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/nf2/props".
19 include "nf2/defs.ma".
24 \forall (c: C).(\forall (n: nat).(nf2 c (TSort n)))
26 \lambda (c: C).(\lambda (n: nat).(\lambda (t2: T).(\lambda (H: (pr2 c (TSort
27 n) t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal
28 T (TSort n)) t2 (pr2_gen_sort c t2 n H))))).
31 \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (b: B).(\forall (v:
32 T).(\forall (t: T).((nf2 (CHead c (Bind b) v) t) \to (nf2 c (THead (Bind
35 \lambda (c: C).(\lambda (u: T).(\lambda (H: ((\forall (t2: T).((pr2 c u t2)
36 \to (eq T u t2))))).(\lambda (b: B).(\lambda (v: T).(\lambda (t: T).(\lambda
37 (H0: ((\forall (t2: T).((pr2 (CHead c (Bind b) v) t t2) \to (eq T t
38 t2))))).(\lambda (t2: T).(\lambda (H1: (pr2 c (THead (Bind Abst) u t)
39 t2)).(let H2 \def (pr2_gen_abst c u t t2 H1) in (ex3_2_ind T T (\lambda (u2:
40 T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2:
41 T).(\lambda (_: T).(pr2 c u u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
42 (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t t3))))) (eq T (THead
43 (Bind Abst) u t) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq T t2
44 (THead (Bind Abst) x0 x1))).(\lambda (H4: (pr2 c u x0)).(\lambda (H5:
45 ((\forall (b0: B).(\forall (u0: T).(pr2 (CHead c (Bind b0) u0) t
46 x1))))).(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(eq T (THead
47 (Bind Abst) u t) t0)) (f_equal3 K T T T THead (Bind Abst) (Bind Abst) u x0 t
48 x1 (refl_equal K (Bind Abst)) (H x0 H4) (H0 x1 (H5 b v))) t2 H3))))))
51 theorem nf2_appl_lref:
52 \forall (c: C).(\forall (u: T).((nf2 c u) \to (\forall (i: nat).((nf2 c
53 (TLRef i)) \to (nf2 c (THead (Flat Appl) u (TLRef i)))))))
55 \lambda (c: C).(\lambda (u: T).(\lambda (H: ((\forall (t2: T).((pr2 c u t2)
56 \to (eq T u t2))))).(\lambda (i: nat).(\lambda (H0: ((\forall (t2: T).((pr2 c
57 (TLRef i) t2) \to (eq T (TLRef i) t2))))).(\lambda (t2: T).(\lambda (H1: (pr2
58 c (THead (Flat Appl) u (TLRef i)) t2)).(let H2 \def (pr2_gen_appl c u (TLRef
59 i) t2 H1) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
60 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u u2)))
61 (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3)))) (ex4_4 T T T T
62 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
63 (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
64 T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3))))))
65 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u
66 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
67 T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))))
68 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
69 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
70 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
71 (_: T).(eq T (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda
72 (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq
73 T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
74 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
75 T).(\lambda (_: T).(pr2 c u u2))))))) (\lambda (_: B).(\lambda (y1:
76 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
77 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
78 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))
79 (eq T (THead (Flat Appl) u (TLRef i)) t2) (\lambda (H3: (ex3_2 T T (\lambda
80 (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
81 T).(\lambda (_: T).(pr2 c u u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
82 (TLRef i) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
83 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u u2)))
84 (\lambda (_: T).(\lambda (t3: T).(pr2 c (TLRef i) t3))) (eq T (THead (Flat
85 Appl) u (TLRef i)) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T
86 t2 (THead (Flat Appl) x0 x1))).(\lambda (H5: (pr2 c u x0)).(\lambda (H6: (pr2
87 c (TLRef i) x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t: T).(eq T
88 (THead (Flat Appl) u (TLRef i)) t)) (let H7 \def (eq_ind_r T x1 (\lambda (t:
89 T).(pr2 c (TLRef i) t)) H6 (TLRef i) (H0 x1 H6)) in (eq_ind T (TLRef i)
90 (\lambda (t: T).(eq T (THead (Flat Appl) u (TLRef i)) (THead (Flat Appl) x0
91 t))) (let H8 \def (eq_ind_r T x0 (\lambda (t: T).(pr2 c u t)) H5 u (H x0 H5))
92 in (eq_ind T u (\lambda (t: T).(eq T (THead (Flat Appl) u (TLRef i)) (THead
93 (Flat Appl) t (TLRef i)))) (refl_equal T (THead (Flat Appl) u (TLRef i))) x0
94 (H x0 H5))) x1 (H0 x1 H6))) t2 H4)))))) H3)) (\lambda (H3: (ex4_4 T T T T
95 (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
96 (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
97 T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3))))))
98 (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u
99 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
100 T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1
101 t3))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
102 T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind Abst) y1 z1)))))) (\lambda
103 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
104 (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
105 T).(\lambda (_: T).(pr2 c u u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
106 (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind
107 b) u0) z1 t3))))))) (eq T (THead (Flat Appl) u (TLRef i)) t2) (\lambda (x0:
108 T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (eq T
109 (TLRef i) (THead (Bind Abst) x0 x1))).(\lambda (H5: (eq T t2 (THead (Bind
110 Abbr) x2 x3))).(\lambda (_: (pr2 c u x2)).(\lambda (_: ((\forall (b:
111 B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) x1 x3))))).(eq_ind_r T (THead
112 (Bind Abbr) x2 x3) (\lambda (t: T).(eq T (THead (Flat Appl) u (TLRef i)) t))
113 (let H8 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return
114 (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
115 \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) x0
116 x1) H4) in (False_ind (eq T (THead (Flat Appl) u (TLRef i)) (THead (Bind
117 Abbr) x2 x3)) H8)) t2 H5))))))))) H3)) (\lambda (H3: (ex6_6 B T T T T T
118 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
119 T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
120 T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
121 (TLRef i) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
122 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
123 t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
124 (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
125 T).(\lambda (_: T).(pr2 c u u2))))))) (\lambda (_: B).(\lambda (y1:
126 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
127 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
128 T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1
129 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
130 (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
131 Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
132 T).(\lambda (_: T).(\lambda (_: T).(eq T (TLRef i) (THead (Bind b) y1
133 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
134 T).(\lambda (u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat
135 Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
136 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u u2)))))))
137 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
138 T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
139 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
140 (CHead c (Bind b) y2) z1 z2))))))) (eq T (THead (Flat Appl) u (TLRef i)) t2)
141 (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda
142 (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H5: (eq
143 T (TLRef i) (THead (Bind x0) x1 x2))).(\lambda (H6: (eq T t2 (THead (Bind x0)
144 x5 (THead (Flat Appl) (lift (S O) O x4) x3)))).(\lambda (_: (pr2 c u
145 x4)).(\lambda (_: (pr2 c x1 x5)).(\lambda (_: (pr2 (CHead c (Bind x0) x5) x2
146 x3)).(eq_ind_r T (THead (Bind x0) x5 (THead (Flat Appl) (lift (S O) O x4)
147 x3)) (\lambda (t: T).(eq T (THead (Flat Appl) u (TLRef i)) t)) (let H10 \def
148 (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_:
149 T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
150 (THead _ _ _) \Rightarrow False])) I (THead (Bind x0) x1 x2) H5) in
151 (False_ind (eq T (THead (Flat Appl) u (TLRef i)) (THead (Bind x0) x5 (THead
152 (Flat Appl) (lift (S O) O x4) x3))) H10)) t2 H6))))))))))))) H3)) H2)))))))).
154 theorem nf2_lref_abst:
155 \forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c
156 (CHead e (Bind Abst) u)) \to (nf2 c (TLRef i))))))
158 \lambda (c: C).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
159 (H: (getl i c (CHead e (Bind Abst) u))).(\lambda (t2: T).(\lambda (H0: (pr2 c
160 (TLRef i) t2)).(let H1 \def (pr2_gen_lref c t2 i H0) in (or_ind (eq T t2
161 (TLRef i)) (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c (CHead d
162 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift (S i) O
163 u0))))) (eq T (TLRef i) t2) (\lambda (H2: (eq T t2 (TLRef i))).(eq_ind_r T
164 (TLRef i) (\lambda (t: T).(eq T (TLRef i) t)) (refl_equal T (TLRef i)) t2
165 H2)) (\lambda (H2: (ex2_2 C T (\lambda (d: C).(\lambda (u0: T).(getl i c
166 (CHead d (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift
167 (S i) O u0)))))).(ex2_2_ind C T (\lambda (d: C).(\lambda (u0: T).(getl i c
168 (CHead d (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t2 (lift
169 (S i) O u0)))) (eq T (TLRef i) t2) (\lambda (x0: C).(\lambda (x1: T).(\lambda
170 (H3: (getl i c (CHead x0 (Bind Abbr) x1))).(\lambda (H4: (eq T t2 (lift (S i)
171 O x1))).(eq_ind_r T (lift (S i) O x1) (\lambda (t: T).(eq T (TLRef i) t))
172 (let H5 \def (eq_ind C (CHead e (Bind Abst) u) (\lambda (c0: C).(getl i c
173 c0)) H (CHead x0 (Bind Abbr) x1) (getl_mono c (CHead e (Bind Abst) u) i H
174 (CHead x0 (Bind Abbr) x1) H3)) in (let H6 \def (eq_ind C (CHead e (Bind Abst)
175 u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort
176 _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return
177 (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return
178 (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True |
179 Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead x0 (Bind
180 Abbr) x1) (getl_mono c (CHead e (Bind Abst) u) i H (CHead x0 (Bind Abbr) x1)
181 H3)) in (False_ind (eq T (TLRef i) (lift (S i) O x1)) H6))) t2 H4))))) H2))
185 \forall (d: C).(\forall (t: T).((nf2 d t) \to (\forall (c: C).(\forall (h:
186 nat).(\forall (i: nat).((drop h i c d) \to (nf2 c (lift h i t))))))))
188 \lambda (d: C).(\lambda (t: T).(\lambda (H: ((\forall (t2: T).((pr2 d t t2)
189 \to (eq T t t2))))).(\lambda (c: C).(\lambda (h: nat).(\lambda (i:
190 nat).(\lambda (H0: (drop h i c d)).(\lambda (t2: T).(\lambda (H1: (pr2 c
191 (lift h i t) t2)).(let H2 \def (pr2_gen_lift c t t2 h i H1 d H0) in (ex2_ind
192 T (\lambda (t3: T).(eq T t2 (lift h i t3))) (\lambda (t3: T).(pr2 d t t3))
193 (eq T (lift h i t) t2) (\lambda (x: T).(\lambda (H3: (eq T t2 (lift h i
194 x))).(\lambda (H4: (pr2 d t x)).(eq_ind_r T (lift h i x) (\lambda (t0: T).(eq
195 T (lift h i t) t0)) (let H_y \def (H x H4) in (let H5 \def (eq_ind_r T x
196 (\lambda (t0: T).(pr2 d t t0)) H4 t H_y) in (eq_ind T t (\lambda (t0: T).(eq
197 T (lift h i t) (lift h i t0))) (refl_equal T (lift h i t)) x H_y))) t2 H3))))