TSTP Solution File: SWC001+1 by SuperZenon---0.0.1
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : SuperZenon---0.0.1
% Problem : SWC001+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 : n032.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 22:06:04 EDT 2022
% Result : Theorem 148.02s 148.36s
% Output : Proof 148.28s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : SWC001+1 : TPTP v8.1.0. Released v2.4.0.
% 0.00/0.10 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s
% 0.09/0.28 % Computer : n032.cluster.edu
% 0.09/0.28 % Model : x86_64 x86_64
% 0.09/0.28 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.28 % Memory : 8042.1875MB
% 0.09/0.28 % OS : Linux 3.10.0-693.el7.x86_64
% 0.09/0.28 % CPULimit : 300
% 0.09/0.28 % WCLimit : 600
% 0.09/0.28 % DateTime : Sun Jun 12 08:30:40 EDT 2022
% 0.13/0.29 % CPUTime :
% 148.02/148.36 % SZS status Theorem
% 148.02/148.36 (* PROOF-FOUND *)
% 148.02/148.36 (* BEGIN-PROOF *)
% 148.02/148.36 % SZS output start Proof
% 148.02/148.36 1. (memberP T_0 T_1) (-. (memberP T_0 T_1)) ### Axiom
% 148.02/148.36 2. (ssItem T_1) (-. (ssItem T_1)) ### Axiom
% 148.02/148.36 3. (ssItem T_1) (-. (ssItem T_1)) ### Axiom
% 148.02/148.36 4. (ssItem T_1) (-. (ssItem T_1)) ### Axiom
% 148.02/148.36 5. (ssItem T_1) (-. (ssItem T_1)) ### Axiom
% 148.02/148.36 6. (ssItem T_1) (-. (ssItem T_1)) ### Axiom
% 148.02/148.36 7. (ssList T_2) (-. (ssList T_2)) ### Axiom
% 148.02/148.36 8. (ssList T_2) (-. (ssList T_2)) ### Axiom
% 148.02/148.36 9. (ssList T_3) (-. (ssList T_3)) ### Axiom
% 148.02/148.36 10. (ssList T_2) (-. (ssList T_2)) ### Axiom
% 148.02/148.36 11. (memberP T_3 T_1) (-. (memberP T_3 T_1)) ### Axiom
% 148.02/148.36 12. (-. ((memberP T_3 T_1) \/ (memberP T_2 T_1))) (memberP T_3 T_1) ### NotOr 11
% 148.02/148.36 13. (T_2 = T_3) (T_3 != T_2) ### Sym(=)
% 148.02/148.36 14. (T_2 != T_2) ### Refl(=)
% 148.02/148.36 15. ((app T_3 T_2) != (app T_2 T_2)) (T_2 = T_3) ### NotEqual 13 14
% 148.02/148.36 16. (-. (T_1 != T_1)) (T_1 != T_1) ### Axiom
% 148.02/148.36 17. (-. (memberP (app T_2 T_2) T_1)) (memberP (app T_3 T_2) T_1) (-. (T_1 != T_1)) (T_2 = T_3) ### P-NotP 15 16
% 148.02/148.36 18. ((memberP (app T_3 T_2) T_1) <=> ((memberP T_3 T_1) \/ (memberP T_2 T_1))) (T_2 = T_3) (-. (T_1 != T_1)) (-. (memberP (app T_2 T_2) T_1)) (memberP T_3 T_1) ### Equiv 12 17
% 148.02/148.36 19. ((ssList T_2) => ((memberP (app T_3 T_2) T_1) <=> ((memberP T_3 T_1) \/ (memberP T_2 T_1)))) (memberP T_3 T_1) (-. (memberP (app T_2 T_2) T_1)) (-. (T_1 != T_1)) (T_2 = T_3) (ssList T_2) ### Imply 10 18
% 148.02/148.36 20. (All W, ((ssList W) => ((memberP (app T_3 W) T_1) <=> ((memberP T_3 T_1) \/ (memberP W T_1))))) (ssList T_2) (T_2 = T_3) (-. (T_1 != T_1)) (-. (memberP (app T_2 T_2) T_1)) (memberP T_3 T_1) ### All 19
% 148.02/148.36 21. ((ssList T_3) => (All W, ((ssList W) => ((memberP (app T_3 W) T_1) <=> ((memberP T_3 T_1) \/ (memberP W T_1)))))) (memberP T_3 T_1) (-. (memberP (app T_2 T_2) T_1)) (-. (T_1 != T_1)) (T_2 = T_3) (ssList T_2) (ssList T_3) ### Imply 9 20
% 148.02/148.36 22. (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_3) (ssList T_2) (T_2 = T_3) (-. (T_1 != T_1)) (-. (memberP (app T_2 T_2) T_1)) (memberP T_3 T_1) ### All 21
% 148.02/148.36 23. (-. (memberP T_2 T_1)) (memberP T_2 T_1) ### Axiom
% 148.02/148.36 24. (-. (memberP T_2 T_1)) (memberP T_2 T_1) ### Axiom
% 148.02/148.36 25. ((memberP T_2 T_1) \/ (memberP T_2 T_1)) (-. (memberP T_2 T_1)) ### Or 23 24
% 148.02/148.36 26. ((memberP (app T_2 T_2) T_1) <=> ((memberP T_2 T_1) \/ (memberP T_2 T_1))) (-. (memberP T_2 T_1)) (memberP T_3 T_1) (-. (T_1 != T_1)) (T_2 = T_3) (ssList T_2) (ssList T_3) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) ### Equiv 22 25
% 148.02/148.36 27. ((ssList T_2) => ((memberP (app T_2 T_2) T_1) <=> ((memberP T_2 T_1) \/ (memberP T_2 T_1)))) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_3) (T_2 = T_3) (-. (T_1 != T_1)) (memberP T_3 T_1) (-. (memberP T_2 T_1)) (ssList T_2) ### Imply 8 26
% 148.02/148.36 28. (All W, ((ssList W) => ((memberP (app T_2 W) T_1) <=> ((memberP T_2 T_1) \/ (memberP W T_1))))) (ssList T_2) (-. (memberP T_2 T_1)) (memberP T_3 T_1) (-. (T_1 != T_1)) (T_2 = T_3) (ssList T_3) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) ### All 27
% 148.02/148.36 29. ((ssList T_2) => (All W, ((ssList W) => ((memberP (app T_2 W) T_1) <=> ((memberP T_2 T_1) \/ (memberP W T_1)))))) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_3) (T_2 = T_3) (-. (T_1 != T_1)) (memberP T_3 T_1) (-. (memberP T_2 T_1)) (ssList T_2) ### Imply 7 28
% 148.02/148.36 30. (ssList T_2) (-. (memberP T_2 T_1)) (memberP T_3 T_1) (-. (T_1 != T_1)) (T_2 = T_3) (ssList T_3) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) ### All 29
% 148.02/148.36 31. (leq T_1 T_1) (-. (leq T_1 T_1)) ### Axiom
% 148.02/148.36 32. (-. ((T_1 != T_1) /\ (leq T_1 T_1))) (leq T_1 T_1) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_3) (T_2 = T_3) (memberP T_3 T_1) (-. (memberP T_2 T_1)) (ssList T_2) ### NotAnd 30 31
% 148.02/148.36 33. (T_1 != T_1) ### Refl(=)
% 148.02/148.36 34. ((T_1 != T_1) /\ (leq T_1 T_1)) ### And 33
% 148.02/148.36 35. ((lt T_1 T_1) <=> ((T_1 != T_1) /\ (leq T_1 T_1))) (ssList T_2) (-. (memberP T_2 T_1)) (memberP T_3 T_1) (T_2 = T_3) (ssList T_3) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (leq T_1 T_1) ### Equiv 32 34
% 148.02/148.36 36. ((ssItem T_1) => ((lt T_1 T_1) <=> ((T_1 != T_1) /\ (leq T_1 T_1)))) (leq T_1 T_1) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_3) (T_2 = T_3) (memberP T_3 T_1) (-. (memberP T_2 T_1)) (ssList T_2) (ssItem T_1) ### Imply 6 35
% 148.02/148.36 37. (All V, ((ssItem V) => ((lt T_1 V) <=> ((T_1 != V) /\ (leq T_1 V))))) (ssItem T_1) (ssList T_2) (-. (memberP T_2 T_1)) (memberP T_3 T_1) (T_2 = T_3) (ssList T_3) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (leq T_1 T_1) ### All 36
% 148.02/148.36 38. ((ssItem T_1) => (All V, ((ssItem V) => ((lt T_1 V) <=> ((T_1 != V) /\ (leq T_1 V)))))) (leq T_1 T_1) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_3) (T_2 = T_3) (memberP T_3 T_1) (-. (memberP T_2 T_1)) (ssList T_2) (ssItem T_1) ### Imply 5 37
% 148.02/148.36 39. (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssItem T_1) (ssList T_2) (-. (memberP T_2 T_1)) (memberP T_3 T_1) (T_2 = T_3) (ssList T_3) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (leq T_1 T_1) ### All 38
% 148.02/148.36 40. (-. (-. (memberP T_3 T_1))) (leq T_1 T_1) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_3) (T_2 = T_3) (-. (memberP T_2 T_1)) (ssList T_2) (ssItem T_1) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) ### NotNot 39
% 148.02/148.36 41. (memberP T_4 T_1) (-. (memberP T_4 T_1)) ### Axiom
% 148.02/148.36 42. (-. ((-. (memberP T_3 T_1)) /\ (memberP T_4 T_1))) (memberP T_4 T_1) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssItem T_1) (ssList T_2) (-. (memberP T_2 T_1)) (T_2 = T_3) (ssList T_3) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (leq T_1 T_1) ### NotAnd 40 41
% 148.02/148.36 43. (-. (-. (memberP T_4 T_1))) (leq T_1 T_1) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_3) (T_2 = T_3) (-. (memberP T_2 T_1)) (ssList T_2) (ssItem T_1) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (-. ((-. (memberP T_3 T_1)) /\ (memberP T_4 T_1))) ### NotNot 42
% 148.02/148.36 44. (-. (memberP T_3 T_1)) (memberP T_3 T_1) ### Axiom
% 148.02/148.36 45. (-. (-. (memberP T_3 T_1))) (-. (memberP T_3 T_1)) ### NotNot 44
% 148.02/148.36 46. (ssItem T_1) (-. (ssItem T_1)) ### Axiom
% 148.02/148.36 47. (ssList T_4) (-. (ssList T_4)) ### Axiom
% 148.02/148.36 48. (ssItem T_1) (-. (ssItem T_1)) ### Axiom
% 148.02/148.36 49. (ssList T_3) (-. (ssList T_3)) ### Axiom
% 148.02/148.36 50. (ssItem T_1) (-. (ssItem T_1)) ### Axiom
% 148.02/148.36 51. (ssList T_0) (-. (ssList T_0)) ### Axiom
% 148.02/148.36 52. (ssList T_3) (-. (ssList T_3)) ### Axiom
% 148.02/148.36 53. (memberP T_0 T_1) (-. (memberP T_0 T_1)) ### Axiom
% 148.02/148.36 54. (-. ((memberP T_0 T_1) \/ (memberP T_3 T_1))) (memberP T_0 T_1) ### NotOr 53
% 148.02/148.36 55. (T_0 = T_4) (T_0 != T_4) ### Axiom
% 148.02/148.36 56. (T_3 != T_3) ### Refl(=)
% 148.02/148.36 57. ((app T_0 T_3) != (app T_4 T_3)) (T_0 = T_4) ### NotEqual 55 56
% 148.02/148.36 58. (-. (T_1 != T_1)) (T_1 != T_1) ### Axiom
% 148.02/148.36 59. (-. (memberP (app T_4 T_3) T_1)) (memberP (app T_0 T_3) T_1) (-. (T_1 != T_1)) (T_0 = T_4) ### P-NotP 57 58
% 148.02/148.36 60. ((memberP (app T_0 T_3) T_1) <=> ((memberP T_0 T_1) \/ (memberP T_3 T_1))) (T_0 = T_4) (-. (T_1 != T_1)) (-. (memberP (app T_4 T_3) T_1)) (memberP T_0 T_1) ### Equiv 54 59
% 148.02/148.36 61. ((ssList T_3) => ((memberP (app T_0 T_3) T_1) <=> ((memberP T_0 T_1) \/ (memberP T_3 T_1)))) (memberP T_0 T_1) (-. (memberP (app T_4 T_3) T_1)) (-. (T_1 != T_1)) (T_0 = T_4) (ssList T_3) ### Imply 52 60
% 148.14/148.42 62. (All W, ((ssList W) => ((memberP (app T_0 W) T_1) <=> ((memberP T_0 T_1) \/ (memberP W T_1))))) (ssList T_3) (T_0 = T_4) (-. (T_1 != T_1)) (-. (memberP (app T_4 T_3) T_1)) (memberP T_0 T_1) ### All 61
% 148.14/148.42 63. ((ssList T_0) => (All W, ((ssList W) => ((memberP (app T_0 W) T_1) <=> ((memberP T_0 T_1) \/ (memberP W T_1)))))) (memberP T_0 T_1) (-. (memberP (app T_4 T_3) T_1)) (-. (T_1 != T_1)) (T_0 = T_4) (ssList T_3) (ssList T_0) ### Imply 51 62
% 148.14/148.42 64. (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_0) (ssList T_3) (T_0 = T_4) (-. (T_1 != T_1)) (-. (memberP (app T_4 T_3) T_1)) (memberP T_0 T_1) ### All 63
% 148.14/148.42 65. (leq T_1 T_1) (-. (leq T_1 T_1)) ### Axiom
% 148.14/148.42 66. (-. ((T_1 != T_1) /\ (leq T_1 T_1))) (leq T_1 T_1) (memberP T_0 T_1) (-. (memberP (app T_4 T_3) T_1)) (T_0 = T_4) (ssList T_3) (ssList T_0) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) ### NotAnd 64 65
% 148.14/148.42 67. (-. (lt T_1 T_1)) (lt T_1 T_1) ### Axiom
% 148.14/148.42 68. ((lt T_1 T_1) <=> ((T_1 != T_1) /\ (leq T_1 T_1))) (-. (lt T_1 T_1)) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_0) (ssList T_3) (T_0 = T_4) (-. (memberP (app T_4 T_3) T_1)) (memberP T_0 T_1) (leq T_1 T_1) ### Equiv 66 67
% 148.14/148.42 69. ((ssItem T_1) => ((lt T_1 T_1) <=> ((T_1 != T_1) /\ (leq T_1 T_1)))) (leq T_1 T_1) (memberP T_0 T_1) (-. (memberP (app T_4 T_3) T_1)) (T_0 = T_4) (ssList T_3) (ssList T_0) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (-. (lt T_1 T_1)) (ssItem T_1) ### Imply 50 68
% 148.14/148.42 70. (All V, ((ssItem V) => ((lt T_1 V) <=> ((T_1 != V) /\ (leq T_1 V))))) (ssItem T_1) (-. (lt T_1 T_1)) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_0) (ssList T_3) (T_0 = T_4) (-. (memberP (app T_4 T_3) T_1)) (memberP T_0 T_1) (leq T_1 T_1) ### All 69
% 148.14/148.42 71. (-. (memberP T_4 T_1)) (memberP T_4 T_1) ### Axiom
% 148.14/148.42 72. (-. (memberP T_3 T_1)) (memberP T_3 T_1) ### Axiom
% 148.14/148.42 73. ((memberP T_4 T_1) \/ (memberP T_3 T_1)) (-. (memberP T_3 T_1)) (-. (memberP T_4 T_1)) ### Or 71 72
% 148.14/148.42 74. ((memberP (app T_4 T_3) T_1) <=> ((memberP T_4 T_1) \/ (memberP T_3 T_1))) (-. (memberP T_4 T_1)) (-. (memberP T_3 T_1)) (leq T_1 T_1) (memberP T_0 T_1) (T_0 = T_4) (ssList T_3) (ssList T_0) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (-. (lt T_1 T_1)) (ssItem T_1) (All V, ((ssItem V) => ((lt T_1 V) <=> ((T_1 != V) /\ (leq T_1 V))))) ### Equiv 70 73
% 148.14/148.42 75. ((ssList T_3) => ((memberP (app T_4 T_3) T_1) <=> ((memberP T_4 T_1) \/ (memberP T_3 T_1)))) (All V, ((ssItem V) => ((lt T_1 V) <=> ((T_1 != V) /\ (leq T_1 V))))) (ssItem T_1) (-. (lt T_1 T_1)) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_0) (T_0 = T_4) (memberP T_0 T_1) (leq T_1 T_1) (-. (memberP T_3 T_1)) (-. (memberP T_4 T_1)) (ssList T_3) ### Imply 49 74
% 148.14/148.42 76. (All W, ((ssList W) => ((memberP (app T_4 W) T_1) <=> ((memberP T_4 T_1) \/ (memberP W T_1))))) (ssList T_3) (-. (memberP T_4 T_1)) (-. (memberP T_3 T_1)) (leq T_1 T_1) (memberP T_0 T_1) (T_0 = T_4) (ssList T_0) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (-. (lt T_1 T_1)) (ssItem T_1) (All V, ((ssItem V) => ((lt T_1 V) <=> ((T_1 != V) /\ (leq T_1 V))))) ### All 75
% 148.14/148.42 77. ((ssItem T_1) => (-. (lt T_1 T_1))) (All V, ((ssItem V) => ((lt T_1 V) <=> ((T_1 != V) /\ (leq T_1 V))))) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_0) (T_0 = T_4) (memberP T_0 T_1) (leq T_1 T_1) (-. (memberP T_3 T_1)) (-. (memberP T_4 T_1)) (ssList T_3) (All W, ((ssList W) => ((memberP (app T_4 W) T_1) <=> ((memberP T_4 T_1) \/ (memberP W T_1))))) (ssItem T_1) ### Imply 48 76
% 148.14/148.42 78. (All U, ((ssItem U) => (-. (lt U U)))) (ssItem T_1) (All W, ((ssList W) => ((memberP (app T_4 W) T_1) <=> ((memberP T_4 T_1) \/ (memberP W T_1))))) (ssList T_3) (-. (memberP T_4 T_1)) (-. (memberP T_3 T_1)) (leq T_1 T_1) (memberP T_0 T_1) (T_0 = T_4) (ssList T_0) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (All V, ((ssItem V) => ((lt T_1 V) <=> ((T_1 != V) /\ (leq T_1 V))))) ### All 77
% 148.14/148.42 79. ((ssList T_4) => (All W, ((ssList W) => ((memberP (app T_4 W) T_1) <=> ((memberP T_4 T_1) \/ (memberP W T_1)))))) (All V, ((ssItem V) => ((lt T_1 V) <=> ((T_1 != V) /\ (leq T_1 V))))) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_0) (T_0 = T_4) (memberP T_0 T_1) (leq T_1 T_1) (-. (memberP T_3 T_1)) (-. (memberP T_4 T_1)) (ssList T_3) (ssItem T_1) (All U, ((ssItem U) => (-. (lt U U)))) (ssList T_4) ### Imply 47 78
% 148.14/148.42 80. (ssList T_4) (All U, ((ssItem U) => (-. (lt U U)))) (ssItem T_1) (ssList T_3) (-. (memberP T_4 T_1)) (-. (memberP T_3 T_1)) (leq T_1 T_1) (memberP T_0 T_1) (T_0 = T_4) (ssList T_0) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (All V, ((ssItem V) => ((lt T_1 V) <=> ((T_1 != V) /\ (leq T_1 V))))) ### All 79
% 148.14/148.42 81. ((ssItem T_1) => (All V, ((ssItem V) => ((lt T_1 V) <=> ((T_1 != V) /\ (leq T_1 V)))))) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_0) (T_0 = T_4) (memberP T_0 T_1) (leq T_1 T_1) (-. (memberP T_3 T_1)) (-. (memberP T_4 T_1)) (ssList T_3) (All U, ((ssItem U) => (-. (lt U U)))) (ssList T_4) (ssItem T_1) ### Imply 46 80
% 148.14/148.42 82. (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssItem T_1) (ssList T_4) (All U, ((ssItem U) => (-. (lt U U)))) (ssList T_3) (-. (memberP T_4 T_1)) (-. (memberP T_3 T_1)) (leq T_1 T_1) (memberP T_0 T_1) (T_0 = T_4) (ssList T_0) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) ### All 81
% 148.14/148.42 83. (-. ((-. (memberP T_3 T_1)) /\ (memberP T_4 T_1))) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_0) (T_0 = T_4) (memberP T_0 T_1) (leq T_1 T_1) (ssList T_3) (All U, ((ssItem U) => (-. (lt U U)))) (ssList T_4) (ssItem T_1) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (-. (memberP T_3 T_1)) ### NotAnd 45 82
% 148.14/148.42 84. (-. ((-. (memberP T_4 T_1)) /\ (memberP T_3 T_1))) (ssList T_4) (All U, ((ssItem U) => (-. (lt U U)))) (memberP T_0 T_1) (T_0 = T_4) (ssList T_0) (-. ((-. (memberP T_3 T_1)) /\ (memberP T_4 T_1))) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssItem T_1) (ssList T_2) (-. (memberP T_2 T_1)) (T_2 = T_3) (ssList T_3) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (leq T_1 T_1) ### NotAnd 43 83
% 148.14/148.42 85. (-. (((-. (memberP T_4 T_1)) /\ (memberP T_3 T_1)) \/ ((-. (memberP T_3 T_1)) /\ (memberP T_4 T_1)))) (leq T_1 T_1) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_3) (T_2 = T_3) (-. (memberP T_2 T_1)) (ssList T_2) (ssItem T_1) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssList T_0) (T_0 = T_4) (memberP T_0 T_1) (All U, ((ssItem U) => (-. (lt U U)))) (ssList T_4) ### NotOr 84
% 148.14/148.42 86. (-. ((ssItem T_1) /\ (((-. (memberP T_4 T_1)) /\ (memberP T_3 T_1)) \/ ((-. (memberP T_3 T_1)) /\ (memberP T_4 T_1))))) (ssList T_4) (All U, ((ssItem U) => (-. (lt U U)))) (memberP T_0 T_1) (T_0 = T_4) (ssList T_0) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssList T_2) (-. (memberP T_2 T_1)) (T_2 = T_3) (ssList T_3) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (leq T_1 T_1) (ssItem T_1) ### NotAnd 4 85
% 148.17/148.47 87. (-. (Ex Y, ((ssItem Y) /\ (((-. (memberP T_4 Y)) /\ (memberP T_3 Y)) \/ ((-. (memberP T_3 Y)) /\ (memberP T_4 Y)))))) (ssItem T_1) (leq T_1 T_1) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_3) (T_2 = T_3) (-. (memberP T_2 T_1)) (ssList T_2) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssList T_0) (T_0 = T_4) (memberP T_0 T_1) (All U, ((ssItem U) => (-. (lt U U)))) (ssList T_4) ### NotExists 86
% 148.17/148.47 88. ((ssItem T_1) => (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1)))))))) (ssList T_4) (All U, ((ssItem U) => (-. (lt U U)))) (memberP T_0 T_1) (T_0 = T_4) (ssList T_0) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssList T_2) (-. (memberP T_2 T_1)) (T_2 = T_3) (ssList T_3) (leq T_1 T_1) (-. (Ex Y, ((ssItem Y) /\ (((-. (memberP T_4 Y)) /\ (memberP T_3 Y)) \/ ((-. (memberP T_3 Y)) /\ (memberP T_4 Y)))))) (ssItem T_1) ### Imply 3 87
% 148.17/148.47 89. (All U, ((ssItem U) => (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) U) <=> ((memberP V U) \/ (memberP W U))))))))) (ssItem T_1) (-. (Ex Y, ((ssItem Y) /\ (((-. (memberP T_4 Y)) /\ (memberP T_3 Y)) \/ ((-. (memberP T_3 Y)) /\ (memberP T_4 Y)))))) (leq T_1 T_1) (ssList T_3) (T_2 = T_3) (-. (memberP T_2 T_1)) (ssList T_2) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssList T_0) (T_0 = T_4) (memberP T_0 T_1) (All U, ((ssItem U) => (-. (lt U U)))) (ssList T_4) ### All 88
% 148.17/148.47 90. ((ssItem T_1) => (leq T_1 T_1)) (ssList T_4) (All U, ((ssItem U) => (-. (lt U U)))) (memberP T_0 T_1) (T_0 = T_4) (ssList T_0) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssList T_2) (-. (memberP T_2 T_1)) (T_2 = T_3) (ssList T_3) (-. (Ex Y, ((ssItem Y) /\ (((-. (memberP T_4 Y)) /\ (memberP T_3 Y)) \/ ((-. (memberP T_3 Y)) /\ (memberP T_4 Y)))))) (All U, ((ssItem U) => (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) U) <=> ((memberP V U) \/ (memberP W U))))))))) (ssItem T_1) ### Imply 2 89
% 148.17/148.47 91. (All U, ((ssItem U) => (leq U U))) (ssItem T_1) (All U, ((ssItem U) => (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) U) <=> ((memberP V U) \/ (memberP W U))))))))) (-. (Ex Y, ((ssItem Y) /\ (((-. (memberP T_4 Y)) /\ (memberP T_3 Y)) \/ ((-. (memberP T_3 Y)) /\ (memberP T_4 Y)))))) (ssList T_3) (T_2 = T_3) (-. (memberP T_2 T_1)) (ssList T_2) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssList T_0) (T_0 = T_4) (memberP T_0 T_1) (All U, ((ssItem U) => (-. (lt U U)))) (ssList T_4) ### All 90
% 148.17/148.47 92. (-. ((memberP T_0 T_1) /\ (memberP T_2 T_1))) (ssList T_4) (All U, ((ssItem U) => (-. (lt U U)))) (T_0 = T_4) (ssList T_0) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssList T_2) (T_2 = T_3) (ssList T_3) (-. (Ex Y, ((ssItem Y) /\ (((-. (memberP T_4 Y)) /\ (memberP T_3 Y)) \/ ((-. (memberP T_3 Y)) /\ (memberP T_4 Y)))))) (All U, ((ssItem U) => (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) U) <=> ((memberP V U) \/ (memberP W U))))))))) (ssItem T_1) (All U, ((ssItem U) => (leq U U))) (memberP T_0 T_1) ### NotAnd 1 91
% 148.17/148.47 93. (-. (-. (memberP T_0 T_1))) (All U, ((ssItem U) => (leq U U))) (ssItem T_1) (All U, ((ssItem U) => (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) U) <=> ((memberP V U) \/ (memberP W U))))))))) (-. (Ex Y, ((ssItem Y) /\ (((-. (memberP T_4 Y)) /\ (memberP T_3 Y)) \/ ((-. (memberP T_3 Y)) /\ (memberP T_4 Y)))))) (ssList T_3) (T_2 = T_3) (ssList T_2) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssList T_0) (T_0 = T_4) (All U, ((ssItem U) => (-. (lt U U)))) (ssList T_4) (-. ((memberP T_0 T_1) /\ (memberP T_2 T_1))) ### NotNot 92
% 148.17/148.47 94. (ssItem T_1) (-. (ssItem T_1)) ### Axiom
% 148.17/148.47 95. (ssItem T_1) (-. (ssItem T_1)) ### Axiom
% 148.17/148.47 96. (ssItem T_1) (-. (ssItem T_1)) ### Axiom
% 148.17/148.47 97. (ssItem T_1) (-. (ssItem T_1)) ### Axiom
% 148.17/148.47 98. (ssItem T_1) (-. (ssItem T_1)) ### Axiom
% 148.17/148.47 99. (ssList T_0) (-. (ssList T_0)) ### Axiom
% 148.17/148.47 100. (ssList T_0) (-. (ssList T_0)) ### Axiom
% 148.17/148.47 101. (ssList T_4) (-. (ssList T_4)) ### Axiom
% 148.17/148.47 102. (ssList T_0) (-. (ssList T_0)) ### Axiom
% 148.17/148.47 103. (memberP T_4 T_1) (-. (memberP T_4 T_1)) ### Axiom
% 148.17/148.47 104. (-. ((memberP T_4 T_1) \/ (memberP T_0 T_1))) (memberP T_4 T_1) ### NotOr 103
% 148.17/148.47 105. (T_0 = T_4) (T_4 != T_0) ### Sym(=)
% 148.17/148.47 106. (T_0 != T_0) ### Refl(=)
% 148.17/148.47 107. ((app T_4 T_0) != (app T_0 T_0)) (T_0 = T_4) ### NotEqual 105 106
% 148.17/148.47 108. (-. (T_1 != T_1)) (T_1 != T_1) ### Axiom
% 148.17/148.47 109. (-. (memberP (app T_0 T_0) T_1)) (memberP (app T_4 T_0) T_1) (-. (T_1 != T_1)) (T_0 = T_4) ### P-NotP 107 108
% 148.17/148.47 110. ((memberP (app T_4 T_0) T_1) <=> ((memberP T_4 T_1) \/ (memberP T_0 T_1))) (T_0 = T_4) (-. (T_1 != T_1)) (-. (memberP (app T_0 T_0) T_1)) (memberP T_4 T_1) ### Equiv 104 109
% 148.17/148.47 111. ((ssList T_0) => ((memberP (app T_4 T_0) T_1) <=> ((memberP T_4 T_1) \/ (memberP T_0 T_1)))) (memberP T_4 T_1) (-. (memberP (app T_0 T_0) T_1)) (-. (T_1 != T_1)) (T_0 = T_4) (ssList T_0) ### Imply 102 110
% 148.17/148.47 112. (All W, ((ssList W) => ((memberP (app T_4 W) T_1) <=> ((memberP T_4 T_1) \/ (memberP W T_1))))) (ssList T_0) (T_0 = T_4) (-. (T_1 != T_1)) (-. (memberP (app T_0 T_0) T_1)) (memberP T_4 T_1) ### All 111
% 148.17/148.47 113. ((ssList T_4) => (All W, ((ssList W) => ((memberP (app T_4 W) T_1) <=> ((memberP T_4 T_1) \/ (memberP W T_1)))))) (memberP T_4 T_1) (-. (memberP (app T_0 T_0) T_1)) (-. (T_1 != T_1)) (T_0 = T_4) (ssList T_0) (ssList T_4) ### Imply 101 112
% 148.17/148.47 114. (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_4) (ssList T_0) (T_0 = T_4) (-. (T_1 != T_1)) (-. (memberP (app T_0 T_0) T_1)) (memberP T_4 T_1) ### All 113
% 148.17/148.47 115. (-. (memberP T_0 T_1)) (memberP T_0 T_1) ### Axiom
% 148.17/148.47 116. (-. (memberP T_0 T_1)) (memberP T_0 T_1) ### Axiom
% 148.17/148.47 117. ((memberP T_0 T_1) \/ (memberP T_0 T_1)) (-. (memberP T_0 T_1)) ### Or 115 116
% 148.17/148.47 118. ((memberP (app T_0 T_0) T_1) <=> ((memberP T_0 T_1) \/ (memberP T_0 T_1))) (-. (memberP T_0 T_1)) (memberP T_4 T_1) (-. (T_1 != T_1)) (T_0 = T_4) (ssList T_0) (ssList T_4) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) ### Equiv 114 117
% 148.17/148.47 119. ((ssList T_0) => ((memberP (app T_0 T_0) T_1) <=> ((memberP T_0 T_1) \/ (memberP T_0 T_1)))) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_4) (T_0 = T_4) (-. (T_1 != T_1)) (memberP T_4 T_1) (-. (memberP T_0 T_1)) (ssList T_0) ### Imply 100 118
% 148.17/148.47 120. (All W, ((ssList W) => ((memberP (app T_0 W) T_1) <=> ((memberP T_0 T_1) \/ (memberP W T_1))))) (ssList T_0) (-. (memberP T_0 T_1)) (memberP T_4 T_1) (-. (T_1 != T_1)) (T_0 = T_4) (ssList T_4) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) ### All 119
% 148.17/148.47 121. ((ssList T_0) => (All W, ((ssList W) => ((memberP (app T_0 W) T_1) <=> ((memberP T_0 T_1) \/ (memberP W T_1)))))) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_4) (T_0 = T_4) (-. (T_1 != T_1)) (memberP T_4 T_1) (-. (memberP T_0 T_1)) (ssList T_0) ### Imply 99 120
% 148.17/148.47 122. (ssList T_0) (-. (memberP T_0 T_1)) (memberP T_4 T_1) (-. (T_1 != T_1)) (T_0 = T_4) (ssList T_4) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) ### All 121
% 148.17/148.47 123. (leq T_1 T_1) (-. (leq T_1 T_1)) ### Axiom
% 148.17/148.47 124. (-. ((T_1 != T_1) /\ (leq T_1 T_1))) (leq T_1 T_1) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_4) (T_0 = T_4) (memberP T_4 T_1) (-. (memberP T_0 T_1)) (ssList T_0) ### NotAnd 122 123
% 148.17/148.47 125. ((lt T_1 T_1) <=> ((T_1 != T_1) /\ (leq T_1 T_1))) (ssList T_0) (-. (memberP T_0 T_1)) (memberP T_4 T_1) (T_0 = T_4) (ssList T_4) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (leq T_1 T_1) ### Equiv 124 34
% 148.17/148.50 126. ((ssItem T_1) => ((lt T_1 T_1) <=> ((T_1 != T_1) /\ (leq T_1 T_1)))) (leq T_1 T_1) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_4) (T_0 = T_4) (memberP T_4 T_1) (-. (memberP T_0 T_1)) (ssList T_0) (ssItem T_1) ### Imply 98 125
% 148.17/148.50 127. (All V, ((ssItem V) => ((lt T_1 V) <=> ((T_1 != V) /\ (leq T_1 V))))) (ssItem T_1) (ssList T_0) (-. (memberP T_0 T_1)) (memberP T_4 T_1) (T_0 = T_4) (ssList T_4) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (leq T_1 T_1) ### All 126
% 148.17/148.50 128. ((ssItem T_1) => (All V, ((ssItem V) => ((lt T_1 V) <=> ((T_1 != V) /\ (leq T_1 V)))))) (leq T_1 T_1) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_4) (T_0 = T_4) (memberP T_4 T_1) (-. (memberP T_0 T_1)) (ssList T_0) (ssItem T_1) ### Imply 97 127
% 148.17/148.50 129. (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssItem T_1) (ssList T_0) (-. (memberP T_0 T_1)) (memberP T_4 T_1) (T_0 = T_4) (ssList T_4) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (leq T_1 T_1) ### All 128
% 148.17/148.50 130. (-. (-. (memberP T_4 T_1))) (leq T_1 T_1) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_4) (T_0 = T_4) (-. (memberP T_0 T_1)) (ssList T_0) (ssItem T_1) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) ### NotNot 129
% 148.17/148.50 131. (ssItem T_1) (-. (ssItem T_1)) ### Axiom
% 148.17/148.50 132. (ssItem T_1) (-. (ssItem T_1)) ### Axiom
% 148.17/148.50 133. (ssList T_3) (-. (ssList T_3)) ### Axiom
% 148.17/148.50 134. (ssList (nil)) (-. (ssList (nil))) ### Axiom
% 148.17/148.50 135. (ssItem T_1) (-. (ssItem T_1)) ### Axiom
% 148.17/148.50 136. (ssItem T_1) (-. (ssItem T_1)) ### Axiom
% 148.17/148.50 137. (ssList T_2) (-. (ssList T_2)) ### Axiom
% 148.17/148.50 138. (ssList (nil)) (-. (ssList (nil))) ### Axiom
% 148.17/148.50 139. (memberP T_2 T_1) (-. (memberP T_2 T_1)) ### Axiom
% 148.17/148.50 140. (-. ((memberP T_2 T_1) \/ (memberP (nil) T_1))) (memberP T_2 T_1) ### NotOr 139
% 148.17/148.50 141. (T_2 = T_3) (T_2 != T_3) ### Axiom
% 148.17/148.50 142. ((nil) != (nil)) ### NotEqual
% 148.17/148.50 143. ((app T_2 (nil)) != (app T_3 (nil))) (T_2 = T_3) ### NotEqual 141 142
% 148.17/148.50 144. (-. (T_1 != T_1)) (T_1 != T_1) ### Axiom
% 148.17/148.50 145. (-. (memberP (app T_3 (nil)) T_1)) (memberP (app T_2 (nil)) T_1) (-. (T_1 != T_1)) (T_2 = T_3) ### P-NotP 143 144
% 148.17/148.50 146. ((memberP (app T_2 (nil)) T_1) <=> ((memberP T_2 T_1) \/ (memberP (nil) T_1))) (T_2 = T_3) (-. (T_1 != T_1)) (-. (memberP (app T_3 (nil)) T_1)) (memberP T_2 T_1) ### Equiv 140 145
% 148.17/148.50 147. ((ssList (nil)) => ((memberP (app T_2 (nil)) T_1) <=> ((memberP T_2 T_1) \/ (memberP (nil) T_1)))) (memberP T_2 T_1) (-. (memberP (app T_3 (nil)) T_1)) (-. (T_1 != T_1)) (T_2 = T_3) (ssList (nil)) ### Imply 138 146
% 148.17/148.50 148. (All W, ((ssList W) => ((memberP (app T_2 W) T_1) <=> ((memberP T_2 T_1) \/ (memberP W T_1))))) (ssList (nil)) (T_2 = T_3) (-. (T_1 != T_1)) (-. (memberP (app T_3 (nil)) T_1)) (memberP T_2 T_1) ### All 147
% 148.17/148.50 149. ((ssList T_2) => (All W, ((ssList W) => ((memberP (app T_2 W) T_1) <=> ((memberP T_2 T_1) \/ (memberP W T_1)))))) (memberP T_2 T_1) (-. (memberP (app T_3 (nil)) T_1)) (-. (T_1 != T_1)) (T_2 = T_3) (ssList (nil)) (ssList T_2) ### Imply 137 148
% 148.17/148.50 150. (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_2) (ssList (nil)) (T_2 = T_3) (-. (T_1 != T_1)) (-. (memberP (app T_3 (nil)) T_1)) (memberP T_2 T_1) ### All 149
% 148.17/148.50 151. (leq T_1 T_1) (-. (leq T_1 T_1)) ### Axiom
% 148.17/148.50 152. (-. ((T_1 != T_1) /\ (leq T_1 T_1))) (leq T_1 T_1) (memberP T_2 T_1) (-. (memberP (app T_3 (nil)) T_1)) (T_2 = T_3) (ssList (nil)) (ssList T_2) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) ### NotAnd 150 151
% 148.17/148.50 153. (-. (lt T_1 T_1)) (lt T_1 T_1) ### Axiom
% 148.17/148.50 154. ((lt T_1 T_1) <=> ((T_1 != T_1) /\ (leq T_1 T_1))) (-. (lt T_1 T_1)) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_2) (ssList (nil)) (T_2 = T_3) (-. (memberP (app T_3 (nil)) T_1)) (memberP T_2 T_1) (leq T_1 T_1) ### Equiv 152 153
% 148.17/148.50 155. ((ssItem T_1) => ((lt T_1 T_1) <=> ((T_1 != T_1) /\ (leq T_1 T_1)))) (leq T_1 T_1) (memberP T_2 T_1) (-. (memberP (app T_3 (nil)) T_1)) (T_2 = T_3) (ssList (nil)) (ssList T_2) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (-. (lt T_1 T_1)) (ssItem T_1) ### Imply 136 154
% 148.17/148.50 156. (All V, ((ssItem V) => ((lt T_1 V) <=> ((T_1 != V) /\ (leq T_1 V))))) (ssItem T_1) (-. (lt T_1 T_1)) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_2) (ssList (nil)) (T_2 = T_3) (-. (memberP (app T_3 (nil)) T_1)) (memberP T_2 T_1) (leq T_1 T_1) ### All 155
% 148.17/148.50 157. ((ssItem T_1) => (-. (lt T_1 T_1))) (leq T_1 T_1) (memberP T_2 T_1) (-. (memberP (app T_3 (nil)) T_1)) (T_2 = T_3) (ssList (nil)) (ssList T_2) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (All V, ((ssItem V) => ((lt T_1 V) <=> ((T_1 != V) /\ (leq T_1 V))))) (ssItem T_1) ### Imply 135 156
% 148.17/148.50 158. (All U, ((ssItem U) => (-. (lt U U)))) (ssItem T_1) (All V, ((ssItem V) => ((lt T_1 V) <=> ((T_1 != V) /\ (leq T_1 V))))) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_2) (ssList (nil)) (T_2 = T_3) (-. (memberP (app T_3 (nil)) T_1)) (memberP T_2 T_1) (leq T_1 T_1) ### All 157
% 148.17/148.50 159. (-. (memberP T_3 T_1)) (memberP T_3 T_1) ### Axiom
% 148.17/148.50 160. (-. (memberP (nil) T_1)) (memberP (nil) T_1) ### Axiom
% 148.17/148.50 161. ((memberP T_3 T_1) \/ (memberP (nil) T_1)) (-. (memberP (nil) T_1)) (-. (memberP T_3 T_1)) ### Or 159 160
% 148.17/148.50 162. ((memberP (app T_3 (nil)) T_1) <=> ((memberP T_3 T_1) \/ (memberP (nil) T_1))) (-. (memberP T_3 T_1)) (-. (memberP (nil) T_1)) (leq T_1 T_1) (memberP T_2 T_1) (T_2 = T_3) (ssList (nil)) (ssList T_2) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (All V, ((ssItem V) => ((lt T_1 V) <=> ((T_1 != V) /\ (leq T_1 V))))) (ssItem T_1) (All U, ((ssItem U) => (-. (lt U U)))) ### Equiv 158 161
% 148.17/148.50 163. ((ssList (nil)) => ((memberP (app T_3 (nil)) T_1) <=> ((memberP T_3 T_1) \/ (memberP (nil) T_1)))) (All U, ((ssItem U) => (-. (lt U U)))) (ssItem T_1) (All V, ((ssItem V) => ((lt T_1 V) <=> ((T_1 != V) /\ (leq T_1 V))))) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_2) (T_2 = T_3) (memberP T_2 T_1) (leq T_1 T_1) (-. (memberP (nil) T_1)) (-. (memberP T_3 T_1)) (ssList (nil)) ### Imply 134 162
% 148.17/148.50 164. (All W, ((ssList W) => ((memberP (app T_3 W) T_1) <=> ((memberP T_3 T_1) \/ (memberP W T_1))))) (ssList (nil)) (-. (memberP T_3 T_1)) (-. (memberP (nil) T_1)) (leq T_1 T_1) (memberP T_2 T_1) (T_2 = T_3) (ssList T_2) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (All V, ((ssItem V) => ((lt T_1 V) <=> ((T_1 != V) /\ (leq T_1 V))))) (ssItem T_1) (All U, ((ssItem U) => (-. (lt U U)))) ### All 163
% 148.17/148.50 165. ((ssList T_3) => (All W, ((ssList W) => ((memberP (app T_3 W) T_1) <=> ((memberP T_3 T_1) \/ (memberP W T_1)))))) (All U, ((ssItem U) => (-. (lt U U)))) (ssItem T_1) (All V, ((ssItem V) => ((lt T_1 V) <=> ((T_1 != V) /\ (leq T_1 V))))) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_2) (T_2 = T_3) (memberP T_2 T_1) (leq T_1 T_1) (-. (memberP (nil) T_1)) (-. (memberP T_3 T_1)) (ssList (nil)) (ssList T_3) ### Imply 133 164
% 148.17/148.50 166. (ssList T_3) (ssList (nil)) (-. (memberP T_3 T_1)) (-. (memberP (nil) T_1)) (leq T_1 T_1) (memberP T_2 T_1) (T_2 = T_3) (ssList T_2) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (All V, ((ssItem V) => ((lt T_1 V) <=> ((T_1 != V) /\ (leq T_1 V))))) (ssItem T_1) (All U, ((ssItem U) => (-. (lt U U)))) ### All 165
% 148.28/148.54 167. ((ssItem T_1) => (All V, ((ssItem V) => ((lt T_1 V) <=> ((T_1 != V) /\ (leq T_1 V)))))) (All U, ((ssItem U) => (-. (lt U U)))) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_2) (T_2 = T_3) (memberP T_2 T_1) (leq T_1 T_1) (-. (memberP (nil) T_1)) (-. (memberP T_3 T_1)) (ssList (nil)) (ssList T_3) (ssItem T_1) ### Imply 132 166
% 148.28/148.54 168. (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssItem T_1) (ssList T_3) (ssList (nil)) (-. (memberP T_3 T_1)) (-. (memberP (nil) T_1)) (leq T_1 T_1) (memberP T_2 T_1) (T_2 = T_3) (ssList T_2) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (All U, ((ssItem U) => (-. (lt U U)))) ### All 167
% 148.28/148.54 169. ((ssItem T_1) => (-. (memberP (nil) T_1))) (All U, ((ssItem U) => (-. (lt U U)))) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_2) (T_2 = T_3) (memberP T_2 T_1) (leq T_1 T_1) (-. (memberP T_3 T_1)) (ssList (nil)) (ssList T_3) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssItem T_1) ### Imply 131 168
% 148.28/148.54 170. (All U, ((ssItem U) => (-. (memberP (nil) U)))) (ssItem T_1) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssList T_3) (ssList (nil)) (-. (memberP T_3 T_1)) (leq T_1 T_1) (memberP T_2 T_1) (T_2 = T_3) (ssList T_2) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (All U, ((ssItem U) => (-. (lt U U)))) ### All 169
% 148.28/148.54 171. (-. ((-. (memberP T_4 T_1)) /\ (memberP T_3 T_1))) (All U, ((ssItem U) => (-. (lt U U)))) (ssList T_2) (T_2 = T_3) (memberP T_2 T_1) (ssList (nil)) (ssList T_3) (All U, ((ssItem U) => (-. (memberP (nil) U)))) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssItem T_1) (ssList T_0) (-. (memberP T_0 T_1)) (T_0 = T_4) (ssList T_4) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (leq T_1 T_1) ### NotAnd 130 170
% 148.28/148.54 172. (-. (((-. (memberP T_4 T_1)) /\ (memberP T_3 T_1)) \/ ((-. (memberP T_3 T_1)) /\ (memberP T_4 T_1)))) (leq T_1 T_1) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_4) (T_0 = T_4) (-. (memberP T_0 T_1)) (ssList T_0) (ssItem T_1) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (All U, ((ssItem U) => (-. (memberP (nil) U)))) (ssList T_3) (ssList (nil)) (memberP T_2 T_1) (T_2 = T_3) (ssList T_2) (All U, ((ssItem U) => (-. (lt U U)))) ### NotOr 171
% 148.28/148.54 173. (-. ((ssItem T_1) /\ (((-. (memberP T_4 T_1)) /\ (memberP T_3 T_1)) \/ ((-. (memberP T_3 T_1)) /\ (memberP T_4 T_1))))) (All U, ((ssItem U) => (-. (lt U U)))) (ssList T_2) (T_2 = T_3) (memberP T_2 T_1) (ssList (nil)) (ssList T_3) (All U, ((ssItem U) => (-. (memberP (nil) U)))) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssList T_0) (-. (memberP T_0 T_1)) (T_0 = T_4) (ssList T_4) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (leq T_1 T_1) (ssItem T_1) ### NotAnd 96 172
% 148.28/148.54 174. (-. (Ex Y, ((ssItem Y) /\ (((-. (memberP T_4 Y)) /\ (memberP T_3 Y)) \/ ((-. (memberP T_3 Y)) /\ (memberP T_4 Y)))))) (ssItem T_1) (leq T_1 T_1) (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1))))))) (ssList T_4) (T_0 = T_4) (-. (memberP T_0 T_1)) (ssList T_0) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (All U, ((ssItem U) => (-. (memberP (nil) U)))) (ssList T_3) (ssList (nil)) (memberP T_2 T_1) (T_2 = T_3) (ssList T_2) (All U, ((ssItem U) => (-. (lt U U)))) ### NotExists 173
% 148.28/148.54 175. ((ssItem T_1) => (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) T_1) <=> ((memberP V T_1) \/ (memberP W T_1)))))))) (All U, ((ssItem U) => (-. (lt U U)))) (ssList T_2) (T_2 = T_3) (memberP T_2 T_1) (ssList (nil)) (ssList T_3) (All U, ((ssItem U) => (-. (memberP (nil) U)))) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssList T_0) (-. (memberP T_0 T_1)) (T_0 = T_4) (ssList T_4) (leq T_1 T_1) (-. (Ex Y, ((ssItem Y) /\ (((-. (memberP T_4 Y)) /\ (memberP T_3 Y)) \/ ((-. (memberP T_3 Y)) /\ (memberP T_4 Y)))))) (ssItem T_1) ### Imply 95 174
% 148.28/148.54 176. (All U, ((ssItem U) => (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) U) <=> ((memberP V U) \/ (memberP W U))))))))) (ssItem T_1) (-. (Ex Y, ((ssItem Y) /\ (((-. (memberP T_4 Y)) /\ (memberP T_3 Y)) \/ ((-. (memberP T_3 Y)) /\ (memberP T_4 Y)))))) (leq T_1 T_1) (ssList T_4) (T_0 = T_4) (-. (memberP T_0 T_1)) (ssList T_0) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (All U, ((ssItem U) => (-. (memberP (nil) U)))) (ssList T_3) (ssList (nil)) (memberP T_2 T_1) (T_2 = T_3) (ssList T_2) (All U, ((ssItem U) => (-. (lt U U)))) ### All 175
% 148.28/148.54 177. ((ssItem T_1) => (leq T_1 T_1)) (All U, ((ssItem U) => (-. (lt U U)))) (ssList T_2) (T_2 = T_3) (memberP T_2 T_1) (ssList (nil)) (ssList T_3) (All U, ((ssItem U) => (-. (memberP (nil) U)))) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssList T_0) (-. (memberP T_0 T_1)) (T_0 = T_4) (ssList T_4) (-. (Ex Y, ((ssItem Y) /\ (((-. (memberP T_4 Y)) /\ (memberP T_3 Y)) \/ ((-. (memberP T_3 Y)) /\ (memberP T_4 Y)))))) (All U, ((ssItem U) => (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) U) <=> ((memberP V U) \/ (memberP W U))))))))) (ssItem T_1) ### Imply 94 176
% 148.28/148.54 178. (All U, ((ssItem U) => (leq U U))) (ssItem T_1) (All U, ((ssItem U) => (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) U) <=> ((memberP V U) \/ (memberP W U))))))))) (-. (Ex Y, ((ssItem Y) /\ (((-. (memberP T_4 Y)) /\ (memberP T_3 Y)) \/ ((-. (memberP T_3 Y)) /\ (memberP T_4 Y)))))) (ssList T_4) (T_0 = T_4) (-. (memberP T_0 T_1)) (ssList T_0) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (All U, ((ssItem U) => (-. (memberP (nil) U)))) (ssList T_3) (ssList (nil)) (memberP T_2 T_1) (T_2 = T_3) (ssList T_2) (All U, ((ssItem U) => (-. (lt U U)))) ### All 177
% 148.28/148.54 179. (memberP T_2 T_1) (-. (memberP T_2 T_1)) ### Axiom
% 148.28/148.54 180. (-. ((memberP T_0 T_1) /\ (memberP T_2 T_1))) (All U, ((ssItem U) => (-. (lt U U)))) (ssList T_2) (T_2 = T_3) (memberP T_2 T_1) (ssList (nil)) (ssList T_3) (All U, ((ssItem U) => (-. (memberP (nil) U)))) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssList T_0) (T_0 = T_4) (ssList T_4) (-. (Ex Y, ((ssItem Y) /\ (((-. (memberP T_4 Y)) /\ (memberP T_3 Y)) \/ ((-. (memberP T_3 Y)) /\ (memberP T_4 Y)))))) (All U, ((ssItem U) => (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) U) <=> ((memberP V U) \/ (memberP W U))))))))) (ssItem T_1) (All U, ((ssItem U) => (leq U U))) ### NotAnd 178 179
% 148.28/148.54 181. (-. (-. (memberP T_2 T_1))) (All U, ((ssItem U) => (leq U U))) (ssItem T_1) (All U, ((ssItem U) => (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) U) <=> ((memberP V U) \/ (memberP W U))))))))) (-. (Ex Y, ((ssItem Y) /\ (((-. (memberP T_4 Y)) /\ (memberP T_3 Y)) \/ ((-. (memberP T_3 Y)) /\ (memberP T_4 Y)))))) (ssList T_4) (T_0 = T_4) (ssList T_0) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (All U, ((ssItem U) => (-. (memberP (nil) U)))) (ssList T_3) (ssList (nil)) (T_2 = T_3) (ssList T_2) (All U, ((ssItem U) => (-. (lt U U)))) (-. ((memberP T_0 T_1) /\ (memberP T_2 T_1))) ### NotNot 180
% 148.28/148.54 182. (-. ((-. (memberP T_0 T_1)) /\ (-. (memberP T_2 T_1)))) (ssList (nil)) (All U, ((ssItem U) => (-. (memberP (nil) U)))) (-. ((memberP T_0 T_1) /\ (memberP T_2 T_1))) (ssList T_4) (All U, ((ssItem U) => (-. (lt U U)))) (T_0 = T_4) (ssList T_0) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssList T_2) (T_2 = T_3) (ssList T_3) (-. (Ex Y, ((ssItem Y) /\ (((-. (memberP T_4 Y)) /\ (memberP T_3 Y)) \/ ((-. (memberP T_3 Y)) /\ (memberP T_4 Y)))))) (All U, ((ssItem U) => (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) U) <=> ((memberP V U) \/ (memberP W U))))))))) (ssItem T_1) (All U, ((ssItem U) => (leq U U))) ### NotAnd 93 181
% 148.28/148.55 183. (-. ((ssItem T_1) => (((-. (memberP T_0 T_1)) /\ (-. (memberP T_2 T_1))) \/ ((memberP T_0 T_1) /\ (memberP T_2 T_1))))) (All U, ((ssItem U) => (leq U U))) (All U, ((ssItem U) => (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) U) <=> ((memberP V U) \/ (memberP W U))))))))) (-. (Ex Y, ((ssItem Y) /\ (((-. (memberP T_4 Y)) /\ (memberP T_3 Y)) \/ ((-. (memberP T_3 Y)) /\ (memberP T_4 Y)))))) (ssList T_3) (T_2 = T_3) (ssList T_2) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssList T_0) (T_0 = T_4) (All U, ((ssItem U) => (-. (lt U U)))) (ssList T_4) (All U, ((ssItem U) => (-. (memberP (nil) U)))) (ssList (nil)) ### ConjTree 182
% 148.28/148.55 184. (-. (All Z, ((ssItem Z) => (((-. (memberP T_0 Z)) /\ (-. (memberP T_2 Z))) \/ ((memberP T_0 Z) /\ (memberP T_2 Z)))))) (ssList (nil)) (All U, ((ssItem U) => (-. (memberP (nil) U)))) (ssList T_4) (All U, ((ssItem U) => (-. (lt U U)))) (T_0 = T_4) (ssList T_0) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssList T_2) (T_2 = T_3) (ssList T_3) (-. (Ex Y, ((ssItem Y) /\ (((-. (memberP T_4 Y)) /\ (memberP T_3 Y)) \/ ((-. (memberP T_3 Y)) /\ (memberP T_4 Y)))))) (All U, ((ssItem U) => (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) U) <=> ((memberP V U) \/ (memberP W U))))))))) (All U, ((ssItem U) => (leq U U))) ### NotAllEx 183
% 148.28/148.55 185. (T_2 = T_3) (T_3 != T_2) ### Sym(=)
% 148.28/148.55 186. (-. (duplicatefreeP T_2)) (duplicatefreeP T_3) (T_2 = T_3) ### P-NotP 185
% 148.28/148.55 187. (-. ((All Z, ((ssItem Z) => (((-. (memberP T_0 Z)) /\ (-. (memberP T_2 Z))) \/ ((memberP T_0 Z) /\ (memberP T_2 Z))))) /\ (duplicatefreeP T_2))) (duplicatefreeP T_3) (All U, ((ssItem U) => (leq U U))) (All U, ((ssItem U) => (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) U) <=> ((memberP V U) \/ (memberP W U))))))))) (-. (Ex Y, ((ssItem Y) /\ (((-. (memberP T_4 Y)) /\ (memberP T_3 Y)) \/ ((-. (memberP T_3 Y)) /\ (memberP T_4 Y)))))) (ssList T_3) (T_2 = T_3) (ssList T_2) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssList T_0) (T_0 = T_4) (All U, ((ssItem U) => (-. (lt U U)))) (ssList T_4) (All U, ((ssItem U) => (-. (memberP (nil) U)))) (ssList (nil)) ### NotAnd 184 186
% 148.28/148.55 188. (-. ((ssList T_4) => ((T_0 != T_4) \/ ((T_2 != T_3) \/ ((-. (duplicatefreeP T_3)) \/ ((Ex Y, ((ssItem Y) /\ (((-. (memberP T_4 Y)) /\ (memberP T_3 Y)) \/ ((-. (memberP T_3 Y)) /\ (memberP T_4 Y))))) \/ ((All Z, ((ssItem Z) => (((-. (memberP T_0 Z)) /\ (-. (memberP T_2 Z))) \/ ((memberP T_0 Z) /\ (memberP T_2 Z))))) /\ (duplicatefreeP T_2)))))))) (ssList (nil)) (All U, ((ssItem U) => (-. (memberP (nil) U)))) (All U, ((ssItem U) => (-. (lt U U)))) (ssList T_0) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssList T_2) (ssList T_3) (All U, ((ssItem U) => (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) U) <=> ((memberP V U) \/ (memberP W U))))))))) (All U, ((ssItem U) => (leq U U))) ### ConjTree 187
% 148.28/148.55 189. (-. (All X, ((ssList X) => ((T_0 != X) \/ ((T_2 != T_3) \/ ((-. (duplicatefreeP T_3)) \/ ((Ex Y, ((ssItem Y) /\ (((-. (memberP X Y)) /\ (memberP T_3 Y)) \/ ((-. (memberP T_3 Y)) /\ (memberP X Y))))) \/ ((All Z, ((ssItem Z) => (((-. (memberP T_0 Z)) /\ (-. (memberP T_2 Z))) \/ ((memberP T_0 Z) /\ (memberP T_2 Z))))) /\ (duplicatefreeP T_2))))))))) (All U, ((ssItem U) => (leq U U))) (All U, ((ssItem U) => (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) U) <=> ((memberP V U) \/ (memberP W U))))))))) (ssList T_3) (ssList T_2) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssList T_0) (All U, ((ssItem U) => (-. (lt U U)))) (All U, ((ssItem U) => (-. (memberP (nil) U)))) (ssList (nil)) ### NotAllEx 188
% 148.28/148.55 190. (-. ((ssList T_3) => (All X, ((ssList X) => ((T_0 != X) \/ ((T_2 != T_3) \/ ((-. (duplicatefreeP T_3)) \/ ((Ex Y, ((ssItem Y) /\ (((-. (memberP X Y)) /\ (memberP T_3 Y)) \/ ((-. (memberP T_3 Y)) /\ (memberP X Y))))) \/ ((All Z, ((ssItem Z) => (((-. (memberP T_0 Z)) /\ (-. (memberP T_2 Z))) \/ ((memberP T_0 Z) /\ (memberP T_2 Z))))) /\ (duplicatefreeP T_2)))))))))) (ssList (nil)) (All U, ((ssItem U) => (-. (memberP (nil) U)))) (All U, ((ssItem U) => (-. (lt U U)))) (ssList T_0) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssList T_2) (All U, ((ssItem U) => (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) U) <=> ((memberP V U) \/ (memberP W U))))))))) (All U, ((ssItem U) => (leq U U))) ### NotImply 189
% 148.28/148.55 191. (-. (All W, ((ssList W) => (All X, ((ssList X) => ((T_0 != X) \/ ((T_2 != W) \/ ((-. (duplicatefreeP W)) \/ ((Ex Y, ((ssItem Y) /\ (((-. (memberP X Y)) /\ (memberP W Y)) \/ ((-. (memberP W Y)) /\ (memberP X Y))))) \/ ((All Z, ((ssItem Z) => (((-. (memberP T_0 Z)) /\ (-. (memberP T_2 Z))) \/ ((memberP T_0 Z) /\ (memberP T_2 Z))))) /\ (duplicatefreeP T_2))))))))))) (All U, ((ssItem U) => (leq U U))) (All U, ((ssItem U) => (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) U) <=> ((memberP V U) \/ (memberP W U))))))))) (ssList T_2) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssList T_0) (All U, ((ssItem U) => (-. (lt U U)))) (All U, ((ssItem U) => (-. (memberP (nil) U)))) (ssList (nil)) ### NotAllEx 190
% 148.28/148.55 192. (-. ((ssList T_0) => (All W, ((ssList W) => (All X, ((ssList X) => ((T_0 != X) \/ ((T_2 != W) \/ ((-. (duplicatefreeP W)) \/ ((Ex Y, ((ssItem Y) /\ (((-. (memberP X Y)) /\ (memberP W Y)) \/ ((-. (memberP W Y)) /\ (memberP X Y))))) \/ ((All Z, ((ssItem Z) => (((-. (memberP T_0 Z)) /\ (-. (memberP T_2 Z))) \/ ((memberP T_0 Z) /\ (memberP T_2 Z))))) /\ (duplicatefreeP T_2)))))))))))) (ssList (nil)) (All U, ((ssItem U) => (-. (memberP (nil) U)))) (All U, ((ssItem U) => (-. (lt U U)))) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (ssList T_2) (All U, ((ssItem U) => (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) U) <=> ((memberP V U) \/ (memberP W U))))))))) (All U, ((ssItem U) => (leq U U))) ### NotImply 191
% 148.28/148.55 193. (-. (All V, ((ssList V) => (All W, ((ssList W) => (All X, ((ssList X) => ((V != X) \/ ((T_2 != W) \/ ((-. (duplicatefreeP W)) \/ ((Ex Y, ((ssItem Y) /\ (((-. (memberP X Y)) /\ (memberP W Y)) \/ ((-. (memberP W Y)) /\ (memberP X Y))))) \/ ((All Z, ((ssItem Z) => (((-. (memberP V Z)) /\ (-. (memberP T_2 Z))) \/ ((memberP V Z) /\ (memberP T_2 Z))))) /\ (duplicatefreeP T_2))))))))))))) (All U, ((ssItem U) => (leq U U))) (All U, ((ssItem U) => (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) U) <=> ((memberP V U) \/ (memberP W U))))))))) (ssList T_2) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (All U, ((ssItem U) => (-. (lt U U)))) (All U, ((ssItem U) => (-. (memberP (nil) U)))) (ssList (nil)) ### NotAllEx 192
% 148.28/148.55 194. (-. ((ssList T_2) => (All V, ((ssList V) => (All W, ((ssList W) => (All X, ((ssList X) => ((V != X) \/ ((T_2 != W) \/ ((-. (duplicatefreeP W)) \/ ((Ex Y, ((ssItem Y) /\ (((-. (memberP X Y)) /\ (memberP W Y)) \/ ((-. (memberP W Y)) /\ (memberP X Y))))) \/ ((All Z, ((ssItem Z) => (((-. (memberP V Z)) /\ (-. (memberP T_2 Z))) \/ ((memberP V Z) /\ (memberP T_2 Z))))) /\ (duplicatefreeP T_2)))))))))))))) (ssList (nil)) (All U, ((ssItem U) => (-. (memberP (nil) U)))) (All U, ((ssItem U) => (-. (lt U U)))) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (All U, ((ssItem U) => (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) U) <=> ((memberP V U) \/ (memberP W U))))))))) (All U, ((ssItem U) => (leq U U))) ### NotImply 193
% 148.28/148.55 195. (-. (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (All X, ((ssList X) => ((V != X) \/ ((U != W) \/ ((-. (duplicatefreeP W)) \/ ((Ex Y, ((ssItem Y) /\ (((-. (memberP X Y)) /\ (memberP W Y)) \/ ((-. (memberP W Y)) /\ (memberP X Y))))) \/ ((All Z, ((ssItem Z) => (((-. (memberP V Z)) /\ (-. (memberP U Z))) \/ ((memberP V Z) /\ (memberP U Z))))) /\ (duplicatefreeP U))))))))))))))) (All U, ((ssItem U) => (leq U U))) (All U, ((ssItem U) => (All V, ((ssList V) => (All W, ((ssList W) => ((memberP (app V W) U) <=> ((memberP V U) \/ (memberP W U))))))))) (All U, ((ssItem U) => (All V, ((ssItem V) => ((lt U V) <=> ((U != V) /\ (leq U V))))))) (All U, ((ssItem U) => (-. (lt U U)))) (All U, ((ssItem U) => (-. (memberP (nil) U)))) (ssList (nil)) ### NotAllEx 194
% 148.28/148.55 % SZS output end Proof
% 148.28/148.55 (* END-PROOF *)
%------------------------------------------------------------------------------