TSTP Solution File: ALG431-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : ALG431-1 : TPTP v8.1.2. Released v4.1.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n024.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 16:43:16 EDT 2023
% Result : Unsatisfiable 101.75s 13.36s
% Output : Proof 102.61s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : ALG431-1 : TPTP v8.1.2. Released v4.1.0.
% 0.07/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.34 % Computer : n024.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Mon Aug 28 05:58:38 EDT 2023
% 0.12/0.35 % CPUTime :
% 101.75/13.36 Command-line arguments: --flatten
% 101.75/13.36
% 101.75/13.36 % SZS status Unsatisfiable
% 101.75/13.36
% 102.61/13.39 % SZS output start Proof
% 102.61/13.39 Take the following subset of the input axioms:
% 102.61/13.40 fof(cls_CHAINED_0, axiom, ~class_Ring__and__Field_Oidom(t_a) | (~class_Int_Oring__char__0(t_a) | c_Polynomial_Odegree(v_p, t_a)=c_HOL_Ozero__class_Ozero(tc_nat))).
% 102.61/13.40 fof(cls_add__diff__cancel_0, axiom, ![T_a, V_a, V_b]: (~class_OrderedGroup_Ogroup__add(T_a) | c_HOL_Ominus__class_Ominus(c_HOL_Oplus__class_Oplus(V_a, V_b, T_a), V_b, T_a)=V_a)).
% 102.61/13.40 fof(cls_conjecture_0, negated_conjecture, ~v_thesis____).
% 102.61/13.40 fof(cls_conjecture_1, negated_conjecture, ![V_x]: (v_p!=c_Polynomial_OpCons(V_x, c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)), t_a) | v_thesis____)).
% 102.61/13.40 fof(cls_diff__0__right_0, axiom, ![T_a2, V_a2]: (~class_OrderedGroup_Ogroup__add(T_a2) | c_HOL_Ominus__class_Ominus(V_a2, c_HOL_Ozero__class_Ozero(T_a2), T_a2)=V_a2)).
% 102.61/13.40 fof(cls_smult__0__right_0, axiom, ![T_a2, V_a2]: (~class_Ring__and__Field_Ocomm__semiring__0(T_a2) | c_Polynomial_Osmult(V_a2, c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a2)), T_a2)=c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a2)))).
% 102.61/13.40 fof(cls_synthetic__div__correct_0, axiom, ![V_c, V_p, T_a2]: (~class_Ring__and__Field_Ocomm__semiring__0(T_a2) | c_HOL_Oplus__class_Oplus(V_p, c_Polynomial_Osmult(V_c, c_Polynomial_Osynthetic__div(V_p, V_c, T_a2), T_a2), tc_Polynomial_Opoly(T_a2))=c_Polynomial_OpCons(c_Polynomial_Opoly(V_p, V_c, T_a2), c_Polynomial_Osynthetic__div(V_p, V_c, T_a2), T_a2))).
% 102.61/13.40 fof(cls_synthetic__div__eq__0__iff_1, axiom, ![T_a2, V_c2, V_p2]: (~class_Ring__and__Field_Ocomm__semiring__0(T_a2) | (c_Polynomial_Odegree(V_p2, T_a2)!=c_HOL_Ozero__class_Ozero(tc_nat) | c_Polynomial_Osynthetic__div(V_p2, V_c2, T_a2)=c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(T_a2))))).
% 102.61/13.40 fof(clsarity_Polynomial__Opoly__Ring__and__Field_Oidom, axiom, ![T_1]: (class_Ring__and__Field_Oidom(tc_Polynomial_Opoly(T_1)) | ~class_Ring__and__Field_Oidom(T_1))).
% 102.61/13.40 fof(clsarity_RealDef__Oreal__Int_Onumber__ring, axiom, class_Int_Onumber__ring(tc_RealDef_Oreal)).
% 102.61/13.40 fof(clsarity_RealDef__Oreal__OrderedGroup_Olordered__ab__group__add__abs, axiom, class_OrderedGroup_Olordered__ab__group__add__abs(tc_RealDef_Oreal)).
% 102.61/13.40 fof(clsarity_RealDef__Oreal__OrderedGroup_Opordered__ab__group__add, axiom, class_OrderedGroup_Opordered__ab__group__add(tc_RealDef_Oreal)).
% 102.61/13.40 fof(clsarity_RealDef__Oreal__RealVector_Oreal__vector, axiom, class_RealVector_Oreal__vector(tc_RealDef_Oreal)).
% 102.61/13.40 fof(clsarity_RealDef__Oreal__Ring__and__Field_Osgn__if, axiom, class_Ring__and__Field_Osgn__if(tc_RealDef_Oreal)).
% 102.61/13.40 fof(clsarity_nat__Divides_Osemiring__div, axiom, class_Divides_Osemiring__div(tc_nat)).
% 102.61/13.40 fof(clsarity_nat__Ring__and__Field_Oordered__semidom, axiom, class_Ring__and__Field_Oordered__semidom(tc_nat)).
% 102.61/13.40 fof(clsrel_Ring__and__Field_Oidom_OrderedGroup_Ogroup__add, axiom, ![T]: (~class_Ring__and__Field_Oidom(T) | class_OrderedGroup_Ogroup__add(T))).
% 102.61/13.40 fof(clsrel_Ring__and__Field_Oidom_Ring__and__Field_Ocomm__semiring__0, axiom, ![T2]: (~class_Ring__and__Field_Oidom(T2) | class_Ring__and__Field_Ocomm__semiring__0(T2))).
% 102.61/13.40 fof(tfree_tcs, negated_conjecture, class_Int_Oring__char__0(t_a)).
% 102.61/13.40 fof(tfree_tcs_01, negated_conjecture, class_Ring__and__Field_Oidom(t_a)).
% 102.61/13.40
% 102.61/13.40 Now clausify the problem and encode Horn clauses using encoding 3 of
% 102.61/13.40 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 102.61/13.40 We repeatedly replace C & s=t => u=v by the two clauses:
% 102.61/13.40 fresh(y, y, x1...xn) = u
% 102.61/13.40 C => fresh(s, t, x1...xn) = v
% 102.61/13.40 where fresh is a fresh function symbol and x1..xn are the free
% 102.61/13.40 variables of u and v.
% 102.61/13.40 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 102.61/13.40 input problem has no model of domain size 1).
% 102.61/13.40
% 102.61/13.40 The encoding turns the above axioms into the following unit equations and goals:
% 102.61/13.40
% 102.61/13.40 Axiom 1 (clsarity_RealDef__Oreal__OrderedGroup_Olordered__ab__group__add__abs): class_OrderedGroup_Olordered__ab__group__add__abs(tc_RealDef_Oreal) = true2.
% 102.61/13.40 Axiom 2 (clsarity_RealDef__Oreal__Ring__and__Field_Osgn__if): class_Ring__and__Field_Osgn__if(tc_RealDef_Oreal) = true2.
% 102.61/13.40 Axiom 3 (clsarity_RealDef__Oreal__RealVector_Oreal__vector): class_RealVector_Oreal__vector(tc_RealDef_Oreal) = true2.
% 102.61/13.40 Axiom 4 (clsarity_nat__Ring__and__Field_Oordered__semidom): class_Ring__and__Field_Oordered__semidom(tc_nat) = true2.
% 102.61/13.40 Axiom 5 (clsarity_RealDef__Oreal__Int_Onumber__ring): class_Int_Onumber__ring(tc_RealDef_Oreal) = true2.
% 102.61/13.40 Axiom 6 (clsarity_RealDef__Oreal__OrderedGroup_Opordered__ab__group__add): class_OrderedGroup_Opordered__ab__group__add(tc_RealDef_Oreal) = true2.
% 102.61/13.40 Axiom 7 (tfree_tcs): class_Int_Oring__char__0(t_a) = true2.
% 102.61/13.40 Axiom 8 (clsarity_nat__Divides_Osemiring__div): class_Divides_Osemiring__div(tc_nat) = true2.
% 102.61/13.40 Axiom 9 (tfree_tcs_01): class_Ring__and__Field_Oidom(t_a) = true2.
% 102.61/13.40 Axiom 10 (cls_conjecture_1): fresh633(X, X) = true2.
% 102.61/13.40 Axiom 11 (cls_CHAINED_0): fresh746(X, X) = c_HOL_Ozero__class_Ozero(tc_nat).
% 102.61/13.40 Axiom 12 (cls_CHAINED_0): fresh745(X, X) = c_Polynomial_Odegree(v_p, t_a).
% 102.61/13.40 Axiom 13 (clsarity_Polynomial__Opoly__Ring__and__Field_Oidom): fresh168(X, X, Y) = true2.
% 102.61/13.40 Axiom 14 (clsrel_Ring__and__Field_Oidom_OrderedGroup_Ogroup__add): fresh131(X, X, Y) = true2.
% 102.61/13.40 Axiom 15 (clsrel_Ring__and__Field_Oidom_Ring__and__Field_Ocomm__semiring__0): fresh125(X, X, Y) = true2.
% 102.61/13.40 Axiom 16 (cls_CHAINED_0): fresh745(class_Int_Oring__char__0(t_a), true2) = fresh746(class_Ring__and__Field_Oidom(t_a), true2).
% 102.61/13.40 Axiom 17 (cls_smult__0__right_0): fresh251(X, X, Y, Z) = c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(Y)).
% 102.61/13.40 Axiom 18 (clsarity_Polynomial__Opoly__Ring__and__Field_Oidom): fresh168(class_Ring__and__Field_Oidom(X), true2, X) = class_Ring__and__Field_Oidom(tc_Polynomial_Opoly(X)).
% 102.61/13.40 Axiom 19 (clsrel_Ring__and__Field_Oidom_OrderedGroup_Ogroup__add): fresh131(class_Ring__and__Field_Oidom(X), true2, X) = class_OrderedGroup_Ogroup__add(X).
% 102.61/13.40 Axiom 20 (clsrel_Ring__and__Field_Oidom_Ring__and__Field_Ocomm__semiring__0): fresh125(class_Ring__and__Field_Oidom(X), true2, X) = class_Ring__and__Field_Ocomm__semiring__0(X).
% 102.61/13.40 Axiom 21 (cls_diff__0__right_0): fresh45(X, X, Y, Z) = Z.
% 102.61/13.40 Axiom 22 (cls_synthetic__div__eq__0__iff_1): fresh225(X, X, Y, Z, W) = c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(Y)).
% 102.61/13.40 Axiom 23 (cls_synthetic__div__eq__0__iff_1): fresh224(X, X, Y, Z, W) = c_Polynomial_Osynthetic__div(Z, W, Y).
% 102.61/13.40 Axiom 24 (cls_add__diff__cancel_0): fresh70(X, X, Y, Z, W) = Z.
% 102.61/13.40 Axiom 25 (cls_diff__0__right_0): fresh45(class_OrderedGroup_Ogroup__add(X), true2, X, Y) = c_HOL_Ominus__class_Ominus(Y, c_HOL_Ozero__class_Ozero(X), X).
% 102.61/13.40 Axiom 26 (cls_smult__0__right_0): fresh251(class_Ring__and__Field_Ocomm__semiring__0(X), true2, X, Y) = c_Polynomial_Osmult(Y, c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(X)), X).
% 102.61/13.40 Axiom 27 (cls_add__diff__cancel_0): fresh70(class_OrderedGroup_Ogroup__add(X), true2, X, Y, Z) = c_HOL_Ominus__class_Ominus(c_HOL_Oplus__class_Oplus(Y, Z, X), Z, X).
% 102.61/13.40 Axiom 28 (cls_conjecture_1): fresh633(v_p, c_Polynomial_OpCons(X, c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)), t_a)) = v_thesis____.
% 102.61/13.40 Axiom 29 (cls_synthetic__div__eq__0__iff_1): fresh224(class_Ring__and__Field_Ocomm__semiring__0(X), true2, X, Y, Z) = fresh225(c_Polynomial_Odegree(Y, X), c_HOL_Ozero__class_Ozero(tc_nat), X, Y, Z).
% 102.61/13.40 Axiom 30 (cls_synthetic__div__correct_0): fresh228(X, X, Y, Z, W) = c_Polynomial_OpCons(c_Polynomial_Opoly(Z, W, Y), c_Polynomial_Osynthetic__div(Z, W, Y), Y).
% 102.61/13.40 Axiom 31 (cls_synthetic__div__correct_0): fresh228(class_Ring__and__Field_Ocomm__semiring__0(X), true2, X, Y, Z) = c_HOL_Oplus__class_Oplus(Y, c_Polynomial_Osmult(Z, c_Polynomial_Osynthetic__div(Y, Z, X), X), tc_Polynomial_Opoly(X)).
% 102.61/13.40
% 102.61/13.40 Lemma 32: class_Ring__and__Field_Osgn__if(tc_RealDef_Oreal) = class_OrderedGroup_Olordered__ab__group__add__abs(tc_RealDef_Oreal).
% 102.61/13.40 Proof:
% 102.61/13.40 class_Ring__and__Field_Osgn__if(tc_RealDef_Oreal)
% 102.61/13.40 = { by axiom 2 (clsarity_RealDef__Oreal__Ring__and__Field_Osgn__if) }
% 102.61/13.40 true2
% 102.61/13.40 = { by axiom 1 (clsarity_RealDef__Oreal__OrderedGroup_Olordered__ab__group__add__abs) R->L }
% 102.61/13.40 class_OrderedGroup_Olordered__ab__group__add__abs(tc_RealDef_Oreal)
% 102.61/13.40
% 102.61/13.40 Lemma 33: class_RealVector_Oreal__vector(tc_RealDef_Oreal) = class_Ring__and__Field_Osgn__if(tc_RealDef_Oreal).
% 102.61/13.40 Proof:
% 102.61/13.40 class_RealVector_Oreal__vector(tc_RealDef_Oreal)
% 102.61/13.40 = { by axiom 3 (clsarity_RealDef__Oreal__RealVector_Oreal__vector) }
% 102.61/13.40 true2
% 102.61/13.40 = { by axiom 1 (clsarity_RealDef__Oreal__OrderedGroup_Olordered__ab__group__add__abs) R->L }
% 102.61/13.40 class_OrderedGroup_Olordered__ab__group__add__abs(tc_RealDef_Oreal)
% 102.61/13.40 = { by lemma 32 R->L }
% 102.61/13.40 class_Ring__and__Field_Osgn__if(tc_RealDef_Oreal)
% 102.61/13.40
% 102.61/13.40 Lemma 34: class_Ring__and__Field_Oordered__semidom(tc_nat) = class_RealVector_Oreal__vector(tc_RealDef_Oreal).
% 102.61/13.40 Proof:
% 102.61/13.40 class_Ring__and__Field_Oordered__semidom(tc_nat)
% 102.61/13.40 = { by axiom 4 (clsarity_nat__Ring__and__Field_Oordered__semidom) }
% 102.61/13.40 true2
% 102.61/13.40 = { by axiom 1 (clsarity_RealDef__Oreal__OrderedGroup_Olordered__ab__group__add__abs) R->L }
% 102.61/13.40 class_OrderedGroup_Olordered__ab__group__add__abs(tc_RealDef_Oreal)
% 102.61/13.40 = { by lemma 32 R->L }
% 102.61/13.40 class_Ring__and__Field_Osgn__if(tc_RealDef_Oreal)
% 102.61/13.40 = { by lemma 33 R->L }
% 102.61/13.40 class_RealVector_Oreal__vector(tc_RealDef_Oreal)
% 102.61/13.40
% 102.61/13.40 Lemma 35: class_Int_Onumber__ring(tc_RealDef_Oreal) = class_Ring__and__Field_Oordered__semidom(tc_nat).
% 102.61/13.40 Proof:
% 102.61/13.40 class_Int_Onumber__ring(tc_RealDef_Oreal)
% 102.61/13.40 = { by axiom 5 (clsarity_RealDef__Oreal__Int_Onumber__ring) }
% 102.61/13.40 true2
% 102.61/13.40 = { by axiom 1 (clsarity_RealDef__Oreal__OrderedGroup_Olordered__ab__group__add__abs) R->L }
% 102.61/13.40 class_OrderedGroup_Olordered__ab__group__add__abs(tc_RealDef_Oreal)
% 102.61/13.40 = { by lemma 32 R->L }
% 102.61/13.40 class_Ring__and__Field_Osgn__if(tc_RealDef_Oreal)
% 102.61/13.40 = { by lemma 33 R->L }
% 102.61/13.40 class_RealVector_Oreal__vector(tc_RealDef_Oreal)
% 102.61/13.40 = { by lemma 34 R->L }
% 102.61/13.40 class_Ring__and__Field_Oordered__semidom(tc_nat)
% 102.61/13.40
% 102.61/13.40 Lemma 36: class_OrderedGroup_Opordered__ab__group__add(tc_RealDef_Oreal) = class_Int_Onumber__ring(tc_RealDef_Oreal).
% 102.61/13.40 Proof:
% 102.61/13.40 class_OrderedGroup_Opordered__ab__group__add(tc_RealDef_Oreal)
% 102.61/13.40 = { by axiom 6 (clsarity_RealDef__Oreal__OrderedGroup_Opordered__ab__group__add) }
% 102.61/13.40 true2
% 102.61/13.40 = { by axiom 1 (clsarity_RealDef__Oreal__OrderedGroup_Olordered__ab__group__add__abs) R->L }
% 102.61/13.40 class_OrderedGroup_Olordered__ab__group__add__abs(tc_RealDef_Oreal)
% 102.61/13.40 = { by lemma 32 R->L }
% 102.61/13.40 class_Ring__and__Field_Osgn__if(tc_RealDef_Oreal)
% 102.61/13.40 = { by lemma 33 R->L }
% 102.61/13.40 class_RealVector_Oreal__vector(tc_RealDef_Oreal)
% 102.61/13.40 = { by lemma 34 R->L }
% 102.61/13.40 class_Ring__and__Field_Oordered__semidom(tc_nat)
% 102.61/13.40 = { by lemma 35 R->L }
% 102.61/13.40 class_Int_Onumber__ring(tc_RealDef_Oreal)
% 102.61/13.40
% 102.61/13.40 Lemma 37: class_Divides_Osemiring__div(tc_nat) = class_OrderedGroup_Opordered__ab__group__add(tc_RealDef_Oreal).
% 102.61/13.40 Proof:
% 102.61/13.40 class_Divides_Osemiring__div(tc_nat)
% 102.61/13.40 = { by axiom 8 (clsarity_nat__Divides_Osemiring__div) }
% 102.61/13.40 true2
% 102.61/13.40 = { by axiom 1 (clsarity_RealDef__Oreal__OrderedGroup_Olordered__ab__group__add__abs) R->L }
% 102.61/13.40 class_OrderedGroup_Olordered__ab__group__add__abs(tc_RealDef_Oreal)
% 102.61/13.40 = { by lemma 32 R->L }
% 102.61/13.40 class_Ring__and__Field_Osgn__if(tc_RealDef_Oreal)
% 102.61/13.40 = { by lemma 33 R->L }
% 102.61/13.40 class_RealVector_Oreal__vector(tc_RealDef_Oreal)
% 102.61/13.40 = { by lemma 34 R->L }
% 102.61/13.40 class_Ring__and__Field_Oordered__semidom(tc_nat)
% 102.61/13.40 = { by lemma 35 R->L }
% 102.61/13.40 class_Int_Onumber__ring(tc_RealDef_Oreal)
% 102.61/13.40 = { by lemma 36 R->L }
% 102.61/13.40 class_OrderedGroup_Opordered__ab__group__add(tc_RealDef_Oreal)
% 102.61/13.40
% 102.61/13.40 Lemma 38: class_Ring__and__Field_Oidom(t_a) = class_Divides_Osemiring__div(tc_nat).
% 102.61/13.40 Proof:
% 102.61/13.40 class_Ring__and__Field_Oidom(t_a)
% 102.61/13.40 = { by axiom 9 (tfree_tcs_01) }
% 102.61/13.40 true2
% 102.61/13.40 = { by axiom 1 (clsarity_RealDef__Oreal__OrderedGroup_Olordered__ab__group__add__abs) R->L }
% 102.61/13.40 class_OrderedGroup_Olordered__ab__group__add__abs(tc_RealDef_Oreal)
% 102.61/13.40 = { by lemma 32 R->L }
% 102.61/13.40 class_Ring__and__Field_Osgn__if(tc_RealDef_Oreal)
% 102.61/13.40 = { by lemma 33 R->L }
% 102.61/13.40 class_RealVector_Oreal__vector(tc_RealDef_Oreal)
% 102.61/13.40 = { by lemma 34 R->L }
% 102.61/13.40 class_Ring__and__Field_Oordered__semidom(tc_nat)
% 102.61/13.40 = { by lemma 35 R->L }
% 102.61/13.40 class_Int_Onumber__ring(tc_RealDef_Oreal)
% 102.61/13.40 = { by lemma 36 R->L }
% 102.61/13.40 class_OrderedGroup_Opordered__ab__group__add(tc_RealDef_Oreal)
% 102.61/13.40 = { by lemma 37 R->L }
% 102.61/13.40 class_Divides_Osemiring__div(tc_nat)
% 102.61/13.40
% 102.61/13.40 Lemma 39: class_Divides_Osemiring__div(tc_nat) = class_Ring__and__Field_Ocomm__semiring__0(t_a).
% 102.61/13.40 Proof:
% 102.61/13.40 class_Divides_Osemiring__div(tc_nat)
% 102.61/13.40 = { by lemma 37 }
% 102.61/13.40 class_OrderedGroup_Opordered__ab__group__add(tc_RealDef_Oreal)
% 102.61/13.40 = { by lemma 36 }
% 102.61/13.40 class_Int_Onumber__ring(tc_RealDef_Oreal)
% 102.61/13.40 = { by lemma 35 }
% 102.61/13.40 class_Ring__and__Field_Oordered__semidom(tc_nat)
% 102.61/13.40 = { by lemma 34 }
% 102.61/13.40 class_RealVector_Oreal__vector(tc_RealDef_Oreal)
% 102.61/13.40 = { by lemma 33 }
% 102.61/13.40 class_Ring__and__Field_Osgn__if(tc_RealDef_Oreal)
% 102.61/13.40 = { by lemma 32 }
% 102.61/13.40 class_OrderedGroup_Olordered__ab__group__add__abs(tc_RealDef_Oreal)
% 102.61/13.40 = { by axiom 1 (clsarity_RealDef__Oreal__OrderedGroup_Olordered__ab__group__add__abs) }
% 102.61/13.40 true2
% 102.61/13.40 = { by axiom 15 (clsrel_Ring__and__Field_Oidom_Ring__and__Field_Ocomm__semiring__0) R->L }
% 102.61/13.40 fresh125(class_Divides_Osemiring__div(tc_nat), class_Divides_Osemiring__div(tc_nat), t_a)
% 102.61/13.40 = { by lemma 38 R->L }
% 102.61/13.40 fresh125(class_Ring__and__Field_Oidom(t_a), class_Divides_Osemiring__div(tc_nat), t_a)
% 102.61/13.40 = { by lemma 37 }
% 102.61/13.40 fresh125(class_Ring__and__Field_Oidom(t_a), class_OrderedGroup_Opordered__ab__group__add(tc_RealDef_Oreal), t_a)
% 102.61/13.40 = { by lemma 36 }
% 102.61/13.40 fresh125(class_Ring__and__Field_Oidom(t_a), class_Int_Onumber__ring(tc_RealDef_Oreal), t_a)
% 102.61/13.40 = { by lemma 35 }
% 102.61/13.40 fresh125(class_Ring__and__Field_Oidom(t_a), class_Ring__and__Field_Oordered__semidom(tc_nat), t_a)
% 102.61/13.40 = { by lemma 34 }
% 102.61/13.40 fresh125(class_Ring__and__Field_Oidom(t_a), class_RealVector_Oreal__vector(tc_RealDef_Oreal), t_a)
% 102.61/13.40 = { by lemma 33 }
% 102.61/13.40 fresh125(class_Ring__and__Field_Oidom(t_a), class_Ring__and__Field_Osgn__if(tc_RealDef_Oreal), t_a)
% 102.61/13.40 = { by lemma 32 }
% 102.61/13.40 fresh125(class_Ring__and__Field_Oidom(t_a), class_OrderedGroup_Olordered__ab__group__add__abs(tc_RealDef_Oreal), t_a)
% 102.61/13.40 = { by axiom 1 (clsarity_RealDef__Oreal__OrderedGroup_Olordered__ab__group__add__abs) }
% 102.61/13.40 fresh125(class_Ring__and__Field_Oidom(t_a), true2, t_a)
% 102.61/13.40 = { by axiom 20 (clsrel_Ring__and__Field_Oidom_Ring__and__Field_Ocomm__semiring__0) }
% 102.61/13.40 class_Ring__and__Field_Ocomm__semiring__0(t_a)
% 102.61/13.40
% 102.61/13.40 Lemma 40: class_OrderedGroup_Ogroup__add(tc_Polynomial_Opoly(t_a)) = class_Divides_Osemiring__div(tc_nat).
% 102.61/13.40 Proof:
% 102.61/13.40 class_OrderedGroup_Ogroup__add(tc_Polynomial_Opoly(t_a))
% 102.61/13.40 = { by axiom 19 (clsrel_Ring__and__Field_Oidom_OrderedGroup_Ogroup__add) R->L }
% 102.61/13.40 fresh131(class_Ring__and__Field_Oidom(tc_Polynomial_Opoly(t_a)), true2, tc_Polynomial_Opoly(t_a))
% 102.61/13.40 = { by axiom 1 (clsarity_RealDef__Oreal__OrderedGroup_Olordered__ab__group__add__abs) R->L }
% 102.61/13.40 fresh131(class_Ring__and__Field_Oidom(tc_Polynomial_Opoly(t_a)), class_OrderedGroup_Olordered__ab__group__add__abs(tc_RealDef_Oreal), tc_Polynomial_Opoly(t_a))
% 102.61/13.40 = { by lemma 32 R->L }
% 102.61/13.40 fresh131(class_Ring__and__Field_Oidom(tc_Polynomial_Opoly(t_a)), class_Ring__and__Field_Osgn__if(tc_RealDef_Oreal), tc_Polynomial_Opoly(t_a))
% 102.61/13.40 = { by lemma 33 R->L }
% 102.61/13.40 fresh131(class_Ring__and__Field_Oidom(tc_Polynomial_Opoly(t_a)), class_RealVector_Oreal__vector(tc_RealDef_Oreal), tc_Polynomial_Opoly(t_a))
% 102.61/13.40 = { by lemma 34 R->L }
% 102.61/13.40 fresh131(class_Ring__and__Field_Oidom(tc_Polynomial_Opoly(t_a)), class_Ring__and__Field_Oordered__semidom(tc_nat), tc_Polynomial_Opoly(t_a))
% 102.61/13.40 = { by lemma 35 R->L }
% 102.61/13.40 fresh131(class_Ring__and__Field_Oidom(tc_Polynomial_Opoly(t_a)), class_Int_Onumber__ring(tc_RealDef_Oreal), tc_Polynomial_Opoly(t_a))
% 102.61/13.40 = { by lemma 36 R->L }
% 102.61/13.40 fresh131(class_Ring__and__Field_Oidom(tc_Polynomial_Opoly(t_a)), class_OrderedGroup_Opordered__ab__group__add(tc_RealDef_Oreal), tc_Polynomial_Opoly(t_a))
% 102.61/13.40 = { by lemma 37 R->L }
% 102.61/13.40 fresh131(class_Ring__and__Field_Oidom(tc_Polynomial_Opoly(t_a)), class_Divides_Osemiring__div(tc_nat), tc_Polynomial_Opoly(t_a))
% 102.61/13.40 = { by axiom 18 (clsarity_Polynomial__Opoly__Ring__and__Field_Oidom) R->L }
% 102.61/13.40 fresh131(fresh168(class_Ring__and__Field_Oidom(t_a), true2, t_a), class_Divides_Osemiring__div(tc_nat), tc_Polynomial_Opoly(t_a))
% 102.61/13.40 = { by axiom 1 (clsarity_RealDef__Oreal__OrderedGroup_Olordered__ab__group__add__abs) R->L }
% 102.61/13.40 fresh131(fresh168(class_Ring__and__Field_Oidom(t_a), class_OrderedGroup_Olordered__ab__group__add__abs(tc_RealDef_Oreal), t_a), class_Divides_Osemiring__div(tc_nat), tc_Polynomial_Opoly(t_a))
% 102.61/13.40 = { by lemma 32 R->L }
% 102.61/13.40 fresh131(fresh168(class_Ring__and__Field_Oidom(t_a), class_Ring__and__Field_Osgn__if(tc_RealDef_Oreal), t_a), class_Divides_Osemiring__div(tc_nat), tc_Polynomial_Opoly(t_a))
% 102.61/13.40 = { by lemma 33 R->L }
% 102.61/13.40 fresh131(fresh168(class_Ring__and__Field_Oidom(t_a), class_RealVector_Oreal__vector(tc_RealDef_Oreal), t_a), class_Divides_Osemiring__div(tc_nat), tc_Polynomial_Opoly(t_a))
% 102.61/13.40 = { by lemma 34 R->L }
% 102.61/13.40 fresh131(fresh168(class_Ring__and__Field_Oidom(t_a), class_Ring__and__Field_Oordered__semidom(tc_nat), t_a), class_Divides_Osemiring__div(tc_nat), tc_Polynomial_Opoly(t_a))
% 102.61/13.40 = { by lemma 35 R->L }
% 102.61/13.40 fresh131(fresh168(class_Ring__and__Field_Oidom(t_a), class_Int_Onumber__ring(tc_RealDef_Oreal), t_a), class_Divides_Osemiring__div(tc_nat), tc_Polynomial_Opoly(t_a))
% 102.61/13.40 = { by lemma 36 R->L }
% 102.61/13.40 fresh131(fresh168(class_Ring__and__Field_Oidom(t_a), class_OrderedGroup_Opordered__ab__group__add(tc_RealDef_Oreal), t_a), class_Divides_Osemiring__div(tc_nat), tc_Polynomial_Opoly(t_a))
% 102.61/13.40 = { by lemma 37 R->L }
% 102.61/13.40 fresh131(fresh168(class_Ring__and__Field_Oidom(t_a), class_Divides_Osemiring__div(tc_nat), t_a), class_Divides_Osemiring__div(tc_nat), tc_Polynomial_Opoly(t_a))
% 102.61/13.40 = { by lemma 38 }
% 102.61/13.40 fresh131(fresh168(class_Divides_Osemiring__div(tc_nat), class_Divides_Osemiring__div(tc_nat), t_a), class_Divides_Osemiring__div(tc_nat), tc_Polynomial_Opoly(t_a))
% 102.61/13.40 = { by axiom 13 (clsarity_Polynomial__Opoly__Ring__and__Field_Oidom) }
% 102.61/13.40 fresh131(true2, class_Divides_Osemiring__div(tc_nat), tc_Polynomial_Opoly(t_a))
% 102.61/13.40 = { by axiom 1 (clsarity_RealDef__Oreal__OrderedGroup_Olordered__ab__group__add__abs) R->L }
% 102.61/13.40 fresh131(class_OrderedGroup_Olordered__ab__group__add__abs(tc_RealDef_Oreal), class_Divides_Osemiring__div(tc_nat), tc_Polynomial_Opoly(t_a))
% 102.61/13.40 = { by lemma 32 R->L }
% 102.61/13.40 fresh131(class_Ring__and__Field_Osgn__if(tc_RealDef_Oreal), class_Divides_Osemiring__div(tc_nat), tc_Polynomial_Opoly(t_a))
% 102.61/13.40 = { by lemma 33 R->L }
% 102.61/13.40 fresh131(class_RealVector_Oreal__vector(tc_RealDef_Oreal), class_Divides_Osemiring__div(tc_nat), tc_Polynomial_Opoly(t_a))
% 102.61/13.40 = { by lemma 34 R->L }
% 102.61/13.40 fresh131(class_Ring__and__Field_Oordered__semidom(tc_nat), class_Divides_Osemiring__div(tc_nat), tc_Polynomial_Opoly(t_a))
% 102.61/13.40 = { by lemma 35 R->L }
% 102.61/13.40 fresh131(class_Int_Onumber__ring(tc_RealDef_Oreal), class_Divides_Osemiring__div(tc_nat), tc_Polynomial_Opoly(t_a))
% 102.61/13.40 = { by lemma 36 R->L }
% 102.61/13.40 fresh131(class_OrderedGroup_Opordered__ab__group__add(tc_RealDef_Oreal), class_Divides_Osemiring__div(tc_nat), tc_Polynomial_Opoly(t_a))
% 102.61/13.40 = { by lemma 37 R->L }
% 102.61/13.40 fresh131(class_Divides_Osemiring__div(tc_nat), class_Divides_Osemiring__div(tc_nat), tc_Polynomial_Opoly(t_a))
% 102.61/13.40 = { by axiom 14 (clsrel_Ring__and__Field_Oidom_OrderedGroup_Ogroup__add) }
% 102.61/13.40 true2
% 102.61/13.40 = { by axiom 1 (clsarity_RealDef__Oreal__OrderedGroup_Olordered__ab__group__add__abs) R->L }
% 102.61/13.40 class_OrderedGroup_Olordered__ab__group__add__abs(tc_RealDef_Oreal)
% 102.61/13.40 = { by lemma 32 R->L }
% 102.61/13.40 class_Ring__and__Field_Osgn__if(tc_RealDef_Oreal)
% 102.61/13.40 = { by lemma 33 R->L }
% 102.61/13.40 class_RealVector_Oreal__vector(tc_RealDef_Oreal)
% 102.61/13.41 = { by lemma 34 R->L }
% 102.61/13.41 class_Ring__and__Field_Oordered__semidom(tc_nat)
% 102.61/13.41 = { by lemma 35 R->L }
% 102.61/13.41 class_Int_Onumber__ring(tc_RealDef_Oreal)
% 102.61/13.41 = { by lemma 36 R->L }
% 102.61/13.41 class_OrderedGroup_Opordered__ab__group__add(tc_RealDef_Oreal)
% 102.61/13.41 = { by lemma 37 R->L }
% 102.61/13.41 class_Divides_Osemiring__div(tc_nat)
% 102.61/13.41
% 102.61/13.41 Lemma 41: c_Polynomial_Osynthetic__div(v_p, X, t_a) = c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)).
% 102.61/13.41 Proof:
% 102.61/13.41 c_Polynomial_Osynthetic__div(v_p, X, t_a)
% 102.61/13.41 = { by axiom 23 (cls_synthetic__div__eq__0__iff_1) R->L }
% 102.61/13.41 fresh224(class_Divides_Osemiring__div(tc_nat), class_Divides_Osemiring__div(tc_nat), t_a, v_p, X)
% 102.61/13.41 = { by lemma 39 }
% 102.61/13.41 fresh224(class_Ring__and__Field_Ocomm__semiring__0(t_a), class_Divides_Osemiring__div(tc_nat), t_a, v_p, X)
% 102.61/13.41 = { by lemma 37 }
% 102.61/13.41 fresh224(class_Ring__and__Field_Ocomm__semiring__0(t_a), class_OrderedGroup_Opordered__ab__group__add(tc_RealDef_Oreal), t_a, v_p, X)
% 102.61/13.41 = { by lemma 36 }
% 102.61/13.41 fresh224(class_Ring__and__Field_Ocomm__semiring__0(t_a), class_Int_Onumber__ring(tc_RealDef_Oreal), t_a, v_p, X)
% 102.61/13.41 = { by lemma 35 }
% 102.61/13.41 fresh224(class_Ring__and__Field_Ocomm__semiring__0(t_a), class_Ring__and__Field_Oordered__semidom(tc_nat), t_a, v_p, X)
% 102.61/13.41 = { by lemma 34 }
% 102.61/13.41 fresh224(class_Ring__and__Field_Ocomm__semiring__0(t_a), class_RealVector_Oreal__vector(tc_RealDef_Oreal), t_a, v_p, X)
% 102.61/13.41 = { by lemma 33 }
% 102.61/13.41 fresh224(class_Ring__and__Field_Ocomm__semiring__0(t_a), class_Ring__and__Field_Osgn__if(tc_RealDef_Oreal), t_a, v_p, X)
% 102.61/13.41 = { by lemma 32 }
% 102.61/13.41 fresh224(class_Ring__and__Field_Ocomm__semiring__0(t_a), class_OrderedGroup_Olordered__ab__group__add__abs(tc_RealDef_Oreal), t_a, v_p, X)
% 102.61/13.41 = { by axiom 1 (clsarity_RealDef__Oreal__OrderedGroup_Olordered__ab__group__add__abs) }
% 102.61/13.41 fresh224(class_Ring__and__Field_Ocomm__semiring__0(t_a), true2, t_a, v_p, X)
% 102.61/13.41 = { by axiom 29 (cls_synthetic__div__eq__0__iff_1) }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), c_HOL_Ozero__class_Ozero(tc_nat), t_a, v_p, X)
% 102.61/13.41 = { by axiom 11 (cls_CHAINED_0) R->L }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), fresh746(class_Divides_Osemiring__div(tc_nat), class_Divides_Osemiring__div(tc_nat)), t_a, v_p, X)
% 102.61/13.41 = { by lemma 37 }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), fresh746(class_Divides_Osemiring__div(tc_nat), class_OrderedGroup_Opordered__ab__group__add(tc_RealDef_Oreal)), t_a, v_p, X)
% 102.61/13.41 = { by lemma 36 }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), fresh746(class_Divides_Osemiring__div(tc_nat), class_Int_Onumber__ring(tc_RealDef_Oreal)), t_a, v_p, X)
% 102.61/13.41 = { by lemma 35 }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), fresh746(class_Divides_Osemiring__div(tc_nat), class_Ring__and__Field_Oordered__semidom(tc_nat)), t_a, v_p, X)
% 102.61/13.41 = { by lemma 34 }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), fresh746(class_Divides_Osemiring__div(tc_nat), class_RealVector_Oreal__vector(tc_RealDef_Oreal)), t_a, v_p, X)
% 102.61/13.41 = { by lemma 33 }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), fresh746(class_Divides_Osemiring__div(tc_nat), class_Ring__and__Field_Osgn__if(tc_RealDef_Oreal)), t_a, v_p, X)
% 102.61/13.41 = { by lemma 32 }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), fresh746(class_Divides_Osemiring__div(tc_nat), class_OrderedGroup_Olordered__ab__group__add__abs(tc_RealDef_Oreal)), t_a, v_p, X)
% 102.61/13.41 = { by axiom 1 (clsarity_RealDef__Oreal__OrderedGroup_Olordered__ab__group__add__abs) }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), fresh746(class_Divides_Osemiring__div(tc_nat), true2), t_a, v_p, X)
% 102.61/13.41 = { by lemma 38 R->L }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), fresh746(class_Ring__and__Field_Oidom(t_a), true2), t_a, v_p, X)
% 102.61/13.41 = { by axiom 16 (cls_CHAINED_0) R->L }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), fresh745(class_Int_Oring__char__0(t_a), true2), t_a, v_p, X)
% 102.61/13.41 = { by axiom 7 (tfree_tcs) }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), fresh745(true2, true2), t_a, v_p, X)
% 102.61/13.41 = { by axiom 1 (clsarity_RealDef__Oreal__OrderedGroup_Olordered__ab__group__add__abs) R->L }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), fresh745(class_OrderedGroup_Olordered__ab__group__add__abs(tc_RealDef_Oreal), true2), t_a, v_p, X)
% 102.61/13.41 = { by lemma 32 R->L }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), fresh745(class_Ring__and__Field_Osgn__if(tc_RealDef_Oreal), true2), t_a, v_p, X)
% 102.61/13.41 = { by lemma 33 R->L }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), fresh745(class_RealVector_Oreal__vector(tc_RealDef_Oreal), true2), t_a, v_p, X)
% 102.61/13.41 = { by lemma 34 R->L }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), fresh745(class_Ring__and__Field_Oordered__semidom(tc_nat), true2), t_a, v_p, X)
% 102.61/13.41 = { by lemma 35 R->L }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), fresh745(class_Int_Onumber__ring(tc_RealDef_Oreal), true2), t_a, v_p, X)
% 102.61/13.41 = { by lemma 36 R->L }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), fresh745(class_OrderedGroup_Opordered__ab__group__add(tc_RealDef_Oreal), true2), t_a, v_p, X)
% 102.61/13.41 = { by lemma 37 R->L }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), fresh745(class_Divides_Osemiring__div(tc_nat), true2), t_a, v_p, X)
% 102.61/13.41 = { by axiom 1 (clsarity_RealDef__Oreal__OrderedGroup_Olordered__ab__group__add__abs) R->L }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), fresh745(class_Divides_Osemiring__div(tc_nat), class_OrderedGroup_Olordered__ab__group__add__abs(tc_RealDef_Oreal)), t_a, v_p, X)
% 102.61/13.41 = { by lemma 32 R->L }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), fresh745(class_Divides_Osemiring__div(tc_nat), class_Ring__and__Field_Osgn__if(tc_RealDef_Oreal)), t_a, v_p, X)
% 102.61/13.41 = { by lemma 33 R->L }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), fresh745(class_Divides_Osemiring__div(tc_nat), class_RealVector_Oreal__vector(tc_RealDef_Oreal)), t_a, v_p, X)
% 102.61/13.41 = { by lemma 34 R->L }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), fresh745(class_Divides_Osemiring__div(tc_nat), class_Ring__and__Field_Oordered__semidom(tc_nat)), t_a, v_p, X)
% 102.61/13.41 = { by lemma 35 R->L }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), fresh745(class_Divides_Osemiring__div(tc_nat), class_Int_Onumber__ring(tc_RealDef_Oreal)), t_a, v_p, X)
% 102.61/13.41 = { by lemma 36 R->L }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), fresh745(class_Divides_Osemiring__div(tc_nat), class_OrderedGroup_Opordered__ab__group__add(tc_RealDef_Oreal)), t_a, v_p, X)
% 102.61/13.41 = { by lemma 37 R->L }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), fresh745(class_Divides_Osemiring__div(tc_nat), class_Divides_Osemiring__div(tc_nat)), t_a, v_p, X)
% 102.61/13.41 = { by axiom 12 (cls_CHAINED_0) }
% 102.61/13.41 fresh225(c_Polynomial_Odegree(v_p, t_a), c_Polynomial_Odegree(v_p, t_a), t_a, v_p, X)
% 102.61/13.41 = { by axiom 22 (cls_synthetic__div__eq__0__iff_1) }
% 102.61/13.41 c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))
% 102.61/13.41
% 102.61/13.41 Goal 1 (cls_conjecture_0): v_thesis____ = true2.
% 102.61/13.41 Proof:
% 102.61/13.41 v_thesis____
% 102.61/13.41 = { by axiom 28 (cls_conjecture_1) R->L }
% 102.61/13.41 fresh633(v_p, c_Polynomial_OpCons(c_Polynomial_Opoly(v_p, X, t_a), c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)), t_a))
% 102.61/13.41 = { by lemma 41 R->L }
% 102.61/13.41 fresh633(v_p, c_Polynomial_OpCons(c_Polynomial_Opoly(v_p, X, t_a), c_Polynomial_Osynthetic__div(v_p, X, t_a), t_a))
% 102.61/13.41 = { by axiom 30 (cls_synthetic__div__correct_0) R->L }
% 102.61/13.41 fresh633(v_p, fresh228(class_Divides_Osemiring__div(tc_nat), class_Divides_Osemiring__div(tc_nat), t_a, v_p, X))
% 102.61/13.41 = { by lemma 39 }
% 102.61/13.41 fresh633(v_p, fresh228(class_Ring__and__Field_Ocomm__semiring__0(t_a), class_Divides_Osemiring__div(tc_nat), t_a, v_p, X))
% 102.61/13.41 = { by lemma 37 }
% 102.61/13.41 fresh633(v_p, fresh228(class_Ring__and__Field_Ocomm__semiring__0(t_a), class_OrderedGroup_Opordered__ab__group__add(tc_RealDef_Oreal), t_a, v_p, X))
% 102.61/13.41 = { by lemma 36 }
% 102.61/13.41 fresh633(v_p, fresh228(class_Ring__and__Field_Ocomm__semiring__0(t_a), class_Int_Onumber__ring(tc_RealDef_Oreal), t_a, v_p, X))
% 102.61/13.41 = { by lemma 35 }
% 102.61/13.41 fresh633(v_p, fresh228(class_Ring__and__Field_Ocomm__semiring__0(t_a), class_Ring__and__Field_Oordered__semidom(tc_nat), t_a, v_p, X))
% 102.61/13.41 = { by lemma 34 }
% 102.61/13.41 fresh633(v_p, fresh228(class_Ring__and__Field_Ocomm__semiring__0(t_a), class_RealVector_Oreal__vector(tc_RealDef_Oreal), t_a, v_p, X))
% 102.61/13.41 = { by lemma 33 }
% 102.61/13.41 fresh633(v_p, fresh228(class_Ring__and__Field_Ocomm__semiring__0(t_a), class_Ring__and__Field_Osgn__if(tc_RealDef_Oreal), t_a, v_p, X))
% 102.61/13.41 = { by lemma 32 }
% 102.61/13.41 fresh633(v_p, fresh228(class_Ring__and__Field_Ocomm__semiring__0(t_a), class_OrderedGroup_Olordered__ab__group__add__abs(tc_RealDef_Oreal), t_a, v_p, X))
% 102.61/13.41 = { by axiom 1 (clsarity_RealDef__Oreal__OrderedGroup_Olordered__ab__group__add__abs) }
% 102.61/13.41 fresh633(v_p, fresh228(class_Ring__and__Field_Ocomm__semiring__0(t_a), true2, t_a, v_p, X))
% 102.61/13.41 = { by axiom 31 (cls_synthetic__div__correct_0) }
% 102.61/13.41 fresh633(v_p, c_HOL_Oplus__class_Oplus(v_p, c_Polynomial_Osmult(X, c_Polynomial_Osynthetic__div(v_p, X, t_a), t_a), tc_Polynomial_Opoly(t_a)))
% 102.61/13.41 = { by lemma 41 }
% 102.61/13.41 fresh633(v_p, c_HOL_Oplus__class_Oplus(v_p, c_Polynomial_Osmult(X, c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)), t_a), tc_Polynomial_Opoly(t_a)))
% 102.61/13.41 = { by axiom 26 (cls_smult__0__right_0) R->L }
% 102.61/13.41 fresh633(v_p, c_HOL_Oplus__class_Oplus(v_p, fresh251(class_Ring__and__Field_Ocomm__semiring__0(t_a), true2, t_a, X), tc_Polynomial_Opoly(t_a)))
% 102.61/13.41 = { by axiom 1 (clsarity_RealDef__Oreal__OrderedGroup_Olordered__ab__group__add__abs) R->L }
% 102.61/13.41 fresh633(v_p, c_HOL_Oplus__class_Oplus(v_p, fresh251(class_Ring__and__Field_Ocomm__semiring__0(t_a), class_OrderedGroup_Olordered__ab__group__add__abs(tc_RealDef_Oreal), t_a, X), tc_Polynomial_Opoly(t_a)))
% 102.61/13.41 = { by lemma 32 R->L }
% 102.61/13.41 fresh633(v_p, c_HOL_Oplus__class_Oplus(v_p, fresh251(class_Ring__and__Field_Ocomm__semiring__0(t_a), class_Ring__and__Field_Osgn__if(tc_RealDef_Oreal), t_a, X), tc_Polynomial_Opoly(t_a)))
% 102.61/13.41 = { by lemma 33 R->L }
% 102.61/13.41 fresh633(v_p, c_HOL_Oplus__class_Oplus(v_p, fresh251(class_Ring__and__Field_Ocomm__semiring__0(t_a), class_RealVector_Oreal__vector(tc_RealDef_Oreal), t_a, X), tc_Polynomial_Opoly(t_a)))
% 102.61/13.41 = { by lemma 34 R->L }
% 102.61/13.41 fresh633(v_p, c_HOL_Oplus__class_Oplus(v_p, fresh251(class_Ring__and__Field_Ocomm__semiring__0(t_a), class_Ring__and__Field_Oordered__semidom(tc_nat), t_a, X), tc_Polynomial_Opoly(t_a)))
% 102.61/13.41 = { by lemma 35 R->L }
% 102.61/13.41 fresh633(v_p, c_HOL_Oplus__class_Oplus(v_p, fresh251(class_Ring__and__Field_Ocomm__semiring__0(t_a), class_Int_Onumber__ring(tc_RealDef_Oreal), t_a, X), tc_Polynomial_Opoly(t_a)))
% 102.61/13.41 = { by lemma 36 R->L }
% 102.61/13.41 fresh633(v_p, c_HOL_Oplus__class_Oplus(v_p, fresh251(class_Ring__and__Field_Ocomm__semiring__0(t_a), class_OrderedGroup_Opordered__ab__group__add(tc_RealDef_Oreal), t_a, X), tc_Polynomial_Opoly(t_a)))
% 102.61/13.41 = { by lemma 37 R->L }
% 102.61/13.41 fresh633(v_p, c_HOL_Oplus__class_Oplus(v_p, fresh251(class_Ring__and__Field_Ocomm__semiring__0(t_a), class_Divides_Osemiring__div(tc_nat), t_a, X), tc_Polynomial_Opoly(t_a)))
% 102.61/13.41 = { by lemma 39 R->L }
% 102.61/13.41 fresh633(v_p, c_HOL_Oplus__class_Oplus(v_p, fresh251(class_Divides_Osemiring__div(tc_nat), class_Divides_Osemiring__div(tc_nat), t_a, X), tc_Polynomial_Opoly(t_a)))
% 102.61/13.41 = { by axiom 17 (cls_smult__0__right_0) }
% 102.61/13.41 fresh633(v_p, c_HOL_Oplus__class_Oplus(v_p, c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)), tc_Polynomial_Opoly(t_a)))
% 102.61/13.41 = { by axiom 21 (cls_diff__0__right_0) R->L }
% 102.61/13.41 fresh633(v_p, fresh45(class_Divides_Osemiring__div(tc_nat), class_Divides_Osemiring__div(tc_nat), tc_Polynomial_Opoly(t_a), c_HOL_Oplus__class_Oplus(v_p, c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)), tc_Polynomial_Opoly(t_a))))
% 102.61/13.41 = { by lemma 40 R->L }
% 102.61/13.41 fresh633(v_p, fresh45(class_OrderedGroup_Ogroup__add(tc_Polynomial_Opoly(t_a)), class_Divides_Osemiring__div(tc_nat), tc_Polynomial_Opoly(t_a), c_HOL_Oplus__class_Oplus(v_p, c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)), tc_Polynomial_Opoly(t_a))))
% 102.61/13.41 = { by lemma 37 }
% 102.61/13.41 fresh633(v_p, fresh45(class_OrderedGroup_Ogroup__add(tc_Polynomial_Opoly(t_a)), class_OrderedGroup_Opordered__ab__group__add(tc_RealDef_Oreal), tc_Polynomial_Opoly(t_a), c_HOL_Oplus__class_Oplus(v_p, c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)), tc_Polynomial_Opoly(t_a))))
% 102.61/13.41 = { by lemma 36 }
% 102.61/13.41 fresh633(v_p, fresh45(class_OrderedGroup_Ogroup__add(tc_Polynomial_Opoly(t_a)), class_Int_Onumber__ring(tc_RealDef_Oreal), tc_Polynomial_Opoly(t_a), c_HOL_Oplus__class_Oplus(v_p, c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)), tc_Polynomial_Opoly(t_a))))
% 102.61/13.41 = { by lemma 35 }
% 102.61/13.41 fresh633(v_p, fresh45(class_OrderedGroup_Ogroup__add(tc_Polynomial_Opoly(t_a)), class_Ring__and__Field_Oordered__semidom(tc_nat), tc_Polynomial_Opoly(t_a), c_HOL_Oplus__class_Oplus(v_p, c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)), tc_Polynomial_Opoly(t_a))))
% 102.61/13.41 = { by lemma 34 }
% 102.61/13.41 fresh633(v_p, fresh45(class_OrderedGroup_Ogroup__add(tc_Polynomial_Opoly(t_a)), class_RealVector_Oreal__vector(tc_RealDef_Oreal), tc_Polynomial_Opoly(t_a), c_HOL_Oplus__class_Oplus(v_p, c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)), tc_Polynomial_Opoly(t_a))))
% 102.61/13.41 = { by lemma 33 }
% 102.61/13.41 fresh633(v_p, fresh45(class_OrderedGroup_Ogroup__add(tc_Polynomial_Opoly(t_a)), class_Ring__and__Field_Osgn__if(tc_RealDef_Oreal), tc_Polynomial_Opoly(t_a), c_HOL_Oplus__class_Oplus(v_p, c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)), tc_Polynomial_Opoly(t_a))))
% 102.61/13.41 = { by lemma 32 }
% 102.61/13.41 fresh633(v_p, fresh45(class_OrderedGroup_Ogroup__add(tc_Polynomial_Opoly(t_a)), class_OrderedGroup_Olordered__ab__group__add__abs(tc_RealDef_Oreal), tc_Polynomial_Opoly(t_a), c_HOL_Oplus__class_Oplus(v_p, c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)), tc_Polynomial_Opoly(t_a))))
% 102.61/13.41 = { by axiom 1 (clsarity_RealDef__Oreal__OrderedGroup_Olordered__ab__group__add__abs) }
% 102.61/13.41 fresh633(v_p, fresh45(class_OrderedGroup_Ogroup__add(tc_Polynomial_Opoly(t_a)), true2, tc_Polynomial_Opoly(t_a), c_HOL_Oplus__class_Oplus(v_p, c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)), tc_Polynomial_Opoly(t_a))))
% 102.61/13.41 = { by axiom 25 (cls_diff__0__right_0) }
% 102.61/13.41 fresh633(v_p, c_HOL_Ominus__class_Ominus(c_HOL_Oplus__class_Oplus(v_p, c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)), tc_Polynomial_Opoly(t_a)), c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a)), tc_Polynomial_Opoly(t_a)))
% 102.61/13.41 = { by axiom 27 (cls_add__diff__cancel_0) R->L }
% 102.61/13.41 fresh633(v_p, fresh70(class_OrderedGroup_Ogroup__add(tc_Polynomial_Opoly(t_a)), true2, tc_Polynomial_Opoly(t_a), v_p, c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))))
% 102.61/13.41 = { by axiom 1 (clsarity_RealDef__Oreal__OrderedGroup_Olordered__ab__group__add__abs) R->L }
% 102.61/13.41 fresh633(v_p, fresh70(class_OrderedGroup_Ogroup__add(tc_Polynomial_Opoly(t_a)), class_OrderedGroup_Olordered__ab__group__add__abs(tc_RealDef_Oreal), tc_Polynomial_Opoly(t_a), v_p, c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))))
% 102.61/13.41 = { by lemma 32 R->L }
% 102.61/13.41 fresh633(v_p, fresh70(class_OrderedGroup_Ogroup__add(tc_Polynomial_Opoly(t_a)), class_Ring__and__Field_Osgn__if(tc_RealDef_Oreal), tc_Polynomial_Opoly(t_a), v_p, c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))))
% 102.61/13.41 = { by lemma 33 R->L }
% 102.61/13.41 fresh633(v_p, fresh70(class_OrderedGroup_Ogroup__add(tc_Polynomial_Opoly(t_a)), class_RealVector_Oreal__vector(tc_RealDef_Oreal), tc_Polynomial_Opoly(t_a), v_p, c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))))
% 102.61/13.41 = { by lemma 34 R->L }
% 102.61/13.41 fresh633(v_p, fresh70(class_OrderedGroup_Ogroup__add(tc_Polynomial_Opoly(t_a)), class_Ring__and__Field_Oordered__semidom(tc_nat), tc_Polynomial_Opoly(t_a), v_p, c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))))
% 102.61/13.41 = { by lemma 35 R->L }
% 102.61/13.41 fresh633(v_p, fresh70(class_OrderedGroup_Ogroup__add(tc_Polynomial_Opoly(t_a)), class_Int_Onumber__ring(tc_RealDef_Oreal), tc_Polynomial_Opoly(t_a), v_p, c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))))
% 102.61/13.41 = { by lemma 36 R->L }
% 102.61/13.41 fresh633(v_p, fresh70(class_OrderedGroup_Ogroup__add(tc_Polynomial_Opoly(t_a)), class_OrderedGroup_Opordered__ab__group__add(tc_RealDef_Oreal), tc_Polynomial_Opoly(t_a), v_p, c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))))
% 102.61/13.41 = { by lemma 37 R->L }
% 102.61/13.41 fresh633(v_p, fresh70(class_OrderedGroup_Ogroup__add(tc_Polynomial_Opoly(t_a)), class_Divides_Osemiring__div(tc_nat), tc_Polynomial_Opoly(t_a), v_p, c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))))
% 102.61/13.41 = { by lemma 40 }
% 102.61/13.41 fresh633(v_p, fresh70(class_Divides_Osemiring__div(tc_nat), class_Divides_Osemiring__div(tc_nat), tc_Polynomial_Opoly(t_a), v_p, c_HOL_Ozero__class_Ozero(tc_Polynomial_Opoly(t_a))))
% 102.61/13.41 = { by axiom 24 (cls_add__diff__cancel_0) }
% 102.61/13.41 fresh633(v_p, v_p)
% 102.61/13.41 = { by axiom 10 (cls_conjecture_1) }
% 102.61/13.41 true2
% 102.61/13.41 % SZS output end Proof
% 102.61/13.41
% 102.61/13.41 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------