%------------------------------------------------------------------------------ % File : SuperZenon---0.0.1 % Problem : SWC203+1 : TPTP v8.1.0. Released v2.4.0. % Transfm : none % Format : tptp:raw % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % Computer : n014.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 : Tue Jul 19 22:07:05 EDT 2022 % Result : Theorem 48.40s 48.68s % Output : Proof 48.70s % Verified : % SZS Type : - % Comments : %------------------------------------------------------------------------------ %----WARNING: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % 0.11/0.12 % Problem : SWC203+1 : TPTP v8.1.0. Released v2.4.0. % 0.11/0.12 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s % 0.13/0.33 % Computer : n014.cluster.edu % 0.13/0.33 % Model : x86_64 x86_64 % 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz % 0.13/0.33 % Memory : 8042.1875MB % 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64 % 0.13/0.33 % CPULimit : 300 % 0.13/0.33 % WCLimit : 600 % 0.13/0.33 % DateTime : Sun Jun 12 17:10:22 EDT 2022 % 0.13/0.33 % CPUTime : % 48.40/48.68 % SZS status Theorem % 48.40/48.68 (* PROOF-FOUND *) % 48.40/48.68 (* BEGIN-PROOF *) % 48.40/48.68 % SZS output start Proof % 48.40/48.68 1. (ssList T_0) (-. (ssList T_0)) ### Axiom % 48.40/48.68 2. (ssList T_1) (-. (ssList T_1)) ### Axiom % 48.40/48.68 3. (ssList T_2) (-. (ssList T_2)) ### Axiom % 48.40/48.68 4. (ssList T_0) (-. (ssList T_0)) ### Axiom % 48.40/48.68 5. (ssList (nil)) (-. (ssList (nil))) ### Axiom % 48.40/48.68 6. (ssList T_2) (-. (ssList T_2)) ### Axiom % 48.40/48.68 7. (-. (frontsegP T_2 (nil))) (frontsegP T_2 (nil)) ### Axiom % 48.40/48.68 8. ((ssList T_2) => (frontsegP T_2 (nil))) (-. (frontsegP T_2 (nil))) (ssList T_2) ### Imply 6 7 % 48.40/48.68 9. (All U, ((ssList U) => (frontsegP U (nil)))) (ssList T_2) (-. (frontsegP T_2 (nil))) ### All 8 % 48.40/48.68 10. (ssList T_2) (-. (ssList T_2)) ### Axiom % 48.40/48.68 11. (ssList T_0) (-. (ssList T_0)) ### Axiom % 48.40/48.68 12. (ssList (nil)) (-. (ssList (nil))) ### Axiom % 48.40/48.68 13. ((nil) != T_0) (T_0 = (nil)) ### Sym(=) % 48.40/48.68 14. (-. (T_0 != (nil))) ((nil) != T_0) ### NotNot 13 % 48.40/48.68 15. (-. (neq T_0 (nil))) (neq T_0 (nil)) ### Axiom % 48.40/48.68 16. ((neq T_0 (nil)) <=> (T_0 != (nil))) (-. (neq T_0 (nil))) ((nil) != T_0) ### Equiv 14 15 % 48.40/48.68 17. ((ssList (nil)) => ((neq T_0 (nil)) <=> (T_0 != (nil)))) ((nil) != T_0) (-. (neq T_0 (nil))) (ssList (nil)) ### Imply 12 16 % 48.40/48.68 18. (All V, ((ssList V) => ((neq T_0 V) <=> (T_0 != V)))) (ssList (nil)) (-. (neq T_0 (nil))) ((nil) != T_0) ### All 17 % 48.40/48.68 19. (ssList T_2) (-. (ssList T_2)) ### Axiom % 48.40/48.68 20. (ssList T_2) (-. (ssList T_2)) ### Axiom % 48.40/48.68 21. ((nil) = T_0) ((nil) != T_0) ### Axiom % 48.40/48.68 22. (-. (T_2 != T_2)) (T_2 != T_2) ### Axiom % 48.40/48.68 23. ((nil) != T_2) (T_0 = T_2) (-. (T_2 != T_2)) ((nil) = T_0) ### Trans 21 22 % 48.40/48.68 24. (T_2 != T_2) ### Refl(=) % 48.40/48.68 25. ((neq T_2 T_2) <=> (T_2 != T_2)) ((nil) = T_0) (T_0 = T_2) ((nil) != T_2) ### Equiv 23 24 % 48.40/48.68 26. ((ssList T_2) => ((neq T_2 T_2) <=> (T_2 != T_2))) ((nil) != T_2) (T_0 = T_2) ((nil) = T_0) (ssList T_2) ### Imply 20 25 % 48.40/48.68 27. (All V, ((ssList V) => ((neq T_2 V) <=> (T_2 != V)))) (ssList T_2) ((nil) = T_0) (T_0 = T_2) ((nil) != T_2) ### All 26 % 48.40/48.68 28. ((ssList T_2) => (All V, ((ssList V) => ((neq T_2 V) <=> (T_2 != V))))) ((nil) != T_2) (T_0 = T_2) ((nil) = T_0) (ssList T_2) ### Imply 19 27 % 48.40/48.68 29. (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (ssList T_2) ((nil) = T_0) (T_0 = T_2) ((nil) != T_2) ### All 28 % 48.40/48.68 30. ((segmentP (nil) T_0) <=> ((nil) = T_0)) ((nil) != T_2) (T_0 = T_2) (ssList T_2) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (-. (neq T_0 (nil))) (ssList (nil)) (All V, ((ssList V) => ((neq T_0 V) <=> (T_0 != V)))) ### Equiv 18 29 % 48.40/48.68 31. ((ssList T_0) => ((segmentP (nil) T_0) <=> ((nil) = T_0))) (All V, ((ssList V) => ((neq T_0 V) <=> (T_0 != V)))) (ssList (nil)) (-. (neq T_0 (nil))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (ssList T_2) (T_0 = T_2) ((nil) != T_2) (ssList T_0) ### Imply 11 30 % 48.40/48.68 32. (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (ssList T_0) ((nil) != T_2) (T_0 = T_2) (ssList T_2) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (-. (neq T_0 (nil))) (ssList (nil)) (All V, ((ssList V) => ((neq T_0 V) <=> (T_0 != V)))) ### All 31 % 48.40/48.68 33. (ssList T_2) (-. (ssList T_2)) ### Axiom % 48.40/48.68 34. ((nil) = T_2) ((nil) != T_2) ### Axiom % 48.40/48.68 35. (-. (frontsegP (nil) T_2)) (frontsegP (nil) T_2) ### Axiom % 48.40/48.68 36. ((frontsegP (nil) T_2) <=> ((nil) = T_2)) (-. (frontsegP (nil) T_2)) ((nil) = T_2) ### Equiv 34 35 % 48.40/48.68 37. ((ssList T_2) => ((frontsegP (nil) T_2) <=> ((nil) = T_2))) ((nil) = T_2) (-. (frontsegP (nil) T_2)) (ssList T_2) ### Imply 33 36 % 48.40/48.68 38. (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (ssList T_2) (-. (frontsegP (nil) T_2)) ((nil) = T_2) ### All 37 % 48.40/48.68 39. ((rearsegP (nil) T_2) <=> ((nil) = T_2)) (-. (frontsegP (nil) T_2)) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All V, ((ssList V) => ((neq T_0 V) <=> (T_0 != V)))) (ssList (nil)) (-. (neq T_0 (nil))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (ssList T_2) (T_0 = T_2) (ssList T_0) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) ### Equiv 32 38 % 48.40/48.68 40. ((ssList T_2) => ((rearsegP (nil) T_2) <=> ((nil) = T_2))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (ssList T_0) (T_0 = T_2) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (-. (neq T_0 (nil))) (ssList (nil)) (All V, ((ssList V) => ((neq T_0 V) <=> (T_0 != V)))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (-. (frontsegP (nil) T_2)) (ssList T_2) ### Imply 10 39 % 48.40/48.68 41. (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_2) (-. (frontsegP (nil) T_2)) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All V, ((ssList V) => ((neq T_0 V) <=> (T_0 != V)))) (ssList (nil)) (-. (neq T_0 (nil))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (T_0 = T_2) (ssList T_0) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) ### All 40 % 48.40/48.68 42. (ssList T_1) (-. (ssList T_1)) ### Axiom % 48.40/48.68 43. (ssList (nil)) (-. (ssList (nil))) ### Axiom % 48.40/48.68 44. (ssList T_1) (-. (ssList T_1)) ### Axiom % 48.40/48.68 45. (-. (segmentP T_1 (nil))) (segmentP T_1 (nil)) ### Axiom % 48.40/48.68 46. ((ssList T_1) => (segmentP T_1 (nil))) (-. (segmentP T_1 (nil))) (ssList T_1) ### Imply 44 45 % 48.40/48.68 47. (All U, ((ssList U) => (segmentP U (nil)))) (ssList T_1) (-. (segmentP T_1 (nil))) ### All 46 % 48.40/48.68 48. (ssList T_1) (-. (ssList T_1)) ### Axiom % 48.40/48.68 49. (T_1 = T_2) (T_1 != T_2) ### Axiom % 48.40/48.68 50. ((nil) != (nil)) ### NotEqual % 48.40/48.68 51. ((nil) != T_1) (T_2 = (nil)) (T_1 = T_2) ### Trans-sym 49 50 % 48.40/48.68 52. (-. (segmentP (nil) T_1)) (segmentP (nil) T_1) ### Axiom % 48.40/48.68 53. ((segmentP (nil) T_1) <=> ((nil) = T_1)) (-. (segmentP (nil) T_1)) (T_1 = T_2) (T_2 = (nil)) ### Equiv 51 52 % 48.40/48.68 54. ((ssList T_1) => ((segmentP (nil) T_1) <=> ((nil) = T_1))) (T_2 = (nil)) (T_1 = T_2) (-. (segmentP (nil) T_1)) (ssList T_1) ### Imply 48 53 % 48.40/48.68 55. (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (ssList T_1) (-. (segmentP (nil) T_1)) (T_1 = T_2) (T_2 = (nil)) ### All 54 % 48.40/48.68 56. (ssList T_2) (-. (ssList T_2)) ### Axiom % 48.40/48.68 57. (ssList T_0) (-. (ssList T_0)) ### Axiom % 48.40/48.68 58. (ssList T_0) (-. (ssList T_0)) ### Axiom % 48.40/48.68 59. ((nil) = T_0) ((nil) != T_0) ### Axiom % 48.40/48.68 60. (ssList T_1) (-. (ssList T_1)) ### Axiom % 48.40/48.68 61. (ssList T_0) (-. (ssList T_0)) ### Axiom % 48.40/48.68 62. (ssList T_0) (-. (ssList T_0)) ### Axiom % 48.40/48.68 63. (ssList T_1) (-. (ssList T_1)) ### Axiom % 48.40/48.68 64. (ssList (nil)) (-. (ssList (nil))) ### Axiom % 48.40/48.68 65. (ssList T_0) (-. (ssList T_0)) ### Axiom % 48.40/48.68 66. (-. (segmentP T_0 (nil))) (segmentP T_0 (nil)) ### Axiom % 48.40/48.68 67. ((ssList T_0) => (segmentP T_0 (nil))) (-. (segmentP T_0 (nil))) (ssList T_0) ### Imply 65 66 % 48.40/48.68 68. (All U, ((ssList U) => (segmentP U (nil)))) (ssList T_0) (-. (segmentP T_0 (nil))) ### All 67 % 48.40/48.68 69. (T_1 = (nil)) ((nil) != T_1) ### Sym(=) % 48.40/48.68 70. (ssList (nil)) (-. (ssList (nil))) ### Axiom % 48.40/48.68 71. (ssList T_1) (-. (ssList T_1)) ### Axiom % 48.40/48.68 72. (segmentP T_0 (nil)) (-. (segmentP T_0 (nil))) ### Axiom % 48.40/48.68 73. (segmentP (nil) T_1) (-. (segmentP (nil) T_1)) ### Axiom % 48.40/48.68 74. (-. (segmentP T_0 T_1)) (segmentP T_0 T_1) ### Axiom % 48.40/48.68 75. ((ssList T_1) => (((segmentP T_0 (nil)) /\ (segmentP (nil) T_1)) => (segmentP T_0 T_1))) (-. (segmentP T_0 T_1)) (segmentP (nil) T_1) (segmentP T_0 (nil)) (ssList T_1) ### DisjTree 71 72 73 74 % 48.40/48.68 76. (All W, ((ssList W) => (((segmentP T_0 (nil)) /\ (segmentP (nil) W)) => (segmentP T_0 W)))) (ssList T_1) (segmentP T_0 (nil)) (segmentP (nil) T_1) (-. (segmentP T_0 T_1)) ### All 75 % 48.40/48.68 77. ((ssList (nil)) => (All W, ((ssList W) => (((segmentP T_0 (nil)) /\ (segmentP (nil) W)) => (segmentP T_0 W))))) (-. (segmentP T_0 T_1)) (segmentP (nil) T_1) (segmentP T_0 (nil)) (ssList T_1) (ssList (nil)) ### Imply 70 76 % 48.40/48.68 78. (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_0 V) /\ (segmentP V W)) => (segmentP T_0 W)))))) (ssList (nil)) (ssList T_1) (segmentP T_0 (nil)) (segmentP (nil) T_1) (-. (segmentP T_0 T_1)) ### All 77 % 48.40/48.68 79. ((segmentP (nil) T_1) <=> ((nil) = T_1)) (-. (segmentP T_0 T_1)) (segmentP T_0 (nil)) (ssList T_1) (ssList (nil)) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_0 V) /\ (segmentP V W)) => (segmentP T_0 W)))))) (T_1 = (nil)) ### Equiv 69 78 % 48.40/48.68 80. ((segmentP T_0 (nil)) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W (nil)) X) = T_0)))))) (T_1 = (nil)) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_0 V) /\ (segmentP V W)) => (segmentP T_0 W)))))) (ssList (nil)) (ssList T_1) (-. (segmentP T_0 T_1)) ((segmentP (nil) T_1) <=> ((nil) = T_1)) (ssList T_0) (All U, ((ssList U) => (segmentP U (nil)))) ### Equiv 68 79 % 48.70/48.90 81. ((ssList (nil)) => ((segmentP T_0 (nil)) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W (nil)) X) = T_0))))))) (All U, ((ssList U) => (segmentP U (nil)))) (ssList T_0) ((segmentP (nil) T_1) <=> ((nil) = T_1)) (-. (segmentP T_0 T_1)) (ssList T_1) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_0 V) /\ (segmentP V W)) => (segmentP T_0 W)))))) (T_1 = (nil)) (ssList (nil)) ### Imply 64 80 % 48.70/48.90 82. (All V, ((ssList V) => ((segmentP T_0 V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = T_0)))))))) (ssList (nil)) (T_1 = (nil)) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_0 V) /\ (segmentP V W)) => (segmentP T_0 W)))))) (ssList T_1) (-. (segmentP T_0 T_1)) ((segmentP (nil) T_1) <=> ((nil) = T_1)) (ssList T_0) (All U, ((ssList U) => (segmentP U (nil)))) ### All 81 % 48.70/48.90 83. ((ssList T_1) => ((segmentP (nil) T_1) <=> ((nil) = T_1))) (All U, ((ssList U) => (segmentP U (nil)))) (ssList T_0) (-. (segmentP T_0 T_1)) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_0 V) /\ (segmentP V W)) => (segmentP T_0 W)))))) (T_1 = (nil)) (ssList (nil)) (All V, ((ssList V) => ((segmentP T_0 V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = T_0)))))))) (ssList T_1) ### Imply 63 82 % 48.70/48.90 84. (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (ssList T_1) (All V, ((ssList V) => ((segmentP T_0 V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = T_0)))))))) (ssList (nil)) (T_1 = (nil)) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_0 V) /\ (segmentP V W)) => (segmentP T_0 W)))))) (-. (segmentP T_0 T_1)) (ssList T_0) (All U, ((ssList U) => (segmentP U (nil)))) ### All 83 % 48.70/48.90 85. ((ssList T_0) => (All V, ((ssList V) => ((segmentP T_0 V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = T_0))))))))) (All U, ((ssList U) => (segmentP U (nil)))) (-. (segmentP T_0 T_1)) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_0 V) /\ (segmentP V W)) => (segmentP T_0 W)))))) (T_1 = (nil)) (ssList (nil)) (ssList T_1) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (ssList T_0) ### Imply 62 84 % 48.70/48.90 86. (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (ssList T_0) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (ssList T_1) (ssList (nil)) (T_1 = (nil)) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_0 V) /\ (segmentP V W)) => (segmentP T_0 W)))))) (-. (segmentP T_0 T_1)) (All U, ((ssList U) => (segmentP U (nil)))) ### All 85 % 48.70/48.90 87. ((ssList T_0) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_0 V) /\ (segmentP V W)) => (segmentP T_0 W))))))) (All U, ((ssList U) => (segmentP U (nil)))) (-. (segmentP T_0 T_1)) (T_1 = (nil)) (ssList (nil)) (ssList T_1) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (ssList T_0) ### Imply 61 86 % 48.70/48.90 88. (ssList T_2) (-. (ssList T_2)) ### Axiom % 48.70/48.90 89. (T_2 = (nil)) ((nil) != T_2) ### Sym(=) % 48.70/48.90 90. (ssList T_2) (-. (ssList T_2)) ### Axiom % 48.70/48.90 91. (ssList T_1) (-. (ssList T_1)) ### Axiom % 48.70/48.90 92. (ssList (nil)) (-. (ssList (nil))) ### Axiom % 48.70/48.90 93. (ssList (nil)) (-. (ssList (nil))) ### Axiom % 48.70/48.90 94. (ssList T_0) (-. (ssList T_0)) ### Axiom % 48.70/48.90 95. (ssList T_0) (-. (ssList T_0)) ### Axiom % 48.70/48.90 96. (ssList T_2) (-. (ssList T_2)) ### Axiom % 48.70/48.90 97. (-. (segmentP T_2 (nil))) (segmentP T_2 (nil)) ### Axiom % 48.70/48.90 98. ((ssList T_2) => (segmentP T_2 (nil))) (-. (segmentP T_2 (nil))) (ssList T_2) ### Imply 96 97 % 48.70/48.90 99. (All U, ((ssList U) => (segmentP U (nil)))) (ssList T_2) (-. (segmentP T_2 (nil))) ### All 98 % 48.70/48.90 100. (segmentP (nil) T_0) (-. (segmentP (nil) T_0)) ### Axiom % 48.70/48.90 101. (-. (segmentP T_2 T_0)) (segmentP T_2 T_0) ### Axiom % 48.70/48.90 102. ((ssList T_0) => (((segmentP T_2 (nil)) /\ (segmentP (nil) T_0)) => (segmentP T_2 T_0))) (-. (segmentP T_2 T_0)) (segmentP (nil) T_0) (ssList T_2) (All U, ((ssList U) => (segmentP U (nil)))) (ssList T_0) ### DisjTree 95 99 100 101 % 48.70/48.90 103. (All W, ((ssList W) => (((segmentP T_2 (nil)) /\ (segmentP (nil) W)) => (segmentP T_2 W)))) (ssList T_0) (All U, ((ssList U) => (segmentP U (nil)))) (ssList T_2) (segmentP (nil) T_0) (-. (segmentP T_2 T_0)) ### All 102 % 48.70/48.90 104. (ssList T_2) (-. (ssList T_2)) ### Axiom % 48.70/48.90 105. (ssList (nil)) (-. (ssList (nil))) ### Axiom % 48.70/48.90 106. (ssList T_0) (-. (ssList T_0)) ### Axiom % 48.70/48.90 107. (ssList T_2) (-. (ssList T_2)) ### Axiom % 48.70/48.90 108. (segmentP T_1 (nil)) (-. (segmentP T_1 (nil))) ### Axiom % 48.70/48.90 109. (segmentP (nil) T_2) (-. (segmentP (nil) T_2)) ### Axiom % 48.70/48.90 110. (-. (segmentP T_1 T_2)) (segmentP T_1 T_2) ### Axiom % 48.70/48.90 111. ((ssList T_2) => (((segmentP T_1 (nil)) /\ (segmentP (nil) T_2)) => (segmentP T_1 T_2))) (-. (segmentP T_1 T_2)) (segmentP (nil) T_2) (segmentP T_1 (nil)) (ssList T_2) ### DisjTree 107 108 109 110 % 48.70/48.90 112. (All W, ((ssList W) => (((segmentP T_1 (nil)) /\ (segmentP (nil) W)) => (segmentP T_1 W)))) (ssList T_2) (segmentP T_1 (nil)) (segmentP (nil) T_2) (-. (segmentP T_1 T_2)) ### All 111 % 48.70/48.90 113. (segmentP T_2 T_0) (-. (segmentP T_2 T_0)) ### Axiom % 48.70/48.90 114. (-. (segmentP T_1 T_0)) (segmentP T_1 T_0) ### Axiom % 48.70/48.90 115. ((ssList T_0) => (((segmentP T_1 T_2) /\ (segmentP T_2 T_0)) => (segmentP T_1 T_0))) (-. (segmentP T_1 T_0)) (segmentP T_2 T_0) (segmentP (nil) T_2) (segmentP T_1 (nil)) (ssList T_2) (All W, ((ssList W) => (((segmentP T_1 (nil)) /\ (segmentP (nil) W)) => (segmentP T_1 W)))) (ssList T_0) ### DisjTree 106 112 113 114 % 48.70/48.90 116. (All W, ((ssList W) => (((segmentP T_1 T_2) /\ (segmentP T_2 W)) => (segmentP T_1 W)))) (ssList T_0) (All W, ((ssList W) => (((segmentP T_1 (nil)) /\ (segmentP (nil) W)) => (segmentP T_1 W)))) (ssList T_2) (segmentP T_1 (nil)) (segmentP (nil) T_2) (segmentP T_2 T_0) (-. (segmentP T_1 T_0)) ### All 115 % 48.70/48.90 117. ((ssList (nil)) => (All W, ((ssList W) => (((segmentP T_1 (nil)) /\ (segmentP (nil) W)) => (segmentP T_1 W))))) (-. (segmentP T_1 T_0)) (segmentP T_2 T_0) (segmentP (nil) T_2) (segmentP T_1 (nil)) (ssList T_2) (ssList T_0) (All W, ((ssList W) => (((segmentP T_1 T_2) /\ (segmentP T_2 W)) => (segmentP T_1 W)))) (ssList (nil)) ### Imply 105 116 % 48.70/48.90 118. (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (ssList (nil)) (All W, ((ssList W) => (((segmentP T_1 T_2) /\ (segmentP T_2 W)) => (segmentP T_1 W)))) (ssList T_0) (ssList T_2) (segmentP T_1 (nil)) (segmentP (nil) T_2) (segmentP T_2 T_0) (-. (segmentP T_1 T_0)) ### All 117 % 48.70/48.90 119. ((ssList T_2) => (All W, ((ssList W) => (((segmentP T_1 T_2) /\ (segmentP T_2 W)) => (segmentP T_1 W))))) (-. (segmentP T_1 T_0)) (segmentP T_2 T_0) (segmentP (nil) T_2) (segmentP T_1 (nil)) (ssList T_0) (ssList (nil)) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (ssList T_2) ### Imply 104 118 % 48.70/48.90 120. (ssList T_2) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (ssList (nil)) (ssList T_0) (segmentP T_1 (nil)) (segmentP (nil) T_2) (segmentP T_2 T_0) (-. (segmentP T_1 T_0)) ### All 119 % 48.70/48.90 121. ((segmentP T_2 T_0) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W T_0) X) = T_2)))))) (-. (segmentP T_1 T_0)) (segmentP (nil) T_2) (segmentP T_1 (nil)) (ssList (nil)) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (segmentP (nil) T_0) (ssList T_2) (All U, ((ssList U) => (segmentP U (nil)))) (ssList T_0) (All W, ((ssList W) => (((segmentP T_2 (nil)) /\ (segmentP (nil) W)) => (segmentP T_2 W)))) ### Equiv 103 120 % 48.70/48.90 122. ((ssList T_0) => ((segmentP T_2 T_0) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W T_0) X) = T_2))))))) (All W, ((ssList W) => (((segmentP T_2 (nil)) /\ (segmentP (nil) W)) => (segmentP T_2 W)))) (All U, ((ssList U) => (segmentP U (nil)))) (ssList T_2) (segmentP (nil) T_0) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (ssList (nil)) (segmentP T_1 (nil)) (segmentP (nil) T_2) (-. (segmentP T_1 T_0)) (ssList T_0) ### Imply 94 121 % 48.70/48.92 123. (All V, ((ssList V) => ((segmentP T_2 V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = T_2)))))))) (ssList T_0) (-. (segmentP T_1 T_0)) (segmentP (nil) T_2) (segmentP T_1 (nil)) (ssList (nil)) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (segmentP (nil) T_0) (ssList T_2) (All U, ((ssList U) => (segmentP U (nil)))) (All W, ((ssList W) => (((segmentP T_2 (nil)) /\ (segmentP (nil) W)) => (segmentP T_2 W)))) ### All 122 % 48.70/48.92 124. ((ssList (nil)) => (All W, ((ssList W) => (((segmentP T_2 (nil)) /\ (segmentP (nil) W)) => (segmentP T_2 W))))) (All U, ((ssList U) => (segmentP U (nil)))) (ssList T_2) (segmentP (nil) T_0) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (segmentP T_1 (nil)) (segmentP (nil) T_2) (-. (segmentP T_1 T_0)) (ssList T_0) (All V, ((ssList V) => ((segmentP T_2 V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = T_2)))))))) (ssList (nil)) ### Imply 93 123 % 48.70/48.92 125. (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_2 V) /\ (segmentP V W)) => (segmentP T_2 W)))))) (ssList (nil)) (All V, ((ssList V) => ((segmentP T_2 V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = T_2)))))))) (ssList T_0) (-. (segmentP T_1 T_0)) (segmentP (nil) T_2) (segmentP T_1 (nil)) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (segmentP (nil) T_0) (ssList T_2) (All U, ((ssList U) => (segmentP U (nil)))) ### All 124 % 48.70/48.92 126. ((segmentP T_1 (nil)) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W (nil)) X) = T_1)))))) (ssList T_2) (segmentP (nil) T_0) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (segmentP (nil) T_2) (-. (segmentP T_1 T_0)) (ssList T_0) (All V, ((ssList V) => ((segmentP T_2 V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = T_2)))))))) (ssList (nil)) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_2 V) /\ (segmentP V W)) => (segmentP T_2 W)))))) (ssList T_1) (All U, ((ssList U) => (segmentP U (nil)))) ### Equiv 47 125 % 48.70/48.92 127. ((ssList (nil)) => ((segmentP T_1 (nil)) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W (nil)) X) = T_1))))))) (All U, ((ssList U) => (segmentP U (nil)))) (ssList T_1) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_2 V) /\ (segmentP V W)) => (segmentP T_2 W)))))) (All V, ((ssList V) => ((segmentP T_2 V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = T_2)))))))) (ssList T_0) (-. (segmentP T_1 T_0)) (segmentP (nil) T_2) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (segmentP (nil) T_0) (ssList T_2) (ssList (nil)) ### Imply 92 126 % 48.70/48.92 128. (All V, ((ssList V) => ((segmentP T_1 V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = T_1)))))))) (ssList (nil)) (ssList T_2) (segmentP (nil) T_0) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (segmentP (nil) T_2) (-. (segmentP T_1 T_0)) (ssList T_0) (All V, ((ssList V) => ((segmentP T_2 V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = T_2)))))))) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_2 V) /\ (segmentP V W)) => (segmentP T_2 W)))))) (ssList T_1) (All U, ((ssList U) => (segmentP U (nil)))) ### All 127 % 48.70/48.92 129. ((ssList T_1) => (All V, ((ssList V) => ((segmentP T_1 V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = T_1))))))))) (All U, ((ssList U) => (segmentP U (nil)))) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_2 V) /\ (segmentP V W)) => (segmentP T_2 W)))))) (All V, ((ssList V) => ((segmentP T_2 V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = T_2)))))))) (ssList T_0) (-. (segmentP T_1 T_0)) (segmentP (nil) T_2) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (segmentP (nil) T_0) (ssList T_2) (ssList (nil)) (ssList T_1) ### Imply 91 128 % 48.70/48.92 130. (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (ssList T_1) (ssList (nil)) (ssList T_2) (segmentP (nil) T_0) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (segmentP (nil) T_2) (-. (segmentP T_1 T_0)) (ssList T_0) (All V, ((ssList V) => ((segmentP T_2 V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = T_2)))))))) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_2 V) /\ (segmentP V W)) => (segmentP T_2 W)))))) (All U, ((ssList U) => (segmentP U (nil)))) ### All 129 % 48.70/48.92 131. ((ssList T_2) => (All V, ((ssList V) => ((segmentP T_2 V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = T_2))))))))) (All U, ((ssList U) => (segmentP U (nil)))) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_2 V) /\ (segmentP V W)) => (segmentP T_2 W)))))) (ssList T_0) (-. (segmentP T_1 T_0)) (segmentP (nil) T_2) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (segmentP (nil) T_0) (ssList (nil)) (ssList T_1) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (ssList T_2) ### Imply 90 130 % 48.70/48.92 132. (ssList T_2) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (ssList T_1) (ssList (nil)) (segmentP (nil) T_0) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (segmentP (nil) T_2) (-. (segmentP T_1 T_0)) (ssList T_0) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_2 V) /\ (segmentP V W)) => (segmentP T_2 W)))))) (All U, ((ssList U) => (segmentP U (nil)))) ### All 131 % 48.70/48.92 133. ((segmentP (nil) T_2) <=> ((nil) = T_2)) (All U, ((ssList U) => (segmentP U (nil)))) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_2 V) /\ (segmentP V W)) => (segmentP T_2 W)))))) (ssList T_0) (-. (segmentP T_1 T_0)) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (segmentP (nil) T_0) (ssList (nil)) (ssList T_1) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (ssList T_2) (T_2 = (nil)) ### Equiv 89 132 % 48.70/48.92 134. ((ssList T_2) => ((segmentP (nil) T_2) <=> ((nil) = T_2))) (T_2 = (nil)) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (ssList T_1) (ssList (nil)) (segmentP (nil) T_0) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (-. (segmentP T_1 T_0)) (ssList T_0) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_2 V) /\ (segmentP V W)) => (segmentP T_2 W)))))) (All U, ((ssList U) => (segmentP U (nil)))) (ssList T_2) ### Imply 88 133 % 48.70/48.92 135. (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (ssList T_2) (All U, ((ssList U) => (segmentP U (nil)))) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_2 V) /\ (segmentP V W)) => (segmentP T_2 W)))))) (ssList T_0) (-. (segmentP T_1 T_0)) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (segmentP (nil) T_0) (ssList (nil)) (ssList T_1) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (T_2 = (nil)) ### All 134 % 48.70/48.93 136. (T_3 != T_3) ### Refl(=) % 48.70/48.93 137. (T_0 = T_1) (T_1 != T_0) ### Sym(=) % 48.70/48.93 138. (T_3 != T_0) (T_3 = T_1) (T_0 = T_1) ### Trans 136 137 % 48.70/48.93 139. ((ssList T_1) => (((segmentP T_0 T_1) /\ (segmentP T_1 T_0)) => (T_0 = T_1))) (T_3 = T_1) (T_3 != T_0) (T_2 = (nil)) (segmentP (nil) T_0) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_2 V) /\ (segmentP V W)) => (segmentP T_2 W)))))) (ssList T_2) (ssList T_0) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (ssList (nil)) (T_1 = (nil)) (All U, ((ssList U) => (segmentP U (nil)))) ((ssList T_0) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_0 V) /\ (segmentP V W)) => (segmentP T_0 W))))))) (ssList T_1) ### DisjTree 60 87 135 138 % 48.70/48.93 140. (All V, ((ssList V) => (((segmentP T_0 V) /\ (segmentP V T_0)) => (T_0 = V)))) (ssList T_1) ((ssList T_0) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_0 V) /\ (segmentP V W)) => (segmentP T_0 W))))))) (All U, ((ssList U) => (segmentP U (nil)))) (T_1 = (nil)) (ssList (nil)) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (ssList T_0) (ssList T_2) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_2 V) /\ (segmentP V W)) => (segmentP T_2 W)))))) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (segmentP (nil) T_0) (T_2 = (nil)) (T_3 != T_0) (T_3 = T_1) ### All 139 % 48.70/48.93 141. ((segmentP (nil) T_0) <=> ((nil) = T_0)) (T_3 = T_1) (T_3 != T_0) (T_2 = (nil)) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_2 V) /\ (segmentP V W)) => (segmentP T_2 W)))))) (ssList T_2) (ssList T_0) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (ssList (nil)) (T_1 = (nil)) (All U, ((ssList U) => (segmentP U (nil)))) ((ssList T_0) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_0 V) /\ (segmentP V W)) => (segmentP T_0 W))))))) (ssList T_1) (All V, ((ssList V) => (((segmentP T_0 V) /\ (segmentP V T_0)) => (T_0 = V)))) ((nil) = T_0) ### Equiv 59 140 % 48.70/48.93 142. ((ssList T_0) => ((segmentP (nil) T_0) <=> ((nil) = T_0))) ((nil) = T_0) (All V, ((ssList V) => (((segmentP T_0 V) /\ (segmentP V T_0)) => (T_0 = V)))) (ssList T_1) ((ssList T_0) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_0 V) /\ (segmentP V W)) => (segmentP T_0 W))))))) (All U, ((ssList U) => (segmentP U (nil)))) (T_1 = (nil)) (ssList (nil)) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (ssList T_2) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_2 V) /\ (segmentP V W)) => (segmentP T_2 W)))))) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (T_2 = (nil)) (T_3 != T_0) (T_3 = T_1) (ssList T_0) ### Imply 58 141 % 48.70/48.93 143. (ssList T_0) (T_3 = T_1) (T_3 != T_0) (T_2 = (nil)) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_2 V) /\ (segmentP V W)) => (segmentP T_2 W)))))) (ssList T_2) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (ssList (nil)) (T_1 = (nil)) (All U, ((ssList U) => (segmentP U (nil)))) ((ssList T_0) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_0 V) /\ (segmentP V W)) => (segmentP T_0 W))))))) (ssList T_1) (All V, ((ssList V) => (((segmentP T_0 V) /\ (segmentP V T_0)) => (T_0 = V)))) ((nil) = T_0) ### All 142 % 48.70/48.93 144. ((rearsegP (nil) T_0) <=> ((nil) = T_0)) (All V, ((ssList V) => (((segmentP T_0 V) /\ (segmentP V T_0)) => (T_0 = V)))) (ssList T_1) ((ssList T_0) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_0 V) /\ (segmentP V W)) => (segmentP T_0 W))))))) (All U, ((ssList U) => (segmentP U (nil)))) (T_1 = (nil)) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (ssList T_2) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_2 V) /\ (segmentP V W)) => (segmentP T_2 W)))))) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (T_2 = (nil)) (T_3 != T_0) (T_3 = T_1) (ssList T_0) (-. (neq T_0 (nil))) (ssList (nil)) (All V, ((ssList V) => ((neq T_0 V) <=> (T_0 != V)))) ### Equiv 18 143 % 48.70/48.93 145. ((ssList T_0) => ((rearsegP (nil) T_0) <=> ((nil) = T_0))) (All V, ((ssList V) => ((neq T_0 V) <=> (T_0 != V)))) (ssList (nil)) (-. (neq T_0 (nil))) (T_3 = T_1) (T_3 != T_0) (T_2 = (nil)) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_2 V) /\ (segmentP V W)) => (segmentP T_2 W)))))) (ssList T_2) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (T_1 = (nil)) (All U, ((ssList U) => (segmentP U (nil)))) ((ssList T_0) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_0 V) /\ (segmentP V W)) => (segmentP T_0 W))))))) (ssList T_1) (All V, ((ssList V) => (((segmentP T_0 V) /\ (segmentP V T_0)) => (T_0 = V)))) (ssList T_0) ### Imply 57 144 % 48.70/48.93 146. (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_0) (All V, ((ssList V) => (((segmentP T_0 V) /\ (segmentP V T_0)) => (T_0 = V)))) (ssList T_1) ((ssList T_0) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_0 V) /\ (segmentP V W)) => (segmentP T_0 W))))))) (All U, ((ssList U) => (segmentP U (nil)))) (T_1 = (nil)) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (ssList T_2) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_2 V) /\ (segmentP V W)) => (segmentP T_2 W)))))) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (T_2 = (nil)) (T_3 != T_0) (T_3 = T_1) (-. (neq T_0 (nil))) (ssList (nil)) (All V, ((ssList V) => ((neq T_0 V) <=> (T_0 != V)))) ### All 145 % 48.70/48.93 147. ((ssList T_2) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_2 V) /\ (segmentP V W)) => (segmentP T_2 W))))))) (All V, ((ssList V) => ((neq T_0 V) <=> (T_0 != V)))) (ssList (nil)) (-. (neq T_0 (nil))) (T_3 = T_1) (T_3 != T_0) (T_2 = (nil)) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (T_1 = (nil)) (All U, ((ssList U) => (segmentP U (nil)))) ((ssList T_0) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_0 V) /\ (segmentP V W)) => (segmentP T_0 W))))))) (ssList T_1) (All V, ((ssList V) => (((segmentP T_0 V) /\ (segmentP V T_0)) => (T_0 = V)))) (ssList T_0) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_2) ### Imply 56 146 % 48.70/48.93 148. (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (ssList T_2) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_0) (All V, ((ssList V) => (((segmentP T_0 V) /\ (segmentP V T_0)) => (T_0 = V)))) (ssList T_1) ((ssList T_0) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_0 V) /\ (segmentP V W)) => (segmentP T_0 W))))))) (All U, ((ssList U) => (segmentP U (nil)))) (T_1 = (nil)) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (T_2 = (nil)) (T_3 != T_0) (T_3 = T_1) (-. (neq T_0 (nil))) (ssList (nil)) (All V, ((ssList V) => ((neq T_0 V) <=> (T_0 != V)))) ### All 147 % 48.70/48.97 149. ((ssList (nil)) => (((segmentP T_1 (nil)) /\ (segmentP (nil) T_1)) => (T_1 = (nil)))) (All V, ((ssList V) => ((neq T_0 V) <=> (T_0 != V)))) (-. (neq T_0 (nil))) (T_3 = T_1) (T_3 != T_0) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) ((ssList T_0) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_0 V) /\ (segmentP V W)) => (segmentP T_0 W))))))) (All V, ((ssList V) => (((segmentP T_0 V) /\ (segmentP V T_0)) => (T_0 = V)))) (ssList T_0) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_2) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (T_2 = (nil)) (T_1 = T_2) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (ssList T_1) (All U, ((ssList U) => (segmentP U (nil)))) (ssList (nil)) ### DisjTree 43 47 55 148 % 48.70/48.97 150. (All V, ((ssList V) => (((segmentP T_1 V) /\ (segmentP V T_1)) => (T_1 = V)))) (ssList (nil)) (All U, ((ssList U) => (segmentP U (nil)))) (ssList T_1) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (T_1 = T_2) (T_2 = (nil)) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (ssList T_2) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_0) (All V, ((ssList V) => (((segmentP T_0 V) /\ (segmentP V T_0)) => (T_0 = V)))) ((ssList T_0) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_0 V) /\ (segmentP V W)) => (segmentP T_0 W))))))) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (T_3 != T_0) (T_3 = T_1) (-. (neq T_0 (nil))) (All V, ((ssList V) => ((neq T_0 V) <=> (T_0 != V)))) ### All 149 % 48.70/48.97 151. (All V, ((ssList V) => ((neq T_0 V) <=> (T_0 != V)))) (-. (neq T_0 (nil))) (T_3 = T_1) (T_3 != T_0) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (All V, ((ssList V) => (((segmentP T_0 V) /\ (segmentP V T_0)) => (T_0 = V)))) (ssList T_0) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_2) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (T_2 = (nil)) (T_1 = T_2) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (ssList T_1) (All U, ((ssList U) => (segmentP U (nil)))) (ssList (nil)) (All V, ((ssList V) => (((segmentP T_1 V) /\ (segmentP V T_1)) => (T_1 = V)))) ### All 150 % 48.70/48.97 152. ((ssList T_1) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W))))))) (All V, ((ssList V) => (((segmentP T_1 V) /\ (segmentP V T_1)) => (T_1 = V)))) (ssList (nil)) (All U, ((ssList U) => (segmentP U (nil)))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (T_1 = T_2) (T_2 = (nil)) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (ssList T_2) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_0) (All V, ((ssList V) => (((segmentP T_0 V) /\ (segmentP V T_0)) => (T_0 = V)))) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (T_3 != T_0) (T_3 = T_1) (-. (neq T_0 (nil))) (All V, ((ssList V) => ((neq T_0 V) <=> (T_0 != V)))) (ssList T_1) ### Imply 42 151 % 48.70/48.97 153. (ssList T_1) (All V, ((ssList V) => ((neq T_0 V) <=> (T_0 != V)))) (-. (neq T_0 (nil))) (T_3 = T_1) (T_3 != T_0) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (All V, ((ssList V) => (((segmentP T_0 V) /\ (segmentP V T_0)) => (T_0 = V)))) (ssList T_0) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_2) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (T_2 = (nil)) (T_1 = T_2) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (segmentP U (nil)))) (ssList (nil)) (All V, ((ssList V) => (((segmentP T_1 V) /\ (segmentP V T_1)) => (T_1 = V)))) ### All 152 % 48.70/48.97 154. ((ssList (nil)) => (((frontsegP T_2 (nil)) /\ (frontsegP (nil) T_2)) => (T_2 = (nil)))) (All V, ((ssList V) => (((segmentP T_1 V) /\ (segmentP V T_1)) => (T_1 = V)))) (All U, ((ssList U) => (segmentP U (nil)))) (T_1 = T_2) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (All V, ((ssList V) => (((segmentP T_0 V) /\ (segmentP V T_0)) => (T_0 = V)))) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (T_3 != T_0) (T_3 = T_1) (ssList T_1) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (ssList T_0) (T_0 = T_2) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (-. (neq T_0 (nil))) (All V, ((ssList V) => ((neq T_0 V) <=> (T_0 != V)))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_2) (All U, ((ssList U) => (frontsegP U (nil)))) (ssList (nil)) ### DisjTree 5 9 41 153 % 48.70/48.97 155. (All V, ((ssList V) => (((frontsegP T_2 V) /\ (frontsegP V T_2)) => (T_2 = V)))) (ssList (nil)) (All U, ((ssList U) => (frontsegP U (nil)))) (ssList T_2) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All V, ((ssList V) => ((neq T_0 V) <=> (T_0 != V)))) (-. (neq T_0 (nil))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (T_0 = T_2) (ssList T_0) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (ssList T_1) (T_3 = T_1) (T_3 != T_0) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (All V, ((ssList V) => (((segmentP T_0 V) /\ (segmentP V T_0)) => (T_0 = V)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (T_1 = T_2) (All U, ((ssList U) => (segmentP U (nil)))) (All V, ((ssList V) => (((segmentP T_1 V) /\ (segmentP V T_1)) => (T_1 = V)))) ### All 154 % 48.70/48.97 156. ((ssList T_0) => (All V, ((ssList V) => ((neq T_0 V) <=> (T_0 != V))))) (All V, ((ssList V) => (((segmentP T_1 V) /\ (segmentP V T_1)) => (T_1 = V)))) (All U, ((ssList U) => (segmentP U (nil)))) (T_1 = T_2) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (All V, ((ssList V) => (((segmentP T_0 V) /\ (segmentP V T_0)) => (T_0 = V)))) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (T_3 != T_0) (T_3 = T_1) (ssList T_1) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (T_0 = T_2) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (-. (neq T_0 (nil))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_2) (All U, ((ssList U) => (frontsegP U (nil)))) (ssList (nil)) (All V, ((ssList V) => (((frontsegP T_2 V) /\ (frontsegP V T_2)) => (T_2 = V)))) (ssList T_0) ### Imply 4 155 % 48.70/48.97 157. (ssList T_0) (All V, ((ssList V) => (((frontsegP T_2 V) /\ (frontsegP V T_2)) => (T_2 = V)))) (ssList (nil)) (All U, ((ssList U) => (frontsegP U (nil)))) (ssList T_2) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (-. (neq T_0 (nil))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (T_0 = T_2) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (ssList T_1) (T_3 = T_1) (T_3 != T_0) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (All V, ((ssList V) => (((segmentP T_0 V) /\ (segmentP V T_0)) => (T_0 = V)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (T_1 = T_2) (All U, ((ssList U) => (segmentP U (nil)))) (All V, ((ssList V) => (((segmentP T_1 V) /\ (segmentP V T_1)) => (T_1 = V)))) ### All 156 % 48.70/48.97 158. ((ssList T_2) => (All V, ((ssList V) => (((frontsegP T_2 V) /\ (frontsegP V T_2)) => (T_2 = V))))) (All V, ((ssList V) => (((segmentP T_1 V) /\ (segmentP V T_1)) => (T_1 = V)))) (All U, ((ssList U) => (segmentP U (nil)))) (T_1 = T_2) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (All V, ((ssList V) => (((segmentP T_0 V) /\ (segmentP V T_0)) => (T_0 = V)))) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (T_3 != T_0) (T_3 = T_1) (ssList T_1) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (T_0 = T_2) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (-. (neq T_0 (nil))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (frontsegP U (nil)))) (ssList (nil)) (ssList T_0) (ssList T_2) ### Imply 3 157 % 48.70/48.97 159. (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) (ssList T_2) (ssList T_0) (ssList (nil)) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (-. (neq T_0 (nil))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (T_0 = T_2) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (ssList T_1) (T_3 = T_1) (T_3 != T_0) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (All V, ((ssList V) => (((segmentP T_0 V) /\ (segmentP V T_0)) => (T_0 = V)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (T_1 = T_2) (All U, ((ssList U) => (segmentP U (nil)))) (All V, ((ssList V) => (((segmentP T_1 V) /\ (segmentP V T_1)) => (T_1 = V)))) ### All 158 % 48.70/48.97 160. ((ssList T_1) => (All V, ((ssList V) => (((segmentP T_1 V) /\ (segmentP V T_1)) => (T_1 = V))))) (All U, ((ssList U) => (segmentP U (nil)))) (T_1 = T_2) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (All V, ((ssList V) => (((segmentP T_0 V) /\ (segmentP V T_0)) => (T_0 = V)))) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (T_3 != T_0) (T_3 = T_1) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (T_0 = T_2) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (-. (neq T_0 (nil))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (frontsegP U (nil)))) (ssList (nil)) (ssList T_0) (ssList T_2) (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) (ssList T_1) ### Imply 2 159 % 48.70/48.97 161. (All U, ((ssList U) => (All V, ((ssList V) => (((segmentP U V) /\ (segmentP V U)) => (U = V)))))) (ssList T_1) (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) (ssList T_2) (ssList T_0) (ssList (nil)) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (-. (neq T_0 (nil))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (T_0 = T_2) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (T_3 = T_1) (T_3 != T_0) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (All V, ((ssList V) => (((segmentP T_0 V) /\ (segmentP V T_0)) => (T_0 = V)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (T_1 = T_2) (All U, ((ssList U) => (segmentP U (nil)))) ### All 160 % 48.70/48.97 162. ((ssList T_0) => (All V, ((ssList V) => (((segmentP T_0 V) /\ (segmentP V T_0)) => (T_0 = V))))) (All U, ((ssList U) => (segmentP U (nil)))) (T_1 = T_2) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (T_3 != T_0) (T_3 = T_1) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (T_0 = T_2) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (-. (neq T_0 (nil))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (frontsegP U (nil)))) (ssList (nil)) (ssList T_2) (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) (ssList T_1) (All U, ((ssList U) => (All V, ((ssList V) => (((segmentP U V) /\ (segmentP V U)) => (U = V)))))) (ssList T_0) ### Imply 1 161 % 48.70/48.97 163. (ssList T_0) (All U, ((ssList U) => (All V, ((ssList V) => (((segmentP U V) /\ (segmentP V U)) => (U = V)))))) (ssList T_1) (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) (ssList T_2) (ssList (nil)) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (-. (neq T_0 (nil))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (T_0 = T_2) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (T_3 = T_1) (T_3 != T_0) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (T_1 = T_2) (All U, ((ssList U) => (segmentP U (nil)))) ### All 162 % 48.70/48.97 164. (neq T_3 (nil)) (All U, ((ssList U) => (segmentP U (nil)))) (T_1 = T_2) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (T_3 = T_1) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (T_0 = T_2) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (-. (neq T_0 (nil))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (frontsegP U (nil)))) (ssList (nil)) (ssList T_2) (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) (ssList T_1) (All U, ((ssList U) => (All V, ((ssList V) => (((segmentP U V) /\ (segmentP V U)) => (U = V)))))) (ssList T_0) ### P-NotP 163 50 % 48.70/48.98 165. (-. ((ssList T_1) => ((T_3 != T_1) \/ ((T_0 != T_2) \/ ((T_1 != T_2) \/ ((-. (neq T_3 (nil))) \/ (neq T_0 (nil)))))))) (ssList T_0) (All U, ((ssList U) => (All V, ((ssList V) => (((segmentP U V) /\ (segmentP V U)) => (U = V)))))) (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) (ssList T_2) (ssList (nil)) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (All U, ((ssList U) => (segmentP U (nil)))) ### ConjTree 164 % 48.70/48.98 166. (-. (All X, ((ssList X) => ((T_3 != X) \/ ((T_0 != T_2) \/ ((X != T_2) \/ ((-. (neq T_3 (nil))) \/ (neq T_0 (nil))))))))) (All U, ((ssList U) => (segmentP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (frontsegP U (nil)))) (ssList (nil)) (ssList T_2) (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) (All U, ((ssList U) => (All V, ((ssList V) => (((segmentP U V) /\ (segmentP V U)) => (U = V)))))) (ssList T_0) ### NotAllEx 165 % 48.70/48.98 167. (-. ((ssList T_2) => (All X, ((ssList X) => ((T_3 != X) \/ ((T_0 != T_2) \/ ((X != T_2) \/ ((-. (neq T_3 (nil))) \/ (neq T_0 (nil)))))))))) (ssList T_0) (All U, ((ssList U) => (All V, ((ssList V) => (((segmentP U V) /\ (segmentP V U)) => (U = V)))))) (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) (ssList (nil)) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (All U, ((ssList U) => (segmentP U (nil)))) ### NotImply 166 % 48.70/48.98 168. (-. (All W, ((ssList W) => (All X, ((ssList X) => ((T_3 != X) \/ ((T_0 != W) \/ ((X != W) \/ ((-. (neq T_3 (nil))) \/ (neq T_0 (nil))))))))))) (All U, ((ssList U) => (segmentP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (frontsegP U (nil)))) (ssList (nil)) (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) (All U, ((ssList U) => (All V, ((ssList V) => (((segmentP U V) /\ (segmentP V U)) => (U = V)))))) (ssList T_0) ### NotAllEx 167 % 48.70/48.98 169. (-. ((ssList T_3) => (All W, ((ssList W) => (All X, ((ssList X) => ((T_3 != X) \/ ((T_0 != W) \/ ((X != W) \/ ((-. (neq T_3 (nil))) \/ (neq T_0 (nil)))))))))))) (ssList T_0) (All U, ((ssList U) => (All V, ((ssList V) => (((segmentP U V) /\ (segmentP V U)) => (U = V)))))) (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) (ssList (nil)) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (All U, ((ssList U) => (segmentP U (nil)))) ### NotImply 168 % 48.70/48.98 170. (-. (All V, ((ssList V) => (All W, ((ssList W) => (All X, ((ssList X) => ((V != X) \/ ((T_0 != W) \/ ((X != W) \/ ((-. (neq V (nil))) \/ (neq T_0 (nil))))))))))))) (All U, ((ssList U) => (segmentP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (frontsegP U (nil)))) (ssList (nil)) (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) (All U, ((ssList U) => (All V, ((ssList V) => (((segmentP U V) /\ (segmentP V U)) => (U = V)))))) (ssList T_0) ### NotAllEx 169 % 48.70/48.98 171. (-. ((ssList T_0) => (All V, ((ssList V) => (All W, ((ssList W) => (All X, ((ssList X) => ((V != X) \/ ((T_0 != W) \/ ((X != W) \/ ((-. (neq V (nil))) \/ (neq T_0 (nil)))))))))))))) (All U, ((ssList U) => (All V, ((ssList V) => (((segmentP U V) /\ (segmentP V U)) => (U = V)))))) (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) (ssList (nil)) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (All U, ((ssList U) => (segmentP U (nil)))) ### NotImply 170 % 48.70/48.98 172. (-. (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (All X, ((ssList X) => ((V != X) \/ ((U != W) \/ ((X != W) \/ ((-. (neq V (nil))) \/ (neq U (nil))))))))))))))) (All U, ((ssList U) => (segmentP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (All U, ((ssList U) => (All V, ((ssList V) => ((segmentP U V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = U)))))))))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (frontsegP U (nil)))) (ssList (nil)) (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) (All U, ((ssList U) => (All V, ((ssList V) => (((segmentP U V) /\ (segmentP V U)) => (U = V)))))) ### NotAllEx 171 % 48.70/48.98 % SZS output end Proof % 48.70/48.98 (* END-PROOF *) %------------------------------------------------------------------------------