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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 set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/flt/props".
19 include "flt/defs.ma".
24 \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c u c
27 \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(lt_le_S
28 (plus (cweight c) (tweight u)) (plus (cweight c) (S (plus (tweight u)
29 (tweight t)))) (plus_le_lt_compat (cweight c) (cweight c) (tweight u) (S
30 (plus (tweight u) (tweight t))) (le_n (cweight c)) (le_lt_n_Sm (tweight u)
31 (plus (tweight u) (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: T).(lt_le_S
38 (plus (cweight c) (tweight t)) (plus (cweight c) (S (plus (tweight u)
39 (tweight t)))) (plus_le_lt_compat (cweight c) (cweight c) (tweight t) (S
40 (plus (tweight u) (tweight t))) (le_n (cweight c)) (le_lt_n_Sm (tweight t)
41 (plus (tweight u) (tweight t)) (le_plus_r (tweight u) (tweight t)))))))).
44 \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt (CHead c
45 k u) t c (THead k u t)))))
47 \lambda (_: K).(\lambda (c: C).(\lambda (u: T).(\lambda (t: T).(eq_ind nat
48 (S (plus (cweight c) (plus (tweight u) (tweight t)))) (\lambda (n: nat).(lt
49 (plus (plus (cweight c) (tweight u)) (tweight t)) n)) (eq_ind_r nat (plus
50 (plus (cweight c) (tweight u)) (tweight t)) (\lambda (n: nat).(lt (plus (plus
51 (cweight c) (tweight u)) (tweight t)) (S n))) (le_n (S (plus (plus (cweight
52 c) (tweight u)) (tweight t)))) (plus (cweight c) (plus (tweight u) (tweight
53 t))) (plus_assoc (cweight c) (tweight u) (tweight t))) (plus (cweight c) (S
54 (plus (tweight u) (tweight t)))) (plus_n_Sm (cweight c) (plus (tweight u)
58 \forall (k: K).(\forall (c: C).(\forall (t: T).(\forall (i: nat).(flt c t
59 (CHead c k t) (TLRef i)))))
61 \lambda (_: K).(\lambda (c: C).(\lambda (t: T).(\lambda (_: nat).(le_S_n (S
62 (plus (cweight c) (tweight t))) (plus (plus (cweight c) (tweight t)) (S O))
63 (lt_le_S (S (plus (cweight c) (tweight t))) (S (plus (plus (cweight c)
64 (tweight t)) (S O))) (lt_n_S (plus (cweight c) (tweight t)) (plus (plus
65 (cweight c) (tweight t)) (S O)) (lt_x_plus_x_Sy (plus (cweight c) (tweight
69 \forall (k1: K).(\forall (c1: C).(\forall (c2: C).(\forall (t1: T).((cle
70 (CHead c1 k1 t1) c2) \to (\forall (k2: K).(\forall (t2: T).(\forall (i:
71 nat).(flt c1 t1 (CHead c2 k2 t2) (TLRef i)))))))))
73 \lambda (_: K).(\lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda
74 (H: (le (plus (cweight c1) (tweight t1)) (cweight c2))).(\lambda (_:
75 K).(\lambda (t2: T).(\lambda (_: nat).(le_lt_trans (plus (cweight c1)
76 (tweight t1)) (cweight c2) (plus (plus (cweight c2) (tweight t2)) (S O)) H
77 (eq_ind_r nat (plus (S O) (plus (cweight c2) (tweight t2))) (\lambda (n:
78 nat).(lt (cweight c2) n)) (le_lt_n_Sm (cweight c2) (plus (cweight c2)
79 (tweight t2)) (le_plus_l (cweight c2) (tweight t2))) (plus (plus (cweight c2)
80 (tweight t2)) (S O)) (plus_comm (plus (cweight c2) (tweight t2)) (S
84 \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (i: nat).((flt c1
85 t1 c2 (TLRef i)) \to (\forall (k2: K).(\forall (t2: T).(\forall (j: nat).(flt
86 c1 t1 (CHead c2 k2 t2) (TLRef j)))))))))
88 \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (_: nat).(\lambda
89 (H: (lt (plus (cweight c1) (tweight t1)) (plus (cweight c2) (S O)))).(\lambda
90 (_: K).(\lambda (t2: T).(\lambda (_: nat).(lt_le_trans (plus (cweight c1)
91 (tweight t1)) (plus (cweight c2) (S O)) (plus (plus (cweight c2) (tweight
92 t2)) (S O)) H (le_S_n (plus (cweight c2) (S O)) (plus (plus (cweight c2)
93 (tweight t2)) (S O)) (lt_le_S (plus (cweight c2) (S O)) (S (plus (plus
94 (cweight c2) (tweight t2)) (S O))) (le_lt_n_Sm (plus (cweight c2) (S O))
95 (plus (plus (cweight c2) (tweight t2)) (S O)) (plus_le_compat (cweight c2)
96 (plus (cweight c2) (tweight t2)) (S O) (S O) (le_plus_l (cweight c2) (tweight
97 t2)) (le_n (S O)))))))))))))).
99 theorem flt_wf__q_ind:
100 \forall (P: ((C \to (T \to Prop)))).(((\forall (n: nat).((\lambda (P0: ((C
101 \to (T \to Prop)))).(\lambda (n0: nat).(\forall (c: C).(\forall (t: T).((eq
102 nat (fweight c t) n0) \to (P0 c t)))))) P n))) \to (\forall (c: C).(\forall
105 let Q \def (\lambda (P: ((C \to (T \to Prop)))).(\lambda (n: nat).(\forall
106 (c: C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t)))))) in (\lambda
107 (P: ((C \to (T \to Prop)))).(\lambda (H: ((\forall (n: nat).(\forall (c:
108 C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t))))))).(\lambda (c:
109 C).(\lambda (t: T).(H (fweight c t) c t (refl_equal nat (fweight c t))))))).
112 \forall (P: ((C \to (T \to Prop)))).(((\forall (c2: C).(\forall (t2:
113 T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1)))))
114 \to (P c2 t2))))) \to (\forall (c: C).(\forall (t: T).(P c t))))
116 let Q \def (\lambda (P: ((C \to (T \to Prop)))).(\lambda (n: nat).(\forall
117 (c: C).(\forall (t: T).((eq nat (fweight c t) n) \to (P c t)))))) in (\lambda
118 (P: ((C \to (T \to Prop)))).(\lambda (H: ((\forall (c2: C).(\forall (t2:
119 T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t2) \to (P c1 t1)))))
120 \to (P c2 t2)))))).(\lambda (c: C).(\lambda (t: T).(flt_wf__q_ind P (\lambda
121 (n: nat).(lt_wf_ind n (Q P) (\lambda (n0: nat).(\lambda (H0: ((\forall (m:
122 nat).((lt m n0) \to (Q P m))))).(\lambda (c0: C).(\lambda (t0: T).(\lambda
123 (H1: (eq nat (fweight c0 t0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1:
124 nat).(\forall (m: nat).((lt m n1) \to (\forall (c1: C).(\forall (t1: T).((eq
125 nat (fweight c1 t1) m) \to (P c1 t1))))))) H0 (fweight c0 t0) H1) in (H c0 t0
126 (\lambda (c1: C).(\lambda (t1: T).(\lambda (H3: (flt c1 t1 c0 t0)).(H2
127 (fweight c1 t1) H3 c1 t1 (refl_equal nat (fweight c1 t1))))))))))))))) c