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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 *********************)
19 include "pc3/props.ma".
21 theorem pc3_ind_left__pc3_left_pr3:
22 \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to
25 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
26 t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t t0))) (\lambda
27 (t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2
28 c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2:
29 (pc3_left c t0 t4)).(pc3_left_ur c t3 t0 H0 t4 H2))))))) t1 t2 H)))).
31 theorem pc3_ind_left__pc3_left_trans:
32 \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to
33 (\forall (t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3))))))
35 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1
36 t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t3:
37 T).((pc3_left c t0 t3) \to (pc3_left c t t3))))) (\lambda (t: T).(\lambda
38 (t3: T).(\lambda (H0: (pc3_left c t t3)).H0))) (\lambda (t0: T).(\lambda (t3:
39 T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3
40 t4)).(\lambda (H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t3
41 t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ur c t0
42 t3 H0 t5 (H2 t5 H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0:
43 (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t0 t4)).(\lambda
44 (H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t0
45 t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ux c t0
46 t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))).
48 theorem pc3_ind_left__pc3_left_sym:
49 \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to
52 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1
53 t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t0 t)))
54 (\lambda (t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda
55 (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3
56 t4)).(\lambda (H2: (pc3_left c t4 t3)).(pc3_ind_left__pc3_left_trans c t4 t3
57 H2 t0 (pc3_left_ux c t0 t3 H0 t0 (pc3_left_r c t0))))))))) (\lambda (t0:
58 T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda
59 (_: (pc3_left c t0 t4)).(\lambda (H2: (pc3_left c t4
60 t0)).(pc3_ind_left__pc3_left_trans c t4 t0 H2 t3 (pc3_left_ur c t0 t3 H0 t3
61 (pc3_left_r c t3))))))))) t1 t2 H)))).
63 theorem pc3_ind_left__pc3_left_pc3:
64 \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to
67 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1
68 t2)).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t:
69 T).(pr3 c t2 t)) (pc3_left c t1 t2) (\lambda (x: T).(\lambda (H1: (pr3 c t1
70 x)).(\lambda (H2: (pr3 c t2 x)).(pc3_ind_left__pc3_left_trans c t1 x
71 (pc3_ind_left__pc3_left_pr3 c t1 x H1) t2 (pc3_ind_left__pc3_left_sym c t2 x
72 (pc3_ind_left__pc3_left_pr3 c t2 x H2)))))) H0))))).
74 theorem pc3_ind_left__pc3_pc3_left:
75 \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to
78 \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1
79 t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3 c t t0))) (\lambda
80 (t: T).(pc3_refl c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c
81 t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 t4)).(\lambda (H2: (pc3
82 c t3 t4)).(pc3_t t3 c t0 (pc3_pr2_r c t0 t3 H0) t4 H2))))))) (\lambda (t0:
83 T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda
84 (_: (pc3_left c t0 t4)).(\lambda (H2: (pc3 c t0 t4)).(pc3_t t0 c t3
85 (pc3_pr2_x c t3 t0 H0) t4 H2))))))) t1 t2 H)))).
88 \forall (c: C).(\forall (P: ((T \to (T \to Prop)))).(((\forall (t: T).(P t
89 t))) \to (((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3:
90 T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) \to (((\forall (t1:
91 T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: T).((pc3 c t1 t3) \to
92 ((P t1 t3) \to (P t2 t3)))))))) \to (\forall (t: T).(\forall (t0: T).((pc3 c
93 t t0) \to (P t t0))))))))
95 \lambda (c: C).(\lambda (P: ((T \to (T \to Prop)))).(\lambda (H: ((\forall
96 (t: T).(P t t)))).(\lambda (H0: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1
97 t2) \to (\forall (t3: T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1
98 t3))))))))).(\lambda (H1: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2)
99 \to (\forall (t3: T).((pc3 c t1 t3) \to ((P t1 t3) \to (P t2
100 t3))))))))).(\lambda (t: T).(\lambda (t0: T).(\lambda (H2: (pc3 c t
101 t0)).(pc3_left_ind c (\lambda (t1: T).(\lambda (t2: T).(P t1 t2))) H (\lambda
102 (t1: T).(\lambda (t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3:
103 T).(\lambda (H4: (pc3_left c t2 t3)).(\lambda (H5: (P t2 t3)).(H0 t1 t2 H3 t3
104 (pc3_ind_left__pc3_pc3_left c t2 t3 H4) H5))))))) (\lambda (t1: T).(\lambda
105 (t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (H4: (pc3_left
106 c t1 t3)).(\lambda (H5: (P t1 t3)).(H1 t1 t2 H3 t3
107 (pc3_ind_left__pc3_pc3_left c t1 t3 H4) H5))))))) t t0
108 (pc3_ind_left__pc3_left_pc3 c t t0 H2))))))))).