TSTP Solution File: SWC105+1 by SuperZenon---0.0.1
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- Process Solution
%------------------------------------------------------------------------------
% File : SuperZenon---0.0.1
% Problem : SWC105+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 : n019.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 22:06:36 EDT 2022
% Result : Theorem 7.74s 7.90s
% Output : Proof 7.90s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.08/0.13 % Problem : SWC105+1 : TPTP v8.1.0. Released v2.4.0.
% 0.08/0.14 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s
% 0.14/0.36 % Computer : n019.cluster.edu
% 0.14/0.36 % Model : x86_64 x86_64
% 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36 % Memory : 8042.1875MB
% 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36 % CPULimit : 300
% 0.14/0.36 % WCLimit : 600
% 0.14/0.36 % DateTime : Sun Jun 12 09:14:25 EDT 2022
% 0.14/0.36 % CPUTime :
% 7.74/7.90 % SZS status Theorem
% 7.74/7.90 (* PROOF-FOUND *)
% 7.74/7.90 (* BEGIN-PROOF *)
% 7.74/7.90 % SZS output start Proof
% 7.74/7.90 1. (ssList T_0) (-. (ssList T_0)) ### Axiom
% 7.74/7.90 2. ((nil) = T_0) ((nil) != T_0) ### Axiom
% 7.74/7.90 3. (ssList T_0) (-. (ssList T_0)) ### Axiom
% 7.74/7.90 4. (ssList (nil)) (-. (ssList (nil))) ### Axiom
% 7.74/7.90 5. (ssList (nil)) (-. (ssList (nil))) ### Axiom
% 7.74/7.90 6. (ssList T_1) (-. (ssList T_1)) ### Axiom
% 7.74/7.90 7. (ssList T_2) (-. (ssList T_2)) ### Axiom
% 7.74/7.90 8. ((nil) = T_2) ((nil) != T_2) ### Axiom
% 7.74/7.90 9. (ssList T_3) (-. (ssList T_3)) ### Axiom
% 7.74/7.90 10. (ssList T_2) (-. (ssList T_2)) ### Axiom
% 7.74/7.90 11. (ssList T_3) (-. (ssList T_3)) ### Axiom
% 7.74/7.90 12. (ssList (nil)) (-. (ssList (nil))) ### Axiom
% 7.74/7.90 13. (ssList (nil)) (-. (ssList (nil))) ### Axiom
% 7.74/7.90 14. (ssList T_3) (-. (ssList T_3)) ### Axiom
% 7.74/7.90 15. (-. (rearsegP T_3 (nil))) (rearsegP T_3 (nil)) ### Axiom
% 7.74/7.90 16. ((ssList T_3) => (rearsegP T_3 (nil))) (-. (rearsegP T_3 (nil))) (ssList T_3) ### Imply 14 15
% 7.74/7.90 17. (All U, ((ssList U) => (rearsegP U (nil)))) (ssList T_3) (-. (rearsegP T_3 (nil))) ### All 16
% 7.74/7.90 18. (ssList (nil)) (-. (ssList (nil))) ### Axiom
% 7.74/7.90 19. (-. (rearsegP (nil) (nil))) (ssList (nil)) ### Extension/test/ax49ctrp 18
% 7.74/7.90 20. (ssList T_2) (-. (ssList T_2)) ### Axiom
% 7.74/7.90 21. (rearsegP T_3 (nil)) (-. (rearsegP T_3 (nil))) ### Axiom
% 7.74/7.90 22. (rearsegP (nil) T_2) (-. (rearsegP (nil) T_2)) ### Axiom
% 7.74/7.90 23. (-. (rearsegP T_3 T_2)) (rearsegP T_3 T_2) ### Axiom
% 7.74/7.90 24. ((ssList T_2) => (((rearsegP T_3 (nil)) /\ (rearsegP (nil) T_2)) => (rearsegP T_3 T_2))) (-. (rearsegP T_3 T_2)) (rearsegP (nil) T_2) (rearsegP T_3 (nil)) (ssList T_2) ### DisjTree 20 21 22 23
% 7.74/7.90 25. (All W, ((ssList W) => (((rearsegP T_3 (nil)) /\ (rearsegP (nil) W)) => (rearsegP T_3 W)))) (ssList T_2) (rearsegP T_3 (nil)) (rearsegP (nil) T_2) (-. (rearsegP T_3 T_2)) ### All 24
% 7.74/7.90 26. ((ssList (nil)) => (((rearsegP T_3 (nil)) /\ (rearsegP (nil) (nil))) => (rearsegP T_3 (nil)))) (-. (rearsegP T_3 T_2)) (rearsegP (nil) T_2) (ssList T_2) (All W, ((ssList W) => (((rearsegP T_3 (nil)) /\ (rearsegP (nil) W)) => (rearsegP T_3 W)))) (ssList T_3) (All U, ((ssList U) => (rearsegP U (nil)))) (ssList (nil)) ### DisjTree 13 17 19 25
% 7.74/7.90 27. (ssList (nil)) (All U, ((ssList U) => (rearsegP U (nil)))) (ssList T_3) (All W, ((ssList W) => (((rearsegP T_3 (nil)) /\ (rearsegP (nil) W)) => (rearsegP T_3 W)))) (ssList T_2) (rearsegP (nil) T_2) (-. (rearsegP T_3 T_2)) ### All 26
% 7.74/7.90 28. ((ssList (nil)) => (All W, ((ssList W) => (((rearsegP T_3 (nil)) /\ (rearsegP (nil) W)) => (rearsegP T_3 W))))) (-. (rearsegP T_3 T_2)) (rearsegP (nil) T_2) (ssList T_2) (ssList T_3) (All U, ((ssList U) => (rearsegP U (nil)))) (ssList (nil)) ### Imply 12 27
% 7.74/7.90 29. (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP T_3 V) /\ (rearsegP V W)) => (rearsegP T_3 W)))))) (ssList (nil)) (All U, ((ssList U) => (rearsegP U (nil)))) (ssList T_3) (ssList T_2) (rearsegP (nil) T_2) (-. (rearsegP T_3 T_2)) ### All 28
% 7.74/7.90 30. ((ssList T_3) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP T_3 V) /\ (rearsegP V W)) => (rearsegP T_3 W))))))) (-. (rearsegP T_3 T_2)) (rearsegP (nil) T_2) (ssList T_2) (All U, ((ssList U) => (rearsegP U (nil)))) (ssList (nil)) (ssList T_3) ### Imply 11 29
% 7.74/7.90 31. (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (ssList T_3) (ssList (nil)) (All U, ((ssList U) => (rearsegP U (nil)))) (ssList T_2) (rearsegP (nil) T_2) (-. (rearsegP T_3 T_2)) ### All 30
% 7.74/7.90 32. (T_0 = T_3) (T_3 != T_0) ### Sym(=)
% 7.74/7.90 33. (T_1 = T_2) (T_2 != T_1) ### Sym(=)
% 7.74/7.90 34. (-. (rearsegP T_0 T_1)) (rearsegP T_3 T_2) (T_1 = T_2) (T_0 = T_3) ### P-NotP 32 33
% 7.74/7.90 35. ((rearsegP T_3 T_2) <=> (Ex W, ((ssList W) /\ ((app W T_2) = T_3)))) (T_0 = T_3) (T_1 = T_2) (-. (rearsegP T_0 T_1)) (rearsegP (nil) T_2) (ssList T_2) (All U, ((ssList U) => (rearsegP U (nil)))) (ssList (nil)) (ssList T_3) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) ### Equiv 31 34
% 7.74/7.90 36. ((ssList T_2) => ((rearsegP T_3 T_2) <=> (Ex W, ((ssList W) /\ ((app W T_2) = T_3))))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (ssList T_3) (ssList (nil)) (All U, ((ssList U) => (rearsegP U (nil)))) (rearsegP (nil) T_2) (-. (rearsegP T_0 T_1)) (T_1 = T_2) (T_0 = T_3) (ssList T_2) ### Imply 10 35
% 7.74/7.90 37. (All V, ((ssList V) => ((rearsegP T_3 V) <=> (Ex W, ((ssList W) /\ ((app W V) = T_3)))))) (ssList T_2) (T_0 = T_3) (T_1 = T_2) (-. (rearsegP T_0 T_1)) (rearsegP (nil) T_2) (All U, ((ssList U) => (rearsegP U (nil)))) (ssList (nil)) (ssList T_3) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) ### All 36
% 7.74/7.90 38. ((ssList T_3) => (All V, ((ssList V) => ((rearsegP T_3 V) <=> (Ex W, ((ssList W) /\ ((app W V) = T_3))))))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (ssList (nil)) (All U, ((ssList U) => (rearsegP U (nil)))) (rearsegP (nil) T_2) (-. (rearsegP T_0 T_1)) (T_1 = T_2) (T_0 = T_3) (ssList T_2) (ssList T_3) ### Imply 9 37
% 7.74/7.90 39. (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (ssList T_3) (ssList T_2) (T_0 = T_3) (T_1 = T_2) (-. (rearsegP T_0 T_1)) (rearsegP (nil) T_2) (All U, ((ssList U) => (rearsegP U (nil)))) (ssList (nil)) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) ### All 38
% 7.74/7.90 40. ((rearsegP (nil) T_2) <=> ((nil) = T_2)) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (ssList (nil)) (All U, ((ssList U) => (rearsegP U (nil)))) (-. (rearsegP T_0 T_1)) (T_1 = T_2) (T_0 = T_3) (ssList T_2) (ssList T_3) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) ((nil) = T_2) ### Equiv 8 39
% 7.74/7.90 41. ((ssList T_2) => ((rearsegP (nil) T_2) <=> ((nil) = T_2))) ((nil) = T_2) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (ssList T_3) (T_0 = T_3) (T_1 = T_2) (-. (rearsegP T_0 T_1)) (All U, ((ssList U) => (rearsegP U (nil)))) (ssList (nil)) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (ssList T_2) ### Imply 7 40
% 7.74/7.90 42. (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_2) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (ssList (nil)) (All U, ((ssList U) => (rearsegP U (nil)))) (-. (rearsegP T_0 T_1)) (T_1 = T_2) (T_0 = T_3) (ssList T_3) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) ((nil) = T_2) ### All 41
% 7.74/7.90 43. (-. ((nil) != T_2)) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (ssList T_3) (T_0 = T_3) (T_1 = T_2) (-. (rearsegP T_0 T_1)) (All U, ((ssList U) => (rearsegP U (nil)))) (ssList (nil)) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (ssList T_2) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) ### NotNot 42
% 7.74/7.90 44. (-. (((nil) != T_3) \/ ((nil) != T_2))) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_2) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (ssList (nil)) (All U, ((ssList U) => (rearsegP U (nil)))) (-. (rearsegP T_0 T_1)) (T_1 = T_2) (T_0 = T_3) (ssList T_3) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) ### NotOr 43
% 7.74/7.90 45. (-. (-. (rearsegP T_3 T_2))) (T_0 = T_3) (T_1 = T_2) (-. (rearsegP T_0 T_1)) ### NotNot 34
% 7.74/7.90 46. (-. ((-. (neq T_2 (nil))) \/ (-. (rearsegP T_3 T_2)))) (-. (rearsegP T_0 T_1)) (T_1 = T_2) (T_0 = T_3) ### NotOr 45
% 7.74/7.90 47. (-. ((((nil) != T_3) \/ ((nil) != T_2)) /\ ((-. (neq T_2 (nil))) \/ (-. (rearsegP T_3 T_2))))) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (ssList T_3) (T_0 = T_3) (T_1 = T_2) (-. (rearsegP T_0 T_1)) (All U, ((ssList U) => (rearsegP U (nil)))) (ssList (nil)) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (ssList T_2) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) ### NotAnd 44 46
% 7.85/8.02 48. (ssList T_1) (-. (ssList T_1)) ### Axiom
% 7.85/8.02 49. (ssList T_0) (-. (ssList T_0)) ### Axiom
% 7.85/8.02 50. (ssList T_1) (-. (ssList T_1)) ### Axiom
% 7.85/8.02 51. (rearsegP (nil) T_0) (-. (rearsegP (nil) T_0)) ### Axiom
% 7.85/8.02 52. (rearsegP T_0 T_1) (-. (rearsegP T_0 T_1)) ### Axiom
% 7.85/8.02 53. (-. (rearsegP (nil) T_1)) (rearsegP (nil) T_1) ### Axiom
% 7.85/8.02 54. ((ssList T_1) => (((rearsegP (nil) T_0) /\ (rearsegP T_0 T_1)) => (rearsegP (nil) T_1))) (-. (rearsegP (nil) T_1)) (rearsegP T_0 T_1) (rearsegP (nil) T_0) (ssList T_1) ### DisjTree 50 51 52 53
% 7.85/8.02 55. (All W, ((ssList W) => (((rearsegP (nil) T_0) /\ (rearsegP T_0 W)) => (rearsegP (nil) W)))) (ssList T_1) (rearsegP (nil) T_0) (rearsegP T_0 T_1) (-. (rearsegP (nil) T_1)) ### All 54
% 7.85/8.02 56. ((ssList T_0) => (All W, ((ssList W) => (((rearsegP (nil) T_0) /\ (rearsegP T_0 W)) => (rearsegP (nil) W))))) (-. (rearsegP (nil) T_1)) (rearsegP T_0 T_1) (rearsegP (nil) T_0) (ssList T_1) (ssList T_0) ### Imply 49 55
% 7.85/8.02 57. (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP (nil) V) /\ (rearsegP V W)) => (rearsegP (nil) W)))))) (ssList T_0) (ssList T_1) (rearsegP (nil) T_0) (rearsegP T_0 T_1) (-. (rearsegP (nil) T_1)) ### All 56
% 7.85/8.02 58. (ssList T_1) (-. (ssList T_1)) ### Axiom
% 7.85/8.02 59. (rearsegP (nil) T_1) (-. (rearsegP (nil) T_1)) ### Axiom
% 7.85/8.02 60. ((nil) != T_1) ((nil) = T_1) ### Axiom
% 7.85/8.02 61. ((rearsegP (nil) T_1) <=> ((nil) = T_1)) ((nil) != T_1) (rearsegP (nil) T_1) ### Equiv 59 60
% 7.85/8.02 62. ((ssList T_1) => ((rearsegP (nil) T_1) <=> ((nil) = T_1))) (rearsegP (nil) T_1) ((nil) != T_1) (ssList T_1) ### Imply 58 61
% 7.85/8.02 63. (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_1) ((nil) != T_1) (rearsegP (nil) T_1) ### All 62
% 7.85/8.02 64. ((rearsegP (nil) T_1) <=> (Ex W, ((ssList W) /\ ((app W T_1) = (nil))))) ((nil) != T_1) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (rearsegP T_0 T_1) (rearsegP (nil) T_0) (ssList T_1) (ssList T_0) (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP (nil) V) /\ (rearsegP V W)) => (rearsegP (nil) W)))))) ### Equiv 57 63
% 7.85/8.02 65. ((ssList T_1) => ((rearsegP (nil) T_1) <=> (Ex W, ((ssList W) /\ ((app W T_1) = (nil)))))) (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP (nil) V) /\ (rearsegP V W)) => (rearsegP (nil) W)))))) (ssList T_0) (rearsegP (nil) T_0) (rearsegP T_0 T_1) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) ((nil) != T_1) (ssList T_1) ### Imply 48 64
% 7.85/8.02 66. (All V, ((ssList V) => ((rearsegP (nil) V) <=> (Ex W, ((ssList W) /\ ((app W V) = (nil))))))) (ssList T_1) ((nil) != T_1) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (rearsegP T_0 T_1) (rearsegP (nil) T_0) (ssList T_0) (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP (nil) V) /\ (rearsegP V W)) => (rearsegP (nil) W)))))) ### All 65
% 7.85/8.02 67. ((rearsegP T_0 T_1) <=> (Ex W, ((ssList W) /\ ((app W T_1) = T_0)))) (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP (nil) V) /\ (rearsegP V W)) => (rearsegP (nil) W)))))) (ssList T_0) (rearsegP (nil) T_0) ((nil) != T_1) (ssList T_1) (All V, ((ssList V) => ((rearsegP (nil) V) <=> (Ex W, ((ssList W) /\ ((app W V) = (nil))))))) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_2) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (ssList (nil)) (All U, ((ssList U) => (rearsegP U (nil)))) (T_1 = T_2) (T_0 = T_3) (ssList T_3) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (-. ((((nil) != T_3) \/ ((nil) != T_2)) /\ ((-. (neq T_2 (nil))) \/ (-. (rearsegP T_3 T_2))))) ### Equiv 47 66
% 7.85/8.02 68. ((ssList T_1) => ((rearsegP T_0 T_1) <=> (Ex W, ((ssList W) /\ ((app W T_1) = T_0))))) (-. ((((nil) != T_3) \/ ((nil) != T_2)) /\ ((-. (neq T_2 (nil))) \/ (-. (rearsegP T_3 T_2))))) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (ssList T_3) (T_0 = T_3) (T_1 = T_2) (All U, ((ssList U) => (rearsegP U (nil)))) (ssList (nil)) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (ssList T_2) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (All V, ((ssList V) => ((rearsegP (nil) V) <=> (Ex W, ((ssList W) /\ ((app W V) = (nil))))))) ((nil) != T_1) (rearsegP (nil) T_0) (ssList T_0) (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP (nil) V) /\ (rearsegP V W)) => (rearsegP (nil) W)))))) (ssList T_1) ### Imply 6 67
% 7.85/8.02 69. (All V, ((ssList V) => ((rearsegP T_0 V) <=> (Ex W, ((ssList W) /\ ((app W V) = T_0)))))) (ssList T_1) (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP (nil) V) /\ (rearsegP V W)) => (rearsegP (nil) W)))))) (ssList T_0) (rearsegP (nil) T_0) ((nil) != T_1) (All V, ((ssList V) => ((rearsegP (nil) V) <=> (Ex W, ((ssList W) /\ ((app W V) = (nil))))))) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_2) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (ssList (nil)) (All U, ((ssList U) => (rearsegP U (nil)))) (T_1 = T_2) (T_0 = T_3) (ssList T_3) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (-. ((((nil) != T_3) \/ ((nil) != T_2)) /\ ((-. (neq T_2 (nil))) \/ (-. (rearsegP T_3 T_2))))) ### All 68
% 7.85/8.02 70. ((ssList (nil)) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP (nil) V) /\ (rearsegP V W)) => (rearsegP (nil) W))))))) (-. ((((nil) != T_3) \/ ((nil) != T_2)) /\ ((-. (neq T_2 (nil))) \/ (-. (rearsegP T_3 T_2))))) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (ssList T_3) (T_0 = T_3) (T_1 = T_2) (All U, ((ssList U) => (rearsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (ssList T_2) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (All V, ((ssList V) => ((rearsegP (nil) V) <=> (Ex W, ((ssList W) /\ ((app W V) = (nil))))))) ((nil) != T_1) (rearsegP (nil) T_0) (ssList T_0) (ssList T_1) (All V, ((ssList V) => ((rearsegP T_0 V) <=> (Ex W, ((ssList W) /\ ((app W V) = T_0)))))) (ssList (nil)) ### Imply 5 69
% 7.85/8.02 71. (ssList (nil)) (All V, ((ssList V) => ((rearsegP T_0 V) <=> (Ex W, ((ssList W) /\ ((app W V) = T_0)))))) (ssList T_1) (ssList T_0) (rearsegP (nil) T_0) ((nil) != T_1) (All V, ((ssList V) => ((rearsegP (nil) V) <=> (Ex W, ((ssList W) /\ ((app W V) = (nil))))))) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_2) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (All U, ((ssList U) => (rearsegP U (nil)))) (T_1 = T_2) (T_0 = T_3) (ssList T_3) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (-. ((((nil) != T_3) \/ ((nil) != T_2)) /\ ((-. (neq T_2 (nil))) \/ (-. (rearsegP T_3 T_2))))) ### All 70
% 7.85/8.02 72. ((ssList (nil)) => (All V, ((ssList V) => ((rearsegP (nil) V) <=> (Ex W, ((ssList W) /\ ((app W V) = (nil)))))))) (-. ((((nil) != T_3) \/ ((nil) != T_2)) /\ ((-. (neq T_2 (nil))) \/ (-. (rearsegP T_3 T_2))))) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (ssList T_3) (T_0 = T_3) (T_1 = T_2) (All U, ((ssList U) => (rearsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (ssList T_2) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) ((nil) != T_1) (rearsegP (nil) T_0) (ssList T_0) (ssList T_1) (All V, ((ssList V) => ((rearsegP T_0 V) <=> (Ex W, ((ssList W) /\ ((app W V) = T_0)))))) (ssList (nil)) ### Imply 4 71
% 7.85/8.04 73. (ssList (nil)) (All V, ((ssList V) => ((rearsegP T_0 V) <=> (Ex W, ((ssList W) /\ ((app W V) = T_0)))))) (ssList T_1) (ssList T_0) (rearsegP (nil) T_0) ((nil) != T_1) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_2) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (All U, ((ssList U) => (rearsegP U (nil)))) (T_1 = T_2) (T_0 = T_3) (ssList T_3) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (-. ((((nil) != T_3) \/ ((nil) != T_2)) /\ ((-. (neq T_2 (nil))) \/ (-. (rearsegP T_3 T_2))))) ### All 72
% 7.85/8.04 74. ((ssList T_0) => (All V, ((ssList V) => ((rearsegP T_0 V) <=> (Ex W, ((ssList W) /\ ((app W V) = T_0))))))) (-. ((((nil) != T_3) \/ ((nil) != T_2)) /\ ((-. (neq T_2 (nil))) \/ (-. (rearsegP T_3 T_2))))) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (ssList T_3) (T_0 = T_3) (T_1 = T_2) (All U, ((ssList U) => (rearsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (ssList T_2) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) ((nil) != T_1) (rearsegP (nil) T_0) (ssList T_1) (ssList (nil)) (ssList T_0) ### Imply 3 73
% 7.85/8.04 75. (ssList T_0) (ssList (nil)) (ssList T_1) (rearsegP (nil) T_0) ((nil) != T_1) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_2) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (All U, ((ssList U) => (rearsegP U (nil)))) (T_1 = T_2) (T_0 = T_3) (ssList T_3) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (-. ((((nil) != T_3) \/ ((nil) != T_2)) /\ ((-. (neq T_2 (nil))) \/ (-. (rearsegP T_3 T_2))))) ### All 74
% 7.85/8.04 76. ((rearsegP (nil) T_0) <=> ((nil) = T_0)) (-. ((((nil) != T_3) \/ ((nil) != T_2)) /\ ((-. (neq T_2 (nil))) \/ (-. (rearsegP T_3 T_2))))) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (ssList T_3) (T_0 = T_3) (T_1 = T_2) (All U, ((ssList U) => (rearsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (ssList T_2) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) ((nil) != T_1) (ssList T_1) (ssList (nil)) (ssList T_0) ((nil) = T_0) ### Equiv 2 75
% 7.85/8.04 77. ((ssList T_0) => ((rearsegP (nil) T_0) <=> ((nil) = T_0))) ((nil) = T_0) (ssList (nil)) (ssList T_1) ((nil) != T_1) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_2) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (All U, ((ssList U) => (rearsegP U (nil)))) (T_1 = T_2) (T_0 = T_3) (ssList T_3) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (-. ((((nil) != T_3) \/ ((nil) != T_2)) /\ ((-. (neq T_2 (nil))) \/ (-. (rearsegP T_3 T_2))))) (ssList T_0) ### Imply 1 76
% 7.85/8.04 78. (ssList T_0) (-. ((((nil) != T_3) \/ ((nil) != T_2)) /\ ((-. (neq T_2 (nil))) \/ (-. (rearsegP T_3 T_2))))) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (ssList T_3) (T_0 = T_3) (T_1 = T_2) (All U, ((ssList U) => (rearsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (ssList T_2) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) ((nil) != T_1) (ssList T_1) (ssList (nil)) ((nil) = T_0) ### All 77
% 7.85/8.04 79. (-. ((nil) != T_0)) (ssList (nil)) (ssList T_1) ((nil) != T_1) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_2) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (All U, ((ssList U) => (rearsegP U (nil)))) (T_1 = T_2) (T_0 = T_3) (ssList T_3) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (-. ((((nil) != T_3) \/ ((nil) != T_2)) /\ ((-. (neq T_2 (nil))) \/ (-. (rearsegP T_3 T_2))))) (ssList T_0) ### NotNot 78
% 7.85/8.04 80. (-. (((nil) != T_0) \/ ((nil) = T_1))) (ssList T_0) (-. ((((nil) != T_3) \/ ((nil) != T_2)) /\ ((-. (neq T_2 (nil))) \/ (-. (rearsegP T_3 T_2))))) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (ssList T_3) (T_0 = T_3) (T_1 = T_2) (All U, ((ssList U) => (rearsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (ssList T_2) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_1) (ssList (nil)) ### NotOr 79
% 7.85/8.04 81. (ssList T_3) (-. (ssList T_3)) ### Axiom
% 7.85/8.04 82. ((nil) = T_3) ((nil) != T_3) ### Axiom
% 7.85/8.04 83. (ssList T_3) (-. (ssList T_3)) ### Axiom
% 7.85/8.04 84. (ssList T_1) (-. (ssList T_1)) ### Axiom
% 7.85/8.04 85. (ssList T_1) (-. (ssList T_1)) ### Axiom
% 7.85/8.04 86. (ssList (nil)) (-. (ssList (nil))) ### Axiom
% 7.85/8.04 87. (ssList T_3) (-. (ssList T_3)) ### Axiom
% 7.85/8.04 88. (ssList T_1) (-. (ssList T_1)) ### Axiom
% 7.85/8.04 89. (T_1 = (nil)) ((nil) != T_1) ### Sym(=)
% 7.85/8.04 90. (ssList (nil)) (-. (ssList (nil))) ### Axiom
% 7.85/8.04 91. (ssList (nil)) (-. (ssList (nil))) ### Axiom
% 7.85/8.04 92. (ssList T_3) (-. (ssList T_3)) ### Axiom
% 7.85/8.04 93. (-. (frontsegP T_3 (nil))) (frontsegP T_3 (nil)) ### Axiom
% 7.85/8.04 94. ((ssList T_3) => (frontsegP T_3 (nil))) (-. (frontsegP T_3 (nil))) (ssList T_3) ### Imply 92 93
% 7.85/8.04 95. (All U, ((ssList U) => (frontsegP U (nil)))) (ssList T_3) (-. (frontsegP T_3 (nil))) ### All 94
% 7.85/8.04 96. (ssList (nil)) (-. (ssList (nil))) ### Axiom
% 7.85/8.04 97. (-. (frontsegP (nil) (nil))) (ssList (nil)) ### Extension/test/ax42ctrp 96
% 7.85/8.04 98. (ssList T_1) (-. (ssList T_1)) ### Axiom
% 7.85/8.04 99. (frontsegP T_3 (nil)) (-. (frontsegP T_3 (nil))) ### Axiom
% 7.85/8.04 100. (frontsegP (nil) T_1) (-. (frontsegP (nil) T_1)) ### Axiom
% 7.85/8.04 101. (-. (frontsegP T_3 T_1)) (frontsegP T_3 T_1) ### Axiom
% 7.85/8.04 102. ((ssList T_1) => (((frontsegP T_3 (nil)) /\ (frontsegP (nil) T_1)) => (frontsegP T_3 T_1))) (-. (frontsegP T_3 T_1)) (frontsegP (nil) T_1) (frontsegP T_3 (nil)) (ssList T_1) ### DisjTree 98 99 100 101
% 7.85/8.04 103. (All W, ((ssList W) => (((frontsegP T_3 (nil)) /\ (frontsegP (nil) W)) => (frontsegP T_3 W)))) (ssList T_1) (frontsegP T_3 (nil)) (frontsegP (nil) T_1) (-. (frontsegP T_3 T_1)) ### All 102
% 7.85/8.04 104. ((ssList (nil)) => (((frontsegP T_3 (nil)) /\ (frontsegP (nil) (nil))) => (frontsegP T_3 (nil)))) (-. (frontsegP T_3 T_1)) (frontsegP (nil) T_1) (ssList T_1) (All W, ((ssList W) => (((frontsegP T_3 (nil)) /\ (frontsegP (nil) W)) => (frontsegP T_3 W)))) (ssList T_3) (All U, ((ssList U) => (frontsegP U (nil)))) (ssList (nil)) ### DisjTree 91 95 97 103
% 7.85/8.04 105. (ssList (nil)) (All U, ((ssList U) => (frontsegP U (nil)))) (ssList T_3) (All W, ((ssList W) => (((frontsegP T_3 (nil)) /\ (frontsegP (nil) W)) => (frontsegP T_3 W)))) (ssList T_1) (frontsegP (nil) T_1) (-. (frontsegP T_3 T_1)) ### All 104
% 7.85/8.04 106. ((ssList (nil)) => (All W, ((ssList W) => (((frontsegP T_3 (nil)) /\ (frontsegP (nil) W)) => (frontsegP T_3 W))))) (-. (frontsegP T_3 T_1)) (frontsegP (nil) T_1) (ssList T_1) (ssList T_3) (All U, ((ssList U) => (frontsegP U (nil)))) (ssList (nil)) ### Imply 90 105
% 7.85/8.04 107. (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP T_3 V) /\ (frontsegP V W)) => (frontsegP T_3 W)))))) (ssList (nil)) (All U, ((ssList U) => (frontsegP U (nil)))) (ssList T_3) (ssList T_1) (frontsegP (nil) T_1) (-. (frontsegP T_3 T_1)) ### All 106
% 7.85/8.04 108. ((frontsegP (nil) T_1) <=> ((nil) = T_1)) (-. (frontsegP T_3 T_1)) (ssList T_1) (ssList T_3) (All U, ((ssList U) => (frontsegP U (nil)))) (ssList (nil)) (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP T_3 V) /\ (frontsegP V W)) => (frontsegP T_3 W)))))) (T_1 = (nil)) ### Equiv 89 107
% 7.85/8.05 109. ((ssList T_1) => ((frontsegP (nil) T_1) <=> ((nil) = T_1))) (T_1 = (nil)) (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP T_3 V) /\ (frontsegP V W)) => (frontsegP T_3 W)))))) (ssList (nil)) (All U, ((ssList U) => (frontsegP U (nil)))) (ssList T_3) (-. (frontsegP T_3 T_1)) (ssList T_1) ### Imply 88 108
% 7.85/8.05 110. (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (ssList T_1) (-. (frontsegP T_3 T_1)) (ssList T_3) (All U, ((ssList U) => (frontsegP U (nil)))) (ssList (nil)) (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP T_3 V) /\ (frontsegP V W)) => (frontsegP T_3 W)))))) (T_1 = (nil)) ### All 109
% 7.85/8.05 111. ((ssList T_3) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP T_3 V) /\ (frontsegP V W)) => (frontsegP T_3 W))))))) (T_1 = (nil)) (ssList (nil)) (All U, ((ssList U) => (frontsegP U (nil)))) (-. (frontsegP T_3 T_1)) (ssList T_1) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (ssList T_3) ### Imply 87 110
% 7.85/8.05 112. (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (ssList T_3) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (ssList T_1) (-. (frontsegP T_3 T_1)) (All U, ((ssList U) => (frontsegP U (nil)))) (ssList (nil)) (T_1 = (nil)) ### All 111
% 7.85/8.05 113. (-. (T_1 != (nil))) (ssList (nil)) (All U, ((ssList U) => (frontsegP U (nil)))) (-. (frontsegP T_3 T_1)) (ssList T_1) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (ssList T_3) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) ### NotNot 112
% 7.85/8.05 114. (-. (neq T_1 (nil))) (neq T_1 (nil)) ### Axiom
% 7.85/8.05 115. ((neq T_1 (nil)) <=> (T_1 != (nil))) (-. (neq T_1 (nil))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (ssList T_3) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (ssList T_1) (-. (frontsegP T_3 T_1)) (All U, ((ssList U) => (frontsegP U (nil)))) (ssList (nil)) ### Equiv 113 114
% 7.85/8.05 116. ((ssList (nil)) => ((neq T_1 (nil)) <=> (T_1 != (nil)))) (All U, ((ssList U) => (frontsegP U (nil)))) (-. (frontsegP T_3 T_1)) (ssList T_1) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (ssList T_3) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (-. (neq T_1 (nil))) (ssList (nil)) ### Imply 86 115
% 7.85/8.05 117. (All V, ((ssList V) => ((neq T_1 V) <=> (T_1 != V)))) (ssList (nil)) (-. (neq T_1 (nil))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (ssList T_3) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (ssList T_1) (-. (frontsegP T_3 T_1)) (All U, ((ssList U) => (frontsegP U (nil)))) ### All 116
% 7.85/8.05 118. (ssList T_1) (-. (ssList T_1)) ### Axiom
% 7.85/8.05 119. (ssList T_1) (-. (ssList T_1)) ### Axiom
% 7.85/8.05 120. (ssList (nil)) (-. (ssList (nil))) ### Axiom
% 7.85/8.05 121. (ssList (nil)) (-. (ssList (nil))) ### Axiom
% 7.85/8.05 122. (ssList T_1) (-. (ssList T_1)) ### Axiom
% 7.85/8.05 123. (-. (frontsegP T_1 (nil))) (frontsegP T_1 (nil)) ### Axiom
% 7.85/8.05 124. ((ssList T_1) => (frontsegP T_1 (nil))) (-. (frontsegP T_1 (nil))) (ssList T_1) ### Imply 122 123
% 7.85/8.05 125. (All U, ((ssList U) => (frontsegP U (nil)))) (ssList T_1) (-. (frontsegP T_1 (nil))) ### All 124
% 7.85/8.05 126. (ssList T_3) (-. (ssList T_3)) ### Axiom
% 7.85/8.05 127. (frontsegP T_1 (nil)) (-. (frontsegP T_1 (nil))) ### Axiom
% 7.85/8.05 128. (frontsegP (nil) T_3) (-. (frontsegP (nil) T_3)) ### Axiom
% 7.85/8.05 129. (-. (frontsegP T_1 T_3)) (frontsegP T_1 T_3) ### Axiom
% 7.85/8.05 130. ((ssList T_3) => (((frontsegP T_1 (nil)) /\ (frontsegP (nil) T_3)) => (frontsegP T_1 T_3))) (-. (frontsegP T_1 T_3)) (frontsegP (nil) T_3) (frontsegP T_1 (nil)) (ssList T_3) ### DisjTree 126 127 128 129
% 7.85/8.05 131. (All W, ((ssList W) => (((frontsegP T_1 (nil)) /\ (frontsegP (nil) W)) => (frontsegP T_1 W)))) (ssList T_3) (frontsegP T_1 (nil)) (frontsegP (nil) T_3) (-. (frontsegP T_1 T_3)) ### All 130
% 7.85/8.05 132. ((frontsegP T_1 (nil)) <=> (Ex W, ((ssList W) /\ ((app (nil) W) = T_1)))) (-. (frontsegP T_1 T_3)) (frontsegP (nil) T_3) (ssList T_3) (All W, ((ssList W) => (((frontsegP T_1 (nil)) /\ (frontsegP (nil) W)) => (frontsegP T_1 W)))) (ssList T_1) (All U, ((ssList U) => (frontsegP U (nil)))) ### Equiv 125 131
% 7.85/8.05 133. ((ssList (nil)) => ((frontsegP T_1 (nil)) <=> (Ex W, ((ssList W) /\ ((app (nil) W) = T_1))))) (All U, ((ssList U) => (frontsegP U (nil)))) (ssList T_1) (All W, ((ssList W) => (((frontsegP T_1 (nil)) /\ (frontsegP (nil) W)) => (frontsegP T_1 W)))) (ssList T_3) (frontsegP (nil) T_3) (-. (frontsegP T_1 T_3)) (ssList (nil)) ### Imply 121 132
% 7.85/8.05 134. (All V, ((ssList V) => ((frontsegP T_1 V) <=> (Ex W, ((ssList W) /\ ((app V W) = T_1)))))) (ssList (nil)) (-. (frontsegP T_1 T_3)) (frontsegP (nil) T_3) (ssList T_3) (All W, ((ssList W) => (((frontsegP T_1 (nil)) /\ (frontsegP (nil) W)) => (frontsegP T_1 W)))) (ssList T_1) (All U, ((ssList U) => (frontsegP U (nil)))) ### All 133
% 7.85/8.05 135. ((ssList (nil)) => (All W, ((ssList W) => (((frontsegP T_1 (nil)) /\ (frontsegP (nil) W)) => (frontsegP T_1 W))))) (All U, ((ssList U) => (frontsegP U (nil)))) (ssList T_1) (ssList T_3) (frontsegP (nil) T_3) (-. (frontsegP T_1 T_3)) (All V, ((ssList V) => ((frontsegP T_1 V) <=> (Ex W, ((ssList W) /\ ((app V W) = T_1)))))) (ssList (nil)) ### Imply 120 134
% 7.85/8.05 136. (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP T_1 V) /\ (frontsegP V W)) => (frontsegP T_1 W)))))) (ssList (nil)) (All V, ((ssList V) => ((frontsegP T_1 V) <=> (Ex W, ((ssList W) /\ ((app V W) = T_1)))))) (-. (frontsegP T_1 T_3)) (frontsegP (nil) T_3) (ssList T_3) (ssList T_1) (All U, ((ssList U) => (frontsegP U (nil)))) ### All 135
% 7.85/8.05 137. ((ssList T_1) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP T_1 V) /\ (frontsegP V W)) => (frontsegP T_1 W))))))) (All U, ((ssList U) => (frontsegP U (nil)))) (ssList T_3) (frontsegP (nil) T_3) (-. (frontsegP T_1 T_3)) (All V, ((ssList V) => ((frontsegP T_1 V) <=> (Ex W, ((ssList W) /\ ((app V W) = T_1)))))) (ssList (nil)) (ssList T_1) ### Imply 119 136
% 7.85/8.05 138. (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (ssList T_1) (ssList (nil)) (All V, ((ssList V) => ((frontsegP T_1 V) <=> (Ex W, ((ssList W) /\ ((app V W) = T_1)))))) (-. (frontsegP T_1 T_3)) (frontsegP (nil) T_3) (ssList T_3) (All U, ((ssList U) => (frontsegP U (nil)))) ### All 137
% 7.85/8.05 139. ((ssList T_1) => (All V, ((ssList V) => ((frontsegP T_1 V) <=> (Ex W, ((ssList W) /\ ((app V W) = T_1))))))) (All U, ((ssList U) => (frontsegP U (nil)))) (ssList T_3) (frontsegP (nil) T_3) (-. (frontsegP T_1 T_3)) (ssList (nil)) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (ssList T_1) ### Imply 118 138
% 7.85/8.05 140. (All U, ((ssList U) => (All V, ((ssList V) => ((frontsegP U V) <=> (Ex W, ((ssList W) /\ ((app V W) = U)))))))) (ssList T_1) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (ssList (nil)) (-. (frontsegP T_1 T_3)) (frontsegP (nil) T_3) (ssList T_3) (All U, ((ssList U) => (frontsegP U (nil)))) ### All 139
% 7.85/8.05 141. (T_0 != T_0) ### Refl(=)
% 7.85/8.05 142. (T_3 = T_1) (T_3 != T_1) ### Axiom
% 7.85/8.05 143. (T_0 != T_1) (T_0 = T_3) (T_3 = T_1) ### Trans 141 142
% 7.85/8.05 144. ((ssList T_1) => (((frontsegP T_3 T_1) /\ (frontsegP T_1 T_3)) => (T_3 = T_1))) (T_0 = T_3) (T_0 != T_1) (frontsegP (nil) T_3) (All U, ((ssList U) => (All V, ((ssList V) => ((frontsegP U V) <=> (Ex W, ((ssList W) /\ ((app V W) = U)))))))) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (ssList T_3) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (-. (neq T_1 (nil))) (ssList (nil)) (All V, ((ssList V) => ((neq T_1 V) <=> (T_1 != V)))) (ssList T_1) ### DisjTree 85 117 140 143
% 7.85/8.05 145. (All V, ((ssList V) => (((frontsegP T_3 V) /\ (frontsegP V T_3)) => (T_3 = 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) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (ssList T_3) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => ((frontsegP U V) <=> (Ex W, ((ssList W) /\ ((app V W) = U)))))))) (frontsegP (nil) T_3) (T_0 != T_1) (T_0 = T_3) ### All 144
% 7.90/8.06 146. ((ssList T_1) => (All V, ((ssList V) => ((neq T_1 V) <=> (T_1 != V))))) (T_0 = T_3) (T_0 != T_1) (frontsegP (nil) T_3) (All U, ((ssList U) => (All V, ((ssList V) => ((frontsegP U V) <=> (Ex W, ((ssList W) /\ ((app V W) = U)))))))) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (ssList T_3) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (-. (neq T_1 (nil))) (ssList (nil)) (All V, ((ssList V) => (((frontsegP T_3 V) /\ (frontsegP V T_3)) => (T_3 = V)))) (ssList T_1) ### Imply 84 145
% 7.90/8.06 147. (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (ssList T_1) (All V, ((ssList V) => (((frontsegP T_3 V) /\ (frontsegP V T_3)) => (T_3 = V)))) (ssList (nil)) (-. (neq T_1 (nil))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (ssList T_3) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => ((frontsegP U V) <=> (Ex W, ((ssList W) /\ ((app V W) = U)))))))) (frontsegP (nil) T_3) (T_0 != T_1) (T_0 = T_3) ### All 146
% 7.90/8.06 148. ((ssList T_3) => (All V, ((ssList V) => (((frontsegP T_3 V) /\ (frontsegP V T_3)) => (T_3 = V))))) (T_0 = T_3) (T_0 != T_1) (frontsegP (nil) T_3) (All U, ((ssList U) => (All V, ((ssList V) => ((frontsegP U V) <=> (Ex W, ((ssList W) /\ ((app V W) = U)))))))) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (-. (neq T_1 (nil))) (ssList (nil)) (ssList T_1) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (ssList T_3) ### Imply 83 147
% 7.90/8.06 149. (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) (ssList T_3) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (ssList T_1) (ssList (nil)) (-. (neq T_1 (nil))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => ((frontsegP U V) <=> (Ex W, ((ssList W) /\ ((app V W) = U)))))))) (frontsegP (nil) T_3) (T_0 != T_1) (T_0 = T_3) ### All 148
% 7.90/8.06 150. ((frontsegP (nil) T_3) <=> ((nil) = T_3)) (T_0 = T_3) (T_0 != T_1) (All U, ((ssList U) => (All V, ((ssList V) => ((frontsegP U V) <=> (Ex W, ((ssList W) /\ ((app V W) = U)))))))) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (-. (neq T_1 (nil))) (ssList (nil)) (ssList T_1) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (ssList T_3) (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) ((nil) = T_3) ### Equiv 82 149
% 7.90/8.06 151. ((ssList T_3) => ((frontsegP (nil) T_3) <=> ((nil) = T_3))) ((nil) = T_3) (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (ssList T_1) (ssList (nil)) (-. (neq T_1 (nil))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => ((frontsegP U V) <=> (Ex W, ((ssList W) /\ ((app V W) = U)))))))) (T_0 != T_1) (T_0 = T_3) (ssList T_3) ### Imply 81 150
% 7.90/8.06 152. (ssList T_3) (T_0 = T_3) (T_0 != T_1) (All U, ((ssList U) => (All V, ((ssList V) => ((frontsegP U V) <=> (Ex W, ((ssList W) /\ ((app V W) = U)))))))) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (-. (neq T_1 (nil))) (ssList (nil)) (ssList T_1) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) ((nil) = T_3) ### All 151
% 7.90/8.06 153. ((nil) != (nil)) ### NotEqual
% 7.90/8.06 154. (neq T_0 (nil)) ((nil) = T_3) (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (ssList T_1) (ssList (nil)) (-. (neq T_1 (nil))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => ((frontsegP U V) <=> (Ex W, ((ssList W) /\ ((app V W) = U)))))))) (T_0 = T_3) (ssList T_3) ### P-NotP 152 153
% 7.90/8.06 155. (-. ((nil) != T_3)) (ssList T_3) (T_0 = T_3) (All U, ((ssList U) => (All V, ((ssList V) => ((frontsegP U V) <=> (Ex W, ((ssList W) /\ ((app V W) = U)))))))) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (-. (neq T_1 (nil))) (ssList (nil)) (ssList T_1) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) (neq T_0 (nil)) ### NotNot 154
% 7.90/8.06 156. (-. (((nil) != T_3) \/ ((nil) != T_2))) (neq T_0 (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) => ((neq U V) <=> (U != V)))))) (ssList T_1) (ssList (nil)) (-. (neq T_1 (nil))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => ((frontsegP U V) <=> (Ex W, ((ssList W) /\ ((app V W) = U)))))))) (T_0 = T_3) (ssList T_3) ### NotOr 155
% 7.90/8.06 157. (T_1 = T_2) (T_2 != T_1) ### Sym(=)
% 7.90/8.06 158. (-. (neq T_1 (nil))) (neq T_2 (nil)) (T_1 = T_2) ### P-NotP 157 153
% 7.90/8.06 159. (-. (-. (neq T_2 (nil)))) (T_1 = T_2) (-. (neq T_1 (nil))) ### NotNot 158
% 7.90/8.06 160. (-. ((-. (neq T_2 (nil))) \/ (-. (rearsegP T_3 T_2)))) (-. (neq T_1 (nil))) (T_1 = T_2) ### NotOr 159
% 7.90/8.06 161. (-. ((((nil) != T_3) \/ ((nil) != T_2)) /\ ((-. (neq T_2 (nil))) \/ (-. (rearsegP T_3 T_2))))) (T_1 = T_2) (ssList T_3) (T_0 = T_3) (All U, ((ssList U) => (All V, ((ssList V) => ((frontsegP U V) <=> (Ex W, ((ssList W) /\ ((app V W) = U)))))))) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (-. (neq T_1 (nil))) (ssList (nil)) (ssList T_1) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) (neq T_0 (nil)) ### NotAnd 156 160
% 7.90/8.06 162. (-. ((neq T_1 (nil)) /\ (rearsegP T_0 T_1))) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_2) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (All U, ((ssList U) => (rearsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (neq T_0 (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) => ((neq U V) <=> (U != V)))))) (ssList T_1) (ssList (nil)) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => ((frontsegP U V) <=> (Ex W, ((ssList W) /\ ((app V W) = U)))))))) (T_0 = T_3) (ssList T_3) (T_1 = T_2) (-. ((((nil) != T_3) \/ ((nil) != T_2)) /\ ((-. (neq T_2 (nil))) \/ (-. (rearsegP T_3 T_2))))) ### NotAnd 161 47
% 7.90/8.06 163. (-. (-. (neq T_0 (nil)))) (-. ((((nil) != T_3) \/ ((nil) != T_2)) /\ ((-. (neq T_2 (nil))) \/ (-. (rearsegP T_3 T_2))))) (T_1 = T_2) (ssList T_3) (T_0 = T_3) (All U, ((ssList U) => (All V, ((ssList V) => ((frontsegP U V) <=> (Ex W, ((ssList W) /\ ((app V W) = U)))))))) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (ssList (nil)) (ssList T_1) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (All U, ((ssList U) => (rearsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (ssList T_2) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (-. ((neq T_1 (nil)) /\ (rearsegP T_0 T_1))) ### NotNot 162
% 7.90/8.06 164. (-. ((-. (neq T_0 (nil))) \/ ((neq T_1 (nil)) /\ (rearsegP T_0 T_1)))) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_2) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (All U, ((ssList U) => (rearsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (ssList T_1) (ssList (nil)) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => ((frontsegP U V) <=> (Ex W, ((ssList W) /\ ((app V W) = U)))))))) (T_0 = T_3) (ssList T_3) (T_1 = T_2) (-. ((((nil) != T_3) \/ ((nil) != T_2)) /\ ((-. (neq T_2 (nil))) \/ (-. (rearsegP T_3 T_2))))) ### NotOr 163
% 7.90/8.06 165. (-. ((((nil) != T_0) \/ ((nil) = T_1)) /\ ((-. (neq T_0 (nil))) \/ ((neq T_1 (nil)) /\ (rearsegP T_0 T_1))))) (All U, ((ssList U) => (All V, ((ssList V) => ((frontsegP U V) <=> (Ex W, ((ssList W) /\ ((app V W) = U)))))))) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) (ssList (nil)) (ssList T_1) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_2) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (All U, ((ssList U) => (rearsegP U (nil)))) (T_1 = T_2) (T_0 = T_3) (ssList T_3) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (-. ((((nil) != T_3) \/ ((nil) != T_2)) /\ ((-. (neq T_2 (nil))) \/ (-. (rearsegP T_3 T_2))))) (ssList T_0) ### NotAnd 80 164
% 7.90/8.06 166. (-. ((ssList T_3) => ((T_0 != T_3) \/ ((T_1 != T_2) \/ (((((nil) != T_0) \/ ((nil) = T_1)) /\ ((-. (neq T_0 (nil))) \/ ((neq T_1 (nil)) /\ (rearsegP T_0 T_1)))) \/ ((((nil) != T_3) \/ ((nil) != T_2)) /\ ((-. (neq T_2 (nil))) \/ (-. (rearsegP T_3 T_2))))))))) (ssList T_0) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (All U, ((ssList U) => (rearsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (ssList T_2) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_1) (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) => ((neq U V) <=> (U != V)))))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => ((frontsegP U V) <=> (Ex W, ((ssList W) /\ ((app V W) = U)))))))) ### ConjTree 165
% 7.90/8.06 167. (-. (All X, ((ssList X) => ((T_0 != X) \/ ((T_1 != T_2) \/ (((((nil) != T_0) \/ ((nil) = T_1)) /\ ((-. (neq T_0 (nil))) \/ ((neq T_1 (nil)) /\ (rearsegP T_0 T_1)))) \/ ((((nil) != X) \/ ((nil) != T_2)) /\ ((-. (neq T_2 (nil))) \/ (-. (rearsegP X T_2)))))))))) (All U, ((ssList U) => (All V, ((ssList V) => ((frontsegP U V) <=> (Ex W, ((ssList W) /\ ((app V W) = U)))))))) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) (ssList (nil)) (ssList T_1) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_2) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (All U, ((ssList U) => (rearsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (ssList T_0) ### NotAllEx 166
% 7.90/8.06 168. (-. ((ssList T_2) => (All X, ((ssList X) => ((T_0 != X) \/ ((T_1 != T_2) \/ (((((nil) != T_0) \/ ((nil) = T_1)) /\ ((-. (neq T_0 (nil))) \/ ((neq T_1 (nil)) /\ (rearsegP T_0 T_1)))) \/ ((((nil) != X) \/ ((nil) != T_2)) /\ ((-. (neq T_2 (nil))) \/ (-. (rearsegP X T_2))))))))))) (ssList T_0) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (All U, ((ssList U) => (rearsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_1) (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) => ((neq U V) <=> (U != V)))))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => ((frontsegP U V) <=> (Ex W, ((ssList W) /\ ((app V W) = U)))))))) ### NotImply 167
% 7.90/8.06 169. (-. (All W, ((ssList W) => (All X, ((ssList X) => ((T_0 != X) \/ ((T_1 != W) \/ (((((nil) != T_0) \/ ((nil) = T_1)) /\ ((-. (neq T_0 (nil))) \/ ((neq T_1 (nil)) /\ (rearsegP T_0 T_1)))) \/ ((((nil) != X) \/ ((nil) != W)) /\ ((-. (neq W (nil))) \/ (-. (rearsegP X W)))))))))))) (All U, ((ssList U) => (All V, ((ssList V) => ((frontsegP U V) <=> (Ex W, ((ssList W) /\ ((app V W) = U)))))))) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) (ssList (nil)) (ssList T_1) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (All U, ((ssList U) => (rearsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (ssList T_0) ### NotAllEx 168
% 7.90/8.06 170. (-. ((ssList T_0) => (All W, ((ssList W) => (All X, ((ssList X) => ((T_0 != X) \/ ((T_1 != W) \/ (((((nil) != T_0) \/ ((nil) = T_1)) /\ ((-. (neq T_0 (nil))) \/ ((neq T_1 (nil)) /\ (rearsegP T_0 T_1)))) \/ ((((nil) != X) \/ ((nil) != W)) /\ ((-. (neq W (nil))) \/ (-. (rearsegP X W))))))))))))) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (All U, ((ssList U) => (rearsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (ssList T_1) (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) => ((neq U V) <=> (U != V)))))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => ((frontsegP U V) <=> (Ex W, ((ssList W) /\ ((app V W) = U)))))))) ### NotImply 169
% 7.90/8.06 171. (-. (All V, ((ssList V) => (All W, ((ssList W) => (All X, ((ssList X) => ((V != X) \/ ((T_1 != W) \/ (((((nil) != V) \/ ((nil) = T_1)) /\ ((-. (neq V (nil))) \/ ((neq T_1 (nil)) /\ (rearsegP V T_1)))) \/ ((((nil) != X) \/ ((nil) != W)) /\ ((-. (neq W (nil))) \/ (-. (rearsegP X W)))))))))))))) (All U, ((ssList U) => (All V, ((ssList V) => ((frontsegP U V) <=> (Ex W, ((ssList W) /\ ((app V W) = U)))))))) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) (ssList (nil)) (ssList T_1) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (All U, ((ssList U) => (rearsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) ### NotAllEx 170
% 7.90/8.06 172. (-. ((ssList T_1) => (All V, ((ssList V) => (All W, ((ssList W) => (All X, ((ssList X) => ((V != X) \/ ((T_1 != W) \/ (((((nil) != V) \/ ((nil) = T_1)) /\ ((-. (neq V (nil))) \/ ((neq T_1 (nil)) /\ (rearsegP V T_1)))) \/ ((((nil) != X) \/ ((nil) != W)) /\ ((-. (neq W (nil))) \/ (-. (rearsegP X W))))))))))))))) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) (All U, ((ssList U) => (rearsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (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) => ((neq U V) <=> (U != V)))))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => ((frontsegP U V) <=> (Ex W, ((ssList W) /\ ((app V W) = U)))))))) ### NotImply 171
% 7.90/8.06 173. (-. (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (All X, ((ssList X) => ((V != X) \/ ((U != W) \/ (((((nil) != V) \/ ((nil) = U)) /\ ((-. (neq V (nil))) \/ ((neq U (nil)) /\ (rearsegP V U)))) \/ ((((nil) != X) \/ ((nil) != W)) /\ ((-. (neq W (nil))) \/ (-. (rearsegP X W)))))))))))))))) (All U, ((ssList U) => (All V, ((ssList V) => ((frontsegP U V) <=> (Ex W, ((ssList W) /\ ((app V W) = U)))))))) (All U, ((ssList U) => (frontsegP U (nil)))) (All U, ((ssList U) => ((frontsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((frontsegP U V) /\ (frontsegP V W)) => (frontsegP U W)))))))) (All U, ((ssList U) => (All V, ((ssList V) => ((neq U V) <=> (U != V)))))) (All U, ((ssList U) => (All V, ((ssList V) => (((frontsegP U V) /\ (frontsegP V U)) => (U = V)))))) (ssList (nil)) (All U, ((ssList U) => ((rearsegP (nil) U) <=> ((nil) = U)))) (All U, ((ssList U) => (All V, ((ssList V) => (All W, ((ssList W) => (((rearsegP U V) /\ (rearsegP V W)) => (rearsegP U W)))))))) (All U, ((ssList U) => (rearsegP U (nil)))) (All U, ((ssList U) => (All V, ((ssList V) => ((rearsegP U V) <=> (Ex W, ((ssList W) /\ ((app W V) = U)))))))) ### NotAllEx 172
% 7.90/8.07 % SZS output end Proof
% 7.90/8.07 (* END-PROOF *)
%------------------------------------------------------------------------------