+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was automatically generated: do not edit *********************)
-
-include "basic_1/pc3/props.ma".
-
-implied rec lemma pc3_left_ind (c: C) (P: (T \to (T \to Prop))) (f: (\forall
-(t: T).(P t t))) (f0: (\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to
-(\forall (t3: T).((pc3_left c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) (f1:
-(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3:
-T).((pc3_left c t1 t3) \to ((P t1 t3) \to (P t2 t3)))))))) (t: T) (t0: T) (p:
-pc3_left c t t0) on p: P t t0 \def match p with [(pc3_left_r t1) \Rightarrow
-(f t1) | (pc3_left_ur t1 t2 p0 t3 p1) \Rightarrow (f0 t1 t2 p0 t3 p1
-((pc3_left_ind c P f f0 f1) t2 t3 p1)) | (pc3_left_ux t1 t2 p0 t3 p1)
-\Rightarrow (f1 t1 t2 p0 t3 p1 ((pc3_left_ind c P f f0 f1) t1 t3 p1))].
-
-fact pc3_ind_left__pc3_left_pr3:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to
-(pc3_left c t1 t2))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1
-t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t t0))) (\lambda
-(t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2
-c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2:
-(pc3_left c t0 t4)).(pc3_left_ur c t3 t0 H0 t4 H2))))))) t1 t2 H)))).
-
-fact pc3_ind_left__pc3_left_trans:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to
-(\forall (t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3))))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1
-t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t3:
-T).((pc3_left c t0 t3) \to (pc3_left c t t3))))) (\lambda (t: T).(\lambda
-(t3: T).(\lambda (H0: (pc3_left c t t3)).H0))) (\lambda (t0: T).(\lambda (t3:
-T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3
-t4)).(\lambda (H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t3
-t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ur c t0
-t3 H0 t5 (H2 t5 H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0:
-(pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t0 t4)).(\lambda
-(H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t0
-t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ux c t0
-t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))).
-
-fact pc3_ind_left__pc3_left_sym:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to
-(pc3_left c t2 t1))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1
-t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t0 t)))
-(\lambda (t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda
-(H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3
-t4)).(\lambda (H2: (pc3_left c t4 t3)).(pc3_ind_left__pc3_left_trans c t4 t3
-H2 t0 (pc3_left_ux c t0 t3 H0 t0 (pc3_left_r c t0))))))))) (\lambda (t0:
-T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda
-(_: (pc3_left c t0 t4)).(\lambda (H2: (pc3_left c t4
-t0)).(pc3_ind_left__pc3_left_trans c t4 t0 H2 t3 (pc3_left_ur c t0 t3 H0 t3
-(pc3_left_r c t3))))))))) t1 t2 H)))).
-
-fact pc3_ind_left__pc3_left_pc3:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to
-(pc3_left c t1 t2))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1
-t2)).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t:
-T).(pr3 c t2 t)) (pc3_left c t1 t2) (\lambda (x: T).(\lambda (H1: (pr3 c t1
-x)).(\lambda (H2: (pr3 c t2 x)).(pc3_ind_left__pc3_left_trans c t1 x
-(pc3_ind_left__pc3_left_pr3 c t1 x H1) t2 (pc3_ind_left__pc3_left_sym c t2 x
-(pc3_ind_left__pc3_left_pr3 c t2 x H2)))))) H0))))).
-
-fact pc3_ind_left__pc3_pc3_left:
- \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to
-(pc3 c t1 t2))))
-\def
- \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1
-t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3 c t t0))) (\lambda
-(t: T).(pc3_refl c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c
-t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 t4)).(\lambda (H2: (pc3
-c t3 t4)).(pc3_t t3 c t0 (pc3_pr2_r c t0 t3 H0) t4 H2))))))) (\lambda (t0:
-T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda
-(_: (pc3_left c t0 t4)).(\lambda (H2: (pc3 c t0 t4)).(pc3_t t0 c t3
-(pc3_pr2_x c t3 t0 H0) t4 H2))))))) t1 t2 H)))).
-
-lemma pc3_ind_left:
- \forall (c: C).(\forall (P: ((T \to (T \to Prop)))).(((\forall (t: T).(P t
-t))) \to (((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3:
-T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) \to (((\forall (t1:
-T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: T).((pc3 c t1 t3) \to
-((P t1 t3) \to (P t2 t3)))))))) \to (\forall (t: T).(\forall (t0: T).((pc3 c
-t t0) \to (P t t0))))))))
-\def
- \lambda (c: C).(\lambda (P: ((T \to (T \to Prop)))).(\lambda (H: ((\forall
-(t: T).(P t t)))).(\lambda (H0: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1
-t2) \to (\forall (t3: T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1
-t3))))))))).(\lambda (H1: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2)
-\to (\forall (t3: T).((pc3 c t1 t3) \to ((P t1 t3) \to (P t2
-t3))))))))).(\lambda (t: T).(\lambda (t0: T).(\lambda (H2: (pc3 c t
-t0)).(pc3_left_ind c (\lambda (t1: T).(\lambda (t2: T).(P t1 t2))) H (\lambda
-(t1: T).(\lambda (t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3:
-T).(\lambda (H4: (pc3_left c t2 t3)).(\lambda (H5: (P t2 t3)).(H0 t1 t2 H3 t3
-(pc3_ind_left__pc3_pc3_left c t2 t3 H4) H5))))))) (\lambda (t1: T).(\lambda
-(t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (H4: (pc3_left
-c t1 t3)).(\lambda (H5: (P t1 t3)).(H1 t1 t2 H3 t3
-(pc3_ind_left__pc3_pc3_left c t1 t3 H4) H5))))))) t t0
-(pc3_ind_left__pc3_left_pc3 c t t0 H2))))))))).
-
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