TSTP Solution File: SWW208+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SWW208+1 : TPTP v8.1.2. Released v5.2.0.
% 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 : n029.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 : Fri Sep 1 00:54:25 EDT 2023
% Result : Theorem 251.77s 32.97s
% Output : Proof 251.77s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SWW208+1 : TPTP v8.1.2. Released v5.2.0.
% 0.00/0.13 % 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 : n029.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 : Sun Aug 27 18:54:41 EDT 2023
% 0.12/0.34 % CPUTime :
% 251.77/32.97 Command-line arguments: --no-flatten-goal
% 251.77/32.97
% 251.77/32.97 % SZS status Theorem
% 251.77/32.97
% 251.77/32.99 % SZS output start Proof
% 251.77/32.99 Take the following subset of the input axioms:
% 251.77/32.99 fof(arity_RealDef__Oreal__Groups_Oordered__ab__group__add__abs, axiom, class_Groups_Oordered__ab__group__add__abs(tc_RealDef_Oreal)).
% 251.77/32.99 fof(arity_RealDef__Oreal__Rings_Ocomm__semiring__1, axiom, class_Rings_Ocomm__semiring__1(tc_RealDef_Oreal)).
% 251.77/32.99 fof(conj_0, conjecture, hBOOL(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal, c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)), hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal), v_r), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)))) & ![B_n]: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal, c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, hAPP(c_Complex_OIm, hAPP(v_s, hAPP(v_f____, hAPP(v_ga____, B_n)))))), hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal), v_r), c_Groups_Oone__class_Oone(tc_RealDef_Oreal))))).
% 251.77/32.99 fof(fact_abs__add__one__gt__zero, axiom, ![V_x]: hBOOL(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal, c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)), hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)), c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, V_x))))).
% 251.77/32.99 fof(fact_abs__of__nonneg, axiom, ![T_a, V_a]: (class_Groups_Oordered__ab__group__add__abs(T_a) => (hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(T_a, c_Groups_Ozero__class_Ozero(T_a)), V_a)) => c_Groups_Oabs__class_Oabs(T_a, V_a)=V_a))).
% 251.77/32.99 fof(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J, axiom, ![V_c, T_a2, V_a2]: (class_Rings_Ocomm__semiring__1(T_a2) => hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a2), V_a2), V_c)=hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a2), V_c), V_a2))).
% 251.77/32.99 fof(fact_rp, axiom, hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal, c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)), v_r))).
% 251.77/32.99 fof(fact_th, axiom, ![B_n2]: hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal, c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, hAPP(c_Complex_OIm, hAPP(v_s, B_n2)))), hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal), v_r), c_Groups_Oone__class_Oone(tc_RealDef_Oreal))))).
% 251.77/32.99
% 251.77/32.99 Now clausify the problem and encode Horn clauses using encoding 3 of
% 251.77/32.99 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 251.77/32.99 We repeatedly replace C & s=t => u=v by the two clauses:
% 251.77/32.99 fresh(y, y, x1...xn) = u
% 251.77/32.99 C => fresh(s, t, x1...xn) = v
% 251.77/32.99 where fresh is a fresh function symbol and x1..xn are the free
% 251.77/32.99 variables of u and v.
% 251.77/32.99 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 251.77/32.99 input problem has no model of domain size 1).
% 251.77/32.99
% 251.77/32.99 The encoding turns the above axioms into the following unit equations and goals:
% 251.77/32.99
% 251.77/32.99 Axiom 1 (arity_RealDef__Oreal__Groups_Oordered__ab__group__add__abs): class_Groups_Oordered__ab__group__add__abs(tc_RealDef_Oreal) = true2.
% 251.77/32.99 Axiom 2 (arity_RealDef__Oreal__Rings_Ocomm__semiring__1): class_Rings_Ocomm__semiring__1(tc_RealDef_Oreal) = true2.
% 251.77/32.99 Axiom 3 (fact_abs__of__nonneg): fresh899(X, X, Y, Z) = c_Groups_Oabs__class_Oabs(Z, Y).
% 251.77/32.99 Axiom 4 (fact_abs__of__nonneg): fresh125(X, X, Y, Z) = Y.
% 251.77/32.99 Axiom 5 (fact_less__iff__diff__less__0_1): fresh616(X, X, Y, Z, W) = hBOOL(hAPP(c_Orderings_Oord__class_Oless(W, Z), Y)).
% 251.77/32.99 Axiom 6 (fact_abs__le__D1): fresh908(X, X, Y, Z, W) = hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(W, Z), Y)).
% 251.77/32.99 Axiom 7 (fact_ln__le__cancel__iff_1): fresh1142(X, X, Y, Z) = hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal, Z), Y)).
% 251.77/32.99 Axiom 8 (fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J): fresh824(X, X, Y, Z, W) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(W), Y), Z).
% 251.77/32.99 Axiom 9 (fact_add__nonneg__eq__0__iff): fresh1589(X, X, Y, Z, W) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(W), Z), Y).
% 251.77/32.99 Axiom 10 (fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J): fresh824(class_Rings_Ocomm__semiring__1(X), true2, Y, Z, X) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(X), Z), Y).
% 251.77/32.99 Axiom 11 (fact_sgn__1__pos): fresh258(X, X, Y, Z) = hBOOL(hAPP(c_Orderings_Oord__class_Oless(Z, c_Groups_Ozero__class_Ozero(Z)), Y)).
% 251.77/32.99 Axiom 12 (fact_rp): hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal, c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)), v_r)) = true2.
% 251.77/32.99 Axiom 13 (fact_abs__of__nonneg): fresh899(class_Groups_Oordered__ab__group__add__abs(X), true2, Y, X) = fresh125(hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(X, c_Groups_Ozero__class_Ozero(X)), Y)), true2, Y, X).
% 251.77/32.99 Axiom 14 (fact_add__nonneg__pos): fresh845(X, X, Y, Z, W) = hBOOL(hAPP(c_Orderings_Oord__class_Oless(W, c_Groups_Ozero__class_Ozero(W)), hAPP(hAPP(c_Groups_Oplus__class_Oplus(W), Z), Y))).
% 251.77/32.99 Axiom 15 (fact_abs__add__one__gt__zero): hBOOL(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal, c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)), hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)), c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, X)))) = true2.
% 251.77/32.99 Axiom 16 (fact_th): hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal, c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, hAPP(c_Complex_OIm, hAPP(v_s, X)))), hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal), v_r), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)))) = true2.
% 251.77/32.99
% 251.77/32.99 Lemma 17: fresh824(X, X, Y, Z, W) = fresh1589(V, V, Z, Y, W).
% 251.77/32.99 Proof:
% 251.77/32.99 fresh824(X, X, Y, Z, W)
% 251.77/32.99 = { by axiom 8 (fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) }
% 251.77/32.99 hAPP(hAPP(c_Groups_Oplus__class_Oplus(W), Y), Z)
% 251.77/32.99 = { by axiom 9 (fact_add__nonneg__eq__0__iff) R->L }
% 251.77/32.99 fresh1589(V, V, Z, Y, W)
% 251.77/32.99
% 251.77/32.99 Lemma 18: hAPP(hAPP(c_Groups_Oplus__class_Oplus(X), Y), Z) = fresh1589(W, W, Z, Y, X).
% 251.77/32.99 Proof:
% 251.77/32.99 hAPP(hAPP(c_Groups_Oplus__class_Oplus(X), Y), Z)
% 251.77/32.99 = { by axiom 8 (fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) R->L }
% 251.77/32.99 fresh824(V, V, Y, Z, X)
% 251.77/32.99 = { by lemma 17 }
% 251.77/32.99 fresh1589(W, W, Z, Y, X)
% 251.77/32.99
% 251.77/32.99 Lemma 19: fresh908(X, X, Y, Z, tc_RealDef_Oreal) = fresh1142(W, W, Y, Z).
% 251.77/32.99 Proof:
% 251.77/32.99 fresh908(X, X, Y, Z, tc_RealDef_Oreal)
% 251.77/32.99 = { by axiom 6 (fact_abs__le__D1) }
% 251.77/32.99 hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal, Z), Y))
% 251.77/32.99 = { by axiom 7 (fact_ln__le__cancel__iff_1) R->L }
% 251.77/32.99 fresh1142(W, W, Y, Z)
% 251.77/32.99
% 251.77/32.99 Lemma 20: fresh616(X, X, Y, c_Groups_Ozero__class_Ozero(Z), Z) = fresh258(W, W, Y, Z).
% 251.77/32.99 Proof:
% 251.77/32.99 fresh616(X, X, Y, c_Groups_Ozero__class_Ozero(Z), Z)
% 251.77/32.99 = { by axiom 5 (fact_less__iff__diff__less__0_1) }
% 251.77/32.99 hBOOL(hAPP(c_Orderings_Oord__class_Oless(Z, c_Groups_Ozero__class_Ozero(Z)), Y))
% 251.77/32.99 = { by axiom 11 (fact_sgn__1__pos) R->L }
% 251.77/32.99 fresh258(W, W, Y, Z)
% 251.77/32.99
% 251.77/32.99 Lemma 21: fresh258(X, X, fresh1589(Y, Y, Z, W, V), V) = fresh845(U, U, Z, W, V).
% 251.77/32.99 Proof:
% 251.77/32.99 fresh258(X, X, fresh1589(Y, Y, Z, W, V), V)
% 251.77/32.99 = { by lemma 17 R->L }
% 251.77/32.99 fresh258(X, X, fresh824(T, T, W, Z, V), V)
% 251.77/32.99 = { by axiom 8 (fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) }
% 251.77/32.99 fresh258(X, X, hAPP(hAPP(c_Groups_Oplus__class_Oplus(V), W), Z), V)
% 251.77/32.99 = { by lemma 20 R->L }
% 251.77/32.99 fresh616(S, S, hAPP(hAPP(c_Groups_Oplus__class_Oplus(V), W), Z), c_Groups_Ozero__class_Ozero(V), V)
% 251.77/32.99 = { by axiom 5 (fact_less__iff__diff__less__0_1) }
% 251.77/32.99 hBOOL(hAPP(c_Orderings_Oord__class_Oless(V, c_Groups_Ozero__class_Ozero(V)), hAPP(hAPP(c_Groups_Oplus__class_Oplus(V), W), Z)))
% 251.77/32.99 = { by axiom 14 (fact_add__nonneg__pos) R->L }
% 251.77/33.00 fresh845(U, U, Z, W, V)
% 251.77/33.00
% 251.77/33.00 Goal 1 (conj_0): tuple2(hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal, c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, hAPP(c_Complex_OIm, hAPP(v_s, hAPP(v_f____, hAPP(v_ga____, b_n)))))), hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal), v_r), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)))), hBOOL(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal, c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)), hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal), v_r), c_Groups_Oone__class_Oone(tc_RealDef_Oreal))))) = tuple2(true2, true2).
% 251.77/33.00 Proof:
% 251.77/33.00 tuple2(hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal, c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, hAPP(c_Complex_OIm, hAPP(v_s, hAPP(v_f____, hAPP(v_ga____, b_n)))))), hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal), v_r), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)))), hBOOL(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal, c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)), hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal), v_r), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)))))
% 251.77/33.00 = { by axiom 5 (fact_less__iff__diff__less__0_1) R->L }
% 251.77/33.00 tuple2(hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal, c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, hAPP(c_Complex_OIm, hAPP(v_s, hAPP(v_f____, hAPP(v_ga____, b_n)))))), hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal), v_r), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)))), fresh616(X, X, hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal), v_r), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)), c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal), tc_RealDef_Oreal))
% 251.77/33.00 = { by lemma 20 }
% 251.77/33.00 tuple2(hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal, c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, hAPP(c_Complex_OIm, hAPP(v_s, hAPP(v_f____, hAPP(v_ga____, b_n)))))), hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal), v_r), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)))), fresh258(Y, Y, hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal), v_r), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)), tc_RealDef_Oreal))
% 251.77/33.00 = { by axiom 6 (fact_abs__le__D1) R->L }
% 251.77/33.00 tuple2(fresh908(Z, Z, hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal), v_r), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)), c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, hAPP(c_Complex_OIm, hAPP(v_s, hAPP(v_f____, hAPP(v_ga____, b_n))))), tc_RealDef_Oreal), fresh258(Y, Y, hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal), v_r), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)), tc_RealDef_Oreal))
% 251.77/33.00 = { by lemma 19 }
% 251.77/33.00 tuple2(fresh1142(W, W, hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal), v_r), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)), c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, hAPP(c_Complex_OIm, hAPP(v_s, hAPP(v_f____, hAPP(v_ga____, b_n)))))), fresh258(Y, Y, hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal), v_r), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)), tc_RealDef_Oreal))
% 251.77/33.00 = { by lemma 18 }
% 251.77/33.00 tuple2(fresh1142(W, W, fresh1589(V, V, c_Groups_Oone__class_Oone(tc_RealDef_Oreal), v_r, tc_RealDef_Oreal), c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, hAPP(c_Complex_OIm, hAPP(v_s, hAPP(v_f____, hAPP(v_ga____, b_n)))))), fresh258(Y, Y, hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal), v_r), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)), tc_RealDef_Oreal))
% 251.77/33.00 = { by lemma 18 }
% 251.77/33.00 tuple2(fresh1142(W, W, fresh1589(V, V, c_Groups_Oone__class_Oone(tc_RealDef_Oreal), v_r, tc_RealDef_Oreal), c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, hAPP(c_Complex_OIm, hAPP(v_s, hAPP(v_f____, hAPP(v_ga____, b_n)))))), fresh258(Y, Y, fresh1589(U, U, c_Groups_Oone__class_Oone(tc_RealDef_Oreal), v_r, tc_RealDef_Oreal), tc_RealDef_Oreal))
% 251.77/33.00 = { by lemma 17 R->L }
% 251.77/33.00 tuple2(fresh1142(W, W, fresh824(T, T, v_r, c_Groups_Oone__class_Oone(tc_RealDef_Oreal), tc_RealDef_Oreal), c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, hAPP(c_Complex_OIm, hAPP(v_s, hAPP(v_f____, hAPP(v_ga____, b_n)))))), fresh258(Y, Y, fresh1589(U, U, c_Groups_Oone__class_Oone(tc_RealDef_Oreal), v_r, tc_RealDef_Oreal), tc_RealDef_Oreal))
% 251.77/33.00 = { by axiom 8 (fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) }
% 251.77/33.00 tuple2(fresh1142(W, W, hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal), v_r), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)), c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, hAPP(c_Complex_OIm, hAPP(v_s, hAPP(v_f____, hAPP(v_ga____, b_n)))))), fresh258(Y, Y, fresh1589(U, U, c_Groups_Oone__class_Oone(tc_RealDef_Oreal), v_r, tc_RealDef_Oreal), tc_RealDef_Oreal))
% 251.77/33.00 = { by lemma 19 R->L }
% 251.77/33.00 tuple2(fresh908(S, S, hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal), v_r), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)), c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, hAPP(c_Complex_OIm, hAPP(v_s, hAPP(v_f____, hAPP(v_ga____, b_n))))), tc_RealDef_Oreal), fresh258(Y, Y, fresh1589(U, U, c_Groups_Oone__class_Oone(tc_RealDef_Oreal), v_r, tc_RealDef_Oreal), tc_RealDef_Oreal))
% 251.77/33.00 = { by axiom 6 (fact_abs__le__D1) }
% 251.77/33.00 tuple2(hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal, c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, hAPP(c_Complex_OIm, hAPP(v_s, hAPP(v_f____, hAPP(v_ga____, b_n)))))), hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal), v_r), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)))), fresh258(Y, Y, fresh1589(U, U, c_Groups_Oone__class_Oone(tc_RealDef_Oreal), v_r, tc_RealDef_Oreal), tc_RealDef_Oreal))
% 251.77/33.00 = { by axiom 16 (fact_th) }
% 251.77/33.00 tuple2(true2, fresh258(Y, Y, fresh1589(U, U, c_Groups_Oone__class_Oone(tc_RealDef_Oreal), v_r, tc_RealDef_Oreal), tc_RealDef_Oreal))
% 251.77/33.00 = { by lemma 17 R->L }
% 251.77/33.00 tuple2(true2, fresh258(Y, Y, fresh824(X2, X2, v_r, c_Groups_Oone__class_Oone(tc_RealDef_Oreal), tc_RealDef_Oreal), tc_RealDef_Oreal))
% 251.77/33.00 = { by axiom 8 (fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) }
% 251.77/33.00 tuple2(true2, fresh258(Y, Y, hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal), v_r), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)), tc_RealDef_Oreal))
% 251.77/33.00 = { by axiom 10 (fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) R->L }
% 251.77/33.00 tuple2(true2, fresh258(Y, Y, fresh824(class_Rings_Ocomm__semiring__1(tc_RealDef_Oreal), true2, c_Groups_Oone__class_Oone(tc_RealDef_Oreal), v_r, tc_RealDef_Oreal), tc_RealDef_Oreal))
% 251.77/33.00 = { by axiom 2 (arity_RealDef__Oreal__Rings_Ocomm__semiring__1) }
% 251.77/33.00 tuple2(true2, fresh258(Y, Y, fresh824(true2, true2, c_Groups_Oone__class_Oone(tc_RealDef_Oreal), v_r, tc_RealDef_Oreal), tc_RealDef_Oreal))
% 251.77/33.00 = { by lemma 17 }
% 251.77/33.00 tuple2(true2, fresh258(Y, Y, fresh1589(Y2, Y2, v_r, c_Groups_Oone__class_Oone(tc_RealDef_Oreal), tc_RealDef_Oreal), tc_RealDef_Oreal))
% 251.77/33.00 = { by lemma 21 }
% 251.77/33.00 tuple2(true2, fresh845(Z2, Z2, v_r, c_Groups_Oone__class_Oone(tc_RealDef_Oreal), tc_RealDef_Oreal))
% 251.77/33.00 = { by axiom 4 (fact_abs__of__nonneg) R->L }
% 251.77/33.00 tuple2(true2, fresh845(Z2, Z2, fresh125(true2, true2, v_r, tc_RealDef_Oreal), c_Groups_Oone__class_Oone(tc_RealDef_Oreal), tc_RealDef_Oreal))
% 251.77/33.00 = { by axiom 12 (fact_rp) R->L }
% 251.77/33.00 tuple2(true2, fresh845(Z2, Z2, fresh125(hBOOL(hAPP(c_Orderings_Oord__class_Oless__eq(tc_RealDef_Oreal, c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)), v_r)), true2, v_r, tc_RealDef_Oreal), c_Groups_Oone__class_Oone(tc_RealDef_Oreal), tc_RealDef_Oreal))
% 251.77/33.00 = { by axiom 13 (fact_abs__of__nonneg) R->L }
% 251.77/33.00 tuple2(true2, fresh845(Z2, Z2, fresh899(class_Groups_Oordered__ab__group__add__abs(tc_RealDef_Oreal), true2, v_r, tc_RealDef_Oreal), c_Groups_Oone__class_Oone(tc_RealDef_Oreal), tc_RealDef_Oreal))
% 251.77/33.00 = { by axiom 1 (arity_RealDef__Oreal__Groups_Oordered__ab__group__add__abs) }
% 251.77/33.00 tuple2(true2, fresh845(Z2, Z2, fresh899(true2, true2, v_r, tc_RealDef_Oreal), c_Groups_Oone__class_Oone(tc_RealDef_Oreal), tc_RealDef_Oreal))
% 251.77/33.00 = { by axiom 3 (fact_abs__of__nonneg) }
% 251.77/33.00 tuple2(true2, fresh845(Z2, Z2, c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r), c_Groups_Oone__class_Oone(tc_RealDef_Oreal), tc_RealDef_Oreal))
% 251.77/33.00 = { by lemma 21 R->L }
% 251.77/33.00 tuple2(true2, fresh258(W2, W2, fresh1589(V2, V2, c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r), c_Groups_Oone__class_Oone(tc_RealDef_Oreal), tc_RealDef_Oreal), tc_RealDef_Oreal))
% 251.77/33.00 = { by lemma 17 R->L }
% 251.77/33.00 tuple2(true2, fresh258(W2, W2, fresh824(U2, U2, c_Groups_Oone__class_Oone(tc_RealDef_Oreal), c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r), tc_RealDef_Oreal), tc_RealDef_Oreal))
% 251.77/33.00 = { by axiom 8 (fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) }
% 251.77/33.00 tuple2(true2, fresh258(W2, W2, hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)), c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r)), tc_RealDef_Oreal))
% 251.77/33.00 = { by lemma 20 R->L }
% 251.77/33.00 tuple2(true2, fresh616(T2, T2, hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)), c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r)), c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal), tc_RealDef_Oreal))
% 251.77/33.00 = { by axiom 5 (fact_less__iff__diff__less__0_1) }
% 251.77/33.00 tuple2(true2, hBOOL(hAPP(c_Orderings_Oord__class_Oless(tc_RealDef_Oreal, c_Groups_Ozero__class_Ozero(tc_RealDef_Oreal)), hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_RealDef_Oreal), c_Groups_Oone__class_Oone(tc_RealDef_Oreal)), c_Groups_Oabs__class_Oabs(tc_RealDef_Oreal, v_r)))))
% 251.77/33.00 = { by axiom 15 (fact_abs__add__one__gt__zero) }
% 251.77/33.00 tuple2(true2, true2)
% 251.77/33.00 % SZS output end Proof
% 251.77/33.00
% 251.77/33.00 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------