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/iso/defs.ma".
19 include "Basic-1/tlist/defs.ma".
22 \forall (u2: T).(\forall (n1: nat).((iso (TSort n1) u2) \to (ex nat (\lambda
23 (n2: nat).(eq T u2 (TSort n2))))))
25 \lambda (u2: T).(\lambda (n1: nat).(\lambda (H: (iso (TSort n1)
26 u2)).(insert_eq T (TSort n1) (\lambda (t: T).(iso t u2)) (\lambda (_: T).(ex
27 nat (\lambda (n2: nat).(eq T u2 (TSort n2))))) (\lambda (y: T).(\lambda (H0:
28 (iso y u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (TSort n1))
29 \to (ex nat (\lambda (n2: nat).(eq T t0 (TSort n2))))))) (\lambda (n0:
30 nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n0) (TSort n1))).(let H2
31 \def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat)
32 with [(TSort n) \Rightarrow n | (TLRef _) \Rightarrow n0 | (THead _ _ _)
33 \Rightarrow n0])) (TSort n0) (TSort n1) H1) in (ex_intro nat (\lambda (n3:
34 nat).(eq T (TSort n2) (TSort n3))) n2 (refl_equal T (TSort n2))))))) (\lambda
35 (i1: nat).(\lambda (i2: nat).(\lambda (H1: (eq T (TLRef i1) (TSort n1))).(let
36 H2 \def (eq_ind T (TLRef i1) (\lambda (ee: T).(match ee in T return (\lambda
37 (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
38 (THead _ _ _) \Rightarrow False])) I (TSort n1) H1) in (False_ind (ex nat
39 (\lambda (n2: nat).(eq T (TLRef i2) (TSort n2)))) H2))))) (\lambda (v1:
40 T).(\lambda (v2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k:
41 K).(\lambda (H1: (eq T (THead k v1 t1) (TSort n1))).(let H2 \def (eq_ind T
42 (THead k v1 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
43 with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
44 _) \Rightarrow True])) I (TSort n1) H1) in (False_ind (ex nat (\lambda (n2:
45 nat).(eq T (THead k v2 t2) (TSort n2)))) H2)))))))) y u2 H0))) H))).
51 \forall (u2: T).(\forall (n1: nat).((iso (TLRef n1) u2) \to (ex nat (\lambda
52 (n2: nat).(eq T u2 (TLRef n2))))))
54 \lambda (u2: T).(\lambda (n1: nat).(\lambda (H: (iso (TLRef n1)
55 u2)).(insert_eq T (TLRef n1) (\lambda (t: T).(iso t u2)) (\lambda (_: T).(ex
56 nat (\lambda (n2: nat).(eq T u2 (TLRef n2))))) (\lambda (y: T).(\lambda (H0:
57 (iso y u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n1))
58 \to (ex nat (\lambda (n2: nat).(eq T t0 (TLRef n2))))))) (\lambda (n0:
59 nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n0) (TLRef n1))).(let H2
60 \def (eq_ind T (TSort n0) (\lambda (ee: T).(match ee in T return (\lambda (_:
61 T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
62 (THead _ _ _) \Rightarrow False])) I (TLRef n1) H1) in (False_ind (ex nat
63 (\lambda (n3: nat).(eq T (TSort n2) (TLRef n3)))) H2))))) (\lambda (i1:
64 nat).(\lambda (i2: nat).(\lambda (H1: (eq T (TLRef i1) (TLRef n1))).(let H2
65 \def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat)
66 with [(TSort _) \Rightarrow i1 | (TLRef n) \Rightarrow n | (THead _ _ _)
67 \Rightarrow i1])) (TLRef i1) (TLRef n1) H1) in (ex_intro nat (\lambda (n2:
68 nat).(eq T (TLRef i2) (TLRef n2))) i2 (refl_equal T (TLRef i2))))))) (\lambda
69 (v1: T).(\lambda (v2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k:
70 K).(\lambda (H1: (eq T (THead k v1 t1) (TLRef n1))).(let H2 \def (eq_ind T
71 (THead k v1 t1) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
72 with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
73 _) \Rightarrow True])) I (TLRef n1) H1) in (False_ind (ex nat (\lambda (n2:
74 nat).(eq T (THead k v2 t2) (TLRef n2)))) H2)))))))) y u2 H0))) H))).
80 \forall (k: K).(\forall (v1: T).(\forall (t1: T).(\forall (u2: T).((iso
81 (THead k v1 t1) u2) \to (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T u2
82 (THead k v2 t2)))))))))
84 \lambda (k: K).(\lambda (v1: T).(\lambda (t1: T).(\lambda (u2: T).(\lambda
85 (H: (iso (THead k v1 t1) u2)).(insert_eq T (THead k v1 t1) (\lambda (t:
86 T).(iso t u2)) (\lambda (_: T).(ex_2 T T (\lambda (v2: T).(\lambda (t2:
87 T).(eq T u2 (THead k v2 t2)))))) (\lambda (y: T).(\lambda (H0: (iso y
88 u2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).((eq T t (THead k v1 t1)) \to
89 (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T t0 (THead k v2 t2))))))))
90 (\lambda (n1: nat).(\lambda (n2: nat).(\lambda (H1: (eq T (TSort n1) (THead k
91 v1 t1))).(let H2 \def (eq_ind T (TSort n1) (\lambda (ee: T).(match ee in T
92 return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _)
93 \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead k v1 t1) H1)
94 in (False_ind (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T (TSort n2)
95 (THead k v2 t2))))) H2))))) (\lambda (i1: nat).(\lambda (i2: nat).(\lambda
96 (H1: (eq T (TLRef i1) (THead k v1 t1))).(let H2 \def (eq_ind T (TLRef i1)
97 (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
98 \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
99 False])) I (THead k v1 t1) H1) in (False_ind (ex_2 T T (\lambda (v2:
100 T).(\lambda (t2: T).(eq T (TLRef i2) (THead k v2 t2))))) H2))))) (\lambda
101 (v0: T).(\lambda (v2: T).(\lambda (t0: T).(\lambda (t2: T).(\lambda (k0:
102 K).(\lambda (H1: (eq T (THead k0 v0 t0) (THead k v1 t1))).(let H2 \def
103 (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
104 [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _)
105 \Rightarrow k1])) (THead k0 v0 t0) (THead k v1 t1) H1) in ((let H3 \def
106 (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
107 [(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ t _)
108 \Rightarrow t])) (THead k0 v0 t0) (THead k v1 t1) H1) in ((let H4 \def
109 (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
110 [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
111 \Rightarrow t])) (THead k0 v0 t0) (THead k v1 t1) H1) in (\lambda (_: (eq T
112 v0 v1)).(\lambda (H6: (eq K k0 k)).(eq_ind_r K k (\lambda (k1: K).(ex_2 T T
113 (\lambda (v3: T).(\lambda (t3: T).(eq T (THead k1 v2 t2) (THead k v3 t3))))))
114 (ex_2_intro T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead k v2 t2)
115 (THead k v3 t3)))) v2 t2 (refl_equal T (THead k v2 t2))) k0 H6)))) H3))
116 H2)))))))) y u2 H0))) H))))).
121 theorem iso_flats_lref_bind_false:
122 \forall (f: F).(\forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall
123 (t: T).(\forall (vs: TList).((iso (THeads (Flat f) vs (TLRef i)) (THead (Bind
124 b) v t)) \to (\forall (P: Prop).P)))))))
126 \lambda (f: F).(\lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda
127 (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: TList).((iso (THeads
128 (Flat f) t0 (TLRef i)) (THead (Bind b) v t)) \to (\forall (P: Prop).P)))
129 (\lambda (H: (iso (TLRef i) (THead (Bind b) v t))).(\lambda (P: Prop).(let
130 H_x \def (iso_gen_lref (THead (Bind b) v t) i H) in (let H0 \def H_x in
131 (ex_ind nat (\lambda (n2: nat).(eq T (THead (Bind b) v t) (TLRef n2))) P
132 (\lambda (x: nat).(\lambda (H1: (eq T (THead (Bind b) v t) (TLRef x))).(let
133 H2 \def (eq_ind T (THead (Bind b) v t) (\lambda (ee: T).(match ee in T return
134 (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
135 \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef x) H1) in
136 (False_ind P H2)))) H0))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda
137 (_: (((iso (THeads (Flat f) t1 (TLRef i)) (THead (Bind b) v t)) \to (\forall
138 (P: Prop).P)))).(\lambda (H0: (iso (THead (Flat f) t0 (THeads (Flat f) t1
139 (TLRef i))) (THead (Bind b) v t))).(\lambda (P: Prop).(let H_x \def
140 (iso_gen_head (Flat f) t0 (THeads (Flat f) t1 (TLRef i)) (THead (Bind b) v t)
141 H0) in (let H1 \def H_x in (ex_2_ind T T (\lambda (v2: T).(\lambda (t2:
142 T).(eq T (THead (Bind b) v t) (THead (Flat f) v2 t2)))) P (\lambda (x0:
143 T).(\lambda (x1: T).(\lambda (H2: (eq T (THead (Bind b) v t) (THead (Flat f)
144 x0 x1))).(let H3 \def (eq_ind T (THead (Bind b) v t) (\lambda (ee: T).(match
145 ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
146 (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
147 (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
148 False])])) I (THead (Flat f) x0 x1) H2) in (False_ind P H3))))) H1))))))))
154 theorem iso_flats_flat_bind_false:
155 \forall (f1: F).(\forall (f2: F).(\forall (b: B).(\forall (v: T).(\forall
156 (v2: T).(\forall (t: T).(\forall (t2: T).(\forall (vs: TList).((iso (THeads
157 (Flat f1) vs (THead (Flat f2) v2 t2)) (THead (Bind b) v t)) \to (\forall (P:
160 \lambda (f1: F).(\lambda (f2: F).(\lambda (b: B).(\lambda (v: T).(\lambda
161 (v2: T).(\lambda (t: T).(\lambda (t2: T).(\lambda (vs: TList).(TList_ind
162 (\lambda (t0: TList).((iso (THeads (Flat f1) t0 (THead (Flat f2) v2 t2))
163 (THead (Bind b) v t)) \to (\forall (P: Prop).P))) (\lambda (H: (iso (THead
164 (Flat f2) v2 t2) (THead (Bind b) v t))).(\lambda (P: Prop).(let H_x \def
165 (iso_gen_head (Flat f2) v2 t2 (THead (Bind b) v t) H) in (let H0 \def H_x in
166 (ex_2_ind T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) v t)
167 (THead (Flat f2) v3 t3)))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1:
168 (eq T (THead (Bind b) v t) (THead (Flat f2) x0 x1))).(let H2 \def (eq_ind T
169 (THead (Bind b) v t) (\lambda (ee: T).(match ee in T return (\lambda (_:
170 T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
171 (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
172 [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
173 f2) x0 x1) H1) in (False_ind P H2))))) H0))))) (\lambda (t0: T).(\lambda (t1:
174 TList).(\lambda (_: (((iso (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))
175 (THead (Bind b) v t)) \to (\forall (P: Prop).P)))).(\lambda (H0: (iso (THead
176 (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) (THead (Bind b) v
177 t))).(\lambda (P: Prop).(let H_x \def (iso_gen_head (Flat f1) t0 (THeads
178 (Flat f1) t1 (THead (Flat f2) v2 t2)) (THead (Bind b) v t) H0) in (let H1
179 \def H_x in (ex_2_ind T T (\lambda (v3: T).(\lambda (t3: T).(eq T (THead
180 (Bind b) v t) (THead (Flat f1) v3 t3)))) P (\lambda (x0: T).(\lambda (x1:
181 T).(\lambda (H2: (eq T (THead (Bind b) v t) (THead (Flat f1) x0 x1))).(let H3
182 \def (eq_ind T (THead (Bind b) v t) (\lambda (ee: T).(match ee in T return
183 (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
184 \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
185 (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
186 False])])) I (THead (Flat f1) x0 x1) H2) in (False_ind P H3))))) H1))))))))