%------------------------------------------------------------------------------ % File : Zenon---0.7.1 % Problem : GEO191+2 : TPTP v8.1.0. Released v3.3.0. % Transfm : none % Format : tptp:raw % Command : run_zenon %s %d % Computer : n010.cluster.edu % Model : x86_64 x86_64 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz % Memory : 8042.1875MB % OS : Linux 3.10.0-693.el7.x86_64 % CPULimit : 300s % WCLimit : 600s % DateTime : Sat Jul 16 07:00:36 EDT 2022 % Result : Theorem 3.45s 3.63s % Output : Proof 3.45s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.03/0.14 % Problem : GEO191+2 : TPTP v8.1.0. Released v3.3.0. % 0.03/0.14 % Command : run_zenon %s %d % 0.15/0.36 % Computer : n010.cluster.edu % 0.15/0.36 % Model : x86_64 x86_64 % 0.15/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.15/0.36 % Memory : 8042.1875MB % 0.15/0.36 % OS : Linux 3.10.0-693.el7.x86_64 % 0.15/0.36 % CPULimit : 300 % 0.15/0.36 % WCLimit : 600 % 0.15/0.36 % DateTime : Sat Jun 18 01:30:38 EDT 2022 % 0.15/0.36 % CPUTime : % 3.45/3.63 (* PROOF-FOUND *) % 3.45/3.63 % SZS status Theorem % 3.45/3.63 (* BEGIN-PROOF *) % 3.45/3.63 % SZS output start Proof % 3.45/3.63 Theorem con : (forall X : zenon_U, (forall Y : zenon_U, (forall U : zenon_U, (forall V : zenon_U, (((convergent_lines X Y)/\((convergent_lines U V)/\((apart_point_and_line (intersection_point X Y) U)\/(apart_point_and_line (intersection_point X Y) V))))->((apart_point_and_line (intersection_point U V) X)\/(apart_point_and_line (intersection_point U V) Y))))))). % 3.45/3.63 Proof. % 3.45/3.63 assert (zenon_L1_ : forall (zenon_TY_p : zenon_U) (zenon_TX_q : zenon_U), (apart_point_and_line (intersection_point zenon_TX_q zenon_TY_p) zenon_TX_q) -> (convergent_lines zenon_TX_q zenon_TY_p) -> False). % 3.45/3.63 do 2 intro. intros zenon_Hd zenon_He. % 3.45/3.63 generalize (con2 zenon_TX_q). zenon_intro zenon_H11. % 3.45/3.63 generalize (zenon_H11 zenon_TY_p). zenon_intro zenon_H12. % 3.45/3.63 generalize (apart1 (intersection_point zenon_TX_q zenon_TY_p)). zenon_intro zenon_H13. % 3.45/3.63 generalize (zenon_H12 (intersection_point zenon_TX_q zenon_TY_p)). zenon_intro zenon_H14. % 3.45/3.63 apply (zenon_imply_s _ _ zenon_H14); [ zenon_intro zenon_H16 | zenon_intro zenon_H15 ]. % 3.45/3.63 exact (zenon_H16 zenon_He). % 3.45/3.63 apply (zenon_imply_s _ _ zenon_H15); [ zenon_intro zenon_H18 | zenon_intro zenon_H17 ]. % 3.45/3.63 apply (zenon_notor_s _ _ zenon_H18). zenon_intro zenon_H1a. zenon_intro zenon_H19. % 3.45/3.63 exact (zenon_H1a zenon_Hd). % 3.45/3.63 exact (zenon_H13 zenon_H17). % 3.45/3.63 (* end of lemma zenon_L1_ *) % 3.45/3.63 assert (zenon_L2_ : forall (zenon_TV_bg : zenon_U) (zenon_TU_bh : zenon_U) (zenon_TY_p : zenon_U) (zenon_TX_q : zenon_U), (forall U : zenon_U, (forall V : zenon_U, (((distinct_points (intersection_point zenon_TX_q zenon_TY_p) (intersection_point zenon_TU_bh zenon_TV_bg))/\(distinct_lines U V))->((apart_point_and_line (intersection_point zenon_TX_q zenon_TY_p) U)\/((apart_point_and_line (intersection_point zenon_TX_q zenon_TY_p) V)\/((apart_point_and_line (intersection_point zenon_TU_bh zenon_TV_bg) U)\/(apart_point_and_line (intersection_point zenon_TU_bh zenon_TV_bg) V))))))) -> (~(apart_point_and_line (intersection_point zenon_TU_bh zenon_TV_bg) zenon_TY_p)) -> (~(apart_point_and_line (intersection_point zenon_TU_bh zenon_TV_bg) zenon_TX_q)) -> (~(apart_point_and_line (intersection_point zenon_TX_q zenon_TY_p) zenon_TY_p)) -> (convergent_lines zenon_TX_q zenon_TY_p) -> (distinct_lines zenon_TX_q zenon_TY_p) -> (distinct_points (intersection_point zenon_TX_q zenon_TY_p) (intersection_point zenon_TU_bh zenon_TV_bg)) -> False). % 3.45/3.63 do 4 intro. intros zenon_H1b zenon_H1c zenon_H1d zenon_H19 zenon_He zenon_H1e zenon_H1f. % 3.45/3.63 generalize (zenon_H1b zenon_TX_q). zenon_intro zenon_H22. % 3.45/3.63 generalize (zenon_H22 zenon_TY_p). zenon_intro zenon_H23. % 3.45/3.63 apply (zenon_imply_s _ _ zenon_H23); [ zenon_intro zenon_H25 | zenon_intro zenon_H24 ]. % 3.45/3.63 apply (zenon_notand_s _ _ zenon_H25); [ zenon_intro zenon_H27 | zenon_intro zenon_H26 ]. % 3.45/3.63 exact (zenon_H27 zenon_H1f). % 3.45/3.63 exact (zenon_H26 zenon_H1e). % 3.45/3.63 apply (zenon_or_s _ _ zenon_H24); [ zenon_intro zenon_Hd | zenon_intro zenon_H28 ]. % 3.45/3.63 apply (zenon_L1_ zenon_TY_p zenon_TX_q); trivial. % 3.45/3.63 apply (zenon_or_s _ _ zenon_H28); [ zenon_intro zenon_H2a | zenon_intro zenon_H29 ]. % 3.45/3.63 exact (zenon_H19 zenon_H2a). % 3.45/3.63 apply (zenon_or_s _ _ zenon_H29); [ zenon_intro zenon_H2c | zenon_intro zenon_H2b ]. % 3.45/3.63 exact (zenon_H1d zenon_H2c). % 3.45/3.63 exact (zenon_H1c zenon_H2b). % 3.45/3.63 (* end of lemma zenon_L2_ *) % 3.45/3.63 assert (zenon_L3_ : forall (zenon_TV_bg : zenon_U) (zenon_TU_bh : zenon_U), (forall Z : zenon_U, ((convergent_lines zenon_TU_bh zenon_TV_bg)->(((apart_point_and_line Z zenon_TU_bh)\/(apart_point_and_line Z zenon_TV_bg))->(distinct_points Z (intersection_point zenon_TU_bh zenon_TV_bg))))) -> (convergent_lines zenon_TU_bh zenon_TV_bg) -> (apart_point_and_line (intersection_point zenon_TU_bh zenon_TV_bg) zenon_TU_bh) -> (~(distinct_points (intersection_point zenon_TU_bh zenon_TV_bg) (intersection_point zenon_TU_bh zenon_TV_bg))) -> False). % 3.45/3.63 do 2 intro. intros zenon_H2d zenon_H2e zenon_H2f zenon_H30. % 3.45/3.63 generalize (zenon_H2d (intersection_point zenon_TU_bh zenon_TV_bg)). zenon_intro zenon_H31. % 3.45/3.63 apply (zenon_imply_s _ _ zenon_H31); [ zenon_intro zenon_H33 | zenon_intro zenon_H32 ]. % 3.45/3.63 exact (zenon_H33 zenon_H2e). % 3.45/3.63 apply (zenon_imply_s _ _ zenon_H32); [ zenon_intro zenon_H35 | zenon_intro zenon_H34 ]. % 3.45/3.63 apply (zenon_notor_s _ _ zenon_H35). zenon_intro zenon_H37. zenon_intro zenon_H36. % 3.45/3.63 exact (zenon_H37 zenon_H2f). % 3.45/3.63 exact (zenon_H30 zenon_H34). % 3.45/3.63 (* end of lemma zenon_L3_ *) % 3.45/3.63 assert (zenon_L4_ : forall (zenon_TY_p : zenon_U) (zenon_TX_q : zenon_U), (apart_point_and_line (intersection_point zenon_TX_q zenon_TY_p) zenon_TY_p) -> (convergent_lines zenon_TX_q zenon_TY_p) -> False). % 3.45/3.63 do 2 intro. intros zenon_H2a zenon_He. % 3.45/3.63 generalize (con2 zenon_TX_q). zenon_intro zenon_H11. % 3.45/3.63 generalize (zenon_H11 zenon_TY_p). zenon_intro zenon_H12. % 3.45/3.63 generalize (apart1 (intersection_point zenon_TX_q zenon_TY_p)). zenon_intro zenon_H13. % 3.45/3.63 generalize (zenon_H12 (intersection_point zenon_TX_q zenon_TY_p)). zenon_intro zenon_H14. % 3.45/3.63 apply (zenon_imply_s _ _ zenon_H14); [ zenon_intro zenon_H16 | zenon_intro zenon_H15 ]. % 3.45/3.63 exact (zenon_H16 zenon_He). % 3.45/3.63 apply (zenon_imply_s _ _ zenon_H15); [ zenon_intro zenon_H18 | zenon_intro zenon_H17 ]. % 3.45/3.63 apply (zenon_notor_s _ _ zenon_H18). zenon_intro zenon_H1a. zenon_intro zenon_H19. % 3.45/3.63 exact (zenon_H19 zenon_H2a). % 3.45/3.63 exact (zenon_H13 zenon_H17). % 3.45/3.63 (* end of lemma zenon_L4_ *) % 3.45/3.63 assert (zenon_L5_ : forall (zenon_TX_q : zenon_U), (convergent_lines zenon_TX_q zenon_TX_q) -> False). % 3.45/3.63 do 1 intro. intros zenon_H38. % 3.45/3.63 generalize (apart3 zenon_TX_q). zenon_intro zenon_H39. % 3.45/3.63 exact (zenon_H39 zenon_H38). % 3.45/3.63 (* end of lemma zenon_L5_ *) % 3.45/3.63 assert (zenon_L6_ : forall (zenon_TV_bg : zenon_U) (zenon_TU_bh : zenon_U), (forall Z : zenon_U, ((convergent_lines zenon_TU_bh zenon_TV_bg)->(((apart_point_and_line Z zenon_TU_bh)\/(apart_point_and_line Z zenon_TV_bg))->(distinct_points Z (intersection_point zenon_TU_bh zenon_TV_bg))))) -> (convergent_lines zenon_TU_bh zenon_TV_bg) -> (apart_point_and_line (intersection_point zenon_TU_bh zenon_TV_bg) zenon_TV_bg) -> (~(distinct_points (intersection_point zenon_TU_bh zenon_TV_bg) (intersection_point zenon_TU_bh zenon_TV_bg))) -> False). % 3.45/3.63 do 2 intro. intros zenon_H2d zenon_H2e zenon_H3a zenon_H30. % 3.45/3.63 generalize (zenon_H2d (intersection_point zenon_TU_bh zenon_TV_bg)). zenon_intro zenon_H31. % 3.45/3.63 apply (zenon_imply_s _ _ zenon_H31); [ zenon_intro zenon_H33 | zenon_intro zenon_H32 ]. % 3.45/3.63 exact (zenon_H33 zenon_H2e). % 3.45/3.63 apply (zenon_imply_s _ _ zenon_H32); [ zenon_intro zenon_H35 | zenon_intro zenon_H34 ]. % 3.45/3.63 apply (zenon_notor_s _ _ zenon_H35). zenon_intro zenon_H37. zenon_intro zenon_H36. % 3.45/3.63 exact (zenon_H36 zenon_H3a). % 3.45/3.63 exact (zenon_H30 zenon_H34). % 3.45/3.63 (* end of lemma zenon_L6_ *) % 3.45/3.63 assert (zenon_L7_ : forall (zenon_TV_bg : zenon_U) (zenon_TU_bh : zenon_U), (apart_point_and_line (intersection_point zenon_TU_bh zenon_TV_bg) zenon_TV_bg) -> (convergent_lines zenon_TU_bh zenon_TV_bg) -> False). % 3.45/3.63 do 2 intro. intros zenon_H3a zenon_H2e. % 3.45/3.63 generalize (apart1 (intersection_point zenon_TU_bh zenon_TV_bg)). zenon_intro zenon_H30. % 3.45/3.63 generalize (con2 zenon_TU_bh). zenon_intro zenon_H3b. % 3.45/3.63 generalize (zenon_H3b zenon_TV_bg). zenon_intro zenon_H2d. % 3.45/3.63 apply (zenon_L6_ zenon_TV_bg zenon_TU_bh); trivial. % 3.45/3.63 (* end of lemma zenon_L7_ *) % 3.45/3.63 assert (zenon_L8_ : forall (zenon_TU_bh : zenon_U) (zenon_TV_bg : zenon_U) (zenon_TX_q : zenon_U) (zenon_TY_p : zenon_U), (forall Y : zenon_U, ((convergent_lines zenon_TY_p Y)->(distinct_lines zenon_TY_p Y))) -> (convergent_lines zenon_TY_p zenon_TX_q) -> (forall Z : zenon_U, ((apart_point_and_line (intersection_point zenon_TX_q zenon_TY_p) zenon_TV_bg)->((distinct_points (intersection_point zenon_TX_q zenon_TY_p) Z)\/(apart_point_and_line Z zenon_TV_bg)))) -> (apart_point_and_line (intersection_point zenon_TX_q zenon_TY_p) zenon_TV_bg) -> (~(apart_point_and_line (intersection_point zenon_TU_bh zenon_TV_bg) zenon_TX_q)) -> (~(apart_point_and_line (intersection_point zenon_TU_bh zenon_TV_bg) zenon_TY_p)) -> (~(apart_point_and_line (intersection_point zenon_TX_q zenon_TY_p) zenon_TX_q)) -> (~(apart_point_and_line (intersection_point zenon_TX_q zenon_TY_p) zenon_TY_p)) -> (~(distinct_lines zenon_TV_bg zenon_TV_bg)) -> (convergent_lines zenon_TU_bh zenon_TV_bg) -> False). % 3.45/3.63 do 4 intro. intros zenon_H3c zenon_H3d zenon_H3e zenon_H3f zenon_H1d zenon_H1c zenon_H1a zenon_H19 zenon_H40 zenon_H2e. % 3.45/3.63 generalize (zenon_H3c zenon_TX_q). zenon_intro zenon_H41. % 3.45/3.63 apply (zenon_imply_s _ _ zenon_H41); [ zenon_intro zenon_H43 | zenon_intro zenon_H42 ]. % 3.45/3.63 exact (zenon_H43 zenon_H3d). % 3.45/3.63 generalize (ceq2 (intersection_point zenon_TU_bh zenon_TV_bg)). zenon_intro zenon_H44. % 3.45/3.63 generalize (zenon_H44 zenon_TV_bg). zenon_intro zenon_H45. % 3.45/3.63 generalize (zenon_H45 zenon_TV_bg). zenon_intro zenon_H46. % 3.45/3.63 apply (zenon_imply_s _ _ zenon_H46); [ zenon_intro zenon_H36 | zenon_intro zenon_H47 ]. % 3.45/3.63 generalize (zenon_H3e (intersection_point zenon_TU_bh zenon_TV_bg)). zenon_intro zenon_H48. % 3.45/3.63 apply (zenon_imply_s _ _ zenon_H48); [ zenon_intro zenon_H4a | zenon_intro zenon_H49 ]. % 3.45/3.63 exact (zenon_H4a zenon_H3f). % 3.45/3.63 apply (zenon_or_s _ _ zenon_H49); [ zenon_intro zenon_H1f | zenon_intro zenon_H3a ]. % 3.45/3.63 generalize (cu1 (intersection_point zenon_TX_q zenon_TY_p)). zenon_intro zenon_H4b. % 3.45/3.63 generalize (zenon_H4b (intersection_point zenon_TU_bh zenon_TV_bg)). zenon_intro zenon_H1b. % 3.45/3.63 generalize (zenon_H1b zenon_TY_p). zenon_intro zenon_H4c. % 3.45/3.63 generalize (zenon_H4c zenon_TX_q). zenon_intro zenon_H4d. % 3.45/3.63 apply (zenon_imply_s _ _ zenon_H4d); [ zenon_intro zenon_H4f | zenon_intro zenon_H4e ]. % 3.45/3.63 apply (zenon_notand_s _ _ zenon_H4f); [ zenon_intro zenon_H27 | zenon_intro zenon_H50 ]. % 3.45/3.63 exact (zenon_H27 zenon_H1f). % 3.45/3.63 exact (zenon_H50 zenon_H42). % 3.45/3.63 apply (zenon_or_s _ _ zenon_H4e); [ zenon_intro zenon_H2a | zenon_intro zenon_H51 ]. % 3.45/3.63 exact (zenon_H19 zenon_H2a). % 3.45/3.63 apply (zenon_or_s _ _ zenon_H51); [ zenon_intro zenon_Hd | zenon_intro zenon_H52 ]. % 3.45/3.63 exact (zenon_H1a zenon_Hd). % 3.45/3.63 apply (zenon_or_s _ _ zenon_H52); [ zenon_intro zenon_H2b | zenon_intro zenon_H2c ]. % 3.45/3.63 exact (zenon_H1c zenon_H2b). % 3.45/3.63 exact (zenon_H1d zenon_H2c). % 3.45/3.63 exact (zenon_H36 zenon_H3a). % 3.45/3.63 apply (zenon_or_s _ _ zenon_H47); [ zenon_intro zenon_H53 | zenon_intro zenon_H3a ]. % 3.45/3.63 exact (zenon_H40 zenon_H53). % 3.45/3.63 apply (zenon_L7_ zenon_TV_bg zenon_TU_bh); trivial. % 3.45/3.63 (* end of lemma zenon_L8_ *) % 3.45/3.63 assert (zenon_L9_ : forall (zenon_TU_bh : zenon_U) (zenon_TV_bg : zenon_U) (zenon_TY_p : zenon_U) (zenon_TX_q : zenon_U), (forall Z : zenon_U, ((convergent_lines zenon_TX_q zenon_TY_p)->((convergent_lines zenon_TX_q Z)\/(convergent_lines zenon_TY_p Z)))) -> (convergent_lines zenon_TX_q zenon_TY_p) -> (forall Y : zenon_U, ((convergent_lines zenon_TY_p Y)->(distinct_lines zenon_TY_p Y))) -> (forall Z : zenon_U, ((apart_point_and_line (intersection_point zenon_TX_q zenon_TY_p) zenon_TV_bg)->((distinct_points (intersection_point zenon_TX_q zenon_TY_p) Z)\/(apart_point_and_line Z zenon_TV_bg)))) -> (apart_point_and_line (intersection_point zenon_TX_q zenon_TY_p) zenon_TV_bg) -> (~(apart_point_and_line (intersection_point zenon_TU_bh zenon_TV_bg) zenon_TX_q)) -> (~(apart_point_and_line (intersection_point zenon_TU_bh zenon_TV_bg) zenon_TY_p)) -> (~(apart_point_and_line (intersection_point zenon_TX_q zenon_TY_p) zenon_TX_q)) -> (~(apart_point_and_line (intersection_point zenon_TX_q zenon_TY_p) zenon_TY_p)) -> (~(distinct_lines zenon_TV_bg zenon_TV_bg)) -> (convergent_lines zenon_TU_bh zenon_TV_bg) -> False). % 3.45/3.63 do 4 intro. intros zenon_H54 zenon_He zenon_H3c zenon_H3e zenon_H3f zenon_H1d zenon_H1c zenon_H1a zenon_H19 zenon_H40 zenon_H2e. % 3.45/3.63 generalize (zenon_H54 zenon_TX_q). zenon_intro zenon_H55. % 3.45/3.63 apply (zenon_imply_s _ _ zenon_H55); [ zenon_intro zenon_H16 | zenon_intro zenon_H56 ]. % 3.45/3.63 exact (zenon_H16 zenon_He). % 3.45/3.63 apply (zenon_or_s _ _ zenon_H56); [ zenon_intro zenon_H38 | zenon_intro zenon_H3d ]. % 3.45/3.63 apply (zenon_L5_ zenon_TX_q); trivial. % 3.45/3.63 apply (zenon_L8_ zenon_TU_bh zenon_TV_bg zenon_TX_q zenon_TY_p); trivial. % 3.45/3.63 (* end of lemma zenon_L9_ *) % 3.45/3.63 assert (zenon_L10_ : forall (zenon_TU_bh : zenon_U) (zenon_TV_bg : zenon_U) (zenon_TY_p : zenon_U) (zenon_TX_q : zenon_U), (forall Z : zenon_U, ((convergent_lines zenon_TX_q zenon_TY_p)->((convergent_lines zenon_TX_q Z)\/(convergent_lines zenon_TY_p Z)))) -> (convergent_lines zenon_TX_q zenon_TY_p) -> (forall Y : zenon_U, ((convergent_lines zenon_TY_p Y)->(distinct_lines zenon_TY_p Y))) -> (apart_point_and_line (intersection_point zenon_TX_q zenon_TY_p) zenon_TV_bg) -> (~(apart_point_and_line (intersection_point zenon_TU_bh zenon_TV_bg) zenon_TX_q)) -> (~(apart_point_and_line (intersection_point zenon_TU_bh zenon_TV_bg) zenon_TY_p)) -> (~(apart_point_and_line (intersection_point zenon_TX_q zenon_TY_p) zenon_TY_p)) -> (~(distinct_lines zenon_TV_bg zenon_TV_bg)) -> (convergent_lines zenon_TU_bh zenon_TV_bg) -> (forall Z : zenon_U, ((apart_point_and_line (intersection_point zenon_TX_q zenon_TY_p) zenon_TX_q)->((distinct_lines zenon_TX_q Z)\/(apart_point_and_line (intersection_point zenon_TX_q zenon_TY_p) Z)))) -> (forall Y : zenon_U, (forall Z : zenon_U, ((apart_point_and_line (intersection_point zenon_TX_q zenon_TY_p) Y)->((distinct_points (intersection_point zenon_TX_q zenon_TY_p) Z)\/(apart_point_and_line Z Y))))) -> False). % 3.45/3.63 do 4 intro. intros zenon_H54 zenon_He zenon_H3c zenon_H3f zenon_H1d zenon_H1c zenon_H19 zenon_H40 zenon_H2e zenon_H57 zenon_H58. % 3.45/3.63 generalize (zenon_H58 zenon_TV_bg). zenon_intro zenon_H3e. % 3.45/3.63 generalize (zenon_H58 zenon_TX_q). zenon_intro zenon_H59. % 3.45/3.63 generalize (zenon_H59 (intersection_point zenon_TU_bh zenon_TV_bg)). zenon_intro zenon_H5a. % 3.45/3.63 apply (zenon_imply_s _ _ zenon_H5a); [ zenon_intro zenon_H1a | zenon_intro zenon_H5b ]. % 3.45/3.63 apply (zenon_L9_ zenon_TU_bh zenon_TV_bg zenon_TY_p zenon_TX_q); trivial. % 3.45/3.63 apply (zenon_or_s _ _ zenon_H5b); [ zenon_intro zenon_H1f | zenon_intro zenon_H2c ]. % 3.45/3.63 generalize (cu1 (intersection_point zenon_TX_q zenon_TY_p)). zenon_intro zenon_H4b. % 3.45/3.63 generalize (zenon_H4b (intersection_point zenon_TU_bh zenon_TV_bg)). zenon_intro zenon_H1b. % 3.45/3.63 generalize (zenon_H57 zenon_TY_p). zenon_intro zenon_H5c. % 3.45/3.63 apply (zenon_imply_s _ _ zenon_H5c); [ zenon_intro zenon_H1a | zenon_intro zenon_H5d ]. % 3.45/3.63 apply (zenon_L9_ zenon_TU_bh zenon_TV_bg zenon_TY_p zenon_TX_q); trivial. % 3.45/3.63 apply (zenon_or_s _ _ zenon_H5d); [ zenon_intro zenon_H1e | zenon_intro zenon_H2a ]. % 3.45/3.63 apply (zenon_L2_ zenon_TV_bg zenon_TU_bh zenon_TY_p zenon_TX_q); trivial. % 3.45/3.63 exact (zenon_H19 zenon_H2a). % 3.45/3.63 exact (zenon_H1d zenon_H2c). % 3.45/3.63 (* end of lemma zenon_L10_ *) % 3.45/3.63 assert (zenon_L11_ : forall (zenon_TU_bh : zenon_U) (zenon_TV_bg : zenon_U) (zenon_TX_q : zenon_U) (zenon_TY_p : zenon_U), (~(distinct_lines zenon_TY_p zenon_TY_p)) -> (forall Z : zenon_U, ((convergent_lines zenon_TX_q zenon_TY_p)->((convergent_lines zenon_TX_q Z)\/(convergent_lines zenon_TY_p Z)))) -> (convergent_lines zenon_TX_q zenon_TY_p) -> (forall Y : zenon_U, ((convergent_lines zenon_TY_p Y)->(distinct_lines zenon_TY_p Y))) -> (apart_point_and_line (intersection_point zenon_TX_q zenon_TY_p) zenon_TV_bg) -> (~(apart_point_and_line (intersection_point zenon_TU_bh zenon_TV_bg) zenon_TX_q)) -> (~(apart_point_and_line (intersection_point zenon_TU_bh zenon_TV_bg) zenon_TY_p)) -> (~(distinct_lines zenon_TV_bg zenon_TV_bg)) -> (convergent_lines zenon_TU_bh zenon_TV_bg) -> False). % 3.45/3.63 do 4 intro. intros zenon_H5e zenon_H54 zenon_He zenon_H3c zenon_H3f zenon_H1d zenon_H1c zenon_H40 zenon_H2e. % 3.45/3.63 generalize (ceq2 (intersection_point zenon_TX_q zenon_TY_p)). zenon_intro zenon_H5f. % 3.45/3.63 generalize (zenon_H5f zenon_TY_p). zenon_intro zenon_H60. % 3.45/3.63 generalize (zenon_H5f zenon_TX_q). zenon_intro zenon_H57. % 3.45/3.63 generalize (zenon_H60 zenon_TY_p). zenon_intro zenon_H61. % 3.45/3.63 apply (zenon_imply_s _ _ zenon_H61); [ zenon_intro zenon_H19 | zenon_intro zenon_H62 ]. % 3.45/3.63 generalize (ceq1 (intersection_point zenon_TX_q zenon_TY_p)). zenon_intro zenon_H58. % 3.45/3.63 apply (zenon_L10_ zenon_TU_bh zenon_TV_bg zenon_TY_p zenon_TX_q); trivial. % 3.45/3.63 apply (zenon_or_s _ _ zenon_H62); [ zenon_intro zenon_H63 | zenon_intro zenon_H2a ]. % 3.45/3.63 exact (zenon_H5e zenon_H63). % 3.45/3.63 apply (zenon_L4_ zenon_TY_p zenon_TX_q); trivial. % 3.45/3.63 (* end of lemma zenon_L11_ *) % 3.45/3.63 apply NNPP. intro zenon_G. % 3.45/3.63 apply (zenon_notallex_s (fun X : zenon_U => (forall Y : zenon_U, (forall U : zenon_U, (forall V : zenon_U, (((convergent_lines X Y)/\((convergent_lines U V)/\((apart_point_and_line (intersection_point X Y) U)\/(apart_point_and_line (intersection_point X Y) V))))->((apart_point_and_line (intersection_point U V) X)\/(apart_point_and_line (intersection_point U V) Y))))))) zenon_G); [ zenon_intro zenon_H64; idtac ]. % 3.45/3.63 elim zenon_H64. zenon_intro zenon_TX_q. zenon_intro zenon_H65. % 3.45/3.63 apply (zenon_notallex_s (fun Y : zenon_U => (forall U : zenon_U, (forall V : zenon_U, (((convergent_lines zenon_TX_q Y)/\((convergent_lines U V)/\((apart_point_and_line (intersection_point zenon_TX_q Y) U)\/(apart_point_and_line (intersection_point zenon_TX_q Y) V))))->((apart_point_and_line (intersection_point U V) zenon_TX_q)\/(apart_point_and_line (intersection_point U V) Y)))))) zenon_H65); [ zenon_intro zenon_H66; idtac ]. % 3.45/3.63 elim zenon_H66. zenon_intro zenon_TY_p. zenon_intro zenon_H67. % 3.45/3.63 apply (zenon_notallex_s (fun U : zenon_U => (forall V : zenon_U, (((convergent_lines zenon_TX_q zenon_TY_p)/\((convergent_lines U V)/\((apart_point_and_line (intersection_point zenon_TX_q zenon_TY_p) U)\/(apart_point_and_line (intersection_point zenon_TX_q zenon_TY_p) V))))->((apart_point_and_line (intersection_point U V) zenon_TX_q)\/(apart_point_and_line (intersection_point U V) zenon_TY_p))))) zenon_H67); [ zenon_intro zenon_H68; idtac ]. % 3.45/3.63 elim zenon_H68. zenon_intro zenon_TU_bh. zenon_intro zenon_H69. % 3.45/3.63 apply (zenon_notallex_s (fun V : zenon_U => (((convergent_lines zenon_TX_q zenon_TY_p)/\((convergent_lines zenon_TU_bh V)/\((apart_point_and_line (intersection_point zenon_TX_q zenon_TY_p) zenon_TU_bh)\/(apart_point_and_line (intersection_point zenon_TX_q zenon_TY_p) V))))->((apart_point_and_line (intersection_point zenon_TU_bh V) zenon_TX_q)\/(apart_point_and_line (intersection_point zenon_TU_bh V) zenon_TY_p)))) zenon_H69); [ zenon_intro zenon_H6a; idtac ]. % 3.45/3.63 elim zenon_H6a. zenon_intro zenon_TV_bg. zenon_intro zenon_H6b. % 3.45/3.63 apply (zenon_notimply_s _ _ zenon_H6b). zenon_intro zenon_H6d. zenon_intro zenon_H6c. % 3.45/3.63 apply (zenon_notor_s _ _ zenon_H6c). zenon_intro zenon_H1d. zenon_intro zenon_H1c. % 3.45/3.63 apply (zenon_and_s _ _ zenon_H6d). zenon_intro zenon_He. zenon_intro zenon_H6e. % 3.45/3.63 apply (zenon_and_s _ _ zenon_H6e). zenon_intro zenon_H2e. zenon_intro zenon_H6f. % 3.45/3.63 apply (zenon_or_s _ _ zenon_H6f); [ zenon_intro zenon_H70 | zenon_intro zenon_H3f ]. % 3.45/3.63 generalize (apart2 zenon_TY_p). zenon_intro zenon_H5e. % 3.45/3.63 generalize (ceq2 (intersection_point zenon_TX_q zenon_TY_p)). zenon_intro zenon_H5f. % 3.45/3.63 generalize (ceq1 (intersection_point zenon_TX_q zenon_TY_p)). zenon_intro zenon_H58. % 3.45/3.63 generalize (zenon_H58 zenon_TU_bh). zenon_intro zenon_H71. % 3.45/3.63 generalize (apart5 zenon_TX_q). zenon_intro zenon_H72. % 3.45/3.63 generalize (zenon_H72 zenon_TY_p). zenon_intro zenon_H73. % 3.45/3.63 generalize (zenon_H73 zenon_TY_p). zenon_intro zenon_H74. % 3.45/3.63 apply (zenon_imply_s _ _ zenon_H74); [ zenon_intro zenon_H26 | zenon_intro zenon_H75 ]. % 3.45/3.63 generalize (ceq3 zenon_TX_q). zenon_intro zenon_H76. % 3.45/3.63 generalize (zenon_H76 zenon_TY_p). zenon_intro zenon_H77. % 3.45/3.63 apply (zenon_imply_s _ _ zenon_H77); [ zenon_intro zenon_H16 | zenon_intro zenon_H1e ]. % 3.45/3.63 exact (zenon_H16 zenon_He). % 3.45/3.63 exact (zenon_H26 zenon_H1e). % 3.45/3.63 apply (zenon_or_s _ _ zenon_H75); [ zenon_intro zenon_H1e | zenon_intro zenon_H63 ]. % 3.45/3.63 generalize (zenon_H5f zenon_TY_p). zenon_intro zenon_H60. % 3.45/3.63 generalize (zenon_H60 zenon_TY_p). zenon_intro zenon_H61. % 3.45/3.63 apply (zenon_imply_s _ _ zenon_H61); [ zenon_intro zenon_H19 | zenon_intro zenon_H62 ]. % 3.45/3.63 generalize (apart2 zenon_TU_bh). zenon_intro zenon_H78. % 3.45/3.63 generalize (ceq2 (intersection_point zenon_TU_bh zenon_TV_bg)). zenon_intro zenon_H44. % 3.45/3.63 generalize (zenon_H44 zenon_TU_bh). zenon_intro zenon_H79. % 3.45/3.63 generalize (zenon_H79 zenon_TU_bh). zenon_intro zenon_H7a. % 3.45/3.63 apply (zenon_imply_s _ _ zenon_H7a); [ zenon_intro zenon_H37 | zenon_intro zenon_H7b ]. % 3.45/3.63 generalize (zenon_H71 (intersection_point zenon_TU_bh zenon_TV_bg)). zenon_intro zenon_H7c. % 3.45/3.63 apply (zenon_imply_s _ _ zenon_H7c); [ zenon_intro zenon_H7e | zenon_intro zenon_H7d ]. % 3.45/3.63 exact (zenon_H7e zenon_H70). % 3.45/3.63 apply (zenon_or_s _ _ zenon_H7d); [ zenon_intro zenon_H1f | zenon_intro zenon_H2f ]. % 3.45/3.63 generalize (cu1 (intersection_point zenon_TX_q zenon_TY_p)). zenon_intro zenon_H4b. % 3.45/3.63 generalize (zenon_H4b (intersection_point zenon_TU_bh zenon_TV_bg)). zenon_intro zenon_H1b. % 3.45/3.63 apply (zenon_L2_ zenon_TV_bg zenon_TU_bh zenon_TY_p zenon_TX_q); trivial. % 3.45/3.63 exact (zenon_H37 zenon_H2f). % 3.45/3.63 apply (zenon_or_s _ _ zenon_H7b); [ zenon_intro zenon_H7f | zenon_intro zenon_H2f ]. % 3.45/3.63 exact (zenon_H78 zenon_H7f). % 3.45/3.63 generalize (con2 zenon_TU_bh). zenon_intro zenon_H3b. % 3.45/3.63 generalize (zenon_H3b zenon_TV_bg). zenon_intro zenon_H2d. % 3.45/3.63 generalize (apart1 (intersection_point zenon_TU_bh zenon_TV_bg)). zenon_intro zenon_H30. % 3.45/3.63 apply (zenon_L3_ zenon_TV_bg zenon_TU_bh); trivial. % 3.45/3.63 apply (zenon_or_s _ _ zenon_H62); [ zenon_intro zenon_H63 | zenon_intro zenon_H2a ]. % 3.45/3.63 exact (zenon_H5e zenon_H63). % 3.45/3.63 apply (zenon_L4_ zenon_TY_p zenon_TX_q); trivial. % 3.45/3.63 exact (zenon_H5e zenon_H63). % 3.45/3.63 generalize (apart2 zenon_TY_p). zenon_intro zenon_H5e. % 3.45/3.63 generalize (apart2 zenon_TV_bg). zenon_intro zenon_H40. % 3.45/3.63 generalize (apart6 zenon_TX_q). zenon_intro zenon_H80. % 3.45/3.63 generalize (zenon_H80 zenon_TY_p). zenon_intro zenon_H54. % 3.45/3.63 generalize (ceq3 zenon_TY_p). zenon_intro zenon_H3c. % 3.45/3.63 apply (zenon_L11_ zenon_TU_bh zenon_TV_bg zenon_TX_q zenon_TY_p); trivial. % 3.45/3.63 Qed. % 3.45/3.63 % SZS output end Proof % 3.45/3.63 (* END-PROOF *) % 3.45/3.63 nodes searched: 362911 % 3.45/3.63 max branch formulas: 5002 % 3.45/3.63 proof nodes created: 6705 % 3.45/3.63 formulas created: 430154 % 3.45/3.63 %------------------------------------------------------------------------------