TSTP Solution File: RNG006-3 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : RNG006-3 : TPTP v8.1.2. Released v1.0.0.
% 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 : n025.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 : Thu Aug 31 13:58:44 EDT 2023
% Result : Unsatisfiable 5.33s 1.12s
% Output : Proof 5.93s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : RNG006-3 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.35 % Computer : n025.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 : Sun Aug 27 03:04:23 EDT 2023
% 0.14/0.35 % CPUTime :
% 5.33/1.12 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 5.33/1.12
% 5.33/1.12 % SZS status Unsatisfiable
% 5.33/1.12
% 5.93/1.14 % SZS output start Proof
% 5.93/1.14 Take the following subset of the input axioms:
% 5.93/1.14 fof(aPb_plus_IaPc, hypothesis, sum(aPb, additive_inverse(aPc), aPb_S_IaPc)).
% 5.93/1.14 fof(a_times_b, hypothesis, product(a, b, aPb)).
% 5.93/1.14 fof(a_times_c, hypothesis, product(a, c, aPc)).
% 5.93/1.14 fof(addition_is_well_defined, axiom, ![X, Y, U, V]: (~sum(X, Y, U) | (~sum(X, Y, V) | U=V))).
% 5.93/1.14 fof(additive_identity2, axiom, ![X2]: sum(X2, additive_identity, X2)).
% 5.93/1.14 fof(associativity_of_addition2, axiom, ![Z, W, X2, Y2, U2, V5]: (~sum(X2, Y2, U2) | (~sum(Y2, Z, V5) | (~sum(X2, V5, W) | sum(U2, Z, W))))).
% 5.93/1.14 fof(b_plus_inverse_c, hypothesis, sum(b, additive_inverse(c), bS_Ic)).
% 5.93/1.14 fof(closure_of_addition, axiom, ![X2, Y2]: sum(X2, Y2, add(X2, Y2))).
% 5.93/1.14 fof(closure_of_multiplication, axiom, ![X2, Y2]: product(X2, Y2, multiply(X2, Y2))).
% 5.93/1.14 fof(commutativity_of_addition, axiom, ![X2, Y2, Z2]: (~sum(X2, Y2, Z2) | sum(Y2, X2, Z2))).
% 5.93/1.14 fof(distributivity1, axiom, ![V1, V2, V3, V4, X2, Y2, Z2]: (~product(X2, Y2, V1) | (~product(X2, Z2, V2) | (~sum(Y2, Z2, V3) | (~product(X2, V3, V4) | sum(V1, V2, V4)))))).
% 5.93/1.14 fof(left_inverse, axiom, ![X2]: sum(additive_inverse(X2), X2, additive_identity)).
% 5.93/1.14 fof(prove_a_times_bS_Ic_is_aPb_S__IaPc, negated_conjecture, ~product(a, bS_Ic, aPb_S_IaPc)).
% 5.93/1.14 fof(right_inverse, axiom, ![X2]: sum(X2, additive_inverse(X2), additive_identity)).
% 5.93/1.14
% 5.93/1.14 Now clausify the problem and encode Horn clauses using encoding 3 of
% 5.93/1.14 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 5.93/1.14 We repeatedly replace C & s=t => u=v by the two clauses:
% 5.93/1.14 fresh(y, y, x1...xn) = u
% 5.93/1.14 C => fresh(s, t, x1...xn) = v
% 5.93/1.14 where fresh is a fresh function symbol and x1..xn are the free
% 5.93/1.14 variables of u and v.
% 5.93/1.14 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 5.93/1.14 input problem has no model of domain size 1).
% 5.93/1.14
% 5.93/1.14 The encoding turns the above axioms into the following unit equations and goals:
% 5.93/1.14
% 5.93/1.14 Axiom 1 (additive_identity2): sum(X, additive_identity, X) = true.
% 5.93/1.14 Axiom 2 (a_times_b): product(a, b, aPb) = true.
% 5.93/1.14 Axiom 3 (a_times_c): product(a, c, aPc) = true.
% 5.93/1.14 Axiom 4 (right_inverse): sum(X, additive_inverse(X), additive_identity) = true.
% 5.93/1.14 Axiom 5 (b_plus_inverse_c): sum(b, additive_inverse(c), bS_Ic) = true.
% 5.93/1.14 Axiom 6 (aPb_plus_IaPc): sum(aPb, additive_inverse(aPc), aPb_S_IaPc) = true.
% 5.93/1.14 Axiom 7 (left_inverse): sum(additive_inverse(X), X, additive_identity) = true.
% 5.93/1.14 Axiom 8 (addition_is_well_defined): fresh3(X, X, Y, Z) = Z.
% 5.93/1.14 Axiom 9 (closure_of_addition): sum(X, Y, add(X, Y)) = true.
% 5.93/1.14 Axiom 10 (closure_of_multiplication): product(X, Y, multiply(X, Y)) = true.
% 5.93/1.14 Axiom 11 (associativity_of_addition2): fresh31(X, X, Y, Z, W) = true.
% 5.93/1.14 Axiom 12 (distributivity1): fresh25(X, X, Y, Z, W) = true.
% 5.93/1.14 Axiom 13 (commutativity_of_addition): fresh5(X, X, Y, Z, W) = true.
% 5.93/1.14 Axiom 14 (addition_is_well_defined): fresh4(X, X, Y, Z, W, V) = W.
% 5.93/1.14 Axiom 15 (associativity_of_addition2): fresh8(X, X, Y, Z, W, V, U) = sum(W, V, U).
% 5.93/1.14 Axiom 16 (associativity_of_addition2): fresh30(X, X, Y, Z, W, V, U, T) = fresh31(sum(Y, Z, W), true, W, V, T).
% 5.93/1.14 Axiom 17 (distributivity1): fresh23(X, X, Y, Z, W, V, U, T) = sum(Z, V, T).
% 5.93/1.14 Axiom 18 (commutativity_of_addition): fresh5(sum(X, Y, Z), true, X, Y, Z) = sum(Y, X, Z).
% 5.93/1.14 Axiom 19 (distributivity1): fresh24(X, X, Y, Z, W, V, U, T, S) = fresh25(sum(Z, V, T), true, W, U, S).
% 5.93/1.14 Axiom 20 (addition_is_well_defined): fresh4(sum(X, Y, Z), true, X, Y, W, Z) = fresh3(sum(X, Y, W), true, W, Z).
% 5.93/1.14 Axiom 21 (associativity_of_addition2): fresh30(sum(X, Y, Z), true, W, X, V, Y, Z, U) = fresh8(sum(W, Z, U), true, W, X, V, Y, U).
% 5.93/1.14 Axiom 22 (distributivity1): fresh22(X, X, Y, Z, W, V, U, T, S) = fresh23(product(Y, Z, W), true, Z, W, V, U, T, S).
% 5.93/1.14 Axiom 23 (distributivity1): fresh22(product(X, Y, Z), true, X, W, V, U, T, Y, Z) = fresh24(product(X, U, T), true, X, W, V, U, T, Y, Z).
% 5.93/1.14
% 5.93/1.14 Lemma 24: fresh3(sum(X, Y, Z), true, Z, add(X, Y)) = Z.
% 5.93/1.14 Proof:
% 5.93/1.14 fresh3(sum(X, Y, Z), true, Z, add(X, Y))
% 5.93/1.14 = { by axiom 20 (addition_is_well_defined) R->L }
% 5.93/1.14 fresh4(sum(X, Y, add(X, Y)), true, X, Y, Z, add(X, Y))
% 5.93/1.14 = { by axiom 9 (closure_of_addition) }
% 5.93/1.14 fresh4(true, true, X, Y, Z, add(X, Y))
% 5.93/1.14 = { by axiom 14 (addition_is_well_defined) }
% 5.93/1.14 Z
% 5.93/1.14
% 5.93/1.14 Lemma 25: sum(X, Y, add(Y, X)) = true.
% 5.93/1.14 Proof:
% 5.93/1.14 sum(X, Y, add(Y, X))
% 5.93/1.14 = { by axiom 18 (commutativity_of_addition) R->L }
% 5.93/1.14 fresh5(sum(Y, X, add(Y, X)), true, Y, X, add(Y, X))
% 5.93/1.14 = { by axiom 9 (closure_of_addition) }
% 5.93/1.14 fresh5(true, true, Y, X, add(Y, X))
% 5.93/1.14 = { by axiom 13 (commutativity_of_addition) }
% 5.93/1.14 true
% 5.93/1.14
% 5.93/1.14 Lemma 26: add(X, Y) = add(Y, X).
% 5.93/1.14 Proof:
% 5.93/1.14 add(X, Y)
% 5.93/1.14 = { by lemma 24 R->L }
% 5.93/1.14 fresh3(sum(Y, X, add(X, Y)), true, add(X, Y), add(Y, X))
% 5.93/1.14 = { by lemma 25 }
% 5.93/1.14 fresh3(true, true, add(X, Y), add(Y, X))
% 5.93/1.14 = { by axiom 8 (addition_is_well_defined) }
% 5.93/1.15 add(Y, X)
% 5.93/1.15
% 5.93/1.15 Lemma 27: fresh30(X, X, Y, Z, add(Y, Z), W, V, U) = true.
% 5.93/1.15 Proof:
% 5.93/1.15 fresh30(X, X, Y, Z, add(Y, Z), W, V, U)
% 5.93/1.15 = { by axiom 16 (associativity_of_addition2) }
% 5.93/1.15 fresh31(sum(Y, Z, add(Y, Z)), true, add(Y, Z), W, U)
% 5.93/1.15 = { by axiom 9 (closure_of_addition) }
% 5.93/1.15 fresh31(true, true, add(Y, Z), W, U)
% 5.93/1.15 = { by axiom 11 (associativity_of_addition2) }
% 5.93/1.15 true
% 5.93/1.15
% 5.93/1.15 Lemma 28: fresh30(sum(X, Y, additive_identity), true, Z, X, W, Y, additive_identity, Z) = sum(W, Y, Z).
% 5.93/1.15 Proof:
% 5.93/1.15 fresh30(sum(X, Y, additive_identity), true, Z, X, W, Y, additive_identity, Z)
% 5.93/1.15 = { by axiom 21 (associativity_of_addition2) }
% 5.93/1.15 fresh8(sum(Z, additive_identity, Z), true, Z, X, W, Y, Z)
% 5.93/1.15 = { by axiom 1 (additive_identity2) }
% 5.93/1.15 fresh8(true, true, Z, X, W, Y, Z)
% 5.93/1.15 = { by axiom 15 (associativity_of_addition2) }
% 5.93/1.15 sum(W, Y, Z)
% 5.93/1.15
% 5.93/1.15 Goal 1 (prove_a_times_bS_Ic_is_aPb_S__IaPc): product(a, bS_Ic, aPb_S_IaPc) = true.
% 5.93/1.15 Proof:
% 5.93/1.15 product(a, bS_Ic, aPb_S_IaPc)
% 5.93/1.15 = { by axiom 8 (addition_is_well_defined) R->L }
% 5.93/1.15 product(a, bS_Ic, fresh3(true, true, add(additive_inverse(aPc), aPb), aPb_S_IaPc))
% 5.93/1.15 = { by lemma 25 R->L }
% 5.93/1.15 product(a, bS_Ic, fresh3(sum(aPb, additive_inverse(aPc), add(additive_inverse(aPc), aPb)), true, add(additive_inverse(aPc), aPb), aPb_S_IaPc))
% 5.93/1.15 = { by axiom 20 (addition_is_well_defined) R->L }
% 5.93/1.15 product(a, bS_Ic, fresh4(sum(aPb, additive_inverse(aPc), aPb_S_IaPc), true, aPb, additive_inverse(aPc), add(additive_inverse(aPc), aPb), aPb_S_IaPc))
% 5.93/1.15 = { by axiom 6 (aPb_plus_IaPc) }
% 5.93/1.15 product(a, bS_Ic, fresh4(true, true, aPb, additive_inverse(aPc), add(additive_inverse(aPc), aPb), aPb_S_IaPc))
% 5.93/1.15 = { by axiom 14 (addition_is_well_defined) }
% 5.93/1.15 product(a, bS_Ic, add(additive_inverse(aPc), aPb))
% 5.93/1.15 = { by lemma 24 R->L }
% 5.93/1.15 product(a, bS_Ic, add(additive_inverse(aPc), fresh3(sum(multiply(a, bS_Ic), aPc, aPb), true, aPb, add(multiply(a, bS_Ic), aPc))))
% 5.93/1.15 = { by axiom 8 (addition_is_well_defined) R->L }
% 5.93/1.15 product(a, bS_Ic, add(additive_inverse(aPc), fresh3(sum(multiply(a, fresh3(true, true, add(b, additive_inverse(c)), bS_Ic)), aPc, aPb), true, aPb, add(multiply(a, bS_Ic), aPc))))
% 5.93/1.15 = { by axiom 9 (closure_of_addition) R->L }
% 5.93/1.15 product(a, bS_Ic, add(additive_inverse(aPc), fresh3(sum(multiply(a, fresh3(sum(b, additive_inverse(c), add(b, additive_inverse(c))), true, add(b, additive_inverse(c)), bS_Ic)), aPc, aPb), true, aPb, add(multiply(a, bS_Ic), aPc))))
% 5.93/1.15 = { by axiom 20 (addition_is_well_defined) R->L }
% 5.93/1.15 product(a, bS_Ic, add(additive_inverse(aPc), fresh3(sum(multiply(a, fresh4(sum(b, additive_inverse(c), bS_Ic), true, b, additive_inverse(c), add(b, additive_inverse(c)), bS_Ic)), aPc, aPb), true, aPb, add(multiply(a, bS_Ic), aPc))))
% 5.93/1.15 = { by axiom 5 (b_plus_inverse_c) }
% 5.93/1.15 product(a, bS_Ic, add(additive_inverse(aPc), fresh3(sum(multiply(a, fresh4(true, true, b, additive_inverse(c), add(b, additive_inverse(c)), bS_Ic)), aPc, aPb), true, aPb, add(multiply(a, bS_Ic), aPc))))
% 5.93/1.15 = { by axiom 14 (addition_is_well_defined) }
% 5.93/1.15 product(a, bS_Ic, add(additive_inverse(aPc), fresh3(sum(multiply(a, add(b, additive_inverse(c))), aPc, aPb), true, aPb, add(multiply(a, bS_Ic), aPc))))
% 5.93/1.15 = { by axiom 17 (distributivity1) R->L }
% 5.93/1.15 product(a, bS_Ic, add(additive_inverse(aPc), fresh3(fresh23(true, true, add(b, additive_inverse(c)), multiply(a, add(b, additive_inverse(c))), c, aPc, b, aPb), true, aPb, add(multiply(a, bS_Ic), aPc))))
% 5.93/1.15 = { by axiom 10 (closure_of_multiplication) R->L }
% 5.93/1.15 product(a, bS_Ic, add(additive_inverse(aPc), fresh3(fresh23(product(a, add(b, additive_inverse(c)), multiply(a, add(b, additive_inverse(c)))), true, add(b, additive_inverse(c)), multiply(a, add(b, additive_inverse(c))), c, aPc, b, aPb), true, aPb, add(multiply(a, bS_Ic), aPc))))
% 5.93/1.15 = { by axiom 22 (distributivity1) R->L }
% 5.93/1.15 product(a, bS_Ic, add(additive_inverse(aPc), fresh3(fresh22(true, true, a, add(b, additive_inverse(c)), multiply(a, add(b, additive_inverse(c))), c, aPc, b, aPb), true, aPb, add(multiply(a, bS_Ic), aPc))))
% 5.93/1.15 = { by axiom 2 (a_times_b) R->L }
% 5.93/1.15 product(a, bS_Ic, add(additive_inverse(aPc), fresh3(fresh22(product(a, b, aPb), true, a, add(b, additive_inverse(c)), multiply(a, add(b, additive_inverse(c))), c, aPc, b, aPb), true, aPb, add(multiply(a, bS_Ic), aPc))))
% 5.93/1.15 = { by axiom 23 (distributivity1) }
% 5.93/1.15 product(a, bS_Ic, add(additive_inverse(aPc), fresh3(fresh24(product(a, c, aPc), true, a, add(b, additive_inverse(c)), multiply(a, add(b, additive_inverse(c))), c, aPc, b, aPb), true, aPb, add(multiply(a, bS_Ic), aPc))))
% 5.93/1.15 = { by axiom 3 (a_times_c) }
% 5.93/1.15 product(a, bS_Ic, add(additive_inverse(aPc), fresh3(fresh24(true, true, a, add(b, additive_inverse(c)), multiply(a, add(b, additive_inverse(c))), c, aPc, b, aPb), true, aPb, add(multiply(a, bS_Ic), aPc))))
% 5.93/1.15 = { by axiom 19 (distributivity1) }
% 5.93/1.15 product(a, bS_Ic, add(additive_inverse(aPc), fresh3(fresh25(sum(add(b, additive_inverse(c)), c, b), true, multiply(a, add(b, additive_inverse(c))), aPc, aPb), true, aPb, add(multiply(a, bS_Ic), aPc))))
% 5.93/1.15 = { by lemma 28 R->L }
% 5.93/1.15 product(a, bS_Ic, add(additive_inverse(aPc), fresh3(fresh25(fresh30(sum(additive_inverse(c), c, additive_identity), true, b, additive_inverse(c), add(b, additive_inverse(c)), c, additive_identity, b), true, multiply(a, add(b, additive_inverse(c))), aPc, aPb), true, aPb, add(multiply(a, bS_Ic), aPc))))
% 5.93/1.15 = { by axiom 7 (left_inverse) }
% 5.93/1.15 product(a, bS_Ic, add(additive_inverse(aPc), fresh3(fresh25(fresh30(true, true, b, additive_inverse(c), add(b, additive_inverse(c)), c, additive_identity, b), true, multiply(a, add(b, additive_inverse(c))), aPc, aPb), true, aPb, add(multiply(a, bS_Ic), aPc))))
% 5.93/1.15 = { by lemma 27 }
% 5.93/1.15 product(a, bS_Ic, add(additive_inverse(aPc), fresh3(fresh25(true, true, multiply(a, add(b, additive_inverse(c))), aPc, aPb), true, aPb, add(multiply(a, bS_Ic), aPc))))
% 5.93/1.15 = { by axiom 12 (distributivity1) }
% 5.93/1.15 product(a, bS_Ic, add(additive_inverse(aPc), fresh3(true, true, aPb, add(multiply(a, bS_Ic), aPc))))
% 5.93/1.15 = { by axiom 8 (addition_is_well_defined) }
% 5.93/1.15 product(a, bS_Ic, add(additive_inverse(aPc), add(multiply(a, bS_Ic), aPc)))
% 5.93/1.15 = { by lemma 26 }
% 5.93/1.15 product(a, bS_Ic, add(additive_inverse(aPc), add(aPc, multiply(a, bS_Ic))))
% 5.93/1.15 = { by lemma 26 R->L }
% 5.93/1.15 product(a, bS_Ic, add(add(aPc, multiply(a, bS_Ic)), additive_inverse(aPc)))
% 5.93/1.15 = { by axiom 8 (addition_is_well_defined) R->L }
% 5.93/1.15 product(a, bS_Ic, fresh3(true, true, multiply(a, bS_Ic), add(add(aPc, multiply(a, bS_Ic)), additive_inverse(aPc))))
% 5.93/1.15 = { by lemma 27 R->L }
% 5.93/1.15 product(a, bS_Ic, fresh3(fresh30(true, true, multiply(a, bS_Ic), aPc, add(multiply(a, bS_Ic), aPc), additive_inverse(aPc), additive_identity, multiply(a, bS_Ic)), true, multiply(a, bS_Ic), add(add(aPc, multiply(a, bS_Ic)), additive_inverse(aPc))))
% 5.93/1.15 = { by axiom 4 (right_inverse) R->L }
% 5.93/1.15 product(a, bS_Ic, fresh3(fresh30(sum(aPc, additive_inverse(aPc), additive_identity), true, multiply(a, bS_Ic), aPc, add(multiply(a, bS_Ic), aPc), additive_inverse(aPc), additive_identity, multiply(a, bS_Ic)), true, multiply(a, bS_Ic), add(add(aPc, multiply(a, bS_Ic)), additive_inverse(aPc))))
% 5.93/1.15 = { by lemma 28 }
% 5.93/1.15 product(a, bS_Ic, fresh3(sum(add(multiply(a, bS_Ic), aPc), additive_inverse(aPc), multiply(a, bS_Ic)), true, multiply(a, bS_Ic), add(add(aPc, multiply(a, bS_Ic)), additive_inverse(aPc))))
% 5.93/1.15 = { by lemma 26 }
% 5.93/1.15 product(a, bS_Ic, fresh3(sum(add(aPc, multiply(a, bS_Ic)), additive_inverse(aPc), multiply(a, bS_Ic)), true, multiply(a, bS_Ic), add(add(aPc, multiply(a, bS_Ic)), additive_inverse(aPc))))
% 5.93/1.15 = { by lemma 24 }
% 5.93/1.15 product(a, bS_Ic, multiply(a, bS_Ic))
% 5.93/1.15 = { by axiom 10 (closure_of_multiplication) }
% 5.93/1.15 true
% 5.93/1.15 % SZS output end Proof
% 5.93/1.15
% 5.93/1.15 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------