TSTP Solution File: RNG039-1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : RNG039-1 : 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 : n007.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 13:58:59 EDT 2023
% Result : Unsatisfiable 0.20s 0.49s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : RNG039-1 : TPTP v8.1.2. Released v1.0.0.
% 0.07/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n007.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 : Sun Aug 27 01:53:12 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.20/0.49 Command-line arguments: --kbo-weight0 --lhs-weight 5 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --goal-heuristic
% 0.20/0.49
% 0.20/0.49 % SZS status Unsatisfiable
% 0.20/0.49
% 0.20/0.50 % SZS output start Proof
% 0.20/0.50 Take the following subset of the input axioms:
% 0.20/0.50 fof(a_times_b, negated_conjecture, product(a, b, c)).
% 0.20/0.50 fof(clause33, axiom, ![A]: add(A, additive_identity)=A).
% 0.20/0.50 fof(clause34, axiom, ![A2]: add(A2, A2)=additive_identity).
% 0.20/0.50 fof(clause35, axiom, ![A2]: multiply(A2, A2)=A2).
% 0.20/0.50 fof(clause37, axiom, multiply(b, a)=d).
% 0.20/0.50 fof(clause44, axiom, ![B, A2]: product(a, multiply(b, A2), multiply(B, A2))).
% 0.20/0.50 fof(clause67, axiom, product(add(a, b), b, add(c, b))).
% 0.20/0.50 fof(clause70, axiom, product(add(a, b), a, add(a, d))).
% 0.20/0.50 fof(clause71, axiom, ![A2]: product(A2, A2, A2)).
% 0.20/0.50 fof(closure_of_multiplication, axiom, ![X, Y]: product(X, Y, multiply(X, Y))).
% 0.20/0.50 fof(multiplication_is_well_defined, axiom, ![U, V, X2, Y2]: (~product(X2, Y2, U) | (~product(X2, Y2, V) | U=V))).
% 0.20/0.50 fof(prove_c_equals_d, negated_conjecture, c!=d).
% 0.20/0.50
% 0.20/0.50 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.50 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.50 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.50 fresh(y, y, x1...xn) = u
% 0.20/0.50 C => fresh(s, t, x1...xn) = v
% 0.20/0.50 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.50 variables of u and v.
% 0.20/0.50 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.50 input problem has no model of domain size 1).
% 0.20/0.50
% 0.20/0.50 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.50
% 0.20/0.50 Axiom 1 (clause35): multiply(X, X) = X.
% 0.20/0.50 Axiom 2 (clause37): multiply(b, a) = d.
% 0.20/0.50 Axiom 3 (clause34): add(X, X) = additive_identity.
% 0.20/0.50 Axiom 4 (clause33): add(X, additive_identity) = X.
% 0.20/0.50 Axiom 5 (clause71): product(X, X, X) = true.
% 0.20/0.50 Axiom 6 (a_times_b): product(a, b, c) = true.
% 0.20/0.50 Axiom 7 (closure_of_multiplication): product(X, Y, multiply(X, Y)) = true.
% 0.20/0.50 Axiom 8 (multiplication_is_well_defined): fresh(X, X, Y, Z) = Z.
% 0.20/0.50 Axiom 9 (clause44): product(a, multiply(b, X), multiply(Y, X)) = true.
% 0.20/0.50 Axiom 10 (clause70): product(add(a, b), a, add(a, d)) = true.
% 0.20/0.50 Axiom 11 (clause67): product(add(a, b), b, add(c, b)) = true.
% 0.20/0.50 Axiom 12 (multiplication_is_well_defined): fresh2(X, X, Y, Z, W, V) = W.
% 0.20/0.50 Axiom 13 (multiplication_is_well_defined): fresh2(product(X, Y, Z), true, X, Y, W, Z) = fresh(product(X, Y, W), true, W, Z).
% 0.20/0.50
% 0.20/0.50 Lemma 14: product(a, multiply(b, X), X) = true.
% 0.20/0.50 Proof:
% 0.20/0.50 product(a, multiply(b, X), X)
% 0.20/0.50 = { by axiom 1 (clause35) R->L }
% 0.20/0.50 product(a, multiply(b, X), multiply(X, X))
% 0.20/0.50 = { by axiom 9 (clause44) }
% 0.20/0.50 true
% 0.20/0.50
% 0.20/0.50 Lemma 15: fresh(product(a, b, X), true, X, c) = X.
% 0.20/0.50 Proof:
% 0.20/0.50 fresh(product(a, b, X), true, X, c)
% 0.20/0.50 = { by axiom 13 (multiplication_is_well_defined) R->L }
% 0.20/0.50 fresh2(product(a, b, c), true, a, b, X, c)
% 0.20/0.50 = { by axiom 6 (a_times_b) }
% 0.20/0.50 fresh2(true, true, a, b, X, c)
% 0.20/0.50 = { by axiom 12 (multiplication_is_well_defined) }
% 0.20/0.50 X
% 0.20/0.50
% 0.20/0.50 Lemma 16: c = b.
% 0.20/0.50 Proof:
% 0.20/0.50 c
% 0.20/0.50 = { by axiom 8 (multiplication_is_well_defined) R->L }
% 0.20/0.50 fresh(true, true, b, c)
% 0.20/0.50 = { by lemma 14 R->L }
% 0.20/0.50 fresh(product(a, multiply(b, b), b), true, b, c)
% 0.20/0.50 = { by axiom 1 (clause35) }
% 0.20/0.50 fresh(product(a, b, b), true, b, c)
% 0.20/0.50 = { by lemma 15 }
% 0.20/0.50 b
% 0.20/0.50
% 0.20/0.50 Lemma 17: additive_identity = b.
% 0.20/0.50 Proof:
% 0.20/0.50 additive_identity
% 0.20/0.50 = { by axiom 3 (clause34) R->L }
% 0.20/0.50 add(b, b)
% 0.20/0.50 = { by lemma 16 R->L }
% 0.20/0.50 add(c, b)
% 0.20/0.50 = { by axiom 12 (multiplication_is_well_defined) R->L }
% 0.20/0.50 fresh2(true, true, add(a, b), b, add(c, b), b)
% 0.20/0.50 = { by axiom 7 (closure_of_multiplication) R->L }
% 0.20/0.50 fresh2(product(add(a, b), b, multiply(add(a, b), b)), true, add(a, b), b, add(c, b), b)
% 0.20/0.50 = { by lemma 15 R->L }
% 0.20/0.50 fresh2(product(add(a, b), b, fresh(product(a, b, multiply(add(a, b), b)), true, multiply(add(a, b), b), c)), true, add(a, b), b, add(c, b), b)
% 0.20/0.50 = { by axiom 1 (clause35) R->L }
% 0.20/0.50 fresh2(product(add(a, b), b, fresh(product(a, multiply(b, b), multiply(add(a, b), b)), true, multiply(add(a, b), b), c)), true, add(a, b), b, add(c, b), b)
% 0.20/0.50 = { by axiom 9 (clause44) }
% 0.20/0.50 fresh2(product(add(a, b), b, fresh(true, true, multiply(add(a, b), b), c)), true, add(a, b), b, add(c, b), b)
% 0.20/0.50 = { by axiom 8 (multiplication_is_well_defined) }
% 0.20/0.50 fresh2(product(add(a, b), b, c), true, add(a, b), b, add(c, b), b)
% 0.20/0.50 = { by lemma 16 }
% 0.20/0.50 fresh2(product(add(a, b), b, b), true, add(a, b), b, add(c, b), b)
% 0.20/0.50 = { by axiom 13 (multiplication_is_well_defined) }
% 0.20/0.50 fresh(product(add(a, b), b, add(c, b)), true, add(c, b), b)
% 0.20/0.50 = { by axiom 11 (clause67) }
% 0.20/0.50 fresh(true, true, add(c, b), b)
% 0.20/0.50 = { by axiom 8 (multiplication_is_well_defined) }
% 0.20/0.50 b
% 0.20/0.50
% 0.20/0.50 Lemma 18: d = a.
% 0.20/0.50 Proof:
% 0.20/0.50 d
% 0.20/0.50 = { by axiom 8 (multiplication_is_well_defined) R->L }
% 0.20/0.50 fresh(true, true, a, d)
% 0.20/0.50 = { by lemma 14 R->L }
% 0.20/0.50 fresh(product(a, multiply(b, a), a), true, a, d)
% 0.20/0.50 = { by axiom 2 (clause37) }
% 0.20/0.50 fresh(product(a, d, a), true, a, d)
% 0.20/0.50 = { by axiom 13 (multiplication_is_well_defined) R->L }
% 0.20/0.50 fresh2(product(a, d, d), true, a, d, a, d)
% 0.20/0.50 = { by axiom 2 (clause37) R->L }
% 0.20/0.50 fresh2(product(a, multiply(b, a), d), true, a, d, a, d)
% 0.20/0.50 = { by axiom 2 (clause37) R->L }
% 0.20/0.50 fresh2(product(a, multiply(b, a), multiply(b, a)), true, a, d, a, d)
% 0.20/0.50 = { by axiom 9 (clause44) }
% 0.20/0.50 fresh2(true, true, a, d, a, d)
% 0.20/0.50 = { by axiom 12 (multiplication_is_well_defined) }
% 0.20/0.50 a
% 0.20/0.50
% 0.20/0.50 Goal 1 (prove_c_equals_d): c = d.
% 0.20/0.50 Proof:
% 0.20/0.50 c
% 0.20/0.50 = { by lemma 16 }
% 0.20/0.50 b
% 0.20/0.50 = { by axiom 12 (multiplication_is_well_defined) R->L }
% 0.20/0.50 fresh2(true, true, a, a, b, a)
% 0.20/0.50 = { by axiom 5 (clause71) R->L }
% 0.20/0.50 fresh2(product(a, a, a), true, a, a, b, a)
% 0.20/0.50 = { by axiom 13 (multiplication_is_well_defined) }
% 0.20/0.50 fresh(product(a, a, b), true, b, a)
% 0.20/0.50 = { by lemma 17 R->L }
% 0.20/0.50 fresh(product(a, a, additive_identity), true, b, a)
% 0.20/0.51 = { by axiom 3 (clause34) R->L }
% 0.20/0.51 fresh(product(a, a, add(a, a)), true, b, a)
% 0.20/0.51 = { by lemma 18 R->L }
% 0.20/0.51 fresh(product(a, a, add(a, d)), true, b, a)
% 0.20/0.51 = { by axiom 4 (clause33) R->L }
% 0.20/0.51 fresh(product(add(a, additive_identity), a, add(a, d)), true, b, a)
% 0.20/0.51 = { by lemma 17 }
% 0.20/0.51 fresh(product(add(a, b), a, add(a, d)), true, b, a)
% 0.20/0.51 = { by axiom 10 (clause70) }
% 0.20/0.51 fresh(true, true, b, a)
% 0.20/0.51 = { by axiom 8 (multiplication_is_well_defined) }
% 0.20/0.51 a
% 0.20/0.51 = { by lemma 18 R->L }
% 0.20/0.51 d
% 0.20/0.51 % SZS output end Proof
% 0.20/0.51
% 0.20/0.51 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------