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7 (* ||T|| The HELM team. *)
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15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/tlist/props".
19 include "tlist/defs.ma".
21 theorem tslt_wf__q_ind:
22 \forall (P: ((TList \to Prop))).(((\forall (n: nat).((\lambda (P0: ((TList
23 \to Prop))).(\lambda (n0: nat).(\forall (ts: TList).((eq nat (tslen ts) n0)
24 \to (P0 ts))))) P n))) \to (\forall (ts: TList).(P ts)))
26 let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts:
27 TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to
28 Prop))).(\lambda (H: ((\forall (n: nat).(\forall (ts: TList).((eq nat (tslen
29 ts) n) \to (P ts)))))).(\lambda (ts: TList).(H (tslen ts) ts (refl_equal nat
33 \forall (P: ((TList \to Prop))).(((\forall (ts2: TList).(((\forall (ts1:
34 TList).((tslt ts1 ts2) \to (P ts1)))) \to (P ts2)))) \to (\forall (ts:
37 let Q \def (\lambda (P: ((TList \to Prop))).(\lambda (n: nat).(\forall (ts:
38 TList).((eq nat (tslen ts) n) \to (P ts))))) in (\lambda (P: ((TList \to
39 Prop))).(\lambda (H: ((\forall (ts2: TList).(((\forall (ts1: TList).((lt
40 (tslen ts1) (tslen ts2)) \to (P ts1)))) \to (P ts2))))).(\lambda (ts:
41 TList).(tslt_wf__q_ind (\lambda (t: TList).(P t)) (\lambda (n:
42 nat).(lt_wf_ind n (Q (\lambda (t: TList).(P t))) (\lambda (n0: nat).(\lambda
43 (H0: ((\forall (m: nat).((lt m n0) \to (Q (\lambda (t: TList).(P t))
44 m))))).(\lambda (ts0: TList).(\lambda (H1: (eq nat (tslen ts0) n0)).(let H2
45 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall (m: nat).((lt m n1) \to
46 (\forall (ts1: TList).((eq nat (tslen ts1) m) \to (P ts1)))))) H0 (tslen ts0)
47 H1) in (H ts0 (\lambda (ts1: TList).(\lambda (H3: (lt (tslen ts1) (tslen
48 ts0))).(H2 (tslen ts1) H3 ts1 (refl_equal nat (tslen ts1))))))))))))) ts)))).
51 \forall (k: K).(\forall (vs: TList).(\forall (v: T).(\forall (t: T).(eq T
52 (THeads k (TApp vs v) t) (THeads k vs (THead k v t))))))
54 \lambda (k: K).(\lambda (vs: TList).(TList_ind (\lambda (t: TList).(\forall
55 (v: T).(\forall (t0: T).(eq T (THeads k (TApp t v) t0) (THeads k t (THead k v
56 t0)))))) (\lambda (v: T).(\lambda (t: T).(refl_equal T (THead k v t))))
57 (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: ((\forall (v: T).(\forall
58 (t1: T).(eq T (THeads k (TApp t0 v) t1) (THeads k t0 (THead k v
59 t1))))))).(\lambda (v: T).(\lambda (t1: T).(eq_ind_r T (THeads k t0 (THead k
60 v t1)) (\lambda (t2: T).(eq T (THead k t t2) (THead k t (THeads k t0 (THead k
61 v t1))))) (refl_equal T (THead k t (THeads k t0 (THead k v t1)))) (THeads k
62 (TApp t0 v) t1) (H v t1))))))) vs)).
64 theorem tcons_tapp_ex:
65 \forall (ts1: TList).(\forall (t1: T).(ex2_2 TList T (\lambda (ts2:
66 TList).(\lambda (t2: T).(eq TList (TCons t1 ts1) (TApp ts2 t2)))) (\lambda
67 (ts2: TList).(\lambda (_: T).(eq nat (tslen ts1) (tslen ts2))))))
69 \lambda (ts1: TList).(TList_ind (\lambda (t: TList).(\forall (t1: T).(ex2_2
70 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t) (TApp
71 ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t) (tslen
72 ts2))))))) (\lambda (t1: T).(ex2_2_intro TList T (\lambda (ts2:
73 TList).(\lambda (t2: T).(eq TList (TCons t1 TNil) (TApp ts2 t2)))) (\lambda
74 (ts2: TList).(\lambda (_: T).(eq nat O (tslen ts2)))) TNil t1 (refl_equal
75 TList (TApp TNil t1)) (refl_equal nat (tslen TNil)))) (\lambda (t:
76 T).(\lambda (t0: TList).(\lambda (H: ((\forall (t1: T).(ex2_2 TList T
77 (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t1 t0) (TApp ts2
78 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0) (tslen
79 ts2)))))))).(\lambda (t1: T).(let H_x \def (H t) in (let H0 \def H_x in
80 (ex2_2_ind TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons t
81 t0) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (tslen t0)
82 (tslen ts2)))) (ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq
83 TList (TCons t1 (TCons t t0)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda
84 (_: T).(eq nat (S (tslen t0)) (tslen ts2))))) (\lambda (x0: TList).(\lambda
85 (x1: T).(\lambda (H1: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H2: (eq
86 nat (tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t2:
87 TList).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t3: T).(eq TList (TCons
88 t1 t2) (TApp ts2 t3)))) (\lambda (ts2: TList).(\lambda (_: T).(eq nat (S
89 (tslen t0)) (tslen ts2)))))) (eq_ind_r nat (tslen x0) (\lambda (n:
90 nat).(ex2_2 TList T (\lambda (ts2: TList).(\lambda (t2: T).(eq TList (TCons
91 t1 (TApp x0 x1)) (TApp ts2 t2)))) (\lambda (ts2: TList).(\lambda (_: T).(eq
92 nat (S n) (tslen ts2)))))) (ex2_2_intro TList T (\lambda (ts2:
93 TList).(\lambda (t2: T).(eq TList (TCons t1 (TApp x0 x1)) (TApp ts2 t2))))
94 (\lambda (ts2: TList).(\lambda (_: T).(eq nat (S (tslen x0)) (tslen ts2))))
95 (TCons t1 x0) x1 (refl_equal TList (TApp (TCons t1 x0) x1)) (refl_equal nat
96 (tslen (TCons t1 x0)))) (tslen t0) H2) (TCons t t0) H1))))) H0))))))) ts1).
98 theorem tlist_ind_rew:
99 \forall (P: ((TList \to Prop))).((P TNil) \to (((\forall (ts:
100 TList).(\forall (t: T).((P ts) \to (P (TApp ts t)))))) \to (\forall (ts:
103 \lambda (P: ((TList \to Prop))).(\lambda (H: (P TNil)).(\lambda (H0:
104 ((\forall (ts: TList).(\forall (t: T).((P ts) \to (P (TApp ts
105 t))))))).(\lambda (ts: TList).(tslt_wf_ind (\lambda (t: TList).(P t))
106 (\lambda (ts2: TList).(TList_ind (\lambda (t: TList).(((\forall (ts1:
107 TList).((tslt ts1 t) \to (P ts1)))) \to (P t))) (\lambda (_: ((\forall (ts1:
108 TList).((tslt ts1 TNil) \to (P ts1))))).H) (\lambda (t: T).(\lambda (t0:
109 TList).(\lambda (_: ((((\forall (ts1: TList).((tslt ts1 t0) \to (P ts1))))
110 \to (P t0)))).(\lambda (H2: ((\forall (ts1: TList).((tslt ts1 (TCons t t0))
111 \to (P ts1))))).(let H_x \def (tcons_tapp_ex t0 t) in (let H3 \def H_x in
112 (ex2_2_ind TList T (\lambda (ts3: TList).(\lambda (t2: T).(eq TList (TCons t
113 t0) (TApp ts3 t2)))) (\lambda (ts3: TList).(\lambda (_: T).(eq nat (tslen t0)
114 (tslen ts3)))) (P (TCons t t0)) (\lambda (x0: TList).(\lambda (x1:
115 T).(\lambda (H4: (eq TList (TCons t t0) (TApp x0 x1))).(\lambda (H5: (eq nat
116 (tslen t0) (tslen x0))).(eq_ind_r TList (TApp x0 x1) (\lambda (t1: TList).(P
117 t1)) (H0 x0 x1 (H2 x0 (eq_ind nat (tslen t0) (\lambda (n: nat).(lt n (tslen
118 (TCons t t0)))) (le_n (tslen (TCons t t0))) (tslen x0) H5))) (TCons t t0)
119 H4))))) H3))))))) ts2)) ts)))).