1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 include "LambdaDelta-1/nf2/defs.ma".
19 include "LambdaDelta-1/pr2/clen.ma".
21 include "LambdaDelta-1/pr2/fwd.ma".
23 include "LambdaDelta-1/pr0/dec.ma".
25 include "LambdaDelta-1/C/props.ma".
28 \forall (c: C).(\forall (t1: T).(or (nf2 c t1) (ex2 T (\lambda (t2: T).((eq
29 T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2)))))
31 \lambda (c: C).(c_tail_ind (\lambda (c0: C).(\forall (t1: T).(or (\forall
32 (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1
33 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)))))) (\lambda
34 (n: nat).(\lambda (t1: T).(let H_x \def (nf0_dec t1) in (let H \def H_x in
35 (or_ind (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2:
36 T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)))
37 (or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T
38 (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
39 T).(pr2 (CSort n) t1 t2)))) (\lambda (H0: ((\forall (t2: T).((pr0 t1 t2) \to
40 (eq T t1 t2))))).(or_introl (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T
41 t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P)))
42 (\lambda (t2: T).(pr2 (CSort n) t1 t2))) (\lambda (t2: T).(\lambda (H1: (pr2
43 (CSort n) t1 t2)).(let H_y \def (pr2_gen_csort t1 t2 n H1) in (H0 t2
44 H_y)))))) (\lambda (H0: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall
45 (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)))).(ex2_ind T (\lambda (t2:
46 T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2))
47 (or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T
48 (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
49 T).(pr2 (CSort n) t1 t2)))) (\lambda (x: T).(\lambda (H1: (((eq T t1 x) \to
50 (\forall (P: Prop).P)))).(\lambda (H2: (pr0 t1 x)).(or_intror (\forall (t2:
51 T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T
52 t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CSort n) t1 t2)))
53 (ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P)))
54 (\lambda (t2: T).(pr2 (CSort n) t1 t2)) x H1 (pr2_free (CSort n) t1 x
55 H2)))))) H0)) H))))) (\lambda (c0: C).(\lambda (H: ((\forall (t1: T).(or
56 (\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2:
57 T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1
58 t2))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (t1: T).(let H_x \def (H
59 t1) in (let H0 \def H_x in (or_ind (\forall (t2: T).((pr2 c0 t1 t2) \to (eq T
60 t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P)))
61 (\lambda (t2: T).(pr2 c0 t1 t2))) (or (\forall (t2: T).((pr2 (CTail k t c0)
62 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall
63 (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)))) (\lambda (H1:
64 ((\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))))).(K_ind (\lambda (k0:
65 K).(or (\forall (t2: T).((pr2 (CTail k0 t c0) t1 t2) \to (eq T t1 t2))) (ex2
66 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
67 T).(pr2 (CTail k0 t c0) t1 t2))))) (\lambda (b: B).(B_ind (\lambda (b0:
68 B).(or (\forall (t2: T).((pr2 (CTail (Bind b0) t c0) t1 t2) \to (eq T t1
69 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P)))
70 (\lambda (t2: T).(pr2 (CTail (Bind b0) t c0) t1 t2))))) (let H_x0 \def
71 (dnf_dec t t1 (clen c0)) in (let H2 \def H_x0 in (ex_ind T (\lambda (v:
72 T).(or (subst0 (clen c0) t t1 (lift (S O) (clen c0) v)) (eq T t1 (lift (S O)
73 (clen c0) v)))) (or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t1 t2)
74 \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P:
75 Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2)))) (\lambda
76 (x: T).(\lambda (H3: (or (subst0 (clen c0) t t1 (lift (S O) (clen c0) x)) (eq
77 T t1 (lift (S O) (clen c0) x)))).(or_ind (subst0 (clen c0) t t1 (lift (S O)
78 (clen c0) x)) (eq T t1 (lift (S O) (clen c0) x)) (or (\forall (t2: T).((pr2
79 (CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2:
80 T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail
81 (Bind Abbr) t c0) t1 t2)))) (\lambda (H4: (subst0 (clen c0) t t1 (lift (S O)
82 (clen c0) x))).(let H_x1 \def (getl_ctail_clen Abbr t c0) in (let H5 \def
83 H_x1 in (ex_ind nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind Abbr) t
84 c0) (CHead (CSort n) (Bind Abbr) t))) (or (\forall (t2: T).((pr2 (CTail (Bind
85 Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2)
86 \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1
87 t2)))) (\lambda (x0: nat).(\lambda (H6: (getl (clen c0) (CTail (Bind Abbr) t
88 c0) (CHead (CSort x0) (Bind Abbr) t))).(or_intror (\forall (t2: T).((pr2
89 (CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2:
90 T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail
91 (Bind Abbr) t c0) t1 t2))) (ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to
92 (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1
93 t2)) (lift (S O) (clen c0) x) (\lambda (H7: (eq T t1 (lift (S O) (clen c0)
94 x))).(\lambda (P: Prop).(let H8 \def (eq_ind T t1 (\lambda (t0: T).(subst0
95 (clen c0) t t0 (lift (S O) (clen c0) x))) H4 (lift (S O) (clen c0) x) H7) in
96 (subst0_gen_lift_false x t (lift (S O) (clen c0) x) (S O) (clen c0) (clen c0)
97 (le_n (clen c0)) (eq_ind_r nat (plus (S O) (clen c0)) (\lambda (n: nat).(lt
98 (clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen c0) (S O)) (plus_sym
99 (clen c0) (S O))) H8 P)))) (pr2_delta (CTail (Bind Abbr) t c0) (CSort x0) t
100 (clen c0) H6 t1 t1 (pr0_refl t1) (lift (S O) (clen c0) x) H4))))) H5))))
101 (\lambda (H4: (eq T t1 (lift (S O) (clen c0) x))).(let H5 \def (eq_ind T t1
102 (\lambda (t0: T).(\forall (t2: T).((pr2 c0 t0 t2) \to (eq T t0 t2)))) H1
103 (lift (S O) (clen c0) x) H4) in (eq_ind_r T (lift (S O) (clen c0) x) (\lambda
104 (t0: T).(or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t0 t2) \to (eq T
105 t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P)))
106 (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t0 t2))))) (or_introl (\forall
107 (t2: T).((pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2) \to (eq T
108 (lift (S O) (clen c0) x) t2))) (ex2 T (\lambda (t2: T).((eq T (lift (S O)
109 (clen c0) x) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail
110 (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2))) (\lambda (t2: T).(\lambda
111 (H6: (pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2)).(let H_x1
112 \def (pr2_gen_ctail (Bind Abbr) c0 t (lift (S O) (clen c0) x) t2 H6) in (let
113 H7 \def H_x1 in (or_ind (pr2 c0 (lift (S O) (clen c0) x) t2) (ex3 T (\lambda
114 (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t0: T).(pr0 (lift (S O)
115 (clen c0) x) t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T (lift
116 (S O) (clen c0) x) t2) (\lambda (H8: (pr2 c0 (lift (S O) (clen c0) x)
117 t2)).(H5 t2 H8)) (\lambda (H8: (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind
118 Abbr))) (\lambda (t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda (t0:
119 T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Bind Abbr)
120 (Bind Abbr))) (\lambda (t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda
121 (t0: T).(subst0 (clen c0) t t0 t2)) (eq T (lift (S O) (clen c0) x) t2)
122 (\lambda (x0: T).(\lambda (_: (eq K (Bind Abbr) (Bind Abbr))).(\lambda (H10:
123 (pr0 (lift (S O) (clen c0) x) x0)).(\lambda (H11: (subst0 (clen c0) t x0
124 t2)).(ex2_ind T (\lambda (t3: T).(eq T x0 (lift (S O) (clen c0) t3)))
125 (\lambda (t3: T).(pr0 x t3)) (eq T (lift (S O) (clen c0) x) t2) (\lambda (x1:
126 T).(\lambda (H12: (eq T x0 (lift (S O) (clen c0) x1))).(\lambda (_: (pr0 x
127 x1)).(let H14 \def (eq_ind T x0 (\lambda (t0: T).(subst0 (clen c0) t t0 t2))
128 H11 (lift (S O) (clen c0) x1) H12) in (subst0_gen_lift_false x1 t t2 (S O)
129 (clen c0) (clen c0) (le_n (clen c0)) (eq_ind_r nat (plus (S O) (clen c0))
130 (\lambda (n: nat).(lt (clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen
131 c0) (S O)) (plus_sym (clen c0) (S O))) H14 (eq T (lift (S O) (clen c0) x)
132 t2)))))) (pr0_gen_lift x x0 (S O) (clen c0) H10)))))) H8)) H7)))))) t1 H4)))
133 H3))) H2))) (or_introl (\forall (t2: T).((pr2 (CTail (Bind Abst) t c0) t1 t2)
134 \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P:
135 Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abst) t c0) t1 t2))) (\lambda
136 (t2: T).(\lambda (H2: (pr2 (CTail (Bind Abst) t c0) t1 t2)).(let H_x0 \def
137 (pr2_gen_ctail (Bind Abst) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind
138 (pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Abst) (Bind Abbr)))
139 (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)))
140 (eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T
141 (\lambda (_: T).(eq K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0))
142 (\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq
143 K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0:
144 T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5:
145 (eq K (Bind Abst) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_:
146 (subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Abst) (\lambda (ee:
147 K).(match ee in K return (\lambda (_: K).Prop) with [(Bind b0) \Rightarrow
148 (match b0 in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False |
149 Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow
150 False])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3))))))
151 (or_introl (\forall (t2: T).((pr2 (CTail (Bind Void) t c0) t1 t2) \to (eq T
152 t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P)))
153 (\lambda (t2: T).(pr2 (CTail (Bind Void) t c0) t1 t2))) (\lambda (t2:
154 T).(\lambda (H2: (pr2 (CTail (Bind Void) t c0) t1 t2)).(let H_x0 \def
155 (pr2_gen_ctail (Bind Void) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind
156 (pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Void) (Bind Abbr)))
157 (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)))
158 (eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T
159 (\lambda (_: T).(eq K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0))
160 (\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq
161 K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0:
162 T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5:
163 (eq K (Bind Void) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_:
164 (subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Void) (\lambda (ee:
165 K).(match ee in K return (\lambda (_: K).Prop) with [(Bind b0) \Rightarrow
166 (match b0 in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow False |
167 Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow
168 False])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3))))))
169 b)) (\lambda (f: F).(or_introl (\forall (t2: T).((pr2 (CTail (Flat f) t c0)
170 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall
171 (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Flat f) t c0) t1 t2))) (\lambda
172 (t2: T).(\lambda (H2: (pr2 (CTail (Flat f) t c0) t1 t2)).(let H_x0 \def
173 (pr2_gen_ctail (Flat f) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind (pr2
174 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0:
175 T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T t1 t2)
176 (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T (\lambda (_:
177 T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0:
178 T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Flat f)
179 (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen
180 c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: (eq K (Flat f)
181 (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: (subst0 (clen c0) t x0
182 t2)).(let H8 \def (eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return
183 (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
184 True])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3)))))))
185 k)) (\lambda (H1: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P:
186 Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)))).(ex2_ind T (\lambda (t2:
187 T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2))
188 (or (\forall (t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T
189 (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2:
190 T).(pr2 (CTail k t c0) t1 t2)))) (\lambda (x: T).(\lambda (H2: (((eq T t1 x)
191 \to (\forall (P: Prop).P)))).(\lambda (H3: (pr2 c0 t1 x)).(or_intror (\forall
192 (t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2:
193 T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t
194 c0) t1 t2))) (ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P:
195 Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)) x H2 (pr2_ctail c0 t1
196 x H3 k t)))))) H1)) H0)))))))) c).