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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/flt/defs.ma".
19 include "Basic-1/C/props.ma".
22 \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c u c
25 \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t:
26 T).(le_lt_plus_plus (cweight c) (cweight c) (tweight u) (S (plus (tweight u)
27 (tweight t))) (le_n (cweight c)) (le_n_S (tweight u) (plus (tweight u)
28 (tweight t)) (le_plus_l (tweight u) (tweight t))))))).
34 \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c t c
37 \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t:
38 T).(le_lt_plus_plus (cweight c) (cweight c) (tweight t) (S (plus (tweight u)
39 (tweight t))) (le_n (cweight c)) (le_n_S (tweight t) (plus (tweight u)
40 (tweight t)) (le_plus_r (tweight u) (tweight t))))))).
46 \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt (CHead c
47 k u) t c (THead k u t)))))
49 \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(eq_ind nat
50 (S (plus (cweight c) (plus (tweight u) (tweight t)))) (\lambda (n: nat).(lt
51 (plus (plus (cweight c) (tweight u)) (tweight t)) n)) (eq_ind_r nat (plus
52 (plus (cweight c) (tweight u)) (tweight t)) (\lambda (n: nat).(lt (plus (plus
53 (cweight c) (tweight u)) (tweight t)) (S n))) (le_n (S (plus (plus (cweight
54 c) (tweight u)) (tweight t)))) (plus (cweight c) (plus (tweight u) (tweight
55 t))) (plus_assoc_l (cweight c) (tweight u) (tweight t))) (plus (cweight c) (S
56 (plus (tweight u) (tweight t)))) (plus_n_Sm (cweight c) (plus (tweight u)
63 \forall (k: K).(\forall (c: C).(\forall (t: T).(\forall (i: nat).(flt c t
64 (CHead c k t) (TLRef i)))))
66 \lambda (_: K).(\lambda (c: C).(\lambda (t: T).(\lambda (_:
67 nat).(lt_x_plus_x_Sy (plus (cweight c) (tweight t)) O)))).
73 \forall (k1: K).(\forall (c1: C).(\forall (c2: C).(\forall (t1: T).((cle
74 (CHead c1 k1 t1) c2) \to (\forall (k2: K).(\forall (t2: T).(\forall (i:
75 nat).(flt c1 t1 (CHead c2 k2 t2) (TLRef i)))))))))
77 \lambda (_: K).(\lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda
78 (H: (le (plus (cweight c1) (tweight t1)) (cweight c2))).(\lambda (_:
79 K).(\lambda (t2: T).(\lambda (_: nat).(le_lt_trans (plus (cweight c1)
80 (tweight t1)) (cweight c2) (plus (plus (cweight c2) (tweight t2)) (S O)) H
81 (eq_ind_r nat (plus (S O) (plus (cweight c2) (tweight t2))) (\lambda (n:
82 nat).(lt (cweight c2) n)) (le_lt_n_Sm (cweight c2) (plus (cweight c2)
83 (tweight t2)) (le_plus_l (cweight c2) (tweight t2))) (plus (plus (cweight c2)
84 (tweight t2)) (S O)) (plus_sym (plus (cweight c2) (tweight t2)) (S
91 \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (i: nat).((flt c1
92 t1 c2 (TLRef i)) \to (\forall (k2: K).(\forall (t2: T).(\forall (j: nat).(flt
93 c1 t1 (CHead c2 k2 t2) (TLRef j)))))))))
95 \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (_: nat).(\lambda
96 (H: (lt (plus (cweight c1) (tweight t1)) (plus (cweight c2) (S O)))).(\lambda
97 (_: K).(\lambda (t2: T).(\lambda (_: nat).(lt_le_trans (plus (cweight c1)
98 (tweight t1)) (plus (cweight c2) (S O)) (plus (plus (cweight c2) (tweight
99 t2)) (S O)) H (le_plus_plus (cweight c2) (plus (cweight c2) (tweight t2)) (S
100 O) (S O) (le_plus_l (cweight c2) (tweight t2)) (le_n (S O))))))))))).
106 \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (t2: T).((flt c1
107 t1 c2 t2) \to (\forall (c3: C).(\forall (t3: T).((flt c2 t2 c3 t3) \to (flt
110 \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
111 (H: (lt (fweight c1 t1) (fweight c2 t2))).(\lambda (c3: C).(\lambda (t3:
112 T).(\lambda (H0: (lt (fweight c2 t2) (fweight c3 t3))).(lt_trans (fweight c1
113 t1) (fweight c2 t2) (fweight c3 t3) H H0)))))))).
118 theorem flt_wf__q_ind:
119 \forall (P: ((C \to (T \to Prop)))).(((\forall (n: nat).((\lambda (P0: ((C
120 \to (T \to Prop)))).(\lambda (n0: nat).(\forall (c: C).(\forall (t: T).((eq
121 nat (fweight c t) n0) \to (P0 c t)))))) P n))) \to (\forall (c: C).(\forall
124 let Q \def (\lambda (P: ((C \to (T \to Prop)))).(\lambda (n: nat).(\forall
125 (c: C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t)))))) in (\lambda
126 (P: ((C \to (T \to Prop)))).(\lambda (H: ((\forall (n: nat).(\forall (c:
127 C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t))))))).(\lambda (c:
128 C).(\lambda (t: T).(H (fweight c t) c t (refl_equal nat (fweight c t))))))).
134 \forall (P: ((C \to (T \to Prop)))).(((\forall (c2: C).(\forall (t2:
135 T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1)))))
136 \to (P c2 t2))))) \to (\forall (c: C).(\forall (t: T).(P c t))))
138 let Q \def (\lambda (P: ((C \to (T \to Prop)))).(\lambda (n: nat).(\forall
139 (c: C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t)))))) in (\lambda
140 (P: ((C \to (T \to Prop)))).(\lambda (H: ((\forall (c2: C).(\forall (t2:
141 T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1)))))
142 \to (P c2 t2)))))).(\lambda (c: C).(\lambda (t: T).(flt_wf__q_ind P (\lambda
143 (n: nat).(lt_wf_ind n (Q P) (\lambda (n0: nat).(\lambda (H0: ((\forall (m:
144 nat).((lt m n0) \to (Q P m))))).(\lambda (c0: C).(\lambda (t0: T).(\lambda
145 (H1: (eq nat (fweight c0 t0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1:
146 nat).(\forall (m: nat).((lt m n1) \to (\forall (c1: C).(\forall (t1: T).((eq
147 nat (fweight c1 t1) m) \to (P c1 t1))))))) H0 (fweight c0 t0) H1) in (H c0 t0
148 (\lambda (c1: C).(\lambda (t1: T).(\lambda (H3: (flt c1 t1 c0 t0)).(H2
149 (fweight c1 t1) H3 c1 t1 (refl_equal nat (fweight c1 t1))))))))))))))) c