TSTP Solution File: SWC213+1 by SuperZenon---0.0.1
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%------------------------------------------------------------------------------
% File : SuperZenon---0.0.1
% Problem : SWC213+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 : n009.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 22:07:08 EDT 2022
% Result : Theorem 5.53s 5.71s
% Output : Proof 5.57s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SWC213+1 : TPTP v8.1.0. Released v2.4.0.
% 0.03/0.12 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s
% 0.12/0.33 % Computer : n009.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 600
% 0.12/0.33 % DateTime : Sun Jun 12 09:59:53 EDT 2022
% 0.12/0.34 % CPUTime :
% 5.53/5.71 % SZS status Theorem
% 5.53/5.71 (* PROOF-FOUND *)
% 5.53/5.71 (* BEGIN-PROOF *)
% 5.53/5.71 % SZS output start Proof
% 5.53/5.71 1. (ssList T_0) (-. (ssList T_0)) ### Axiom
% 5.53/5.71 2. ((nil) = T_0) ((nil) != T_0) ### Axiom
% 5.53/5.71 3. (ssList T_1) (-. (ssList T_1)) ### Axiom
% 5.53/5.71 4. (ssList T_0) (-. (ssList T_0)) ### Axiom
% 5.53/5.71 5. (ssList T_1) (-. (ssList T_1)) ### Axiom
% 5.53/5.71 6. (ssList (nil)) (-. (ssList (nil))) ### Axiom
% 5.53/5.71 7. (ssList T_1) (-. (ssList T_1)) ### Axiom
% 5.53/5.71 8. (T_1 = (nil)) ((nil) != T_1) ### Sym(=)
% 5.53/5.71 9. (ssList T_0) (-. (ssList T_0)) ### Axiom
% 5.53/5.71 10. (ssList (nil)) (-. (ssList (nil))) ### Axiom
% 5.53/5.71 11. (ssList (nil)) (-. (ssList (nil))) ### Axiom
% 5.53/5.71 12. (ssList T_0) (-. (ssList T_0)) ### Axiom
% 5.53/5.71 13. (-. (segmentP T_0 (nil))) (segmentP T_0 (nil)) ### Axiom
% 5.53/5.71 14. ((ssList T_0) => (segmentP T_0 (nil))) (-. (segmentP T_0 (nil))) (ssList T_0) ### Imply 12 13
% 5.53/5.71 15. (All U, ((ssList U) => (segmentP U (nil)))) (ssList T_0) (-. (segmentP T_0 (nil))) ### All 14
% 5.53/5.71 16. (ssList (nil)) (-. (ssList (nil))) ### Axiom
% 5.53/5.71 17. (-. (segmentP (nil) (nil))) (ssList (nil)) ### Extension/test/ax55ctrp 16
% 5.53/5.71 18. (ssList T_1) (-. (ssList T_1)) ### Axiom
% 5.53/5.71 19. (segmentP T_0 (nil)) (-. (segmentP T_0 (nil))) ### Axiom
% 5.53/5.71 20. (segmentP (nil) T_1) (-. (segmentP (nil) T_1)) ### Axiom
% 5.53/5.71 21. (-. (segmentP T_0 T_1)) (segmentP T_0 T_1) ### Axiom
% 5.53/5.71 22. ((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 18 19 20 21
% 5.53/5.71 23. (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 22
% 5.53/5.71 24. ((ssList (nil)) => (((segmentP T_0 (nil)) /\ (segmentP (nil) (nil))) => (segmentP T_0 (nil)))) (-. (segmentP T_0 T_1)) (segmentP (nil) T_1) (ssList T_1) (All W, ((ssList W) => (((segmentP T_0 (nil)) /\ (segmentP (nil) W)) => (segmentP T_0 W)))) (ssList T_0) (All U, ((ssList U) => (segmentP U (nil)))) (ssList (nil)) ### DisjTree 11 15 17 23
% 5.53/5.71 25. (ssList (nil)) (All U, ((ssList U) => (segmentP U (nil)))) (ssList T_0) (All W, ((ssList W) => (((segmentP T_0 (nil)) /\ (segmentP (nil) W)) => (segmentP T_0 W)))) (ssList T_1) (segmentP (nil) T_1) (-. (segmentP T_0 T_1)) ### All 24
% 5.53/5.71 26. ((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) (ssList T_1) (ssList T_0) (All U, ((ssList U) => (segmentP U (nil)))) (ssList (nil)) ### Imply 10 25
% 5.53/5.71 27. (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_0 V) /\ (segmentP V W)) => (segmentP T_0 W)))))) (ssList (nil)) (All U, ((ssList U) => (segmentP U (nil)))) (ssList T_0) (ssList T_1) (segmentP (nil) T_1) (-. (segmentP T_0 T_1)) ### All 26
% 5.53/5.71 28. ((ssList T_0) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_0 V) /\ (segmentP V W)) => (segmentP T_0 W))))))) (-. (segmentP T_0 T_1)) (segmentP (nil) T_1) (ssList T_1) (All U, ((ssList U) => (segmentP U (nil)))) (ssList (nil)) (ssList T_0) ### Imply 9 27
% 5.53/5.71 29. (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (ssList T_0) (ssList (nil)) (All U, ((ssList U) => (segmentP U (nil)))) (ssList T_1) (segmentP (nil) T_1) (-. (segmentP T_0 T_1)) ### All 28
% 5.53/5.71 30. ((segmentP (nil) T_1) <=> ((nil) = T_1)) (-. (segmentP T_0 T_1)) (ssList T_1) (All U, ((ssList U) => (segmentP U (nil)))) (ssList (nil)) (ssList T_0) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (T_1 = (nil)) ### Equiv 8 29
% 5.53/5.71 31. ((ssList T_1) => ((segmentP (nil) T_1) <=> ((nil) = T_1))) (T_1 = (nil)) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (ssList T_0) (ssList (nil)) (All U, ((ssList U) => (segmentP U (nil)))) (-. (segmentP T_0 T_1)) (ssList T_1) ### Imply 7 30
% 5.53/5.71 32. (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (ssList T_1) (-. (segmentP T_0 T_1)) (All U, ((ssList U) => (segmentP U (nil)))) (ssList (nil)) (ssList T_0) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (T_1 = (nil)) ### All 31
% 5.53/5.71 33. (-. (T_1 != (nil))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (ssList T_0) (ssList (nil)) (All U, ((ssList U) => (segmentP U (nil)))) (-. (segmentP T_0 T_1)) (ssList T_1) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) ### NotNot 32
% 5.53/5.71 34. (-. (neq T_1 (nil))) (neq T_1 (nil)) ### Axiom
% 5.53/5.71 35. ((neq T_1 (nil)) <=> (T_1 != (nil))) (-. (neq T_1 (nil))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (ssList T_1) (-. (segmentP T_0 T_1)) (All U, ((ssList U) => (segmentP U (nil)))) (ssList (nil)) (ssList T_0) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) ### Equiv 33 34
% 5.53/5.71 36. ((ssList (nil)) => ((neq T_1 (nil)) <=> (T_1 != (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (ssList T_0) (All U, ((ssList U) => (segmentP U (nil)))) (-. (segmentP T_0 T_1)) (ssList T_1) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (-. (neq T_1 (nil))) (ssList (nil)) ### Imply 6 35
% 5.53/5.71 37. (All V, ((ssList V) => ((neq T_1 V) <=> (T_1 != V)))) (ssList (nil)) (-. (neq T_1 (nil))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (ssList T_1) (-. (segmentP T_0 T_1)) (All U, ((ssList U) => (segmentP U (nil)))) (ssList T_0) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) ### All 36
% 5.53/5.71 38. (ssList T_1) (-. (ssList T_1)) ### Axiom
% 5.53/5.71 39. (ssList T_1) (-. (ssList T_1)) ### Axiom
% 5.53/5.71 40. (ssList (nil)) (-. (ssList (nil))) ### Axiom
% 5.53/5.71 41. (ssList T_1) (-. (ssList T_1)) ### Axiom
% 5.53/5.71 42. (-. (segmentP T_1 (nil))) (segmentP T_1 (nil)) ### Axiom
% 5.53/5.71 43. ((ssList T_1) => (segmentP T_1 (nil))) (-. (segmentP T_1 (nil))) (ssList T_1) ### Imply 41 42
% 5.53/5.71 44. (All U, ((ssList U) => (segmentP U (nil)))) (ssList T_1) (-. (segmentP T_1 (nil))) ### All 43
% 5.53/5.71 45. (ssList (nil)) (-. (ssList (nil))) ### Axiom
% 5.53/5.71 46. (ssList T_0) (-. (ssList T_0)) ### Axiom
% 5.53/5.71 47. (segmentP T_1 (nil)) (-. (segmentP T_1 (nil))) ### Axiom
% 5.53/5.71 48. (segmentP (nil) T_0) (-. (segmentP (nil) T_0)) ### Axiom
% 5.53/5.71 49. (-. (segmentP T_1 T_0)) (segmentP T_1 T_0) ### Axiom
% 5.53/5.71 50. ((ssList T_0) => (((segmentP T_1 (nil)) /\ (segmentP (nil) T_0)) => (segmentP T_1 T_0))) (-. (segmentP T_1 T_0)) (segmentP (nil) T_0) (segmentP T_1 (nil)) (ssList T_0) ### DisjTree 46 47 48 49
% 5.53/5.71 51. (All W, ((ssList W) => (((segmentP T_1 (nil)) /\ (segmentP (nil) W)) => (segmentP T_1 W)))) (ssList T_0) (segmentP T_1 (nil)) (segmentP (nil) T_0) (-. (segmentP T_1 T_0)) ### All 50
% 5.53/5.71 52. ((ssList (nil)) => (All W, ((ssList W) => (((segmentP T_1 (nil)) /\ (segmentP (nil) W)) => (segmentP T_1 W))))) (-. (segmentP T_1 T_0)) (segmentP (nil) T_0) (segmentP T_1 (nil)) (ssList T_0) (ssList (nil)) ### Imply 45 51
% 5.53/5.71 53. (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_0) (-. (segmentP T_1 T_0)) ### All 52
% 5.53/5.71 54. ((segmentP T_1 (nil)) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W (nil)) X) = T_1)))))) (-. (segmentP T_1 T_0)) (segmentP (nil) T_0) (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_1) (All U, ((ssList U) => (segmentP U (nil)))) ### Equiv 44 53
% 5.53/5.71 55. ((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_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (ssList T_0) (segmentP (nil) T_0) (-. (segmentP T_1 T_0)) (ssList (nil)) ### Imply 40 54
% 5.57/5.80 56. (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)) (-. (segmentP T_1 T_0)) (segmentP (nil) T_0) (ssList T_0) (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W)))))) (ssList T_1) (All U, ((ssList U) => (segmentP U (nil)))) ### All 55
% 5.57/5.80 57. ((ssList T_1) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP T_1 V) /\ (segmentP V W)) => (segmentP T_1 W))))))) (All U, ((ssList U) => (segmentP U (nil)))) (ssList T_0) (segmentP (nil) T_0) (-. (segmentP T_1 T_0)) (ssList (nil)) (All V, ((ssList V) => ((segmentP T_1 V) <=> (Ex W, ((ssList W) /\ (Ex X, ((ssList X) /\ ((app (app W V) X) = T_1)))))))) (ssList T_1) ### Imply 39 56
% 5.57/5.80 58. (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (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)))))))) (ssList (nil)) (-. (segmentP T_1 T_0)) (segmentP (nil) T_0) (ssList T_0) (All U, ((ssList U) => (segmentP U (nil)))) ### All 57
% 5.57/5.80 59. ((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)))) (ssList T_0) (segmentP (nil) T_0) (-. (segmentP T_1 T_0)) (ssList (nil)) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (ssList T_1) ### Imply 38 58
% 5.57/5.80 60. (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) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (ssList (nil)) (-. (segmentP T_1 T_0)) (segmentP (nil) T_0) (ssList T_0) (All U, ((ssList U) => (segmentP U (nil)))) ### All 59
% 5.57/5.80 61. (T_2 != T_2) ### Refl(=)
% 5.57/5.80 62. (T_0 = T_1) (T_0 != T_1) ### Axiom
% 5.57/5.80 63. (T_2 != T_1) (T_2 = T_0) (T_0 = T_1) ### Trans 61 62
% 5.57/5.80 64. ((ssList T_1) => (((segmentP T_0 T_1) /\ (segmentP T_1 T_0)) => (T_0 = T_1))) (T_2 = T_0) (T_2 != T_1) (segmentP (nil) 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)))))))) (ssList T_0) (All U, ((ssList U) => (segmentP U (nil)))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (-. (neq T_1 (nil))) (ssList (nil)) (All V, ((ssList V) => ((neq T_1 V) <=> (T_1 != V)))) (ssList T_1) ### DisjTree 5 37 60 63
% 5.57/5.80 65. (All V, ((ssList V) => (((segmentP T_0 V) /\ (segmentP V T_0)) => (T_0 = V)))) (ssList T_1) (All V, ((ssList V) => ((neq T_1 V) <=> (T_1 != V)))) (ssList (nil)) (-. (neq T_1 (nil))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (segmentP U (nil)))) (ssList T_0) (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)))))))))) (segmentP (nil) T_0) (T_2 != T_1) (T_2 = T_0) ### All 64
% 5.57/5.80 66. ((ssList T_0) => (All V, ((ssList V) => (((segmentP T_0 V) /\ (segmentP V T_0)) => (T_0 = V))))) (T_2 = T_0) (T_2 != T_1) (segmentP (nil) 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)))))))) (All U, ((ssList U) => (segmentP U (nil)))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (-. (neq T_1 (nil))) (ssList (nil)) (All V, ((ssList V) => ((neq T_1 V) <=> (T_1 != V)))) (ssList T_1) (ssList T_0) ### Imply 4 65
% 5.57/5.80 67. (All U, ((ssList U) => (All V, ((ssList V) => (((segmentP U V) /\ (segmentP V U)) => (U = V)))))) (ssList T_0) (ssList T_1) (All V, ((ssList V) => ((neq T_1 V) <=> (T_1 != V)))) (ssList (nil)) (-. (neq T_1 (nil))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (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)))))))))) (segmentP (nil) T_0) (T_2 != T_1) (T_2 = T_0) ### All 66
% 5.57/5.80 68. ((ssList T_1) => (All V, ((ssList V) => ((neq T_1 V) <=> (T_1 != V))))) (T_2 = T_0) (T_2 != T_1) (segmentP (nil) 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)))))))) (All U, ((ssList U) => (segmentP U (nil)))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (-. (neq T_1 (nil))) (ssList (nil)) (ssList T_0) (All U, ((ssList U) => (All V, ((ssList V) => (((segmentP U V) /\ (segmentP V U)) => (U = V)))))) (ssList T_1) ### Imply 3 67
% 5.57/5.80 69. (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (ssList T_1) (All U, ((ssList U) => (All V, ((ssList V) => (((segmentP U V) /\ (segmentP V U)) => (U = V)))))) (ssList T_0) (ssList (nil)) (-. (neq T_1 (nil))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (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)))))))))) (segmentP (nil) T_0) (T_2 != T_1) (T_2 = T_0) ### All 68
% 5.57/5.80 70. ((segmentP (nil) T_0) <=> ((nil) = T_0)) (T_2 = T_0) (T_2 != 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)))))))))) (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)))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (-. (neq T_1 (nil))) (ssList (nil)) (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) => ((neq U V) <=> (U != V)))))) ((nil) = T_0) ### Equiv 2 69
% 5.57/5.80 71. ((ssList T_0) => ((segmentP (nil) T_0) <=> ((nil) = T_0))) ((nil) = T_0) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (ssList T_1) (All U, ((ssList U) => (All V, ((ssList V) => (((segmentP U V) /\ (segmentP V U)) => (U = V)))))) (ssList (nil)) (-. (neq T_1 (nil))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (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)))))))))) (T_2 != T_1) (T_2 = T_0) (ssList T_0) ### Imply 1 70
% 5.57/5.80 72. (ssList T_0) (T_2 = T_0) (T_2 != 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)))))))))) (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)))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (-. (neq T_1 (nil))) (ssList (nil)) (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) => ((neq U V) <=> (U != V)))))) ((nil) = T_0) ### All 71
% 5.57/5.81 73. ((nil) != (nil)) ### NotEqual
% 5.57/5.81 74. (neq T_2 (nil)) ((nil) = T_0) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (ssList T_1) (All U, ((ssList U) => (All V, ((ssList V) => (((segmentP U V) /\ (segmentP V U)) => (U = V)))))) (ssList (nil)) (-. (neq T_1 (nil))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (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)))))))))) (T_2 = T_0) (ssList T_0) ### P-NotP 72 73
% 5.57/5.81 75. (-. ((nil) != T_0)) (ssList T_0) (T_2 = 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)))))))) (All U, ((ssList U) => (segmentP U (nil)))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (-. (neq T_1 (nil))) (ssList (nil)) (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) => ((neq U V) <=> (U != V)))))) (neq T_2 (nil)) ### NotNot 74
% 5.57/5.81 76. (ssList T_1) (-. (ssList T_1)) ### Axiom
% 5.57/5.81 77. (ssList (nil)) (-. (ssList (nil))) ### Axiom
% 5.57/5.81 78. ((nil) != T_1) (T_1 = (nil)) ### Sym(=)
% 5.57/5.81 79. (-. (T_1 != (nil))) ((nil) != T_1) ### NotNot 78
% 5.57/5.81 80. (-. (neq T_1 (nil))) (neq T_1 (nil)) ### Axiom
% 5.57/5.81 81. ((neq T_1 (nil)) <=> (T_1 != (nil))) (-. (neq T_1 (nil))) ((nil) != T_1) ### Equiv 79 80
% 5.57/5.81 82. ((ssList (nil)) => ((neq T_1 (nil)) <=> (T_1 != (nil)))) ((nil) != T_1) (-. (neq T_1 (nil))) (ssList (nil)) ### Imply 77 81
% 5.57/5.81 83. (All V, ((ssList V) => ((neq T_1 V) <=> (T_1 != V)))) (ssList (nil)) (-. (neq T_1 (nil))) ((nil) != T_1) ### All 82
% 5.57/5.81 84. ((ssList T_1) => (All V, ((ssList V) => ((neq T_1 V) <=> (T_1 != V))))) ((nil) != T_1) (-. (neq T_1 (nil))) (ssList (nil)) (ssList T_1) ### Imply 76 83
% 5.57/5.81 85. (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (ssList T_1) (ssList (nil)) (-. (neq T_1 (nil))) ((nil) != T_1) ### All 84
% 5.57/5.81 86. (T_3 != T_3) ### Refl(=)
% 5.57/5.81 87. ((nil) != T_3) (T_1 = T_3) (-. (neq T_1 (nil))) (ssList (nil)) (ssList T_1) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) ### Trans 85 86
% 5.57/5.81 88. (-. (((nil) != T_0) /\ ((nil) = T_3))) (T_1 = T_3) (neq T_2 (nil)) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (ssList T_1) (All U, ((ssList U) => (All V, ((ssList V) => (((segmentP U V) /\ (segmentP V U)) => (U = V)))))) (ssList (nil)) (-. (neq T_1 (nil))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (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)))))))))) (T_2 = T_0) (ssList T_0) ### NotAnd 75 87
% 5.57/5.81 89. (-. ((ssList T_0) => ((T_2 != T_0) \/ ((T_1 != T_3) \/ ((-. (neq T_2 (nil))) \/ ((All Y, ((ssList Y) => (All Z, ((ssList Z) => (((app (app Y T_3) Z) != T_0) \/ ((-. (equalelemsP T_3)) \/ ((Ex X1, ((ssItem X1) /\ (Ex X2, ((ssList X2) /\ (((app X2 (cons X1 (nil))) = Y) /\ (Ex X3, ((ssList X3) /\ ((app (cons X1 (nil)) X3) = T_3)))))))) \/ (Ex X4, ((ssItem X4) /\ (Ex X5, ((ssList X5) /\ (((app (cons X4 (nil)) X5) = Z) /\ (Ex X6, ((ssList X6) /\ ((app X6 (cons X4 (nil))) = T_3))))))))))))))) \/ ((neq T_1 (nil)) \/ (((nil) != T_0) /\ ((nil) = T_3))))))))) (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)))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (ssList (nil)) (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) => ((neq U V) <=> (U != V)))))) ### ConjTree 88
% 5.57/5.81 90. (-. (All X, ((ssList X) => ((T_2 != X) \/ ((T_1 != T_3) \/ ((-. (neq T_2 (nil))) \/ ((All Y, ((ssList Y) => (All Z, ((ssList Z) => (((app (app Y T_3) Z) != X) \/ ((-. (equalelemsP T_3)) \/ ((Ex X1, ((ssItem X1) /\ (Ex X2, ((ssList X2) /\ (((app X2 (cons X1 (nil))) = Y) /\ (Ex X3, ((ssList X3) /\ ((app (cons X1 (nil)) X3) = T_3)))))))) \/ (Ex X4, ((ssItem X4) /\ (Ex X5, ((ssList X5) /\ (((app (cons X4 (nil)) X5) = Z) /\ (Ex X6, ((ssList X6) /\ ((app X6 (cons X4 (nil))) = T_3))))))))))))))) \/ ((neq T_1 (nil)) \/ (((nil) != X) /\ ((nil) = T_3)))))))))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (ssList T_1) (All U, ((ssList U) => (All V, ((ssList V) => (((segmentP U V) /\ (segmentP V U)) => (U = V)))))) (ssList (nil)) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (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)))))))))) ### NotAllEx 89
% 5.57/5.81 91. (-. ((ssList T_3) => (All X, ((ssList X) => ((T_2 != X) \/ ((T_1 != T_3) \/ ((-. (neq T_2 (nil))) \/ ((All Y, ((ssList Y) => (All Z, ((ssList Z) => (((app (app Y T_3) Z) != X) \/ ((-. (equalelemsP T_3)) \/ ((Ex X1, ((ssItem X1) /\ (Ex X2, ((ssList X2) /\ (((app X2 (cons X1 (nil))) = Y) /\ (Ex X3, ((ssList X3) /\ ((app (cons X1 (nil)) X3) = T_3)))))))) \/ (Ex X4, ((ssItem X4) /\ (Ex X5, ((ssList X5) /\ (((app (cons X4 (nil)) X5) = Z) /\ (Ex X6, ((ssList X6) /\ ((app X6 (cons X4 (nil))) = T_3))))))))))))))) \/ ((neq T_1 (nil)) \/ (((nil) != X) /\ ((nil) = T_3))))))))))) (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)))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (ssList (nil)) (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) => ((neq U V) <=> (U != V)))))) ### NotImply 90
% 5.57/5.81 92. (-. (All W, ((ssList W) => (All X, ((ssList X) => ((T_2 != X) \/ ((T_1 != W) \/ ((-. (neq T_2 (nil))) \/ ((All Y, ((ssList Y) => (All Z, ((ssList Z) => (((app (app Y W) Z) != X) \/ ((-. (equalelemsP W)) \/ ((Ex X1, ((ssItem X1) /\ (Ex X2, ((ssList X2) /\ (((app X2 (cons X1 (nil))) = Y) /\ (Ex X3, ((ssList X3) /\ ((app (cons X1 (nil)) X3) = W)))))))) \/ (Ex X4, ((ssItem X4) /\ (Ex X5, ((ssList X5) /\ (((app (cons X4 (nil)) X5) = Z) /\ (Ex X6, ((ssList X6) /\ ((app X6 (cons X4 (nil))) = W))))))))))))))) \/ ((neq T_1 (nil)) \/ (((nil) != X) /\ ((nil) = W)))))))))))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (ssList T_1) (All U, ((ssList U) => (All V, ((ssList V) => (((segmentP U V) /\ (segmentP V U)) => (U = V)))))) (ssList (nil)) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (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)))))))))) ### NotAllEx 91
% 5.57/5.81 93. (-. ((ssList T_2) => (All W, ((ssList W) => (All X, ((ssList X) => ((T_2 != X) \/ ((T_1 != W) \/ ((-. (neq T_2 (nil))) \/ ((All Y, ((ssList Y) => (All Z, ((ssList Z) => (((app (app Y W) Z) != X) \/ ((-. (equalelemsP W)) \/ ((Ex X1, ((ssItem X1) /\ (Ex X2, ((ssList X2) /\ (((app X2 (cons X1 (nil))) = Y) /\ (Ex X3, ((ssList X3) /\ ((app (cons X1 (nil)) X3) = W)))))))) \/ (Ex X4, ((ssItem X4) /\ (Ex X5, ((ssList X5) /\ (((app (cons X4 (nil)) X5) = Z) /\ (Ex X6, ((ssList X6) /\ ((app X6 (cons X4 (nil))) = W))))))))))))))) \/ ((neq T_1 (nil)) \/ (((nil) != X) /\ ((nil) = 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) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (All U, ((ssList U) => (segmentP U (nil)))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (ssList (nil)) (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) => ((neq U V) <=> (U != V)))))) ### NotImply 92
% 5.57/5.82 94. (-. (All V, ((ssList V) => (All W, ((ssList W) => (All X, ((ssList X) => ((V != X) \/ ((T_1 != W) \/ ((-. (neq V (nil))) \/ ((All Y, ((ssList Y) => (All Z, ((ssList Z) => (((app (app Y W) Z) != X) \/ ((-. (equalelemsP W)) \/ ((Ex X1, ((ssItem X1) /\ (Ex X2, ((ssList X2) /\ (((app X2 (cons X1 (nil))) = Y) /\ (Ex X3, ((ssList X3) /\ ((app (cons X1 (nil)) X3) = W)))))))) \/ (Ex X4, ((ssItem X4) /\ (Ex X5, ((ssList X5) /\ (((app (cons X4 (nil)) X5) = Z) /\ (Ex X6, ((ssList X6) /\ ((app X6 (cons X4 (nil))) = W))))))))))))))) \/ ((neq T_1 (nil)) \/ (((nil) != X) /\ ((nil) = W)))))))))))))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (ssList T_1) (All U, ((ssList U) => (All V, ((ssList V) => (((segmentP U V) /\ (segmentP V U)) => (U = V)))))) (ssList (nil)) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (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)))))))))) ### NotAllEx 93
% 5.57/5.82 95. (-. ((ssList T_1) => (All V, ((ssList V) => (All W, ((ssList W) => (All X, ((ssList X) => ((V != X) \/ ((T_1 != W) \/ ((-. (neq V (nil))) \/ ((All Y, ((ssList Y) => (All Z, ((ssList Z) => (((app (app Y W) Z) != X) \/ ((-. (equalelemsP W)) \/ ((Ex X1, ((ssItem X1) /\ (Ex X2, ((ssList X2) /\ (((app X2 (cons X1 (nil))) = Y) /\ (Ex X3, ((ssList X3) /\ ((app (cons X1 (nil)) X3) = W)))))))) \/ (Ex X4, ((ssItem X4) /\ (Ex X5, ((ssList X5) /\ (((app (cons X4 (nil)) X5) = Z) /\ (Ex X6, ((ssList X6) /\ ((app X6 (cons X4 (nil))) = W))))))))))))))) \/ ((neq T_1 (nil)) \/ (((nil) != X) /\ ((nil) = 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) => (All V, ((ssList V) => (All W, ((ssList W) => (((segmentP U V) /\ (segmentP V W)) => (segmentP U W)))))))) (All U, ((ssList U) => (segmentP U (nil)))) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (ssList (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) => ((neq U V) <=> (U != V)))))) ### NotImply 94
% 5.57/5.82 96. (-. (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (All X, ((ssList X) => ((V != X) \/ ((U != W) \/ ((-. (neq V (nil))) \/ ((All Y, ((ssList Y) => (All Z, ((ssList Z) => (((app (app Y W) Z) != X) \/ ((-. (equalelemsP W)) \/ ((Ex X1, ((ssItem X1) /\ (Ex X2, ((ssList X2) /\ (((app X2 (cons X1 (nil))) = Y) /\ (Ex X3, ((ssList X3) /\ ((app (cons X1 (nil)) X3) = W)))))))) \/ (Ex X4, ((ssItem X4) /\ (Ex X5, ((ssList X5) /\ (((app (cons X4 (nil)) X5) = Z) /\ (Ex X6, ((ssList X6) /\ ((app X6 (cons X4 (nil))) = W))))))))))))))) \/ ((neq U (nil)) \/ (((nil) != X) /\ ((nil) = W)))))))))))))))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (All U, ((ssList U) => (All V, ((ssList V) => (((segmentP U V) /\ (segmentP V U)) => (U = V)))))) (ssList (nil)) (All U, ((ssList U) => ((segmentP (nil) U) <=> ((nil) = U)))) (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)))))))))) ### NotAllEx 95
% 5.57/5.82 % SZS output end Proof
% 5.57/5.82 (* END-PROOF *)
%------------------------------------------------------------------------------