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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 *)
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15 (* This file was automatically generated: do not edit *********************)
17 include "Basic-1/tlist/defs.ma".
19 theorem tslt_wf__q_ind:
20 \forall (P: ((TList \to Prop))).(((\forall (n: nat).((\lambda (P0: ((TList
21 \to Prop))).(\lambda (n0: nat).(\forall (ts: TList).((eq nat (tslen ts) n0)
22 \to (P0 ts))))) P n))) \to (\forall (ts: TList).(P ts)))
24 let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts:
25 TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to
26 Prop))).(\lambda (H: ((\forall (n: nat).(\forall (ts: TList).((eq nat (tslen
27 ts) n) \to (P ts)))))).(\lambda (ts: TList).(H (tslen ts) ts (refl_equal nat
34 \forall (P: ((TList \to Prop))).(((\forall (ts2: TList).(((\forall (ts1:
35 TList).((tslt ts1 ts2) \to (P ts1)))) \to (P ts2)))) \to (\forall (ts:
38 let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts:
39 TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to
40 Prop))).(\lambda (H: ((\forall (ts2: TList).(((\forall (ts1: TList).((lt
41 (tslen ts1) (tslen ts2)) \to (P ts1)))) \to (P ts2))))).(\lambda (ts:
42 TList).(tslt_wf__q_ind (\lambda (t: TList).(P t)) (\lambda (n:
43 nat).(lt_wf_ind n (Q (\lambda (t: TList).(P t))) (\lambda (n0: nat).(\lambda
44 (H0: ((\forall (m: nat).((lt m n0) \to (Q (\lambda (t: TList).(P t))
45 m))))).(\lambda (ts0: TList).(\lambda (H1: (eq nat (tslen ts0) n0)).(let H2
46 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall (m: nat).((lt m n1) \to
47 (\forall (ts1: TList).((eq nat (tslen ts1) m) \to (P ts1)))))) H0 (tslen ts0)
48 H1) in (H ts0 (\lambda (ts1: TList).(\lambda (H3: (lt (tslen ts1) (tslen
49 ts0))).(H2 (tslen ts1) H3 ts1 (refl_equal nat (tslen ts1))))))))))))) ts)))).
55 \forall (k: K).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).(eq T
56 (THeads k (TApp vs v) t) (THeads k vs (THead k v t))))))
58 \lambda (k: K).(\lambda (v: T).(\lambda (t: T).(\lambda (vs:
59 TList).(TList_ind (\lambda (t0: TList).(eq T (THeads k (TApp t0 v) t) (THeads
60 k t0 (THead k v t)))) (refl_equal T (THead k v t)) (\lambda (t0: T).(\lambda
61 (t1: TList).(\lambda (H: (eq T (THeads k (TApp t1 v) t) (THeads k t1 (THead k
62 v t)))).(eq_ind T (THeads k (TApp t1 v) t) (\lambda (t2: T).(eq T (THead k t0
63 (THeads k (TApp t1 v) t)) (THead k t0 t2))) (refl_equal T (THead k t0 (THeads
64 k (TApp t1 v) t))) (THeads k t1 (THead k v t)) H)))) vs)))).
69 theorem tcons_tapp_ex:
70 \forall (ts1: TList).(\forall (t1: T).(ex2_2 TList T (\lambda (ts2:
71 TList).(\lambda (t2: T).(eq TList (TCons t1 ts1) (TApp ts2 t2)))) (\lambda
72 (ts2: TList).(\lambda (_: T).(eq nat (tslen ts1) (tslen ts2))))))
74 \lambda (ts1: TList).(TList_ind (\lambda (t: TList).(\forall (t1: T).(ex2_2
75 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t) (TApp
76 ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t) (tslen
77 ts2))))))) (\lambda (t1: T).(ex2_2_intro TList T (\lambda (ts2:
78 TList).(\lambda (t2: T).(eq TList (TCons t1 TNil) (TApp ts2 t2)))) (\lambda
79 (ts2: TList).(\lambda (_: T).(eq nat O (tslen ts2)))) TNil t1 (refl_equal
80 TList (TApp TNil t1)) (refl_equal nat (tslen TNil)))) (\lambda (t:
81 T).(\lambda (t0: TList).(\lambda (H: ((\forall (t1: T).(ex2_2 TList T
82 (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t0) (TApp ts2
83 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0) (tslen
84 ts2)))))))).(\lambda (t1: T).(let H_x \def (H t) in (let H0 \def H_x in
85 (ex2_2_ind TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t
86 t0) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0)
87 (tslen ts2)))) (ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq
88 TList (TCons t1 (TCons t t0)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda
89 (_: T).(eq nat (S (tslen t0)) (tslen ts2))))) (\lambda (x0: TList).(\lambda
90 (x1: T).(\lambda (H1: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H2: (eq
91 nat (tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t2:
92 TList).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t3: T).(eq TList (TCons
93 t1 t2) (TApp ts2 t3)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (S
94 (tslen t0)) (tslen ts2)))))) (eq_ind_r nat (tslen x0) (\lambda (n:
95 nat).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons
96 t1 (TApp x0 x1)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq
97 nat (S n) (tslen ts2)))))) (ex2_2_intro TList T (\lambda (ts2:
98 TList).(\lambda (t2: T).(eq TList (TCons t1 (TApp x0 x1)) (TApp ts2 t2))))
99 (\lambda (ts2: TList).(\lambda (_: T).(eq nat (S (tslen x0)) (tslen ts2))))
100 (TCons t1 x0) x1 (refl_equal TList (TApp (TCons t1 x0) x1)) (refl_equal nat
101 (tslen (TCons t1 x0)))) (tslen t0) H2) (TCons t t0) H1))))) H0))))))) ts1).
106 theorem tlist_ind_rev:
107 \forall (P: ((TList \to Prop))).((P TNil) \to (((\forall (ts:
108 TList).(\forall (t: T).((P ts) \to (P (TApp ts t)))))) \to (\forall (ts:
111 \lambda (P: ((TList \to Prop))).(\lambda (H: (P TNil)).(\lambda (H0:
112 ((\forall (ts: TList).(\forall (t: T).((P ts) \to (P (TApp ts
113 t))))))).(\lambda (ts: TList).(tslt_wf_ind (\lambda (t: TList).(P t))
114 (\lambda (ts2: TList).(TList_ind (\lambda (t: TList).(((\forall (ts1:
115 TList).((tslt ts1 t) \to (P ts1)))) \to (P t))) (\lambda (_: ((\forall (ts1:
116 TList).((tslt ts1 TNil) \to (P ts1))))).H) (\lambda (t: T).(\lambda (t0:
117 TList).(\lambda (_: ((((\forall (ts1: TList).((tslt ts1 t0) \to (P ts1))))
118 \to (P t0)))).(\lambda (H2: ((\forall (ts1: TList).((tslt ts1 (TCons t t0))
119 \to (P ts1))))).(let H_x \def (tcons_tapp_ex t0 t) in (let H3 \def H_x in
120 (ex2_2_ind TList T (\lambda (ts3: TList).(\lambda (t2: T).(eq TList (TCons t
121 t0) (TApp ts3 t2)))) (\lambda (ts3: TList).(\lambda (_: T).(eq nat (tslen t0)
122 (tslen ts3)))) (P (TCons t t0)) (\lambda (x0: TList).(\lambda (x1:
123 T).(\lambda (H4: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H5: (eq nat
124 (tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t1: TList).(P
125 t1)) (H0 x0 x1 (H2 x0 (eq_ind nat (tslen t0) (\lambda (n: nat).(lt n (tslen
126 (TCons t t0)))) (le_n (tslen (TCons t t0))) (tslen x0) H5))) (TCons t t0)
127 H4))))) H3))))))) ts2)) ts)))).