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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 *)
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11 (* v GNU General Public License Version 2 *)
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15 (* This file was automatically generated: do not edit *********************)
17 include "LambdaDelta-1/pc3/props.ma".
19 theorem pc3_ind_left__pc3_left_pr3:
20 \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to
23 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
24 t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t t0))) (\lambda
25 (t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2
26 c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2:
27 (pc3_left c t0 t4)).(pc3_left_ur c t3 t0 H0 t4 H2))))))) t1 t2 H)))).
29 theorem pc3_ind_left__pc3_left_trans:
30 \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to
31 (\forall (t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3))))))
33 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1
34 t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t3:
35 T).((pc3_left c t0 t3) \to (pc3_left c t t3))))) (\lambda (t: T).(\lambda
36 (t3: T).(\lambda (H0: (pc3_left c t t3)).H0))) (\lambda (t0: T).(\lambda (t3:
37 T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3
38 t4)).(\lambda (H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t3
39 t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ur c t0
40 t3 H0 t5 (H2 t5 H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0:
41 (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t0 t4)).(\lambda
42 (H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t0
43 t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ux c t0
44 t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))).
46 theorem pc3_ind_left__pc3_left_sym:
47 \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to
50 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1
51 t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t0 t)))
52 (\lambda (t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda
53 (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3
54 t4)).(\lambda (H2: (pc3_left c t4 t3)).(pc3_ind_left__pc3_left_trans c t4 t3
55 H2 t0 (pc3_left_ux c t0 t3 H0 t0 (pc3_left_r c t0))))))))) (\lambda (t0:
56 T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda
57 (_: (pc3_left c t0 t4)).(\lambda (H2: (pc3_left c t4
58 t0)).(pc3_ind_left__pc3_left_trans c t4 t0 H2 t3 (pc3_left_ur c t0 t3 H0 t3
59 (pc3_left_r c t3))))))))) t1 t2 H)))).
61 theorem pc3_ind_left__pc3_left_pc3:
62 \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to
65 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1
66 t2)).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t:
67 T).(pr3 c t2 t)) (pc3_left c t1 t2) (\lambda (x: T).(\lambda (H1: (pr3 c t1
68 x)).(\lambda (H2: (pr3 c t2 x)).(pc3_ind_left__pc3_left_trans c t1 x
69 (pc3_ind_left__pc3_left_pr3 c t1 x H1) t2 (pc3_ind_left__pc3_left_sym c t2 x
70 (pc3_ind_left__pc3_left_pr3 c t2 x H2)))))) H0))))).
72 theorem pc3_ind_left__pc3_pc3_left:
73 \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to
76 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1
77 t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3 c t t0))) (\lambda
78 (t: T).(pc3_refl c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c
79 t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 t4)).(\lambda (H2: (pc3
80 c t3 t4)).(pc3_t t3 c t0 (pc3_pr2_r c t0 t3 H0) t4 H2))))))) (\lambda (t0:
81 T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda
82 (_: (pc3_left c t0 t4)).(\lambda (H2: (pc3 c t0 t4)).(pc3_t t0 c t3
83 (pc3_pr2_x c t3 t0 H0) t4 H2))))))) t1 t2 H)))).
86 \forall (c: C).(\forall (P: ((T \to (T \to Prop)))).(((\forall (t: T).(P t
87 t))) \to (((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3:
88 T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) \to (((\forall (t1:
89 T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: T).((pc3 c t1 t3) \to
90 ((P t1 t3) \to (P t2 t3)))))))) \to (\forall (t: T).(\forall (t0: T).((pc3 c
91 t t0) \to (P t t0))))))))
93 \lambda (c: C).(\lambda (P: ((T \to (T \to Prop)))).(\lambda (H: ((\forall
94 (t: T).(P t t)))).(\lambda (H0: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1
95 t2) \to (\forall (t3: T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1
96 t3))))))))).(\lambda (H1: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2)
97 \to (\forall (t3: T).((pc3 c t1 t3) \to ((P t1 t3) \to (P t2
98 t3))))))))).(\lambda (t: T).(\lambda (t0: T).(\lambda (H2: (pc3 c t
99 t0)).(pc3_left_ind c (\lambda (t1: T).(\lambda (t2: T).(P t1 t2))) H (\lambda
100 (t1: T).(\lambda (t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3:
101 T).(\lambda (H4: (pc3_left c t2 t3)).(\lambda (H5: (P t2 t3)).(H0 t1 t2 H3 t3
102 (pc3_ind_left__pc3_pc3_left c t2 t3 H4) H5))))))) (\lambda (t1: T).(\lambda
103 (t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (H4: (pc3_left
104 c t1 t3)).(\lambda (H5: (P t1 t3)).(H1 t1 t2 H3 t3
105 (pc3_ind_left__pc3_pc3_left c t1 t3 H4) H5))))))) t t0
106 (pc3_ind_left__pc3_left_pc3 c t t0 H2))))))))).