TSTP Solution File: ANA031-2 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : ANA031-2 : TPTP v8.1.2. Released v3.2.0.
% 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 : n027.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 17:21:08 EDT 2023
% Result : Unsatisfiable 0.21s 0.48s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.13 % Problem : ANA031-2 : TPTP v8.1.2. Released v3.2.0.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n027.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Aug 25 18:42:34 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.21/0.48 Command-line arguments: --flip-ordering --lhs-weight 1 --depth-weight 60 --distributivity-heuristic
% 0.21/0.48
% 0.21/0.48 % SZS status Unsatisfiable
% 0.21/0.48
% 0.21/0.49 % SZS output start Proof
% 0.21/0.49 Take the following subset of the input axioms:
% 0.21/0.49 fof(cls_OrderedGroup_Oabs__ge__zero_0, axiom, ![T_a, V_a]: (~class_OrderedGroup_Olordered__ab__group__abs(T_a) | c_lessequals(c_0, c_HOL_Oabs(V_a, T_a), T_a))).
% 0.21/0.49 fof(cls_OrderedGroup_Omult__ac__1_0, axiom, ![V_b, V_c, T_a2, V_a2]: (~class_OrderedGroup_Osemigroup__mult(T_a2) | c_times(c_times(V_a2, V_b, T_a2), V_c, T_a2)=c_times(V_a2, c_times(V_b, V_c, T_a2), T_a2))).
% 0.21/0.49 fof(cls_OrderedGroup_Omult__ac__2_0, axiom, ![T_a2, V_a2, V_b2]: (~class_OrderedGroup_Oab__semigroup__mult(T_a2) | c_times(V_a2, V_b2, T_a2)=c_times(V_b2, V_a2, T_a2))).
% 0.21/0.49 fof(cls_Ring__and__Field_Opordered__semiring__class_Omult__left__mono_0, axiom, ![T_a2, V_a2, V_b2, V_c2]: (~class_Ring__and__Field_Opordered__semiring(T_a2) | (~c_lessequals(V_a2, V_b2, T_a2) | (~c_lessequals(c_0, V_c2, T_a2) | c_lessequals(c_times(V_c2, V_a2, T_a2), c_times(V_c2, V_b2, T_a2), T_a2))))).
% 0.21/0.49 fof(cls_conjecture_0, negated_conjecture, ![V_U]: c_lessequals(c_HOL_Oabs(v_b(V_U), t_b), c_times(v_c, c_HOL_Oabs(v_g(V_U), t_b), t_b), t_b)).
% 0.21/0.49 fof(cls_conjecture_1, negated_conjecture, ![V_U2]: ~c_lessequals(c_times(c_HOL_Oabs(v_b(v_x(V_U2)), t_b), c_HOL_Oabs(v_f(v_x(V_U2)), t_b), t_b), c_times(V_U2, c_times(c_HOL_Oabs(v_f(v_x(V_U2)), t_b), c_HOL_Oabs(v_g(v_x(V_U2)), t_b), t_b), t_b), t_b)).
% 0.21/0.49 fof(clsrel_Ring__and__Field_Oordered__idom_17, axiom, ![T]: (~class_Ring__and__Field_Oordered__idom(T) | class_OrderedGroup_Oab__semigroup__mult(T))).
% 0.21/0.49 fof(clsrel_Ring__and__Field_Oordered__idom_21, axiom, ![T2]: (~class_Ring__and__Field_Oordered__idom(T2) | class_OrderedGroup_Osemigroup__mult(T2))).
% 0.21/0.49 fof(clsrel_Ring__and__Field_Oordered__idom_42, axiom, ![T2]: (~class_Ring__and__Field_Oordered__idom(T2) | class_Ring__and__Field_Opordered__semiring(T2))).
% 0.21/0.49 fof(clsrel_Ring__and__Field_Oordered__idom_50, axiom, ![T2]: (~class_Ring__and__Field_Oordered__idom(T2) | class_OrderedGroup_Olordered__ab__group__abs(T2))).
% 0.21/0.49 fof(tfree_tcs, negated_conjecture, class_Ring__and__Field_Oordered__idom(t_b)).
% 0.21/0.49
% 0.21/0.49 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.49 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.49 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.49 fresh(y, y, x1...xn) = u
% 0.21/0.49 C => fresh(s, t, x1...xn) = v
% 0.21/0.49 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.49 variables of u and v.
% 0.21/0.49 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.49 input problem has no model of domain size 1).
% 0.21/0.49
% 0.21/0.49 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.49
% 0.21/0.49 Axiom 1 (tfree_tcs): class_Ring__and__Field_Oordered__idom(t_b) = true2.
% 0.21/0.49 Axiom 2 (clsrel_Ring__and__Field_Oordered__idom_50): fresh(X, X, Y) = true2.
% 0.21/0.49 Axiom 3 (clsrel_Ring__and__Field_Oordered__idom_17): fresh4(X, X, Y) = true2.
% 0.21/0.49 Axiom 4 (clsrel_Ring__and__Field_Oordered__idom_21): fresh3(X, X, Y) = true2.
% 0.21/0.49 Axiom 5 (clsrel_Ring__and__Field_Oordered__idom_42): fresh2(X, X, Y) = true2.
% 0.21/0.49 Axiom 6 (clsrel_Ring__and__Field_Oordered__idom_50): fresh(class_Ring__and__Field_Oordered__idom(X), true2, X) = class_OrderedGroup_Olordered__ab__group__abs(X).
% 0.21/0.49 Axiom 7 (cls_OrderedGroup_Oabs__ge__zero_0): fresh7(X, X, Y, Z) = true2.
% 0.21/0.49 Axiom 8 (clsrel_Ring__and__Field_Oordered__idom_17): fresh4(class_Ring__and__Field_Oordered__idom(X), true2, X) = class_OrderedGroup_Oab__semigroup__mult(X).
% 0.21/0.49 Axiom 9 (clsrel_Ring__and__Field_Oordered__idom_21): fresh3(class_Ring__and__Field_Oordered__idom(X), true2, X) = class_OrderedGroup_Osemigroup__mult(X).
% 0.21/0.49 Axiom 10 (clsrel_Ring__and__Field_Oordered__idom_42): fresh2(class_Ring__and__Field_Oordered__idom(X), true2, X) = class_Ring__and__Field_Opordered__semiring(X).
% 0.21/0.49 Axiom 11 (cls_OrderedGroup_Oabs__ge__zero_0): fresh7(class_OrderedGroup_Olordered__ab__group__abs(X), true2, X, Y) = c_lessequals(c_0, c_HOL_Oabs(Y, X), X).
% 0.21/0.49 Axiom 12 (cls_OrderedGroup_Omult__ac__2_0): fresh6(X, X, Y, Z, W) = c_times(W, Z, Y).
% 0.21/0.49 Axiom 13 (cls_Ring__and__Field_Opordered__semiring__class_Omult__left__mono_0): fresh10(X, X, Y, Z, W, V) = true2.
% 0.21/0.49 Axiom 14 (cls_OrderedGroup_Omult__ac__1_0): fresh8(X, X, Y, Z, W, V) = c_times(Z, c_times(W, V, Y), Y).
% 0.21/0.49 Axiom 15 (cls_OrderedGroup_Omult__ac__2_0): fresh6(class_OrderedGroup_Oab__semigroup__mult(X), true2, X, Y, Z) = c_times(Y, Z, X).
% 0.21/0.49 Axiom 16 (cls_OrderedGroup_Omult__ac__1_0): fresh8(class_OrderedGroup_Osemigroup__mult(X), true2, X, Y, Z, W) = c_times(c_times(Y, Z, X), W, X).
% 0.21/0.49 Axiom 17 (cls_Ring__and__Field_Opordered__semiring__class_Omult__left__mono_0): fresh5(X, X, Y, Z, W, V) = c_lessequals(c_times(W, Z, Y), c_times(W, V, Y), Y).
% 0.21/0.49 Axiom 18 (cls_Ring__and__Field_Opordered__semiring__class_Omult__left__mono_0): fresh9(X, X, Y, Z, W, V) = fresh10(c_lessequals(Z, V, Y), true2, Y, Z, W, V).
% 0.21/0.49 Axiom 19 (cls_Ring__and__Field_Opordered__semiring__class_Omult__left__mono_0): fresh9(class_Ring__and__Field_Opordered__semiring(X), true2, X, Y, Z, W) = fresh5(c_lessequals(c_0, Z, X), true2, X, Y, Z, W).
% 0.21/0.49 Axiom 20 (cls_conjecture_0): c_lessequals(c_HOL_Oabs(v_b(X), t_b), c_times(v_c, c_HOL_Oabs(v_g(X), t_b), t_b), t_b) = true2.
% 0.21/0.49
% 0.21/0.49 Lemma 21: c_times(Y, X, t_b) = c_times(X, Y, t_b).
% 0.21/0.49 Proof:
% 0.21/0.49 c_times(Y, X, t_b)
% 0.21/0.49 = { by axiom 12 (cls_OrderedGroup_Omult__ac__2_0) R->L }
% 0.21/0.49 fresh6(true2, true2, t_b, X, Y)
% 0.21/0.49 = { by axiom 3 (clsrel_Ring__and__Field_Oordered__idom_17) R->L }
% 0.21/0.49 fresh6(fresh4(true2, true2, t_b), true2, t_b, X, Y)
% 0.21/0.49 = { by axiom 1 (tfree_tcs) R->L }
% 0.21/0.49 fresh6(fresh4(class_Ring__and__Field_Oordered__idom(t_b), true2, t_b), true2, t_b, X, Y)
% 0.21/0.49 = { by axiom 8 (clsrel_Ring__and__Field_Oordered__idom_17) }
% 0.21/0.49 fresh6(class_OrderedGroup_Oab__semigroup__mult(t_b), true2, t_b, X, Y)
% 0.21/0.49 = { by axiom 15 (cls_OrderedGroup_Omult__ac__2_0) }
% 0.21/0.50 c_times(X, Y, t_b)
% 0.21/0.50
% 0.21/0.50 Goal 1 (cls_conjecture_1): c_lessequals(c_times(c_HOL_Oabs(v_b(v_x(X)), t_b), c_HOL_Oabs(v_f(v_x(X)), t_b), t_b), c_times(X, c_times(c_HOL_Oabs(v_f(v_x(X)), t_b), c_HOL_Oabs(v_g(v_x(X)), t_b), t_b), t_b), t_b) = true2.
% 0.21/0.50 The goal is true when:
% 0.21/0.50 X = v_c
% 0.21/0.50
% 0.21/0.50 Proof:
% 0.21/0.50 c_lessequals(c_times(c_HOL_Oabs(v_b(v_x(v_c)), t_b), c_HOL_Oabs(v_f(v_x(v_c)), t_b), t_b), c_times(v_c, c_times(c_HOL_Oabs(v_f(v_x(v_c)), t_b), c_HOL_Oabs(v_g(v_x(v_c)), t_b), t_b), t_b), t_b)
% 0.21/0.50 = { by lemma 21 R->L }
% 0.21/0.50 c_lessequals(c_times(c_HOL_Oabs(v_f(v_x(v_c)), t_b), c_HOL_Oabs(v_b(v_x(v_c)), t_b), t_b), c_times(v_c, c_times(c_HOL_Oabs(v_f(v_x(v_c)), t_b), c_HOL_Oabs(v_g(v_x(v_c)), t_b), t_b), t_b), t_b)
% 0.21/0.50 = { by lemma 21 }
% 0.21/0.50 c_lessequals(c_times(c_HOL_Oabs(v_f(v_x(v_c)), t_b), c_HOL_Oabs(v_b(v_x(v_c)), t_b), t_b), c_times(v_c, c_times(c_HOL_Oabs(v_g(v_x(v_c)), t_b), c_HOL_Oabs(v_f(v_x(v_c)), t_b), t_b), t_b), t_b)
% 0.21/0.50 = { by axiom 14 (cls_OrderedGroup_Omult__ac__1_0) R->L }
% 0.21/0.50 c_lessequals(c_times(c_HOL_Oabs(v_f(v_x(v_c)), t_b), c_HOL_Oabs(v_b(v_x(v_c)), t_b), t_b), fresh8(true2, true2, t_b, v_c, c_HOL_Oabs(v_g(v_x(v_c)), t_b), c_HOL_Oabs(v_f(v_x(v_c)), t_b)), t_b)
% 0.21/0.50 = { by axiom 4 (clsrel_Ring__and__Field_Oordered__idom_21) R->L }
% 0.21/0.50 c_lessequals(c_times(c_HOL_Oabs(v_f(v_x(v_c)), t_b), c_HOL_Oabs(v_b(v_x(v_c)), t_b), t_b), fresh8(fresh3(true2, true2, t_b), true2, t_b, v_c, c_HOL_Oabs(v_g(v_x(v_c)), t_b), c_HOL_Oabs(v_f(v_x(v_c)), t_b)), t_b)
% 0.21/0.50 = { by axiom 1 (tfree_tcs) R->L }
% 0.21/0.50 c_lessequals(c_times(c_HOL_Oabs(v_f(v_x(v_c)), t_b), c_HOL_Oabs(v_b(v_x(v_c)), t_b), t_b), fresh8(fresh3(class_Ring__and__Field_Oordered__idom(t_b), true2, t_b), true2, t_b, v_c, c_HOL_Oabs(v_g(v_x(v_c)), t_b), c_HOL_Oabs(v_f(v_x(v_c)), t_b)), t_b)
% 0.21/0.50 = { by axiom 9 (clsrel_Ring__and__Field_Oordered__idom_21) }
% 0.21/0.50 c_lessequals(c_times(c_HOL_Oabs(v_f(v_x(v_c)), t_b), c_HOL_Oabs(v_b(v_x(v_c)), t_b), t_b), fresh8(class_OrderedGroup_Osemigroup__mult(t_b), true2, t_b, v_c, c_HOL_Oabs(v_g(v_x(v_c)), t_b), c_HOL_Oabs(v_f(v_x(v_c)), t_b)), t_b)
% 0.21/0.50 = { by axiom 16 (cls_OrderedGroup_Omult__ac__1_0) }
% 0.21/0.50 c_lessequals(c_times(c_HOL_Oabs(v_f(v_x(v_c)), t_b), c_HOL_Oabs(v_b(v_x(v_c)), t_b), t_b), c_times(c_times(v_c, c_HOL_Oabs(v_g(v_x(v_c)), t_b), t_b), c_HOL_Oabs(v_f(v_x(v_c)), t_b), t_b), t_b)
% 0.21/0.50 = { by lemma 21 R->L }
% 0.21/0.50 c_lessequals(c_times(c_HOL_Oabs(v_f(v_x(v_c)), t_b), c_HOL_Oabs(v_b(v_x(v_c)), t_b), t_b), c_times(c_HOL_Oabs(v_f(v_x(v_c)), t_b), c_times(v_c, c_HOL_Oabs(v_g(v_x(v_c)), t_b), t_b), t_b), t_b)
% 0.21/0.50 = { by lemma 21 R->L }
% 0.21/0.50 c_lessequals(c_times(c_HOL_Oabs(v_f(v_x(v_c)), t_b), c_HOL_Oabs(v_b(v_x(v_c)), t_b), t_b), c_times(c_HOL_Oabs(v_f(v_x(v_c)), t_b), c_times(c_HOL_Oabs(v_g(v_x(v_c)), t_b), v_c, t_b), t_b), t_b)
% 0.21/0.50 = { by axiom 17 (cls_Ring__and__Field_Opordered__semiring__class_Omult__left__mono_0) R->L }
% 0.21/0.50 fresh5(true2, true2, t_b, c_HOL_Oabs(v_b(v_x(v_c)), t_b), c_HOL_Oabs(v_f(v_x(v_c)), t_b), c_times(c_HOL_Oabs(v_g(v_x(v_c)), t_b), v_c, t_b))
% 0.21/0.50 = { by axiom 7 (cls_OrderedGroup_Oabs__ge__zero_0) R->L }
% 0.21/0.50 fresh5(fresh7(true2, true2, t_b, v_f(v_x(v_c))), true2, t_b, c_HOL_Oabs(v_b(v_x(v_c)), t_b), c_HOL_Oabs(v_f(v_x(v_c)), t_b), c_times(c_HOL_Oabs(v_g(v_x(v_c)), t_b), v_c, t_b))
% 0.21/0.50 = { by axiom 2 (clsrel_Ring__and__Field_Oordered__idom_50) R->L }
% 0.21/0.50 fresh5(fresh7(fresh(true2, true2, t_b), true2, t_b, v_f(v_x(v_c))), true2, t_b, c_HOL_Oabs(v_b(v_x(v_c)), t_b), c_HOL_Oabs(v_f(v_x(v_c)), t_b), c_times(c_HOL_Oabs(v_g(v_x(v_c)), t_b), v_c, t_b))
% 0.21/0.50 = { by axiom 1 (tfree_tcs) R->L }
% 0.21/0.50 fresh5(fresh7(fresh(class_Ring__and__Field_Oordered__idom(t_b), true2, t_b), true2, t_b, v_f(v_x(v_c))), true2, t_b, c_HOL_Oabs(v_b(v_x(v_c)), t_b), c_HOL_Oabs(v_f(v_x(v_c)), t_b), c_times(c_HOL_Oabs(v_g(v_x(v_c)), t_b), v_c, t_b))
% 0.21/0.50 = { by axiom 6 (clsrel_Ring__and__Field_Oordered__idom_50) }
% 0.21/0.50 fresh5(fresh7(class_OrderedGroup_Olordered__ab__group__abs(t_b), true2, t_b, v_f(v_x(v_c))), true2, t_b, c_HOL_Oabs(v_b(v_x(v_c)), t_b), c_HOL_Oabs(v_f(v_x(v_c)), t_b), c_times(c_HOL_Oabs(v_g(v_x(v_c)), t_b), v_c, t_b))
% 0.21/0.50 = { by axiom 11 (cls_OrderedGroup_Oabs__ge__zero_0) }
% 0.21/0.50 fresh5(c_lessequals(c_0, c_HOL_Oabs(v_f(v_x(v_c)), t_b), t_b), true2, t_b, c_HOL_Oabs(v_b(v_x(v_c)), t_b), c_HOL_Oabs(v_f(v_x(v_c)), t_b), c_times(c_HOL_Oabs(v_g(v_x(v_c)), t_b), v_c, t_b))
% 0.21/0.50 = { by axiom 19 (cls_Ring__and__Field_Opordered__semiring__class_Omult__left__mono_0) R->L }
% 0.21/0.50 fresh9(class_Ring__and__Field_Opordered__semiring(t_b), true2, t_b, c_HOL_Oabs(v_b(v_x(v_c)), t_b), c_HOL_Oabs(v_f(v_x(v_c)), t_b), c_times(c_HOL_Oabs(v_g(v_x(v_c)), t_b), v_c, t_b))
% 0.21/0.50 = { by axiom 10 (clsrel_Ring__and__Field_Oordered__idom_42) R->L }
% 0.21/0.50 fresh9(fresh2(class_Ring__and__Field_Oordered__idom(t_b), true2, t_b), true2, t_b, c_HOL_Oabs(v_b(v_x(v_c)), t_b), c_HOL_Oabs(v_f(v_x(v_c)), t_b), c_times(c_HOL_Oabs(v_g(v_x(v_c)), t_b), v_c, t_b))
% 0.21/0.50 = { by axiom 1 (tfree_tcs) }
% 0.21/0.50 fresh9(fresh2(true2, true2, t_b), true2, t_b, c_HOL_Oabs(v_b(v_x(v_c)), t_b), c_HOL_Oabs(v_f(v_x(v_c)), t_b), c_times(c_HOL_Oabs(v_g(v_x(v_c)), t_b), v_c, t_b))
% 0.21/0.50 = { by axiom 5 (clsrel_Ring__and__Field_Oordered__idom_42) }
% 0.21/0.50 fresh9(true2, true2, t_b, c_HOL_Oabs(v_b(v_x(v_c)), t_b), c_HOL_Oabs(v_f(v_x(v_c)), t_b), c_times(c_HOL_Oabs(v_g(v_x(v_c)), t_b), v_c, t_b))
% 0.21/0.50 = { by lemma 21 R->L }
% 0.21/0.50 fresh9(true2, true2, t_b, c_HOL_Oabs(v_b(v_x(v_c)), t_b), c_HOL_Oabs(v_f(v_x(v_c)), t_b), c_times(v_c, c_HOL_Oabs(v_g(v_x(v_c)), t_b), t_b))
% 0.21/0.50 = { by axiom 18 (cls_Ring__and__Field_Opordered__semiring__class_Omult__left__mono_0) }
% 0.21/0.50 fresh10(c_lessequals(c_HOL_Oabs(v_b(v_x(v_c)), t_b), c_times(v_c, c_HOL_Oabs(v_g(v_x(v_c)), t_b), t_b), t_b), true2, t_b, c_HOL_Oabs(v_b(v_x(v_c)), t_b), c_HOL_Oabs(v_f(v_x(v_c)), t_b), c_times(v_c, c_HOL_Oabs(v_g(v_x(v_c)), t_b), t_b))
% 0.21/0.50 = { by axiom 20 (cls_conjecture_0) }
% 0.21/0.50 fresh10(true2, true2, t_b, c_HOL_Oabs(v_b(v_x(v_c)), t_b), c_HOL_Oabs(v_f(v_x(v_c)), t_b), c_times(v_c, c_HOL_Oabs(v_g(v_x(v_c)), t_b), t_b))
% 0.21/0.50 = { by axiom 13 (cls_Ring__and__Field_Opordered__semiring__class_Omult__left__mono_0) }
% 0.21/0.50 true2
% 0.21/0.50 % SZS output end Proof
% 0.21/0.50
% 0.21/0.50 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------