TSTP Solution File: ANA023-10 by Toma---0.4
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%------------------------------------------------------------------------------
% File : Toma---0.4
% Problem : ANA023-10 : TPTP v8.1.2. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : toma --casc %s
% Computer : n014.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Aug 30 17:20:36 EDT 2023
% Result : Unsatisfiable 1.97s 2.26s
% Output : CNFRefutation 1.97s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.13/0.13 % Problem : ANA023-10 : TPTP v8.1.2. Released v7.5.0.
% 0.14/0.14 % Command : toma --casc %s
% 0.14/0.35 % Computer : n014.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Fri Aug 25 18:30:18 EDT 2023
% 0.14/0.35 % CPUTime :
% 1.97/2.26 % SZS status Unsatisfiable
% 1.97/2.26 % SZS output start Proof
% 1.97/2.26 original problem:
% 1.97/2.26 axioms:
% 1.97/2.26 ifeq2(A, A, B, C) = B
% 1.97/2.26 ifeq(A, A, B, C) = B
% 1.97/2.26 ifeq2(class_OrderedGroup_Ocomm__monoid__add(T_a), true(), c_plus(c_0(), V_y, T_a), V_y) = V_y
% 1.97/2.26 ifeq(class_OrderedGroup_Opordered__ab__group__add(T_a), true(), ifeq(c_lessequals(V_a, c_minus(V_c, V_b, T_a), T_a), true(), c_lessequals(c_plus(V_a, V_b, T_a), V_c, T_a), true()), true()) = true()
% 1.97/2.26 ifeq(class_OrderedGroup_Opordered__ab__group__add(T_a), true(), ifeq(c_lessequals(c_plus(V_a, V_b, T_a), V_c, T_a), true(), c_lessequals(V_a, c_minus(V_c, V_b, T_a), T_a), true()), true()) = true()
% 1.97/2.26 ifeq(class_Orderings_Oorder(T_a), true(), ifeq(c_lessequals(V_x, V_y, T_a), true(), ifeq(c_lessequals(V_y, V_z, T_a), true(), c_lessequals(V_x, V_z, T_a), true()), true()), true()) = true()
% 1.97/2.26 c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()) = true()
% 1.97/2.26 c_lessequals(v_k(v_x()), v_f(v_x()), t_b()) = true()
% 1.97/2.26 ifeq(class_Orderings_Olinorder(T), true(), class_Orderings_Oorder(T), true()) = true()
% 1.97/2.26 ifeq(class_Ring__and__Field_Oordered__idom(T), true(), class_OrderedGroup_Ocomm__monoid__add(T), true()) = true()
% 1.97/2.26 ifeq(class_Ring__and__Field_Oordered__idom(T), true(), class_Orderings_Olinorder(T), true()) = true()
% 1.97/2.26 ifeq(class_Ring__and__Field_Oordered__idom(T), true(), class_OrderedGroup_Opordered__ab__group__add(T), true()) = true()
% 1.97/2.26 class_Ring__and__Field_Oordered__idom(t_b()) = true()
% 1.97/2.26 goal:
% 1.97/2.26 c_lessequals(c_0(), c_minus(v_f(v_x()), v_g(v_x()), t_b()), t_b()) != true()
% 1.97/2.26 To show the unsatisfiability of the original goal,
% 1.97/2.26 it suffices to show that c_lessequals(c_0(), c_minus(v_f(v_x()), v_g(v_x()), t_b()), t_b()) = true() (skolemized goal) is valid under the axioms.
% 1.97/2.26 Here is an equational proof:
% 1.97/2.26 0: ifeq2(X0, X0, X1, X2) = X1.
% 1.97/2.26 Proof: Axiom.
% 1.97/2.26
% 1.97/2.26 1: ifeq(X0, X0, X1, X2) = X1.
% 1.97/2.26 Proof: Axiom.
% 1.97/2.26
% 1.97/2.26 2: ifeq2(class_OrderedGroup_Ocomm__monoid__add(X3), true(), c_plus(c_0(), X4, X3), X4) = X4.
% 1.97/2.26 Proof: Axiom.
% 1.97/2.26
% 1.97/2.26 3: ifeq(class_OrderedGroup_Opordered__ab__group__add(X3), true(), ifeq(c_lessequals(X5, c_minus(X6, X7, X3), X3), true(), c_lessequals(c_plus(X5, X7, X3), X6, X3), true()), true()) = true().
% 1.97/2.26 Proof: Axiom.
% 1.97/2.26
% 1.97/2.26 4: ifeq(class_OrderedGroup_Opordered__ab__group__add(X3), true(), ifeq(c_lessequals(c_plus(X5, X7, X3), X6, X3), true(), c_lessequals(X5, c_minus(X6, X7, X3), X3), true()), true()) = true().
% 1.97/2.26 Proof: Axiom.
% 1.97/2.26
% 1.97/2.26 5: ifeq(class_Orderings_Oorder(X3), true(), ifeq(c_lessequals(X8, X4, X3), true(), ifeq(c_lessequals(X4, X9, X3), true(), c_lessequals(X8, X9, X3), true()), true()), true()) = true().
% 1.97/2.26 Proof: Axiom.
% 1.97/2.26
% 1.97/2.26 6: c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()) = true().
% 1.97/2.26 Proof: Axiom.
% 1.97/2.26
% 1.97/2.26 7: c_lessequals(v_k(v_x()), v_f(v_x()), t_b()) = true().
% 1.97/2.26 Proof: Axiom.
% 1.97/2.26
% 1.97/2.26 8: ifeq(class_Orderings_Olinorder(X10), true(), class_Orderings_Oorder(X10), true()) = true().
% 1.97/2.26 Proof: Axiom.
% 1.97/2.26
% 1.97/2.26 9: ifeq(class_Ring__and__Field_Oordered__idom(X10), true(), class_OrderedGroup_Ocomm__monoid__add(X10), true()) = true().
% 1.97/2.26 Proof: Axiom.
% 1.97/2.26
% 1.97/2.26 10: ifeq(class_Ring__and__Field_Oordered__idom(X10), true(), class_Orderings_Olinorder(X10), true()) = true().
% 1.97/2.26 Proof: Axiom.
% 1.97/2.26
% 1.97/2.26 11: ifeq(class_Ring__and__Field_Oordered__idom(X10), true(), class_OrderedGroup_Opordered__ab__group__add(X10), true()) = true().
% 1.97/2.26 Proof: Axiom.
% 1.97/2.26
% 1.97/2.26 12: class_Ring__and__Field_Oordered__idom(t_b()) = true().
% 1.97/2.26 Proof: Axiom.
% 1.97/2.26
% 1.97/2.26 13: ifeq2(class_OrderedGroup_Ocomm__monoid__add(X3), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()), c_plus(c_0(), X4, X3), X4) = X4.
% 1.97/2.26 Proof: Rewrite equation 2,
% 1.97/2.26 lhs with equations [6]
% 1.97/2.26 rhs with equations [].
% 1.97/2.26
% 1.97/2.26 14: ifeq(class_OrderedGroup_Opordered__ab__group__add(X3), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()), ifeq(c_lessequals(X5, c_minus(X6, X7, X3), X3), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()), c_lessequals(c_plus(X5, X7, X3), X6, X3), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b())), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b())) = c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()).
% 1.97/2.26 Proof: Rewrite equation 3,
% 1.97/2.26 lhs with equations [6,6,6,6]
% 1.97/2.26 rhs with equations [6].
% 1.97/2.26
% 1.97/2.26 15: ifeq(class_OrderedGroup_Opordered__ab__group__add(X3), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()), ifeq(c_lessequals(c_plus(X5, X7, X3), X6, X3), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()), c_lessequals(X5, c_minus(X6, X7, X3), X3), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b())), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b())) = c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()).
% 1.97/2.26 Proof: Rewrite equation 4,
% 1.97/2.26 lhs with equations [6,6,6,6]
% 1.97/2.26 rhs with equations [6].
% 1.97/2.26
% 1.97/2.26 16: ifeq(class_Orderings_Oorder(X3), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()), ifeq(c_lessequals(X8, X4, X3), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()), ifeq(c_lessequals(X4, X9, X3), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()), c_lessequals(X8, X9, X3), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b())), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b())), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b())) = c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()).
% 1.97/2.26 Proof: Rewrite equation 5,
% 1.97/2.26 lhs with equations [6,6,6,6,6,6]
% 1.97/2.26 rhs with equations [6].
% 1.97/2.26
% 1.97/2.26 17: c_lessequals(v_k(v_x()), v_f(v_x()), t_b()) = c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()).
% 1.97/2.26 Proof: Rewrite equation 7,
% 1.97/2.26 lhs with equations []
% 1.97/2.26 rhs with equations [6].
% 1.97/2.26
% 1.97/2.26 18: ifeq(class_Orderings_Olinorder(X10), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()), class_Orderings_Oorder(X10), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b())) = c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()).
% 1.97/2.26 Proof: Rewrite equation 8,
% 1.97/2.26 lhs with equations [6,6]
% 1.97/2.26 rhs with equations [6].
% 1.97/2.26
% 1.97/2.26 19: ifeq(class_Ring__and__Field_Oordered__idom(X10), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()), class_OrderedGroup_Ocomm__monoid__add(X10), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b())) = c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()).
% 1.97/2.26 Proof: Rewrite equation 9,
% 1.97/2.26 lhs with equations [6,6]
% 1.97/2.26 rhs with equations [6].
% 1.97/2.26
% 1.97/2.26 20: ifeq(class_Ring__and__Field_Oordered__idom(X10), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()), class_Orderings_Olinorder(X10), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b())) = c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()).
% 1.97/2.26 Proof: Rewrite equation 10,
% 1.97/2.26 lhs with equations [6,6]
% 1.97/2.26 rhs with equations [6].
% 1.97/2.26
% 1.97/2.26 21: ifeq(class_Ring__and__Field_Oordered__idom(X10), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()), class_OrderedGroup_Opordered__ab__group__add(X10), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b())) = c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()).
% 1.97/2.26 Proof: Rewrite equation 11,
% 1.97/2.26 lhs with equations [6,6]
% 1.97/2.26 rhs with equations [6].
% 1.97/2.26
% 1.97/2.26 22: class_Ring__and__Field_Oordered__idom(t_b()) = c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()).
% 1.97/2.26 Proof: Rewrite equation 12,
% 1.97/2.26 lhs with equations []
% 1.97/2.26 rhs with equations [6].
% 1.97/2.26
% 1.97/2.26 23: c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()) = ifeq(class_OrderedGroup_Opordered__ab__group__add(t_b()), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()), c_lessequals(c_plus(c_0(), v_g(v_x()), t_b()), v_k(v_x()), t_b()), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b())).
% 1.97/2.26 Proof: A critical pair between equations 14 and 1.
% 1.97/2.26
% 1.97/2.26 24: c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()) = ifeq(c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()), class_OrderedGroup_Ocomm__monoid__add(t_b()), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b())).
% 1.97/2.26 Proof: A critical pair between equations 19 and 22.
% 1.97/2.26
% 1.97/2.26 25: c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()) = ifeq(c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()), class_Orderings_Olinorder(t_b()), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b())).
% 1.97/2.26 Proof: A critical pair between equations 20 and 22.
% 1.97/2.26
% 1.97/2.26 26: c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()) = ifeq(c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()), class_OrderedGroup_Opordered__ab__group__add(t_b()), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b())).
% 1.97/2.26 Proof: A critical pair between equations 21 and 22.
% 1.97/2.26
% 1.97/2.26 30: c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()) = ifeq(class_Orderings_Oorder(t_b()), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()), ifeq(c_lessequals(X8, v_k(v_x()), t_b()), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()), ifeq(c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()), c_lessequals(X8, v_f(v_x()), t_b()), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b())), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b())), c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b())).
% 1.97/2.26 Proof: A critical pair between equations 16 and 17.
% 1.97/2.26
% 1.97/2.26 33: true() = ifeq(class_Orderings_Oorder(t_b()), true(), ifeq(c_lessequals(X8, v_k(v_x()), t_b()), true(), c_lessequals(X8, v_f(v_x()), t_b()), true()), true()).
% 1.97/2.26 Proof: Rewrite equation 30,
% 1.97/2.26 lhs with equations [6]
% 1.97/2.26 rhs with equations [6,6,6,6,6,1,6,6].
% 1.97/2.26
% 1.97/2.26 37: true() = class_OrderedGroup_Opordered__ab__group__add(t_b()).
% 1.97/2.26 Proof: Rewrite equation 26,
% 1.97/2.26 lhs with equations [6]
% 1.97/2.26 rhs with equations [6,6,6,1].
% 1.97/2.26
% 1.97/2.26 38: true() = class_Orderings_Olinorder(t_b()).
% 1.97/2.26 Proof: Rewrite equation 25,
% 1.97/2.26 lhs with equations [6]
% 1.97/2.26 rhs with equations [6,6,6,1].
% 1.97/2.26
% 1.97/2.26 39: true() = class_OrderedGroup_Ocomm__monoid__add(t_b()).
% 1.97/2.26 Proof: Rewrite equation 24,
% 1.97/2.26 lhs with equations [6]
% 1.97/2.26 rhs with equations [6,6,6,1].
% 1.97/2.26
% 1.97/2.26 40: class_OrderedGroup_Ocomm__monoid__add(t_b()) = c_lessequals(c_plus(c_0(), v_g(v_x()), t_b()), v_k(v_x()), t_b()).
% 1.97/2.26 Proof: Rewrite equation 23,
% 1.97/2.26 lhs with equations [6,39]
% 1.97/2.26 rhs with equations [37,39,6,39,6,39,1].
% 1.97/2.26
% 1.97/2.26 45: ifeq(class_Orderings_Olinorder(X10), class_OrderedGroup_Ocomm__monoid__add(t_b()), class_Orderings_Oorder(X10), class_OrderedGroup_Ocomm__monoid__add(t_b())) = class_OrderedGroup_Ocomm__monoid__add(t_b()).
% 1.97/2.26 Proof: Rewrite equation 18,
% 1.97/2.26 lhs with equations [6,39,6,39]
% 1.97/2.26 rhs with equations [6,39].
% 1.97/2.26
% 1.97/2.26 47: c_lessequals(c_0(), c_minus(v_k(v_x()), v_g(v_x()), t_b()), t_b()) = class_OrderedGroup_Ocomm__monoid__add(t_b()).
% 1.97/2.26 Proof: Rewrite equation 6,
% 1.97/2.26 lhs with equations []
% 1.97/2.26 rhs with equations [39].
% 1.97/2.26
% 1.97/2.26 49: ifeq(class_OrderedGroup_Opordered__ab__group__add(X3), class_OrderedGroup_Ocomm__monoid__add(t_b()), ifeq(c_lessequals(c_plus(X5, X7, X3), X6, X3), class_OrderedGroup_Ocomm__monoid__add(t_b()), c_lessequals(X5, c_minus(X6, X7, X3), X3), class_OrderedGroup_Ocomm__monoid__add(t_b())), class_OrderedGroup_Ocomm__monoid__add(t_b())) = class_OrderedGroup_Ocomm__monoid__add(t_b()).
% 1.97/2.26 Proof: Rewrite equation 15,
% 1.97/2.26 lhs with equations [47,47,47,47]
% 1.97/2.26 rhs with equations [47].
% 1.97/2.26
% 1.97/2.26 51: ifeq2(class_OrderedGroup_Ocomm__monoid__add(X3), class_OrderedGroup_Ocomm__monoid__add(t_b()), c_plus(c_0(), X4, X3), X4) = X4.
% 1.97/2.26 Proof: Rewrite equation 13,
% 1.97/2.26 lhs with equations [47]
% 1.97/2.26 rhs with equations [].
% 1.97/2.26
% 1.97/2.26 52: X7 = c_plus(c_0(), X7, t_b()).
% 1.97/2.26 Proof: A critical pair between equations 51 and 0.
% 1.97/2.26
% 1.97/2.26 53: class_OrderedGroup_Ocomm__monoid__add(t_b()) = ifeq(true(), class_OrderedGroup_Ocomm__monoid__add(t_b()), class_Orderings_Oorder(t_b()), class_OrderedGroup_Ocomm__monoid__add(t_b())).
% 1.97/2.26 Proof: A critical pair between equations 45 and 38.
% 1.97/2.26
% 1.97/2.26 69: c_lessequals(v_g(v_x()), v_k(v_x()), t_b()) = class_Orderings_Oorder(t_b()).
% 1.97/2.26 Proof: Rewrite equation 53,
% 1.97/2.26 lhs with equations [40,52]
% 1.97/2.26 rhs with equations [39,40,52,40,52,40,52,1].
% 1.97/2.26
% 1.97/2.26 72: ifeq(class_OrderedGroup_Opordered__ab__group__add(X3), c_lessequals(v_g(v_x()), v_k(v_x()), t_b()), ifeq(c_lessequals(c_plus(X5, X7, X3), X6, X3), c_lessequals(v_g(v_x()), v_k(v_x()), t_b()), c_lessequals(X5, c_minus(X6, X7, X3), X3), c_lessequals(v_g(v_x()), v_k(v_x()), t_b())), c_lessequals(v_g(v_x()), v_k(v_x()), t_b())) = c_lessequals(v_g(v_x()), v_k(v_x()), t_b()).
% 1.97/2.26 Proof: Rewrite equation 49,
% 1.97/2.26 lhs with equations [40,52,40,52,40,52,40,52]
% 1.97/2.26 rhs with equations [40,52].
% 1.97/2.26
% 1.97/2.26 81: class_OrderedGroup_Ocomm__monoid__add(t_b()) = c_lessequals(v_g(v_x()), v_k(v_x()), t_b()).
% 1.97/2.26 Proof: Rewrite equation 40,
% 1.97/2.26 lhs with equations []
% 1.97/2.26 rhs with equations [52].
% 1.97/2.26
% 1.97/2.26 82: true() = c_lessequals(v_g(v_x()), v_k(v_x()), t_b()).
% 1.97/2.26 Proof: Rewrite equation 39,
% 1.97/2.26 lhs with equations []
% 1.97/2.26 rhs with equations [81].
% 1.97/2.26
% 1.97/2.26 84: c_lessequals(v_g(v_x()), v_k(v_x()), t_b()) = class_OrderedGroup_Opordered__ab__group__add(t_b()).
% 1.97/2.26 Proof: Rewrite equation 37,
% 1.97/2.26 lhs with equations [82]
% 1.97/2.26 rhs with equations [].
% 1.97/2.26
% 1.97/2.26 88: c_lessequals(v_g(v_x()), v_k(v_x()), t_b()) = ifeq(c_lessequals(X8, v_k(v_x()), t_b()), c_lessequals(v_g(v_x()), v_k(v_x()), t_b()), c_lessequals(X8, v_f(v_x()), t_b()), c_lessequals(v_g(v_x()), v_k(v_x()), t_b())).
% 1.97/2.26 Proof: Rewrite equation 33,
% 1.97/2.26 lhs with equations [82]
% 1.97/2.26 rhs with equations [69,82,82,82,82,1].
% 1.97/2.26
% 1.97/2.26 90: c_lessequals(v_g(v_x()), v_f(v_x()), t_b()) = c_lessequals(v_g(v_x()), v_k(v_x()), t_b()).
% 1.97/2.26 Proof: A critical pair between equations 1 and 88.
% 1.97/2.26
% 1.97/2.26 98: c_lessequals(v_g(v_x()), v_k(v_x()), t_b()) = ifeq(class_OrderedGroup_Opordered__ab__group__add(t_b()), c_lessequals(v_g(v_x()), v_k(v_x()), t_b()), ifeq(c_lessequals(X8, X6, t_b()), c_lessequals(v_g(v_x()), v_k(v_x()), t_b()), c_lessequals(c_0(), c_minus(X6, X8, t_b()), t_b()), c_lessequals(v_g(v_x()), v_k(v_x()), t_b())), c_lessequals(v_g(v_x()), v_k(v_x()), t_b())).
% 1.97/2.26 Proof: A critical pair between equations 72 and 52.
% 1.97/2.26
% 1.97/2.26 105: true() = ifeq(c_lessequals(X8, X6, t_b()), true(), c_lessequals(c_0(), c_minus(X6, X8, t_b()), t_b()), true()).
% 1.97/2.26 Proof: Rewrite equation 98,
% 1.97/2.26 lhs with equations [82]
% 1.97/2.26 rhs with equations [84,82,82,82,82,82,1].
% 1.97/2.26
% 1.97/2.26 110: c_lessequals(v_g(v_x()), v_f(v_x()), t_b()) = true().
% 1.97/2.26 Proof: Rewrite equation 90,
% 1.97/2.26 lhs with equations []
% 1.97/2.26 rhs with equations [82].
% 1.97/2.26
% 1.97/2.26 130: true() = class_Orderings_Oorder(t_b()).
% 1.97/2.26 Proof: Rewrite equation 69,
% 1.97/2.26 lhs with equations [82]
% 1.97/2.26 rhs with equations [].
% 1.97/2.26
% 1.97/2.26 132: true() = ifeq(true(), true(), c_lessequals(c_0(), c_minus(v_f(v_x()), v_g(v_x()), t_b()), t_b()), true()).
% 1.97/2.26 Proof: A critical pair between equations 105 and 110.
% 1.97/2.26
% 1.97/2.26 153: class_Orderings_Oorder(t_b()) = c_lessequals(c_0(), c_minus(v_f(v_x()), v_g(v_x()), t_b()), t_b()).
% 1.97/2.26 Proof: Rewrite equation 132,
% 1.97/2.26 lhs with equations [130]
% 1.97/2.26 rhs with equations [130,130,130,1].
% 1.97/2.26
% 1.97/2.26 184: c_lessequals(c_0(), c_minus(v_f(v_x()), v_g(v_x()), t_b()), t_b()) = true().
% 1.97/2.26 Proof: Rewrite lhs with equations [153]
% 1.97/2.26 rhs with equations [130].
% 1.97/2.26
% 1.97/2.26 % SZS output end Proof
%------------------------------------------------------------------------------