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/subst0/dec.ma".
23 include "LambdaDelta-1/T/props.ma".
26 \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c
27 (CHead d (Bind Abbr) u)) \to ((nf2 c (TLRef i)) \to (\forall (P: Prop).P))))))
29 \lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
30 (H: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H0: ((\forall (t2: T).((pr2
31 c (TLRef i) t2) \to (eq T (TLRef i) t2))))).(\lambda (P:
32 Prop).(lift_gen_lref_false (S i) O i (le_O_n i) (le_n (plus O (S i))) u (H0
33 (lift (S i) O u) (pr2_delta c d u i H (TLRef i) (TLRef i) (pr0_refl (TLRef
34 i)) (lift (S i) O u) (subst0_lref u i))) P))))))).
37 \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abst) u
38 t)) \to (land (nf2 c u) (nf2 (CHead c (Bind Abst) u) t)))))
40 \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2:
41 T).((pr2 c (THead (Bind Abst) u t) t2) \to (eq T (THead (Bind Abst) u t)
42 t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall (t2:
43 T).((pr2 (CHead c (Bind Abst) u) t t2) \to (eq T t t2))) (\lambda (t2:
44 T).(\lambda (H0: (pr2 c u t2)).(let H1 \def (f_equal T T (\lambda (e:
45 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u |
46 (TLRef _) \Rightarrow u | (THead _ t0 _) \Rightarrow t0])) (THead (Bind Abst)
47 u t) (THead (Bind Abst) t2 t) (H (THead (Bind Abst) t2 t) (pr2_head_1 c u t2
48 H0 (Bind Abst) t))) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr2 c u
49 t0)) H0 u H1) in (eq_ind T u (\lambda (t0: T).(eq T u t0)) (refl_equal T u)
50 t2 H1))))) (\lambda (t2: T).(\lambda (H0: (pr2 (CHead c (Bind Abst) u) t
51 t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
52 (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _
53 _ t0) \Rightarrow t0])) (THead (Bind Abst) u t) (THead (Bind Abst) u t2) (H
54 (THead (Bind Abst) u t2) (let H_y \def (pr2_gen_cbind Abst c u t t2 H0) in
55 H_y))) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr2 (CHead c (Bind
56 Abst) u) t t0)) H0 t H1) in (eq_ind T t (\lambda (t0: T).(eq T t t0))
57 (refl_equal T t) t2 H1))))))))).
60 \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Flat Cast) u
61 t)) \to (\forall (P: Prop).P))))
63 \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: (nf2 c (THead
64 (Flat Cast) u t))).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) u t (H t
65 (pr2_free c (THead (Flat Cast) u t) t (pr0_epsilon t t (pr0_refl t) u)))
69 \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((nf2 c
70 (THead (Flat Appl) u (THead (Bind Abst) v t))) \to (\forall (P: Prop).P)))))
72 \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H:
73 ((\forall (t2: T).((pr2 c (THead (Flat Appl) u (THead (Bind Abst) v t)) t2)
74 \to (eq T (THead (Flat Appl) u (THead (Bind Abst) v t)) t2))))).(\lambda (P:
75 Prop).(let H0 \def (eq_ind T (THead (Flat Appl) u (THead (Bind Abst) v t))
76 (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
77 \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
78 (match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
79 (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u t) (H (THead (Bind
80 Abbr) u t) (pr2_free c (THead (Flat Appl) u (THead (Bind Abst) v t)) (THead
81 (Bind Abbr) u t) (pr0_beta v u u (pr0_refl u) t t (pr0_refl t))))) in
82 (False_ind P H0))))))).
85 \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c
86 (THead (Flat f) u t)) \to (land (nf2 c u) (nf2 c t))))))
88 \lambda (f: F).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H:
89 ((\forall (t2: T).((pr2 c (THead (Flat f) u t) t2) \to (eq T (THead (Flat f)
90 u t) t2))))).(conj (\forall (t2: T).((pr2 c u t2) \to (eq T u t2))) (\forall
91 (t2: T).((pr2 c t t2) \to (eq T t t2))) (\lambda (t2: T).(\lambda (H0: (pr2 c
92 u t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e in T return
93 (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u |
94 (THead _ t0 _) \Rightarrow t0])) (THead (Flat f) u t) (THead (Flat f) t2 t)
95 (H (THead (Flat f) t2 t) (pr2_head_1 c u t2 H0 (Flat f) t))) in H1)))
96 (\lambda (t2: T).(\lambda (H0: (pr2 c t t2)).(let H1 \def (f_equal T T
97 (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
98 \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0) \Rightarrow t0]))
99 (THead (Flat f) u t) (THead (Flat f) u t2) (H (THead (Flat f) u t2)
100 (pr2_head_2 c u t t2 (Flat f) (pr2_cflat c t t2 H0 f u)))) in H1)))))))).
102 theorem nf2_gen__nf2_gen_aux:
103 \forall (b: B).(\forall (x: T).(\forall (u: T).(\forall (d: nat).((eq T
104 (THead (Bind b) u (lift (S O) d x)) x) \to (\forall (P: Prop).P)))))
106 \lambda (b: B).(\lambda (x: T).(T_ind (\lambda (t: T).(\forall (u:
107 T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to
108 (\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (d:
109 nat).(\lambda (H: (eq T (THead (Bind b) u (lift (S O) d (TSort n))) (TSort
110 n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead (Bind b) u (lift (S O)
111 d (TSort n))) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
112 with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
113 _) \Rightarrow True])) I (TSort n) H) in (False_ind P H0))))))) (\lambda (n:
114 nat).(\lambda (u: T).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u
115 (lift (S O) d (TLRef n))) (TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind
116 T (THead (Bind b) u (lift (S O) d (TLRef n))) (\lambda (ee: T).(match ee in T
117 return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
118 \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H) in
119 (False_ind P H0))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: ((\forall
120 (u: T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to
121 (\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: ((\forall (u:
122 T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t0)) t0) \to
123 (\forall (P: Prop).P)))))).(\lambda (u: T).(\lambda (d: nat).(\lambda (H1:
124 (eq T (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t
125 t0))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: T).(match e
126 in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef
127 _) \Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u
128 (lift (S O) d (THead k t t0))) (THead k t t0) H1) in ((let H3 \def (f_equal T
129 T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
130 \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t1 _) \Rightarrow t1]))
131 (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t t0) H1) in ((let
132 H4 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
133 with [(TSort _) \Rightarrow (THead k ((let rec lref_map (f: ((nat \to nat)))
134 (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort n) \Rightarrow (TSort
135 n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i
136 | false \Rightarrow (f i)])) | (THead k0 u0 t2) \Rightarrow (THead k0
137 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in lref_map) (\lambda (x0:
138 nat).(plus x0 (S O))) d t) ((let rec lref_map (f: ((nat \to nat))) (d0: nat)
139 (t1: T) on t1: T \def (match t1 with [(TSort n) \Rightarrow (TSort n) |
140 (TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i |
141 false \Rightarrow (f i)])) | (THead k0 u0 t2) \Rightarrow (THead k0 (lref_map
142 f d0 u0) (lref_map f (s k0 d0) t2))]) in lref_map) (\lambda (x0: nat).(plus
143 x0 (S O))) (s k d) t0)) | (TLRef _) \Rightarrow (THead k ((let rec lref_map
144 (f: ((nat \to nat))) (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort
145 n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0)
146 with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u0 t2)
147 \Rightarrow (THead k0 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in
148 lref_map) (\lambda (x0: nat).(plus x0 (S O))) d t) ((let rec lref_map (f:
149 ((nat \to nat))) (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort n)
150 \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with
151 [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u0 t2)
152 \Rightarrow (THead k0 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in
153 lref_map) (\lambda (x0: nat).(plus x0 (S O))) (s k d) t0)) | (THead _ _ t1)
154 \Rightarrow t1])) (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t
155 t0) H1) in (\lambda (_: (eq T u t)).(\lambda (H6: (eq K (Bind b) k)).(let H7
156 \def (eq_ind_r K k (\lambda (k0: K).(eq T (lift (S O) d (THead k0 t t0)) t0))
157 H4 (Bind b) H6) in (let H8 \def (eq_ind T (lift (S O) d (THead (Bind b) t
158 t0)) (\lambda (t1: T).(eq T t1 t0)) H7 (THead (Bind b) (lift (S O) d t) (lift
159 (S O) (S d) t0)) (lift_bind b t t0 (S O) d)) in (H0 (lift (S O) d t) (S d) H8
160 P)))))) H3)) H2))))))))))) x)).
162 theorem nf2_gen_abbr:
163 \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abbr) u
164 t)) \to (\forall (P: Prop).P))))
166 \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2:
167 T).((pr2 c (THead (Bind Abbr) u t) t2) \to (eq T (THead (Bind Abbr) u t)
168 t2))))).(\lambda (P: Prop).(let H_x \def (dnf_dec u t O) in (let H0 \def H_x
169 in (ex_ind T (\lambda (v: T).(or (subst0 O u t (lift (S O) O v)) (eq T t
170 (lift (S O) O v)))) P (\lambda (x: T).(\lambda (H1: (or (subst0 O u t (lift
171 (S O) O x)) (eq T t (lift (S O) O x)))).(or_ind (subst0 O u t (lift (S O) O
172 x)) (eq T t (lift (S O) O x)) P (\lambda (H2: (subst0 O u t (lift (S O) O
173 x))).(let H3 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
174 (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _
175 _ t0) \Rightarrow t0])) (THead (Bind Abbr) u t) (THead (Bind Abbr) u (lift (S
176 O) O x)) (H (THead (Bind Abbr) u (lift (S O) O x)) (pr2_free c (THead (Bind
177 Abbr) u t) (THead (Bind Abbr) u (lift (S O) O x)) (pr0_delta u u (pr0_refl u)
178 t t (pr0_refl t) (lift (S O) O x) H2)))) in (let H4 \def (eq_ind T t (\lambda
179 (t0: T).(subst0 O u t0 (lift (S O) O x))) H2 (lift (S O) O x) H3) in
180 (subst0_refl u (lift (S O) O x) O H4 P)))) (\lambda (H2: (eq T t (lift (S O)
181 O x))).(let H3 \def (eq_ind T t (\lambda (t0: T).(\forall (t2: T).((pr2 c
182 (THead (Bind Abbr) u t0) t2) \to (eq T (THead (Bind Abbr) u t0) t2)))) H
183 (lift (S O) O x) H2) in (nf2_gen__nf2_gen_aux Abbr x u O (H3 x (pr2_free c
184 (THead (Bind Abbr) u (lift (S O) O x)) x (pr0_zeta Abbr not_abbr_abst x x
185 (pr0_refl x) u))) P))) H1))) H0))))))).
187 theorem nf2_gen_void:
188 \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Void) u
189 (lift (S O) O t))) \to (\forall (P: Prop).P))))
191 \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2:
192 T).((pr2 c (THead (Bind Void) u (lift (S O) O t)) t2) \to (eq T (THead (Bind
193 Void) u (lift (S O) O t)) t2))))).(\lambda (P: Prop).(nf2_gen__nf2_gen_aux
194 Void t u O (H t (pr2_free c (THead (Bind Void) u (lift (S O) O t)) t
195 (pr0_zeta Void not_void_abst t t (pr0_refl t) u))) P))))).