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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 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 include "Basic-1/C/defs.ma".
19 include "Basic-1/T/props.ma".
22 \forall (c: C).(\forall (d: C).((clt c d) \to (\forall (k: K).(\forall (t:
23 T).(clt (CHead c k t) (CHead d k t))))))
25 \lambda (c: C).(\lambda (d: C).(\lambda (H: (lt (cweight c) (cweight
26 d))).(\lambda (_: K).(\lambda (t: T).(lt_reg_r (cweight c) (cweight d)
33 \forall (k: K).(\forall (c: C).(\forall (u: T).(clt c (CHead c k u))))
35 \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(eq_ind_r nat (plus (cweight
36 c) O) (\lambda (n: nat).(lt n (plus (cweight c) (tweight u))))
37 (le_lt_plus_plus (cweight c) (cweight c) O (tweight u) (le_n (cweight c))
38 (tweight_lt u)) (cweight c) (plus_n_O (cweight c))))).
43 theorem clt_wf__q_ind:
44 \forall (P: ((C \to Prop))).(((\forall (n: nat).((\lambda (P0: ((C \to
45 Prop))).(\lambda (n0: nat).(\forall (c: C).((eq nat (cweight c) n0) \to (P0
46 c))))) P n))) \to (\forall (c: C).(P c)))
48 let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c:
49 C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to
50 Prop))).(\lambda (H: ((\forall (n: nat).(\forall (c: C).((eq nat (cweight c)
51 n) \to (P c)))))).(\lambda (c: C).(H (cweight c) c (refl_equal nat (cweight
58 \forall (P: ((C \to Prop))).(((\forall (c: C).(((\forall (d: C).((clt d c)
59 \to (P d)))) \to (P c)))) \to (\forall (c: C).(P c)))
61 let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c:
62 C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to
63 Prop))).(\lambda (H: ((\forall (c: C).(((\forall (d: C).((lt (cweight d)
64 (cweight c)) \to (P d)))) \to (P c))))).(\lambda (c: C).(clt_wf__q_ind
65 (\lambda (c0: C).(P c0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (c0:
66 C).(P c0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0)
67 \to (Q (\lambda (c0: C).(P c0)) m))))).(\lambda (c0: C).(\lambda (H1: (eq nat
68 (cweight c0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall
69 (m: nat).((lt m n1) \to (\forall (c1: C).((eq nat (cweight c1) m) \to (P
70 c1)))))) H0 (cweight c0) H1) in (H c0 (\lambda (d: C).(\lambda (H3: (lt
71 (cweight d) (cweight c0))).(H2 (cweight d) H3 d (refl_equal nat (cweight
78 \forall (c: C).(\forall (t: T).(\forall (k: K).(ex_3 K C T (\lambda (h:
79 K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c k t) (CTail h u d))))))))
81 \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (t: T).(\forall (k: K).(ex_3
82 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c0 k t)
83 (CTail h u d))))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (k:
84 K).(ex_3_intro K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C
85 (CHead (CSort n) k t) (CTail h u d))))) k (CSort n) t (refl_equal C (CHead
86 (CSort n) k t)))))) (\lambda (c0: C).(\lambda (H: ((\forall (t: T).(\forall
87 (k: K).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C
88 (CHead c0 k t) (CTail h u d)))))))))).(\lambda (k: K).(\lambda (t:
89 T).(\lambda (t0: T).(\lambda (k0: K).(let H_x \def (H t k) in (let H0 \def
90 H_x in (ex_3_ind K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C
91 (CHead c0 k t) (CTail h u d))))) (ex_3 K C T (\lambda (h: K).(\lambda (d:
92 C).(\lambda (u: T).(eq C (CHead (CHead c0 k t) k0 t0) (CTail h u d))))))
93 (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H1: (eq C (CHead
94 c0 k t) (CTail x0 x2 x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c1:
95 C).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead
96 c1 k0 t0) (CTail h u d))))))) (ex_3_intro K C T (\lambda (h: K).(\lambda (d:
97 C).(\lambda (u: T).(eq C (CHead (CTail x0 x2 x1) k0 t0) (CTail h u d))))) x0
98 (CHead x1 k0 t0) x2 (refl_equal C (CHead (CTail x0 x2 x1) k0 t0))) (CHead c0
99 k t) H1))))) H0))))))))) c).
105 \forall (k: K).(\forall (u: T).(\forall (c: C).(clt c (CTail k u c))))
107 \lambda (k: K).(\lambda (u: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(clt
108 c0 (CTail k u c0))) (\lambda (n: nat).(clt_head k (CSort n) u)) (\lambda (c0:
109 C).(\lambda (H: (clt c0 (CTail k u c0))).(\lambda (k0: K).(\lambda (t:
110 T).(clt_cong c0 (CTail k u c0) H k0 t))))) c))).
116 \forall (P: ((C \to Prop))).(((\forall (n: nat).(P (CSort n)))) \to
117 (((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: T).(P (CTail k t
118 c))))))) \to (\forall (c: C).(P c))))
120 \lambda (P: ((C \to Prop))).(\lambda (H: ((\forall (n: nat).(P (CSort
121 n))))).(\lambda (H0: ((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t:
122 T).(P (CTail k t c)))))))).(\lambda (c: C).(clt_wf_ind (\lambda (c0: C).(P
123 c0)) (\lambda (c0: C).(C_ind (\lambda (c1: C).(((\forall (d: C).((clt d c1)
124 \to (P d)))) \to (P c1))) (\lambda (n: nat).(\lambda (_: ((\forall (d:
125 C).((clt d (CSort n)) \to (P d))))).(H n))) (\lambda (c1: C).(\lambda (_:
126 ((((\forall (d: C).((clt d c1) \to (P d)))) \to (P c1)))).(\lambda (k:
127 K).(\lambda (t: T).(\lambda (H2: ((\forall (d: C).((clt d (CHead c1 k t)) \to
128 (P d))))).(let H_x \def (chead_ctail c1 t k) in (let H3 \def H_x in (ex_3_ind
129 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c1 k t)
130 (CTail h u d))))) (P (CHead c1 k t)) (\lambda (x0: K).(\lambda (x1:
131 C).(\lambda (x2: T).(\lambda (H4: (eq C (CHead c1 k t) (CTail x0 x2
132 x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c2: C).(P c2)) (let H5 \def
133 (eq_ind C (CHead c1 k t) (\lambda (c2: C).(\forall (d: C).((clt d c2) \to (P
134 d)))) H2 (CTail x0 x2 x1) H4) in (H0 x1 (H5 x1 (clt_thead x0 x2 x1)) x0 x2))
135 (CHead c1 k t) H4))))) H3)))))))) c0)) c)))).