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15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/llt/props".
19 include "llt/defs.ma".
21 include "leq/defs.ma".
24 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (eq nat
25 (lweight a1) (lweight a2)))))
27 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
28 a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(eq nat (lweight a) (lweight
29 a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2:
30 nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort h1 n1) k) (aplus g
31 (ASort h2 n2) k))).(refl_equal nat O))))))) (\lambda (a0: A).(\lambda (a3:
32 A).(\lambda (_: (leq g a0 a3)).(\lambda (H1: (eq nat (lweight a0) (lweight
33 a3))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leq g a4 a5)).(\lambda
34 (H3: (eq nat (lweight a4) (lweight a5))).(f_equal nat nat S (plus (lweight
35 a0) (lweight a4)) (plus (lweight a3) (lweight a5)) (f_equal2 nat nat nat plus
36 (lweight a0) (lweight a3) (lweight a4) (lweight a5) H1 H3)))))))))) a1 a2
40 \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall
41 (a3: A).((llt a1 a3) \to (llt a2 a3))))))
43 \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1
44 a2)).(\lambda (a3: A).(\lambda (H0: (lt (lweight a1) (lweight a3))).(let H1
45 \def (eq_ind nat (lweight a1) (\lambda (n: nat).(lt n (lweight a3))) H0
46 (lweight a2) (lweight_repl g a1 a2 H)) in H1)))))).
49 \forall (a1: A).(\forall (a2: A).(\forall (a3: A).((llt a1 a2) \to ((llt a2
50 a3) \to (llt a1 a3)))))
52 \lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda (H: (lt (lweight
53 a1) (lweight a2))).(\lambda (H0: (lt (lweight a2) (lweight a3))).(lt_trans
54 (lweight a1) (lweight a2) (lweight a3) H H0))))).
57 \forall (a1: A).(\forall (a2: A).(llt a1 (AHead a1 a2)))
59 \lambda (a1: A).(\lambda (a2: A).(le_S_n (S (lweight a1)) (S (plus (lweight
60 a1) (lweight a2))) (le_n_S (S (lweight a1)) (S (plus (lweight a1) (lweight
61 a2))) (le_n_S (lweight a1) (plus (lweight a1) (lweight a2)) (le_plus_l
62 (lweight a1) (lweight a2)))))).
65 \forall (a1: A).(\forall (a2: A).(llt a2 (AHead a1 a2)))
67 \lambda (a1: A).(\lambda (a2: A).(le_S_n (S (lweight a2)) (S (plus (lweight
68 a1) (lweight a2))) (le_n_S (S (lweight a2)) (S (plus (lweight a1) (lweight
69 a2))) (le_n_S (lweight a2) (plus (lweight a1) (lweight a2)) (le_plus_r
70 (lweight a1) (lweight a2)))))).
72 theorem llt_wf__q_ind:
73 \forall (P: ((A \to Prop))).(((\forall (n: nat).((\lambda (P0: ((A \to
74 Prop))).(\lambda (n0: nat).(\forall (a: A).((eq nat (lweight a) n0) \to (P0
75 a))))) P n))) \to (\forall (a: A).(P a)))
77 let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a:
78 A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to
79 Prop))).(\lambda (H: ((\forall (n: nat).(\forall (a: A).((eq nat (lweight a)
80 n) \to (P a)))))).(\lambda (a: A).(H (lweight a) a (refl_equal nat (lweight
84 \forall (P: ((A \to Prop))).(((\forall (a2: A).(((\forall (a1: A).((llt a1
85 a2) \to (P a1)))) \to (P a2)))) \to (\forall (a: A).(P a)))
87 let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a:
88 A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to
89 Prop))).(\lambda (H: ((\forall (a2: A).(((\forall (a1: A).((lt (lweight a1)
90 (lweight a2)) \to (P a1)))) \to (P a2))))).(\lambda (a: A).(llt_wf__q_ind
91 (\lambda (a0: A).(P a0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (a0:
92 A).(P a0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0)
93 \to (Q (\lambda (a0: A).(P a0)) m))))).(\lambda (a0: A).(\lambda (H1: (eq nat
94 (lweight a0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall
95 (m: nat).((lt m n1) \to (\forall (a1: A).((eq nat (lweight a1) m) \to (P
96 a1)))))) H0 (lweight a0) H1) in (H a0 (\lambda (a1: A).(\lambda (H3: (lt
97 (lweight a1) (lweight a0))).(H2 (lweight a1) H3 a1 (refl_equal nat (lweight
98 a1))))))))))))) a)))).