eq T t0 (TSort n))))).(ex_ind nat (\lambda (n:
439 nat).(eq T t0 (TSort n))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0:
440 T).(eq T (THead (Flat Appl) u t0) (THead (Bind Abst) w u0)))) (\lambda (w:
441 T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead
442 c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u
443 t0) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i:
444 nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Flat Appl) u t0) (THeads
445 (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda
446 (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
447 TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))
448 (\lambda (x: nat).(\lambda (H10: (eq T t0 (TSort x))).(let H11 \def (eq_ind T
449 t0 (\lambda (t1: T).(nf2 c0 (THead (Flat Appl) u t1))) H4 (TSort x) H10) in
450 (let H12 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1 a2))) H2
451 (TSort x) H10) in (eq_ind_r T (TSort x) (\lambda (t1: T).(or3 (ex3_2 T T
452 (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t1) (THead (Bind
453 Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
454 T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n:
455 nat).(eq T (THead (Flat Appl) u t1) (TSort n)))) (ex3_4 TList nat C T
456 (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T
457 (THead (Flat Appl) u t1) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
458 TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
459 (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
460 C).(\lambda (_: T).(nfs2 c0 ws)))))))) (let H13 \def (match (arity_gen_sort g
461 c0 x (AHead a1 a2) H12) in leq return (\lambda (a0: A).(\lambda (a3:
462 A).(\lambda (_: (leq ? a0 a3)).((eq A a0 (AHead a1 a2)) \to ((eq A a3 (ASort
463 O x)) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat
464 Appl) u (TSort x)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_:
465 T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst)
466 w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (TSort x))
467 (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i:
468 nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Flat Appl) u (TSort x))
469 (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i:
470 nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v))))))
471 (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2
472 c0 ws)))))))))))) with [(leq_sort h1 h2 n1 n2 k H13) \Rightarrow (\lambda
473 (H14: (eq A (ASort h1 n1) (AHead a1 a2))).(\lambda (H15: (eq A (ASort h2 n2)
474 (ASort O x))).((let H16 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e
475 in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead
476 _ _) \Rightarrow False])) I (AHead a1 a2) H14) in (False_ind ((eq A (ASort h2
477 n2) (ASort O x)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2)
478 k)) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat
479 Appl) u (TSort x)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_:
480 T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst)
481 w) u0)))) (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (TSort x))
482 (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i:
483 nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Flat Appl) u (TSort x))
484 (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i:
485 nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v))))))
486 (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2
487 c0 ws))))))))) H16)) H15 H13))) | (leq_head a0 a3 H13 a4 a5 H14) \Rightarrow
488 (\lambda (H15: (eq A (AHead a0 a4) (AHead a1 a2))).(\lambda (H16: (eq A
489 (AHead a3 a5) (ASort O x))).((let H17 \def (f_equal A A (\lambda (e:
490 A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 |
491 (AHead _ a6) \Rightarrow a6])) (AHead a0 a4) (AHead a1 a2) H15) in ((let H18
492 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A)
493 with [(ASort _ _) \Rightarrow a0 | (AHead a6 _) \Rightarrow a6])) (AHead a0
494 a4) (AHead a1 a2) H15) in (eq_ind A a1 (\lambda (a6: A).((eq A a4 a2) \to
495 ((eq A (AHead a3 a5) (ASort O x)) \to ((leq g a6 a3) \to ((leq g a4 a5) \to
496 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u
497 (TSort x)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2
498 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))
499 (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n))))
500 (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_:
501 C).(\lambda (_: T).(eq T (THead (Flat Appl) u (TSort x)) (THeads (Flat Appl)
502 ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d:
503 C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
504 TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0
505 ws)))))))))))) (\lambda (H19: (eq A a4 a2)).(eq_ind A a2 (\lambda (a6:
506 A).((eq A (AHead a3 a5) (ASort O x)) \to ((leq g a1 a3) \to ((leq g a6 a5)
507 \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl)
508 u (TSort x)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2
509 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))
510 (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n))))
511 (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_:
512 C).(\lambda (_: T).(eq T (THead (Flat Appl) u (TSort x)) (THeads (Flat Appl)
513 ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d:
514 C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
515 TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0
516 ws))))))))))) (\lambda (H20: (eq A (AHead a3 a5) (ASort O x))).(let H21 \def
517 (eq_ind A (AHead a3 a5) (\lambda (e: A).(match e in A return (\lambda (_:
518 A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow
519 True])) I (ASort O x) H20) in (False_ind ((leq g a1 a3) \to ((leq g a2 a5)
520 \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl)
521 u (TSort x)) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2
522 c0 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))
523 (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (TSort x)) (TSort n))))
524 (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_:
525 C).(\lambda (_: T).(eq T (THead (Flat Appl) u (TSort x)) (THeads (Flat Appl)
526 ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d:
527 C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
528 TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))))
529 H21))) a4 (sym_eq A a4 a2 H19))) a0 (sym_eq A a0 a1 H18))) H17)) H16 H13
530 H14)))]) in (H13 (refl_equal A (AHead a1 a2)) (refl_equal A (ASort O x)))) t0
531 H10))))) H9)) (\lambda (H9: (ex3_4 TList nat C T (\lambda (ws:
532 TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads
533 (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda
534 (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
535 TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0
536 ws))))))).(ex3_4_ind TList nat C T (\lambda (ws: TList).(\lambda (i:
537 nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads (Flat Appl) ws (TLRef
538 i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v:
539 T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
540 nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))) (or3 (ex3_2 T T
541 (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t0) (THead (Bind
542 Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
543 T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n:
544 nat).(eq T (THead (Flat Appl) u t0) (TSort n)))) (ex3_4 TList nat C T
545 (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T
546 (THead (Flat Appl) u t0) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
547 TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
548 (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
549 C).(\lambda (_: T).(nfs2 c0 ws))))))) (\lambda (x0: TList).(\lambda (x1:
550 nat).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H10: (eq T t0 (THeads (Flat
551 Appl) x0 (TLRef x1)))).(\lambda (H11: (getl x1 c0 (CHead x2 (Bind Abst)
552 x3))).(\lambda (H12: (nfs2 c0 x0)).(let H13 \def (eq_ind T t0 (\lambda (t1:
553 T).(nf2 c0 (THead (Flat Appl) u t1))) H4 (THeads (Flat Appl) x0 (TLRef x1))
554 H10) in (let H14 \def (eq_ind T t0 (\lambda (t1: T).(arity g c0 t1 (AHead a1
555 a2))) H2 (THeads (Flat Appl) x0 (TLRef x1)) H10) in (eq_ind_r T (THeads (Flat
556 Appl) x0 (TLRef x1)) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w:
557 T).(\lambda (u0: T).(eq T (THead (Flat Appl) u t1) (THead (Bind Abst) w
558 u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
559 (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
560 (THead (Flat Appl) u t1) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws:
561 TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Flat
562 Appl) u t1) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
563 TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
564 (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
565 C).(\lambda (_: T).(nfs2 c0 ws)))))))) (or3_intro2 (ex3_2 T T (\lambda (w:
566 T).(\lambda (u0: T).(eq T (THead (Flat Appl) u (THeads (Flat Appl) x0 (TLRef
567 x1))) (THead (Bind Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0
568 w))) (\lambda (w: T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0))))
569 (ex nat (\lambda (n: nat).(eq T (THead (Flat Appl) u (THeads (Flat Appl) x0
570 (TLRef x1))) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda
571 (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead (Flat Appl) u (THeads
572 (Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda
573 (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0
574 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda
575 (_: C).(\lambda (_: T).(nfs2 c0 ws)))))) (ex3_4_intro TList nat C T (\lambda
576 (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (THead
577 (Flat Appl) u (THeads (Flat Appl) x0 (TLRef x1))) (THeads (Flat Appl) ws
578 (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d:
579 C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
580 TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))
581 (TCons u x0) x1 x2 x3 (refl_equal T (THead (Flat Appl) u (THeads (Flat Appl)
582 x0 (TLRef x1)))) H11 (conj (nf2 c0 u) (nfs2 c0 x0) H6 H12))) t0 H10))))))))))
583 H9)) H8))))) H5)))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a0:
584 A).(\lambda (_: (arity g c0 u (asucc g a0))).(\lambda (_: (((nf2 c0 u) \to
585 (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T u (THead (Bind Abst) w
586 u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
587 (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
588 u (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i:
589 nat).(\lambda (_: C).(\lambda (_: T).(eq T u (THeads (Flat Appl) ws (TLRef
590 i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v:
591 T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
592 nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))))).(\lambda (t0:
593 T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (_: (((nf2 c0 t0) \to (or3
594 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T t0 (THead (Bind Abst) w
595 u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
596 (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n: nat).(eq T
597 t0 (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i:
598 nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads (Flat Appl) ws (TLRef
599 i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v:
600 T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
601 nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))))).(\lambda (H4: (nf2
602 c0 (THead (Flat Cast) u t0))).(nf2_gen_cast c0 u t0 H4 (or3 (ex3_2 T T
603 (\lambda (w: T).(\lambda (u0: T).(eq T (THead (Flat Cast) u t0) (THead (Bind
604 Abst) w u0)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
605 T).(\lambda (u0: T).(nf2 (CHead c0 (Bind Abst) w) u0)))) (ex nat (\lambda (n:
606 nat).(eq T (THead (Flat Cast) u t0) (TSort n)))) (ex3_4 TList nat C T
607 (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T
608 (THead (Flat Cast) u t0) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
609 TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
610 (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
611 C).(\lambda (_: T).(nfs2 c0 ws))))))))))))))))) (\lambda (c0: C).(\lambda
612 (t0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 t0 a1)).(\lambda (H1:
613 (((nf2 c0 t0) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0
614 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
615 (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat
616 (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_4 TList nat C T (\lambda (ws:
617 TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads
618 (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda
619 (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
620 TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0
621 ws)))))))))).(\lambda (a2: A).(\lambda (_: (leq g a1 a2)).(\lambda (H3: (nf2
622 c0 t0)).(let H_x \def (H1 H3) in (let H4 \def H_x in (or3_ind (ex3_2 T T
623 (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda
624 (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2
625 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort
626 n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda
627 (_: C).(\lambda (_: T).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))))
628 (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i
629 c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
630 nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))) (or3 (ex3_2 T T
631 (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda
632 (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2
633 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort
634 n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda
635 (_: C).(\lambda (_: T).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))))
636 (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i
637 c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
638 nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))) (\lambda (H5: (ex3_2
639 T T (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u))))
640 (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u:
641 T).(nf2 (CHead c0 (Bind Abst) w) u))))).(ex3_2_ind T T (\lambda (w:
642 T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda (w:
643 T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead
644 c0 (Bind Abst) w) u))) (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T
645 t0 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w)))
646 (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat
647 (\lambda (n: nat).(eq T t0 (TSort n)))) (ex3_4 TList nat C T (\lambda (ws:
648 TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads
649 (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda
650 (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
651 TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))
652 (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t0 (THead (Bind Abst)
653 x0 x1))).(\lambda (H7: (nf2 c0 x0)).(\lambda (H8: (nf2 (CHead c0 (Bind Abst)
654 x0) x1)).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t1: T).(or3 (ex3_2 T
655 T (\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind Abst) w u))))
656 (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u:
657 T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t1
658 (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i:
659 nat).(\lambda (_: C).(\lambda (_: T).(eq T t1 (THeads (Flat Appl) ws (TLRef
660 i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v:
661 T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
662 nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws)))))))) (or3_intro0 (ex3_2 T
663 T (\lambda (w: T).(\lambda (u: T).(eq T (THead (Bind Abst) x0 x1) (THead
664 (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
665 T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n:
666 nat).(eq T (THead (Bind Abst) x0 x1) (TSort n)))) (ex3_4 TList nat C T
667 (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T
668 (THead (Bind Abst) x0 x1) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
669 TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
670 (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
671 C).(\lambda (_: T).(nfs2 c0 ws)))))) (ex3_2_intro T T (\lambda (w:
672 T).(\lambda (u: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) w u))))
673 (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u:
674 T).(nf2 (CHead c0 (Bind Abst) w) u))) x0 x1 (refl_equal T (THead (Bind Abst)
675 x0 x1)) H7 H8)) t0 H6)))))) H5)) (\lambda (H5: (ex nat (\lambda (n: nat).(eq
676 T t0 (TSort n))))).(ex_ind nat (\lambda (n: nat).(eq T t0 (TSort n))) (or3
677 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w
678 u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda
679 (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0
680 (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i:
681 nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads (Flat Appl) ws (TLRef
682 i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v:
683 T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
684 nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))) (\lambda (x:
685 nat).(\lambda (H6: (eq T t0 (TSort x))).(eq_ind_r T (TSort x) (\lambda (t1:
686 T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind
687 Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
688 T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n:
689 nat).(eq T t1 (TSort n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda
690 (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t1 (THeads (Flat Appl) ws
691 (TLRef i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d:
692 C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws:
693 TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))))
694 (or3_intro1 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T (TSort x) (THead
695 (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w:
696 T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n:
697 nat).(eq T (TSort x) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws:
698 TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (TSort x)
699 (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i:
700 nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v))))))
701 (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2
702 c0 ws)))))) (ex_intro nat (\lambda (n: nat).(eq T (TSort x) (TSort n))) x
703 (refl_equal T (TSort x)))) t0 H6))) H5)) (\lambda (H5: (ex3_4 TList nat C T
704 (\lambda (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T
705 t0 (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i:
706 nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v))))))
707 (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2
708 c0 ws))))))).(ex3_4_ind TList nat C T (\lambda (ws: TList).(\lambda (i:
709 nat).(\lambda (_: C).(\lambda (_: T).(eq T t0 (THeads (Flat Appl) ws (TLRef
710 i))))))) (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v:
711 T).(getl i c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
712 nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))) (or3 (ex3_2 T T
713 (\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind Abst) w u)))) (\lambda
714 (w: T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2
715 (CHead c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T t0 (TSort
716 n)))) (ex3_4 TList nat C T (\lambda (ws: TList).(\lambda (i: nat).(\lambda
717 (_: C).(\lambda (_: T).(eq T t0 (THeads (Flat Appl) ws (TLRef i)))))))
718 (\lambda (_: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i
719 c0 (CHead d (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_:
720 nat).(\lambda (_: C).(\lambda (_: T).(nfs2 c0 ws))))))) (\lambda (x0:
721 TList).(\lambda (x1: nat).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H6: (eq
722 T t0 (THeads (Flat Appl) x0 (TLRef x1)))).(\lambda (H7: (getl x1 c0 (CHead x2
723 (Bind Abst) x3))).(\lambda (H8: (nfs2 c0 x0)).(eq_ind_r T (THeads (Flat Appl)
724 x0 (TLRef x1)) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (w: T).(\lambda (u:
725 T).(eq T t1 (THead (Bind Abst) w u)))) (\lambda (w: T).(\lambda (_: T).(nf2
726 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead c0 (Bind Abst) w) u))))
727 (ex nat (\lambda (n: nat).(eq T t1 (TSort n)))) (ex3_4 TList nat C T (\lambda
728 (ws: TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T t1
729 (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_: TList).(\lambda (i:
730 nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d (Bind Abst) v))))))
731 (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_: C).(\lambda (_: T).(nfs2
732 c0 ws)))))))) (or3_intro2 (ex3_2 T T (\lambda (w: T).(\lambda (u: T).(eq T
733 (THeads (Flat Appl) x0 (TLRef x1)) (THead (Bind Abst) w u)))) (\lambda (w:
734 T).(\lambda (_: T).(nf2 c0 w))) (\lambda (w: T).(\lambda (u: T).(nf2 (CHead
735 c0 (Bind Abst) w) u)))) (ex nat (\lambda (n: nat).(eq T (THeads (Flat Appl)
736 x0 (TLRef x1)) (TSort n)))) (ex3_4 TList nat C T (\lambda (ws:
737 TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (THeads (Flat
738 Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
739 TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
740 (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
741 C).(\lambda (_: T).(nfs2 c0 ws)))))) (ex3_4_intro TList nat C T (\lambda (ws:
742 TList).(\lambda (i: nat).(\lambda (_: C).(\lambda (_: T).(eq T (THeads (Flat
743 Appl) x0 (TLRef x1)) (THeads (Flat Appl) ws (TLRef i))))))) (\lambda (_:
744 TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(getl i c0 (CHead d
745 (Bind Abst) v)))))) (\lambda (ws: TList).(\lambda (_: nat).(\lambda (_:
746 C).(\lambda (_: T).(nfs2 c0 ws))))) x0 x1 x2 x3 (refl_equal T (THeads (Flat
747 Appl) x0 (TLRef x1))) H7 H8)) t0 H6)))))))) H5)) H4))))))))))) c t a H))))).
749 theorem pc3_gen_sort_abst:
750 \forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (n: nat).((pc3 c
751 (TSort n) (THead (Bind Abst) u t)) \to (\forall (P: Prop).P)))))
753 \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (n: nat).(\lambda
754 (H: (pc3 c (TSort n) (THead (Bind Abst) u t))).(\lambda (P: Prop).(let H0
755 \def H in (ex2_ind T (\lambda (t0: T).(pr3 c (TSort n) t0)) (\lambda (t0:
756 T).(pr3 c (THead (Bind Abst) u t) t0)) P (\lambda (x: T).(\lambda (H1: (pr3 c
757 (TSort n) x)).(\lambda (H2: (pr3 c (THead (Bind Abst) u t) x)).(let H3 \def
758 (pr3_gen_abst c u t x H2) in (ex3_2_ind T T (\lambda (u2: T).(\lambda (t2:
759 T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3
760 c u u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u0:
761 T).(pr3 (CHead c (Bind b) u0) t t2))))) P (\lambda (x0: T).(\lambda (x1:
762 T).(\lambda (H4: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (_: (pr3 c u
763 x0)).(\lambda (_: ((\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b)
764 u0) t x1))))).(let H7 \def (eq_ind T x (\lambda (t0: T).(eq T t0 (TSort n)))
765 (pr3_gen_sort c x n H1) (THead (Bind Abst) x0 x1) H4) in (let H8 \def (eq_ind
766 T (THead (Bind Abst) x0 x1) (\lambda (ee: T).(match ee in T return (\lambda
767 (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
768 | (THead _ _ _) \Rightarrow True])) I (TSort n) H7) in (False_ind P
769 H8)))))))) H3))))) H0))))))).
771 theorem ty3_gen_abst_abst:
772 \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall
773 (t2: T).((ty3 g c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2)) \to (ex2
774 T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst)
777 \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda
778 (t2: T).(\lambda (H: (ty3 g c (THead (Bind Abst) u t1) (THead (Bind Abst) u
779 t2))).(ex_ind T (\lambda (t: T).(ty3 g c (THead (Bind Abst) u t2) t)) (ex2 T
780 (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u)
781 t1 t2))) (\lambda (x: T).(\lambda (H0: (ty3 g c (THead (Bind Abst) u t2)
782 x)).(ex4_3_ind T T T (\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(pc3 c
783 (THead (Bind Abst) u t3) x)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_:
784 T).(ty3 g c u t)))) (\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(ty3 g
785 (CHead c (Bind Abst) u) t2 t3)))) (\lambda (t3: T).(\lambda (_: T).(\lambda
786 (t0: T).(ty3 g (CHead c (Bind Abst) u) t3 t0)))) (ex2 T (\lambda (w: T).(ty3
787 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2))) (\lambda
788 (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c (THead (Bind
789 Abst) u x0) x)).(\lambda (_: (ty3 g c u x1)).(\lambda (H3: (ty3 g (CHead c
790 (Bind Abst) u) t2 x0)).(\lambda (_: (ty3 g (CHead c (Bind Abst) u) x0
791 x2)).(ex4_3_ind T T T (\lambda (t3: T).(\lambda (_: T).(\lambda (_: T).(pc3 c
792 (THead (Bind Abst) u t3) (THead (Bind Abst) u t2))))) (\lambda (_:
793 T).(\lambda (t: T).(\lambda (_: T).(ty3 g c u t)))) (\lambda (t3: T).(\lambda
794 (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t3)))) (\lambda (t3:
795 T).(\lambda (_: T).(\lambda (t0: T).(ty3 g (CHead c (Bind Abst) u) t3 t0))))
796 (ex2 T (\lambda (w: T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind
797 Abst) u) t1 t2))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda
798 (H5: (pc3 c (THead (Bind Abst) u x3) (THead (Bind Abst) u t2))).(\lambda (H6:
799 (ty3 g c u x4)).(\lambda (H7: (ty3 g (CHead c (Bind Abst) u) t1 x3)).(\lambda
800 (_: (ty3 g (CHead c (Bind Abst) u) x3 x5)).(and_ind (pc3 c u u) (\forall (b:
801 B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) x3 t2))) (ex2 T (\lambda (w:
802 T).(ty3 g c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2)))
803 (\lambda (_: (pc3 c u u)).(\lambda (H10: ((\forall (b: B).(\forall (u0:
804 T).(pc3 (CHead c (Bind b) u0) x3 t2))))).(ex_intro2 T (\lambda (w: T).(ty3 g
805 c u w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) u) t1 t2)) x4 H6
806 (ty3_conv g (CHead c (Bind Abst) u) t2 x0 H3 t1 x3 H7 (H10 Abst u)))))
807 (pc3_gen_abst c u u x3 t2 H5))))))))) (ty3_gen_bind g Abst c u t1 (THead
808 (Bind Abst) u t2) H))))))))) (ty3_gen_bind g Abst c u t2 x H0))))
809 (ty3_correct g c (THead (Bind Abst) u t1) (THead (Bind Abst) u t2) H))))))).
811 theorem ty3_typecheck:
812 \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (v: T).((ty3 g c t
813 v) \to (ex T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u)))))))
815 \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (v: T).(\lambda (H:
816 (ty3 g c t v)).(ex_ind T (\lambda (t0: T).(ty3 g c v t0)) (ex T (\lambda (u:
817 T).(ty3 g c (THead (Flat Cast) v t) u))) (\lambda (x: T).(\lambda (H0: (ty3 g
818 c v x)).(ex_intro T (\lambda (u: T).(ty3 g c (THead (Flat Cast) v t) u)) v
819 (ty3_cast g c t v H x H0)))) (ty3_correct g c t v H)))))).
821 inductive sort: T \to Prop \def
822 | sort_sort: \forall (n: nat).(sort (TSort n))
823 | sort_abst: \forall (u: T).((sort u) \to (\forall (t: T).((sort t) \to (sort
824 (THead (Bind Abst) u t))))).
827 \forall (t: T).((sort t) \to (\forall (c: C).(nf2 c t)))
829 \lambda (t: T).(\lambda (H: (sort t)).(sort_ind (\lambda (t0: T).(\forall
830 (c: C).(nf2 c t0))) (\lambda (n: nat).(\lambda (c: C).(nf2_sort c n)))
831 (\lambda (u: T).(\lambda (_: (sort u)).(\lambda (H1: ((\forall (c: C).(nf2 c
832 u)))).(\lambda (t0: T).(\lambda (_: (sort t0)).(\lambda (H3: ((\forall (c:
833 C).(nf2 c t0)))).(\lambda (c: C).(let H_y \def (H3 (CHead c (Bind Abst) u))
834 in (nf2_abst c u (H1 c) Abst u t0 H_y))))))))) t H)).
837 \forall (t1: T).((sort t1) \to (\forall (t2: T).((sort t2) \to (\forall (c:
838 C).((pc3 c t1 t2) \to (eq T t1 t2))))))
840 \lambda (t1: T).(\lambda (H: (sort t1)).(sort_ind (\lambda (t: T).(\forall
841 (t2: T).((sort t2) \to (\forall (c: C).((pc3 c t t2) \to (eq T t t2))))))
842 (\lambda (n: nat).(\lambda (t2: T).(\lambda (H0: (sort t2)).(sort_ind
843 (\lambda (t: T).(\forall (c: C).((pc3 c (TSort n) t) \to (eq T (TSort n)
844 t)))) (\lambda (n0: nat).(\lambda (c: C).(\lambda (H1: (pc3 c (TSort n)
845 (TSort n0))).(eq_ind nat n (\lambda (n1: nat).(eq T (TSort n) (TSort n1)))
846 (refl_equal T (TSort n)) n0 (pc3_gen_sort c n n0 H1))))) (\lambda (u:
847 T).(\lambda (_: (sort u)).(\lambda (_: ((\forall (c: C).((pc3 c (TSort n) u)
848 \to (eq T (TSort n) u))))).(\lambda (t: T).(\lambda (_: (sort t)).(\lambda
849 (_: ((\forall (c: C).((pc3 c (TSort n) t) \to (eq T (TSort n) t))))).(\lambda
850 (c: C).(\lambda (H5: (pc3 c (TSort n) (THead (Bind Abst) u
851 t))).(pc3_gen_sort_abst c u t n H5 (eq T (TSort n) (THead (Bind Abst) u
852 t))))))))))) t2 H0)))) (\lambda (u: T).(\lambda (_: (sort u)).(\lambda (H1:
853 ((\forall (t2: T).((sort t2) \to (\forall (c: C).((pc3 c u t2) \to (eq T u
854 t2))))))).(\lambda (t: T).(\lambda (_: (sort t)).(\lambda (H3: ((\forall (t2:
855 T).((sort t2) \to (\forall (c: C).((pc3 c t t2) \to (eq T t
856 t2))))))).(\lambda (t2: T).(\lambda (H4: (sort t2)).(sort_ind (\lambda (t0:
857 T).(\forall (c: C).((pc3 c (THead (Bind Abst) u t) t0) \to (eq T (THead (Bind
858 Abst) u t) t0)))) (\lambda (n: nat).(\lambda (c: C).(\lambda (H5: (pc3 c
859 (THead (Bind Abst) u t) (TSort n))).(pc3_gen_sort_abst c u t n (pc3_s c
860 (TSort n) (THead (Bind Abst) u t) H5) (eq T (THead (Bind Abst) u t) (TSort
861 n)))))) (\lambda (u0: T).(\lambda (H5: (sort u0)).(\lambda (_: ((\forall (c:
862 C).((pc3 c (THead (Bind Abst) u t) u0) \to (eq T (THead (Bind Abst) u t)
863 u0))))).(\lambda (t0: T).(\lambda (H7: (sort t0)).(\lambda (_: ((\forall (c:
864 C).((pc3 c (THead (Bind Abst) u t) t0) \to (eq T (THead (Bind Abst) u t)
865 t0))))).(\lambda (c: C).(\lambda (H9: (pc3 c (THead (Bind Abst) u t) (THead
866 (Bind Abst) u0 t0))).(and_ind (pc3 c u u0) (\forall (b: B).(\forall (u1:
867 T).(pc3 (CHead c (Bind b) u1) t t0))) (eq T (THead (Bind Abst) u t) (THead
868 (Bind Abst) u0 t0)) (\lambda (H10: (pc3 c u u0)).(\lambda (H11: ((\forall (b:
869 B).(\forall (u1: T).(pc3 (CHead c (Bind b) u1) t t0))))).(let H_y \def (H11
870 Abbr u) in (let H_y0 \def (H1 u0 H5 c H10) in (let H_y1 \def (H3 t0 H7 (CHead
871 c (Bind Abbr) u) H_y) in (let H12 \def (eq_ind_r T t0 (\lambda (t3: T).(pc3
872 (CHead c (Bind Abbr) u) t t3)) H_y t H_y1) in (let H13 \def (eq_ind_r T t0
873 (\lambda (t3: T).(sort t3)) H7 t H_y1) in (eq_ind T t (\lambda (t3: T).(eq T
874 (THead (Bind Abst) u t) (THead (Bind Abst) u0 t3))) (let H14 \def (eq_ind_r T
875 u0 (\lambda (t3: T).(pc3 c u t3)) H10 u H_y0) in (let H15 \def (eq_ind_r T u0
876 (\lambda (t3: T).(sort t3)) H5 u H_y0) in (eq_ind T u (\lambda (t3: T).(eq T
877 (THead (Bind Abst) u t) (THead (Bind Abst) t3 t))) (refl_equal T (THead (Bind
878 Abst) u t)) u0 H_y0))) t0 H_y1)))))))) (pc3_gen_abst c u u0 t t0 H9))))))))))
879 t2 H4))))))))) t1 H)).
881 theorem sort_correct:
882 \forall (g: G).(\forall (t1: T).((sort t1) \to (\forall (c: C).(ex3 T
883 (\lambda (t2: T).(tau0 g c t1 t2)) (\lambda (t2: T).(ty3 g c t1 t2)) (\lambda
884 (t2: T).(sort t2))))))
886 \lambda (g: G).(\lambda (t1: T).(\lambda (H: (sort t1)).(sort_ind (\lambda
887 (t: T).(\forall (c: C).(ex3 T (\lambda (t2: T).(tau0 g c t t2)) (\lambda (t2:
888 T).(ty3 g c t t2)) (\lambda (t2: T).(sort t2))))) (\lambda (n: nat).(\lambda
889 (c: C).(ex3_intro T (\lambda (t2: T).(tau0 g c (TSort n) t2)) (\lambda (t2:
890 T).(ty3 g c (TSort n) t2)) (\lambda (t2: T).(sort t2)) (TSort (next g n))
891 (tau0_sort g c n) (ty3_sort g c n) (sort_sort (next g n))))) (\lambda (u:
892 T).(\lambda (H0: (sort u)).(\lambda (H1: ((\forall (c: C).(ex3 T (\lambda
893 (t2: T).(tau0 g c u t2)) (\lambda (t2: T).(ty3 g c u t2)) (\lambda (t2:
894 T).(sort t2)))))).(\lambda (t: T).(\lambda (_: (sort t)).(\lambda (H3:
895 ((\forall (c: C).(ex3 T (\lambda (t2: T).(tau0 g c t t2)) (\lambda (t2:
896 T).(ty3 g c t t2)) (\lambda (t2: T).(sort t2)))))).(\lambda (c: C).(let H_x
897 \def (H1 c) in (let H4 \def H_x in (ex3_ind T (\lambda (t2: T).(tau0 g c u
898 t2)) (\lambda (t2: T).(ty3 g c u t2)) (\lambda (t2: T).(sort t2)) (ex3 T
899 (\lambda (t2: T).(tau0 g c (THead (Bind Abst) u t) t2)) (\lambda (t2: T).(ty3
900 g c (THead (Bind Abst) u t) t2)) (\lambda (t2: T).(sort t2))) (\lambda (x0:
901 T).(\lambda (_: (tau0 g c u x0)).(\lambda (H6: (ty3 g c u x0)).(\lambda (_:
902 (sort x0)).(let H_x0 \def (H3 (CHead c (Bind Abst) u)) in (let H8 \def H_x0
903 in (ex3_ind T (\lambda (t2: T).(tau0 g (CHead c (Bind Abst) u) t t2))
904 (\lambda (t2: T).(ty3 g (CHead c (Bind Abst) u) t t2)) (\lambda (t2: T).(sort
905 t2)) (ex3 T (\lambda (t2: T).(tau0 g c (THead (Bind Abst) u t) t2)) (\lambda
906 (t2: T).(ty3 g c (THead (Bind Abst) u t) t2)) (\lambda (t2: T).(sort t2)))
907 (\lambda (x1: T).(\lambda (H9: (tau0 g (CHead c (Bind Abst) u) t
908 x1)).(\lambda (H10: (ty3 g (CHead c (Bind Abst) u) t x1)).(\lambda (H11:
909 (sort x1)).(ex_ind T (\lambda (t0: T).(ty3 g (CHead c (Bind Abst) u) x1 t0))
910 (ex3 T (\lambda (t2: T).(tau0 g c (THead (Bind Abst) u t) t2)) (\lambda (t2:
911 T).(ty3 g c (THead (Bind Abst) u t) t2)) (\lambda (t2: T).(sort t2)))
912 (\lambda (x: T).(\lambda (H12: (ty3 g (CHead c (Bind Abst) u) x1
913 x)).(ex3_intro T (\lambda (t2: T).(tau0 g c (THead (Bind Abst) u t) t2))
914 (\lambda (t2: T).(ty3 g c (THead (Bind Abst) u t) t2)) (\lambda (t2: T).(sort
915 t2)) (THead (Bind Abst) u x1) (tau0_bind g Abst c u t x1 H9) (ty3_bind g c u
916 x0 H6 Abst t x1 H10 x H12) (sort_abst u H0 x1 H11)))) (ty3_correct g (CHead c
917 (Bind Abst) u) t x1 H10)))))) H8))))))) H4)))))))))) t1 H))).
919 definition pchurch_context:
922 \lambda (t: T).(\lambda (u: T).(THead (Bind Abst) t (THead (Bind Abst)
923 (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) u))).
928 \lambda (t: T).(pchurch_context t (lift (S (S O)) O t)).
930 definition church_body:
933 let rec church_body (n: nat) on n: T \def (match n with [O \Rightarrow
934 (TLRef (S O)) | (S n0) \Rightarrow (THead (Flat Appl) (church_body n0) (TLRef
935 O))]) in church_body.
940 \lambda (t: T).(\lambda (n: nat).(pchurch_context t (church_body n))).
942 theorem pnat_props__lift_SSO_O:
943 \forall (t: T).(eq T (lift (S (S O)) O t) (lift (S O) O (lift (S O) O t)))
945 \lambda (t: T).(eq_ind_r T (lift (plus (S O) (S O)) O t) (\lambda (t0:
946 T).(eq T (lift (S (S O)) O t) t0)) (refl_equal T (lift (plus (S O) (S O)) O
947 t)) (lift (S O) O (lift (S O) O t)) (lift_free t (S O) (S O) O O (le_O_n
948 (plus O (S O))) (le_n O))).
950 theorem pnat_props__lift_SO_SO:
951 \forall (t: T).(eq T (lift (S O) (S O) (lift (S O) O t)) (lift (S O) O (lift
954 \lambda (t: T).(eq_ind nat (plus (S O) O) (\lambda (n: nat).(eq T (lift (S
955 O) n (lift (S O) O t)) (lift (S O) O (lift (S O) O t)))) (eq_ind_r T (lift (S
956 O) O (lift (S O) O t)) (\lambda (t0: T).(eq T t0 (lift (S O) O (lift (S O) O
957 t)))) (refl_equal T (lift (S O) O (lift (S O) O t))) (lift (S O) (plus (S O)
958 O) (lift (S O) O t)) (lift_d t (S O) (S O) O O (le_n O))) (S O) (refl_equal
962 \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t
963 u) \to (\forall (n: nat).(ty3 g c (pchurch t n) (pnat t)))))))
965 \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (u: T).(\lambda (H:
966 (ty3 g c t u)).(ex_ind T (\lambda (t0: T).(ty3 g c u t0)) (\forall (n:
967 nat).(ty3 g c (THead (Bind Abst) t (THead (Bind Abst) (THead (Bind Abst)
968 (lift (S O) O t) (lift (S (S O)) O t)) (church_body n))) (THead (Bind Abst) t
969 (THead (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))
970 (lift (S (S O)) O t))))) (\lambda (x: T).(\lambda (H0: (ty3 g c u
971 x)).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(ty3 g c (THead (Bind Abst)
972 t (THead (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O
973 t)) (church_body n0))) (THead (Bind Abst) t (THead (Bind Abst) (THead (Bind
974 Abst) (lift (S O) O t) (lift (S (S O)) O t)) (lift (S (S O)) O t)))))
975 (ty3_bind g c t u H Abst (THead (Bind Abst) (THead (Bind Abst) (lift (S O) O
976 t) (lift (S (S O)) O t)) (TLRef (S O))) (THead (Bind Abst) (THead (Bind Abst)
977 (lift (S O) O t) (lift (S (S O)) O t)) (lift (S (S O)) O t)) (ty3_bind g
978 (CHead c (Bind Abst) t) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O
979 t)) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O u)) (ty3_bind g
980 (CHead c (Bind Abst) t) (lift (S O) O t) (lift (S O) O u) (ty3_lift g c t u H
981 (CHead c (Bind Abst) t) O (S O) (drop_drop (Bind Abst) O c c (drop_refl c)
982 t)) Abst (lift (S (S O)) O t) (lift (S (S O)) O u) (ty3_lift g c t u H (CHead
983 (CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t)) O (S (S O)) (drop_S
984 Abst (CHead (CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t)) c t (S O)
985 (drop_drop (Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t)
986 (drop_refl (CHead c (Bind Abst) t)) (lift (S O) O t)))) (lift (S (S O)) O x)
987 (ty3_lift g c u x H0 (CHead (CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O
988 t)) O (S (S O)) (drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind Abst) (lift
989 (S O) O t)) c t (S O) (drop_drop (Bind Abst) O (CHead c (Bind Abst) t) (CHead
990 c (Bind Abst) t) (drop_refl (CHead c (Bind Abst) t)) (lift (S O) O t)))))
991 Abst (TLRef (S O)) (lift (S (S O)) O t) (ty3_abst g (S O) (CHead (CHead c
992 (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S
993 O)) O t))) c t (getl_head (Bind Abst) O (CHead c (Bind Abst) t) (CHead c
994 (Bind Abst) t) (getl_refl Abst c t) (THead (Bind Abst) (lift (S O) O t) (lift
995 (S (S O)) O t))) u H) (lift (S (S O)) O u) (ty3_lift g c t u H (CHead (CHead
996 c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S
997 O)) O t))) O (S (S O)) (drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind
998 Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) c t (S O)
999 (drop_drop (Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t)
1000 (drop_refl (CHead c (Bind Abst) t)) (THead (Bind Abst) (lift (S O) O t) (lift
1001 (S (S O)) O t)))))) (THead (Bind Abst) (THead (Bind Abst) (lift (S O) O t)
1002 (lift (S (S O)) O t)) (lift (S (S O)) O u)) (ty3_bind g (CHead c (Bind Abst)
1003 t) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (THead (Bind
1004 Abst) (lift (S O) O t) (lift (S (S O)) O u)) (ty3_bind g (CHead c (Bind Abst)
1005 t) (lift (S O) O t) (lift (S O) O u) (ty3_lift g c t u H (CHead c (Bind Abst)
1006 t) O (S O) (drop_drop (Bind Abst) O c c (drop_refl c) t)) Abst (lift (S (S
1007 O)) O t) (lift (S (S O)) O u) (ty3_lift g c t u H (CHead (CHead c (Bind Abst)
1008 t) (Bind Abst) (lift (S O) O t)) O (S (S O)) (drop_S Abst (CHead (CHead c
1009 (Bind Abst) t) (Bind Abst) (lift (S O) O t)) c t (S O) (drop_drop (Bind Abst)
1010 O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t) (drop_refl (CHead c (Bind
1011 Abst) t)) (lift (S O) O t)))) (lift (S (S O)) O x) (ty3_lift g c u x H0
1012 (CHead (CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t)) O (S (S O))
1013 (drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t)) c t
1014 (S O) (drop_drop (Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst)
1015 t) (drop_refl (CHead c (Bind Abst) t)) (lift (S O) O t))))) Abst (lift (S (S
1016 O)) O t) (lift (S (S O)) O u) (ty3_lift g c t u H (CHead (CHead c (Bind Abst)
1017 t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) O
1018 (S (S O)) (drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind Abst) (THead
1019 (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) c t (S O) (drop_drop
1020 (Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t) (drop_refl
1021 (CHead c (Bind Abst) t)) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O))
1022 O t))))) (lift (S (S O)) O x) (ty3_lift g c u x H0 (CHead (CHead c (Bind
1023 Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O
1024 t))) O (S (S O)) (drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind Abst)
1025 (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) c t (S O)
1026 (drop_drop (Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t)
1027 (drop_refl (CHead c (Bind Abst) t)) (THead (Bind Abst) (lift (S O) O t) (lift
1028 (S (S O)) O t))))))) (\lambda (n0: nat).(\lambda (H1: (ty3 g c (THead (Bind
1029 Abst) t (THead (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S
1030 O)) O t)) (church_body n0))) (THead (Bind Abst) t (THead (Bind Abst) (THead
1031 (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (lift (S (S O)) O
1032 t))))).(let H_x \def (ty3_gen_abst_abst g c t (THead (Bind Abst) (THead (Bind
1033 Abst) (lift (S O) O t) (lift (S (S O)) O t)) (church_body n0)) (THead (Bind
1034 Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (lift (S (S
1035 O)) O t)) H1) in (let H2 \def H_x in (ex2_ind T (\lambda (w: T).(ty3 g c t
1036 w)) (\lambda (_: T).(ty3 g (CHead c (Bind Abst) t) (THead (Bind Abst) (THead
1037 (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (church_body n0)) (THead
1038 (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (lift
1039 (S (S O)) O t)))) (ty3 g c (THead (Bind Abst) t (THead (Bind Abst) (THead
1040 (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (THead (Flat Appl)
1041 (church_body n0) (TLRef O)))) (THead (Bind Abst) t (THead (Bind Abst) (THead
1042 (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (lift (S (S O)) O t))))
1043 (\lambda (x0: T).(\lambda (_: (ty3 g c t x0)).(\lambda (H4: (ty3 g (CHead c
1044 (Bind Abst) t) (THead (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift
1045 (S (S O)) O t)) (church_body n0)) (THead (Bind Abst) (THead (Bind Abst) (lift
1046 (S O) O t) (lift (S (S O)) O t)) (lift (S (S O)) O t)))).(let H_x0 \def
1047 (ty3_gen_abst_abst g (CHead c (Bind Abst) t) (THead (Bind Abst) (lift (S O) O
1048 t) (lift (S (S O)) O t)) (church_body n0) (lift (S (S O)) O t) H4) in (let H5
1049 \def H_x0 in (ex2_ind T (\lambda (w: T).(ty3 g (CHead c (Bind Abst) t) (THead
1050 (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) w)) (\lambda (_: T).(ty3 g
1051 (CHead (CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O
1052 t) (lift (S (S O)) O t))) (church_body n0) (lift (S (S O)) O t))) (ty3 g c
1053 (THead (Bind Abst) t (THead (Bind Abst) (THead (Bind Abst) (lift (S O) O t)
1054 (lift (S (S O)) O t)) (THead (Flat Appl) (church_body n0) (TLRef O)))) (THead
1055 (Bind Abst) t (THead (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S
1056 (S O)) O t)) (lift (S (S O)) O t)))) (\lambda (x1: T).(\lambda (H6: (ty3 g
1057 (CHead c (Bind Abst) t) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O
1058 t)) x1)).(\lambda (H7: (ty3 g (CHead (CHead c (Bind Abst) t) (Bind Abst)
1059 (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) (church_body n0)
1060 (lift (S (S O)) O t))).(ty3_bind g c t u H Abst (THead (Bind Abst) (THead
1061 (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (THead (Flat Appl)
1062 (church_body n0) (TLRef O))) (THead (Bind Abst) (THead (Bind Abst) (lift (S
1063 O) O t) (lift (S (S O)) O t)) (lift (S (S O)) O t)) (ty3_bind g (CHead c
1064 (Bind Abst) t) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))
1065 (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O u)) (ty3_bind g (CHead
1066 c (Bind Abst) t) (lift (S O) O t) (lift (S O) O u) (ty3_lift g c t u H (CHead
1067 c (Bind Abst) t) O (S O) (drop_drop (Bind Abst) O c c (drop_refl c) t)) Abst
1068 (lift (S (S O)) O t) (lift (S (S O)) O u) (ty3_lift g c t u H (CHead (CHead c
1069 (Bind Abst) t) (Bind Abst) (lift (S O) O t)) O (S (S O)) (drop_S Abst (CHead
1070 (CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t)) c t (S O) (drop_drop
1071 (Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t) (drop_refl
1072 (CHead c (Bind Abst) t)) (lift (S O) O t)))) (lift (S (S O)) O x) (ty3_lift g
1073 c u x H0 (CHead (CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t)) O (S (S
1074 O)) (drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t))
1075 c t (S O) (drop_drop (Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind
1076 Abst) t) (drop_refl (CHead c (Bind Abst) t)) (lift (S O) O t))))) Abst (THead
1077 (Flat Appl) (church_body n0) (TLRef O)) (lift (S (S O)) O t) (ex_ind T
1078 (\lambda (t0: T).(ty3 g (CHead (CHead c (Bind Abst) t) (Bind Abst) (THead
1079 (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) (lift (S (S O)) O t) t0))
1080 (ty3 g (CHead (CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S
1081 O) O t) (lift (S (S O)) O t))) (THead (Flat Appl) (church_body n0) (TLRef O))
1082 (lift (S (S O)) O t)) (\lambda (x2: T).(\lambda (H8: (ty3 g (CHead (CHead c
1083 (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S
1084 O)) O t))) (lift (S (S O)) O t) x2)).(ty3_conv g (CHead (CHead c (Bind Abst)
1085 t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)))
1086 (lift (S (S O)) O t) x2 H8 (THead (Flat Appl) (church_body n0) (TLRef O))
1087 (THead (Flat Appl) (church_body n0) (THead (Bind Abst) (lift (S (S O)) O t)
1088 (lift (S O) (S O) (lift (S O) O (lift (S O) O t))))) (ty3_appl g (CHead
1089 (CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift
1090 (S (S O)) O t))) (church_body n0) (lift (S (S O)) O t) H7 (TLRef O) (lift (S
1091 O) (S O) (lift (S O) O (lift (S O) O t))) (eq_ind_r T (lift (S O) O (lift (S
1092 O) O t)) (\lambda (t0: T).(ty3 g (CHead (CHead c (Bind Abst) t) (Bind Abst)
1093 (THead (Bind Abst) (lift (S O) O t) t0)) (TLRef O) (THead (Bind Abst) t0
1094 (lift (S O) (S O) (lift (S O) O (lift (S O) O t)))))) (let H9 \def (eq_ind T
1095 (lift (S (S O)) O t) (\lambda (t0: T).(ty3 g (CHead c (Bind Abst) t) (THead
1096 (Bind Abst) (lift (S O) O t) t0) x1)) H6 (lift (S O) O (lift (S O) O t))
1097 (pnat_props__lift_SSO_O t)) in (eq_ind T (lift (S O) O (THead (Bind Abst)
1098 (lift (S O) O t) (lift (S O) O (lift (S O) O t)))) (\lambda (t0: T).(ty3 g
1099 (CHead (CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O
1100 t) (lift (S O) O (lift (S O) O t)))) (TLRef O) t0)) (ty3_abst g O (CHead
1101 (CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift
1102 (S O) O (lift (S O) O t)))) (CHead c (Bind Abst) t) (THead (Bind Abst) (lift
1103 (S O) O t) (lift (S O) O (lift (S O) O t))) (getl_refl Abst (CHead c (Bind
1104 Abst) t) (THead (Bind Abst) (lift (S O) O t) (lift (S O) O (lift (S O) O
1105 t)))) x1 H9) (THead (Bind Abst) (lift (S O) O (lift (S O) O t)) (lift (S O)
1106 (S O) (lift (S O) O (lift (S O) O t)))) (lift_bind Abst (lift (S O) O t)
1107 (lift (S O) O (lift (S O) O t)) (S O) O))) (lift (S (S O)) O t)
1108 (pnat_props__lift_SSO_O t))) (eq_ind_r T (lift (S O) O (lift (S O) O (lift (S
1109 O) O t))) (\lambda (t0: T).(pc3 (CHead (CHead c (Bind Abst) t) (Bind Abst)
1110 (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) (THead (Flat Appl)
1111 (church_body n0) (THead (Bind Abst) (lift (S (S O)) O t) t0)) (lift (S (S O))
1112 O t))) (eq_ind_r T (lift (S O) O (lift (S O) O t)) (\lambda (t0: T).(pc3
1113 (CHead (CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O
1114 t) t0)) (THead (Flat Appl) (church_body n0) (THead (Bind Abst) t0 (lift (S O)
1115 O (lift (S O) O (lift (S O) O t))))) t0)) (pc3_pr3_r (CHead (CHead c (Bind
1116 Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S O) O (lift
1117 (S O) O t)))) (THead (Flat Appl) (church_body n0) (THead (Bind Abst) (lift (S
1118 O) O (lift (S O) O t)) (lift (S O) O (lift (S O) O (lift (S O) O t))))) (lift
1119 (S O) O (lift (S O) O t)) (pr3_t (THead (Bind Abbr) (church_body n0) (lift (S
1120 O) O (lift (S O) O (lift (S O) O t)))) (THead (Flat Appl) (church_body n0)
1121 (THead (Bind Abst) (lift (S O) O (lift (S O) O t)) (lift (S O) O (lift (S O)
1122 O (lift (S O) O t))))) (CHead (CHead c (Bind Abst) t) (Bind Abst) (THead
1123 (Bind Abst) (lift (S O) O t) (lift (S O) O (lift (S O) O t)))) (pr3_pr2
1124 (CHead (CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O
1125 t) (lift (S O) O (lift (S O) O t)))) (THead (Flat Appl) (church_body n0)
1126 (THead (Bind Abst) (lift (S O) O (lift (S O) O t)) (lift (S O) O (lift (S O)
1127 O (lift (S O) O t))))) (THead (Bind Abbr) (church_body n0) (lift (S O) O
1128 (lift (S O) O (lift (S O) O t)))) (pr2_free (CHead (CHead c (Bind Abst) t)
1129 (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S O) O (lift (S O) O
1130 t)))) (THead (Flat Appl) (church_body n0) (THead (Bind Abst) (lift (S O) O
1131 (lift (S O) O t)) (lift (S O) O (lift (S O) O (lift (S O) O t))))) (THead
1132 (Bind Abbr) (church_body n0) (lift (S O) O (lift (S O) O (lift (S O) O t))))
1133 (pr0_beta (lift (S O) O (lift (S O) O t)) (church_body n0) (church_body n0)
1134 (pr0_refl (church_body n0)) (lift (S O) O (lift (S O) O (lift (S O) O t)))
1135 (lift (S O) O (lift (S O) O (lift (S O) O t))) (pr0_refl (lift (S O) O (lift
1136 (S O) O (lift (S O) O t))))))) (lift (S O) O (lift (S O) O t)) (pr3_pr2
1137 (CHead (CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O
1138 t) (lift (S O) O (lift (S O) O t)))) (THead (Bind Abbr) (church_body n0)
1139 (lift (S O) O (lift (S O) O (lift (S O) O t)))) (lift (S O) O (lift (S O) O
1140 t)) (pr2_free (CHead (CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst)
1141 (lift (S O) O t) (lift (S O) O (lift (S O) O t)))) (THead (Bind Abbr)
1142 (church_body n0) (lift (S O) O (lift (S O) O (lift (S O) O t)))) (lift (S O)
1143 O (lift (S O) O t)) (pr0_zeta Abbr not_abbr_abst (lift (S O) O (lift (S O) O
1144 t)) (lift (S O) O (lift (S O) O t)) (pr0_refl (lift (S O) O (lift (S O) O
1145 t))) (church_body n0)))))) (lift (S (S O)) O t) (pnat_props__lift_SSO_O t))
1146 (lift (S O) (S O) (lift (S O) O (lift (S O) O t))) (pnat_props__lift_SO_SO
1147 (lift (S O) O t)))))) (ty3_correct g (CHead (CHead c (Bind Abst) t) (Bind
1148 Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) (church_body
1149 n0) (lift (S (S O)) O t) H7)) (lift (S (S O)) O u) (ty3_lift g c t u H (CHead
1150 (CHead c (Bind Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift
1151 (S (S O)) O t))) O (S (S O)) (drop_S Abst (CHead (CHead c (Bind Abst) t)
1152 (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) c t (S
1153 O) (drop_drop (Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t)
1154 (drop_refl (CHead c (Bind Abst) t)) (THead (Bind Abst) (lift (S O) O t) (lift
1155 (S (S O)) O t)))))) (THead (Bind Abst) (THead (Bind Abst) (lift (S O) O t)
1156 (lift (S (S O)) O t)) (lift (S (S O)) O u)) (ty3_bind g (CHead c (Bind Abst)
1157 t) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t)) (THead (Bind
1158 Abst) (lift (S O) O t) (lift (S (S O)) O u)) (ty3_bind g (CHead c (Bind Abst)
1159 t) (lift (S O) O t) (lift (S O) O u) (ty3_lift g c t u H (CHead c (Bind Abst)
1160 t) O (S O) (drop_drop (Bind Abst) O c c (drop_refl c) t)) Abst (lift (S (S
1161 O)) O t) (lift (S (S O)) O u) (ty3_lift g c t u H (CHead (CHead c (Bind Abst)
1162 t) (Bind Abst) (lift (S O) O t)) O (S (S O)) (drop_S Abst (CHead (CHead c
1163 (Bind Abst) t) (Bind Abst) (lift (S O) O t)) c t (S O) (drop_drop (Bind Abst)
1164 O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t) (drop_refl (CHead c (Bind
1165 Abst) t)) (lift (S O) O t)))) (lift (S (S O)) O x) (ty3_lift g c u x H0
1166 (CHead (CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t)) O (S (S O))
1167 (drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind Abst) (lift (S O) O t)) c t
1168 (S O) (drop_drop (Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst)
1169 t) (drop_refl (CHead c (Bind Abst) t)) (lift (S O) O t))))) Abst (lift (S (S
1170 O)) O t) (lift (S (S O)) O u) (ty3_lift g c t u H (CHead (CHead c (Bind Abst)
1171 t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) O
1172 (S (S O)) (drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind Abst) (THead
1173 (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) c t (S O) (drop_drop
1174 (Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t) (drop_refl
1175 (CHead c (Bind Abst) t)) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O))
1176 O t))))) (lift (S (S O)) O x) (ty3_lift g c u x H0 (CHead (CHead c (Bind
1177 Abst) t) (Bind Abst) (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O
1178 t))) O (S (S O)) (drop_S Abst (CHead (CHead c (Bind Abst) t) (Bind Abst)
1179 (THead (Bind Abst) (lift (S O) O t) (lift (S (S O)) O t))) c t (S O)
1180 (drop_drop (Bind Abst) O (CHead c (Bind Abst) t) (CHead c (Bind Abst) t)
1181 (drop_refl (CHead c (Bind Abst) t)) (THead (Bind Abst) (lift (S O) O t) (lift
1182 (S (S O)) O t)))))))))) H5)))))) H2))))) n)))) (ty3_correct g c t u H)))))).