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 "basic_1/nf2/pr3.ma".
19 include "basic_1/iso/props.ma".
21 theorem nf2_iso_appls_lref:
22 \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs:
23 TList).(\forall (u: T).((pr3 c (THeads (Flat Appl) vs (TLRef i)) u) \to (iso
24 (THeads (Flat Appl) vs (TLRef i)) u))))))
26 \lambda (c: C).(\lambda (i: nat).(\lambda (H: (nf2 c (TLRef i))).(\lambda
27 (vs: TList).(let TMP_4 \def (\lambda (t: TList).(\forall (u: T).((pr3 c
28 (THeads (Flat Appl) t (TLRef i)) u) \to (let TMP_1 \def (Flat Appl) in (let
29 TMP_2 \def (TLRef i) in (let TMP_3 \def (THeads TMP_1 t TMP_2) in (iso TMP_3
30 u))))))) in (let TMP_14 \def (\lambda (u: T).(\lambda (H0: (pr3 c (TLRef i)
31 u)).(let TMP_5 \def (TLRef i) in (let H_y \def (nf2_pr3_unfold c TMP_5 u H0
32 H) in (let TMP_7 \def (\lambda (t: T).(let TMP_6 \def (TLRef i) in (pr3 c
33 TMP_6 t))) in (let TMP_8 \def (TLRef i) in (let H1 \def (eq_ind_r T u TMP_7
34 H0 TMP_8 H_y) in (let TMP_9 \def (TLRef i) in (let TMP_11 \def (\lambda (t:
35 T).(let TMP_10 \def (TLRef i) in (iso TMP_10 t))) in (let TMP_12 \def (TLRef
36 i) in (let TMP_13 \def (iso_refl TMP_12) in (eq_ind T TMP_9 TMP_11 TMP_13 u
37 H_y)))))))))))) in (let TMP_162 \def (\lambda (t: T).(\lambda (t0:
38 TList).(\lambda (H0: ((\forall (u: T).((pr3 c (THeads (Flat Appl) t0 (TLRef
39 i)) u) \to (iso (THeads (Flat Appl) t0 (TLRef i)) u))))).(\lambda (u:
40 T).(\lambda (H1: (pr3 c (THead (Flat Appl) t (THeads (Flat Appl) t0 (TLRef
41 i))) u)).(let TMP_15 \def (Flat Appl) in (let TMP_16 \def (TLRef i) in (let
42 TMP_17 \def (THeads TMP_15 t0 TMP_16) in (let H2 \def (pr3_gen_appl c t
43 TMP_17 u H1) in (let TMP_20 \def (\lambda (u2: T).(\lambda (t2: T).(let
44 TMP_18 \def (Flat Appl) in (let TMP_19 \def (THead TMP_18 u2 t2) in (eq T u
45 TMP_19))))) in (let TMP_21 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c t
46 u2))) in (let TMP_25 \def (\lambda (_: T).(\lambda (t2: T).(let TMP_22 \def
47 (Flat Appl) in (let TMP_23 \def (TLRef i) in (let TMP_24 \def (THeads TMP_22
48 t0 TMP_23) in (pr3 c TMP_24 t2)))))) in (let TMP_26 \def (ex3_2 T T TMP_20
49 TMP_21 TMP_25) in (let TMP_29 \def (\lambda (_: T).(\lambda (_: T).(\lambda
50 (u2: T).(\lambda (t2: T).(let TMP_27 \def (Bind Abbr) in (let TMP_28 \def
51 (THead TMP_27 u2 t2) in (pr3 c TMP_28 u))))))) in (let TMP_30 \def (\lambda
52 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) in
53 (let TMP_36 \def (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
54 (_: T).(let TMP_31 \def (Flat Appl) in (let TMP_32 \def (TLRef i) in (let
55 TMP_33 \def (THeads TMP_31 t0 TMP_32) in (let TMP_34 \def (Bind Abst) in (let
56 TMP_35 \def (THead TMP_34 y1 z1) in (pr3 c TMP_33 TMP_35)))))))))) in (let
57 TMP_39 \def (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2:
58 T).(\forall (b: B).(\forall (u0: T).(let TMP_37 \def (Bind b) in (let TMP_38
59 \def (CHead c TMP_37 u0) in (pr3 TMP_38 z1 t2))))))))) in (let TMP_40 \def
60 (ex4_4 T T T T TMP_29 TMP_30 TMP_36 TMP_39) in (let TMP_42 \def (\lambda (b:
61 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
62 (_: T).(let TMP_41 \def (eq B b Abst) in (not TMP_41)))))))) in (let TMP_48
63 \def (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
64 T).(\lambda (_: T).(\lambda (_: T).(let TMP_43 \def (Flat Appl) in (let
65 TMP_44 \def (TLRef i) in (let TMP_45 \def (THeads TMP_43 t0 TMP_44) in (let
66 TMP_46 \def (Bind b) in (let TMP_47 \def (THead TMP_46 y1 z1) in (pr3 c
67 TMP_45 TMP_47)))))))))))) in (let TMP_55 \def (\lambda (b: B).(\lambda (_:
68 T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(let
69 TMP_49 \def (Bind b) in (let TMP_50 \def (Flat Appl) in (let TMP_51 \def (S
70 O) in (let TMP_52 \def (lift TMP_51 O u2) in (let TMP_53 \def (THead TMP_50
71 TMP_52 z2) in (let TMP_54 \def (THead TMP_49 y2 TMP_53) in (pr3 c TMP_54
72 u))))))))))))) in (let TMP_56 \def (\lambda (_: B).(\lambda (_: T).(\lambda
73 (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))))) in
74 (let TMP_57 \def (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda
75 (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1 y2))))))) in (let TMP_60
76 \def (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
77 T).(\lambda (_: T).(\lambda (y2: T).(let TMP_58 \def (Bind b) in (let TMP_59
78 \def (CHead c TMP_58 y2) in (pr3 TMP_59 z1 z2))))))))) in (let TMP_61 \def
79 (ex6_6 B T T T T T TMP_42 TMP_48 TMP_55 TMP_56 TMP_57 TMP_60) in (let TMP_62
80 \def (Flat Appl) in (let TMP_63 \def (Flat Appl) in (let TMP_64 \def (TLRef
81 i) in (let TMP_65 \def (THeads TMP_63 t0 TMP_64) in (let TMP_66 \def (THead
82 TMP_62 t TMP_65) in (let TMP_67 \def (iso TMP_66 u) in (let TMP_96 \def
83 (\lambda (H3: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T u (THead
84 (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2)))
85 (\lambda (_: T).(\lambda (t2: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i))
86 t2))))).(let TMP_70 \def (\lambda (u2: T).(\lambda (t2: T).(let TMP_68 \def
87 (Flat Appl) in (let TMP_69 \def (THead TMP_68 u2 t2) in (eq T u TMP_69)))))
88 in (let TMP_71 \def (\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))) in (let
89 TMP_75 \def (\lambda (_: T).(\lambda (t2: T).(let TMP_72 \def (Flat Appl) in
90 (let TMP_73 \def (TLRef i) in (let TMP_74 \def (THeads TMP_72 t0 TMP_73) in
91 (pr3 c TMP_74 t2)))))) in (let TMP_76 \def (Flat Appl) in (let TMP_77 \def
92 (Flat Appl) in (let TMP_78 \def (TLRef i) in (let TMP_79 \def (THeads TMP_77
93 t0 TMP_78) in (let TMP_80 \def (THead TMP_76 t TMP_79) in (let TMP_81 \def
94 (iso TMP_80 u) in (let TMP_95 \def (\lambda (x0: T).(\lambda (x1: T).(\lambda
95 (H4: (eq T u (THead (Flat Appl) x0 x1))).(\lambda (_: (pr3 c t x0)).(\lambda
96 (_: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) x1)).(let TMP_82 \def (Flat
97 Appl) in (let TMP_83 \def (THead TMP_82 x0 x1) in (let TMP_89 \def (\lambda
98 (t1: T).(let TMP_84 \def (Flat Appl) in (let TMP_85 \def (Flat Appl) in (let
99 TMP_86 \def (TLRef i) in (let TMP_87 \def (THeads TMP_85 t0 TMP_86) in (let
100 TMP_88 \def (THead TMP_84 t TMP_87) in (iso TMP_88 t1))))))) in (let TMP_90
101 \def (Flat Appl) in (let TMP_91 \def (TLRef i) in (let TMP_92 \def (THeads
102 TMP_90 t0 TMP_91) in (let TMP_93 \def (Flat Appl) in (let TMP_94 \def
103 (iso_head t x0 TMP_92 x1 TMP_93) in (eq_ind_r T TMP_83 TMP_89 TMP_94 u
104 H4)))))))))))))) in (ex3_2_ind T T TMP_70 TMP_71 TMP_75 TMP_81 TMP_95
105 H3)))))))))))) in (let TMP_125 \def (\lambda (H3: (ex4_4 T T T T (\lambda (_:
106 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(pr3 c (THead (Bind
107 Abbr) u2 t2) u))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
108 (_: T).(pr3 c t u2))))) (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
109 T).(\lambda (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind
110 Abst) y1 z1)))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
111 (t2: T).(\forall (b: B).(\forall (u0: T).(pr3 (CHead c (Bind b) u0) z1
112 t2))))))))).(let TMP_99 \def (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
113 T).(\lambda (t2: T).(let TMP_97 \def (Bind Abbr) in (let TMP_98 \def (THead
114 TMP_97 u2 t2) in (pr3 c TMP_98 u))))))) in (let TMP_100 \def (\lambda (_:
115 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2))))) in (let
116 TMP_106 \def (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
117 T).(let TMP_101 \def (Flat Appl) in (let TMP_102 \def (TLRef i) in (let
118 TMP_103 \def (THeads TMP_101 t0 TMP_102) in (let TMP_104 \def (Bind Abst) in
119 (let TMP_105 \def (THead TMP_104 y1 z1) in (pr3 c TMP_103 TMP_105))))))))))
120 in (let TMP_109 \def (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
121 T).(\lambda (t2: T).(\forall (b: B).(\forall (u0: T).(let TMP_107 \def (Bind
122 b) in (let TMP_108 \def (CHead c TMP_107 u0) in (pr3 TMP_108 z1 t2)))))))))
123 in (let TMP_110 \def (Flat Appl) in (let TMP_111 \def (Flat Appl) in (let
124 TMP_112 \def (TLRef i) in (let TMP_113 \def (THeads TMP_111 t0 TMP_112) in
125 (let TMP_114 \def (THead TMP_110 t TMP_113) in (let TMP_115 \def (iso TMP_114
126 u) in (let TMP_124 \def (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2:
127 T).(\lambda (x3: T).(\lambda (_: (pr3 c (THead (Bind Abbr) x2 x3)
128 u)).(\lambda (_: (pr3 c t x2)).(\lambda (H6: (pr3 c (THeads (Flat Appl) t0
129 (TLRef i)) (THead (Bind Abst) x0 x1))).(\lambda (_: ((\forall (b: B).(\forall
130 (u0: T).(pr3 (CHead c (Bind b) u0) x1 x3))))).(let TMP_116 \def (Bind Abst)
131 in (let TMP_117 \def (THead TMP_116 x0 x1) in (let H_y \def (H0 TMP_117 H6)
132 in (let TMP_118 \def (Flat Appl) in (let TMP_119 \def (Flat Appl) in (let
133 TMP_120 \def (TLRef i) in (let TMP_121 \def (THeads TMP_119 t0 TMP_120) in
134 (let TMP_122 \def (THead TMP_118 t TMP_121) in (let TMP_123 \def (iso TMP_122
135 u) in (iso_flats_lref_bind_false Appl Abst i x0 x1 t0 H_y
136 TMP_123)))))))))))))))))) in (ex4_4_ind T T T T TMP_99 TMP_100 TMP_106
137 TMP_109 TMP_115 TMP_124 H3))))))))))))) in (let TMP_161 \def (\lambda (H3:
138 (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
139 (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
140 B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
141 (_: T).(pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead (Bind b) y1 z1))))))))
142 (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
143 (u2: T).(\lambda (y2: T).(pr3 c (THead (Bind b) y2 (THead (Flat Appl) (lift
144 (S O) O u2) z2)) u))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
145 T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr3 c t u2)))))))
146 (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
147 T).(\lambda (y2: T).(pr3 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
148 T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr3
149 (CHead c (Bind b) y2) z1 z2))))))))).(let TMP_127 \def (\lambda (b:
150 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
151 (_: T).(let TMP_126 \def (eq B b Abst) in (not TMP_126)))))))) in (let
152 TMP_133 \def (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
153 T).(\lambda (_: T).(\lambda (_: T).(let TMP_128 \def (Flat Appl) in (let
154 TMP_129 \def (TLRef i) in (let TMP_130 \def (THeads TMP_128 t0 TMP_129) in
155 (let TMP_131 \def (Bind b) in (let TMP_132 \def (THead TMP_131 y1 z1) in (pr3
156 c TMP_130 TMP_132)))))))))))) in (let TMP_140 \def (\lambda (b: B).(\lambda
157 (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2:
158 T).(let TMP_134 \def (Bind b) in (let TMP_135 \def (Flat Appl) in (let
159 TMP_136 \def (S O) in (let TMP_137 \def (lift TMP_136 O u2) in (let TMP_138
160 \def (THead TMP_135 TMP_137 z2) in (let TMP_139 \def (THead TMP_134 y2
161 TMP_138) in (pr3 c TMP_139 u))))))))))))) in (let TMP_141 \def (\lambda (_:
162 B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
163 (_: T).(pr3 c t u2))))))) in (let TMP_142 \def (\lambda (_: B).(\lambda (y1:
164 T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr3 c y1
165 y2))))))) in (let TMP_145 \def (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
166 T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(let TMP_143 \def (Bind
167 b) in (let TMP_144 \def (CHead c TMP_143 y2) in (pr3 TMP_144 z1 z2)))))))))
168 in (let TMP_146 \def (Flat Appl) in (let TMP_147 \def (Flat Appl) in (let
169 TMP_148 \def (TLRef i) in (let TMP_149 \def (THeads TMP_147 t0 TMP_148) in
170 (let TMP_150 \def (THead TMP_146 t TMP_149) in (let TMP_151 \def (iso TMP_150
171 u) in (let TMP_160 \def (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2:
172 T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B
173 x0 Abst))).(\lambda (H5: (pr3 c (THeads (Flat Appl) t0 (TLRef i)) (THead
174 (Bind x0) x1 x2))).(\lambda (_: (pr3 c (THead (Bind x0) x5 (THead (Flat Appl)
175 (lift (S O) O x4) x3)) u)).(\lambda (_: (pr3 c t x4)).(\lambda (_: (pr3 c x1
176 x5)).(\lambda (_: (pr3 (CHead c (Bind x0) x5) x2 x3)).(let TMP_152 \def (Bind
177 x0) in (let TMP_153 \def (THead TMP_152 x1 x2) in (let H_y \def (H0 TMP_153
178 H5) in (let TMP_154 \def (Flat Appl) in (let TMP_155 \def (Flat Appl) in (let
179 TMP_156 \def (TLRef i) in (let TMP_157 \def (THeads TMP_155 t0 TMP_156) in
180 (let TMP_158 \def (THead TMP_154 t TMP_157) in (let TMP_159 \def (iso TMP_158
181 u) in (iso_flats_lref_bind_false Appl x0 i x1 x2 t0 H_y
182 TMP_159)))))))))))))))))))))) in (ex6_6_ind B T T T T T TMP_127 TMP_133
183 TMP_140 TMP_141 TMP_142 TMP_145 TMP_151 TMP_160 H3))))))))))))))) in (or3_ind
184 TMP_26 TMP_40 TMP_61 TMP_67 TMP_96 TMP_125 TMP_161
185 H2))))))))))))))))))))))))))))))))))) in (TList_ind TMP_4 TMP_14 TMP_162