TSTP Solution File: HEN006-6 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : HEN006-6 : 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 : n021.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 01:56:56 EDT 2023
% Result : Unsatisfiable 0.20s 0.42s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : HEN006-6 : TPTP v8.1.2. Released v1.0.0.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.35 % Computer : n021.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Thu Aug 24 13:38:10 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.20/0.42 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.20/0.42
% 0.20/0.42 % SZS status Unsatisfiable
% 0.20/0.42
% 0.20/0.44 % SZS output start Proof
% 0.20/0.44 Take the following subset of the input axioms:
% 0.20/0.44 fof(a_divide_b_LE_d, hypothesis, less_equal(divide(a, b), d)).
% 0.20/0.44 fof(less_equal_and_equal, axiom, ![X, Y]: (~less_equal(X, Y) | (~less_equal(Y, X) | X=Y))).
% 0.20/0.44 fof(prove_a_divide_d_LE_b, negated_conjecture, ~less_equal(divide(a, d), b)).
% 0.20/0.44 fof(quotient_less_equal1, axiom, ![X2, Y2]: (~less_equal(X2, Y2) | divide(X2, Y2)=zero)).
% 0.20/0.44 fof(quotient_less_equal2, axiom, ![X2, Y2]: (divide(X2, Y2)!=zero | less_equal(X2, Y2))).
% 0.20/0.44 fof(quotient_property, axiom, ![Z, X2, Y2]: less_equal(divide(divide(X2, Z), divide(Y2, Z)), divide(divide(X2, Y2), Z))).
% 0.20/0.44 fof(quotient_smaller_than_numerator, axiom, ![X2, Y2]: less_equal(divide(X2, Y2), X2)).
% 0.20/0.44 fof(x_divide_x_is_zero, axiom, ![X2]: divide(X2, X2)=zero).
% 0.20/0.44 fof(zero_divide_anything_is_zero, axiom, ![X2]: divide(zero, X2)=zero).
% 0.20/0.44 fof(zero_is_smallest, axiom, ![X2]: less_equal(zero, X2)).
% 0.20/0.44
% 0.20/0.44 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.44 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.44 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.44 fresh(y, y, x1...xn) = u
% 0.20/0.44 C => fresh(s, t, x1...xn) = v
% 0.20/0.44 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.44 variables of u and v.
% 0.20/0.44 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.44 input problem has no model of domain size 1).
% 0.20/0.44
% 0.20/0.44 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.44
% 0.20/0.44 Axiom 1 (zero_is_smallest): less_equal(zero, X) = true.
% 0.20/0.44 Axiom 2 (x_divide_x_is_zero): divide(X, X) = zero.
% 0.20/0.44 Axiom 3 (zero_divide_anything_is_zero): divide(zero, X) = zero.
% 0.20/0.44 Axiom 4 (less_equal_and_equal): fresh(X, X, Y, Z) = Z.
% 0.20/0.44 Axiom 5 (quotient_less_equal2): fresh4(X, X, Y, Z) = true.
% 0.20/0.44 Axiom 6 (quotient_less_equal1): fresh3(X, X, Y, Z) = zero.
% 0.20/0.44 Axiom 7 (less_equal_and_equal): fresh2(X, X, Y, Z) = Y.
% 0.20/0.44 Axiom 8 (quotient_smaller_than_numerator): less_equal(divide(X, Y), X) = true.
% 0.20/0.44 Axiom 9 (a_divide_b_LE_d): less_equal(divide(a, b), d) = true.
% 0.20/0.44 Axiom 10 (quotient_less_equal2): fresh4(divide(X, Y), zero, X, Y) = less_equal(X, Y).
% 0.20/0.44 Axiom 11 (quotient_less_equal1): fresh3(less_equal(X, Y), true, X, Y) = divide(X, Y).
% 0.20/0.44 Axiom 12 (less_equal_and_equal): fresh2(less_equal(X, Y), true, Y, X) = fresh(less_equal(Y, X), true, Y, X).
% 0.20/0.44 Axiom 13 (quotient_property): less_equal(divide(divide(X, Y), divide(Z, Y)), divide(divide(X, Z), Y)) = true.
% 0.20/0.44
% 0.20/0.44 Lemma 14: divide(X, zero) = X.
% 0.20/0.44 Proof:
% 0.20/0.44 divide(X, zero)
% 0.20/0.44 = { by axiom 7 (less_equal_and_equal) R->L }
% 0.20/0.44 fresh2(true, true, divide(X, zero), X)
% 0.20/0.44 = { by axiom 5 (quotient_less_equal2) R->L }
% 0.20/0.44 fresh2(fresh4(zero, zero, X, divide(X, zero)), true, divide(X, zero), X)
% 0.20/0.44 = { by axiom 7 (less_equal_and_equal) R->L }
% 0.20/0.44 fresh2(fresh4(fresh2(true, true, zero, divide(X, divide(X, zero))), zero, X, divide(X, zero)), true, divide(X, zero), X)
% 0.20/0.44 = { by axiom 5 (quotient_less_equal2) R->L }
% 0.20/0.44 fresh2(fresh4(fresh2(fresh4(zero, zero, divide(X, divide(X, zero)), zero), true, zero, divide(X, divide(X, zero))), zero, X, divide(X, zero)), true, divide(X, zero), X)
% 0.20/0.44 = { by axiom 7 (less_equal_and_equal) R->L }
% 0.20/0.44 fresh2(fresh4(fresh2(fresh4(fresh2(true, true, zero, divide(divide(X, divide(X, zero)), zero)), zero, divide(X, divide(X, zero)), zero), true, zero, divide(X, divide(X, zero))), zero, X, divide(X, zero)), true, divide(X, zero), X)
% 0.20/0.44 = { by axiom 13 (quotient_property) R->L }
% 0.20/0.44 fresh2(fresh4(fresh2(fresh4(fresh2(less_equal(divide(divide(X, divide(X, zero)), divide(zero, divide(X, zero))), divide(divide(X, zero), divide(X, zero))), true, zero, divide(divide(X, divide(X, zero)), zero)), zero, divide(X, divide(X, zero)), zero), true, zero, divide(X, divide(X, zero))), zero, X, divide(X, zero)), true, divide(X, zero), X)
% 0.20/0.44 = { by axiom 3 (zero_divide_anything_is_zero) }
% 0.20/0.44 fresh2(fresh4(fresh2(fresh4(fresh2(less_equal(divide(divide(X, divide(X, zero)), zero), divide(divide(X, zero), divide(X, zero))), true, zero, divide(divide(X, divide(X, zero)), zero)), zero, divide(X, divide(X, zero)), zero), true, zero, divide(X, divide(X, zero))), zero, X, divide(X, zero)), true, divide(X, zero), X)
% 0.20/0.44 = { by axiom 2 (x_divide_x_is_zero) }
% 0.20/0.44 fresh2(fresh4(fresh2(fresh4(fresh2(less_equal(divide(divide(X, divide(X, zero)), zero), zero), true, zero, divide(divide(X, divide(X, zero)), zero)), zero, divide(X, divide(X, zero)), zero), true, zero, divide(X, divide(X, zero))), zero, X, divide(X, zero)), true, divide(X, zero), X)
% 0.20/0.44 = { by axiom 12 (less_equal_and_equal) }
% 0.20/0.44 fresh2(fresh4(fresh2(fresh4(fresh(less_equal(zero, divide(divide(X, divide(X, zero)), zero)), true, zero, divide(divide(X, divide(X, zero)), zero)), zero, divide(X, divide(X, zero)), zero), true, zero, divide(X, divide(X, zero))), zero, X, divide(X, zero)), true, divide(X, zero), X)
% 0.20/0.44 = { by axiom 1 (zero_is_smallest) }
% 0.20/0.44 fresh2(fresh4(fresh2(fresh4(fresh(true, true, zero, divide(divide(X, divide(X, zero)), zero)), zero, divide(X, divide(X, zero)), zero), true, zero, divide(X, divide(X, zero))), zero, X, divide(X, zero)), true, divide(X, zero), X)
% 0.20/0.44 = { by axiom 4 (less_equal_and_equal) }
% 0.20/0.45 fresh2(fresh4(fresh2(fresh4(divide(divide(X, divide(X, zero)), zero), zero, divide(X, divide(X, zero)), zero), true, zero, divide(X, divide(X, zero))), zero, X, divide(X, zero)), true, divide(X, zero), X)
% 0.20/0.45 = { by axiom 10 (quotient_less_equal2) }
% 0.20/0.45 fresh2(fresh4(fresh2(less_equal(divide(X, divide(X, zero)), zero), true, zero, divide(X, divide(X, zero))), zero, X, divide(X, zero)), true, divide(X, zero), X)
% 0.20/0.45 = { by axiom 12 (less_equal_and_equal) }
% 0.20/0.45 fresh2(fresh4(fresh(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), zero, X, divide(X, zero)), true, divide(X, zero), X)
% 0.20/0.45 = { by axiom 1 (zero_is_smallest) }
% 0.20/0.45 fresh2(fresh4(fresh(true, true, zero, divide(X, divide(X, zero))), zero, X, divide(X, zero)), true, divide(X, zero), X)
% 0.20/0.45 = { by axiom 4 (less_equal_and_equal) }
% 0.20/0.45 fresh2(fresh4(divide(X, divide(X, zero)), zero, X, divide(X, zero)), true, divide(X, zero), X)
% 0.20/0.45 = { by axiom 10 (quotient_less_equal2) }
% 0.20/0.45 fresh2(less_equal(X, divide(X, zero)), true, divide(X, zero), X)
% 0.20/0.45 = { by axiom 12 (less_equal_and_equal) }
% 0.20/0.45 fresh(less_equal(divide(X, zero), X), true, divide(X, zero), X)
% 0.20/0.45 = { by axiom 8 (quotient_smaller_than_numerator) }
% 0.20/0.45 fresh(true, true, divide(X, zero), X)
% 0.20/0.45 = { by axiom 4 (less_equal_and_equal) }
% 0.20/0.45 X
% 0.20/0.45
% 0.20/0.45 Goal 1 (prove_a_divide_d_LE_b): less_equal(divide(a, d), b) = true.
% 0.20/0.45 Proof:
% 0.20/0.45 less_equal(divide(a, d), b)
% 0.20/0.45 = { by axiom 10 (quotient_less_equal2) R->L }
% 0.20/0.45 fresh4(divide(divide(a, d), b), zero, divide(a, d), b)
% 0.20/0.45 = { by lemma 14 R->L }
% 0.20/0.45 fresh4(divide(divide(divide(a, d), b), zero), zero, divide(a, d), b)
% 0.20/0.45 = { by axiom 11 (quotient_less_equal1) R->L }
% 0.20/0.45 fresh4(fresh3(less_equal(divide(divide(a, d), b), zero), true, divide(divide(a, d), b), zero), zero, divide(a, d), b)
% 0.20/0.45 = { by axiom 3 (zero_divide_anything_is_zero) R->L }
% 0.20/0.45 fresh4(fresh3(less_equal(divide(divide(a, d), b), divide(zero, b)), true, divide(divide(a, d), b), zero), zero, divide(a, d), b)
% 0.20/0.45 = { by axiom 7 (less_equal_and_equal) R->L }
% 0.20/0.45 fresh4(fresh3(less_equal(divide(divide(a, d), b), divide(fresh2(true, true, zero, divide(divide(a, d), divide(b, d))), b)), true, divide(divide(a, d), b), zero), zero, divide(a, d), b)
% 0.20/0.45 = { by axiom 13 (quotient_property) R->L }
% 0.20/0.45 fresh4(fresh3(less_equal(divide(divide(a, d), b), divide(fresh2(less_equal(divide(divide(a, d), divide(b, d)), divide(divide(a, b), d)), true, zero, divide(divide(a, d), divide(b, d))), b)), true, divide(divide(a, d), b), zero), zero, divide(a, d), b)
% 0.20/0.45 = { by axiom 11 (quotient_less_equal1) R->L }
% 0.20/0.45 fresh4(fresh3(less_equal(divide(divide(a, d), b), divide(fresh2(less_equal(divide(divide(a, d), divide(b, d)), fresh3(less_equal(divide(a, b), d), true, divide(a, b), d)), true, zero, divide(divide(a, d), divide(b, d))), b)), true, divide(divide(a, d), b), zero), zero, divide(a, d), b)
% 0.20/0.45 = { by axiom 9 (a_divide_b_LE_d) }
% 0.20/0.45 fresh4(fresh3(less_equal(divide(divide(a, d), b), divide(fresh2(less_equal(divide(divide(a, d), divide(b, d)), fresh3(true, true, divide(a, b), d)), true, zero, divide(divide(a, d), divide(b, d))), b)), true, divide(divide(a, d), b), zero), zero, divide(a, d), b)
% 0.20/0.45 = { by axiom 6 (quotient_less_equal1) }
% 0.20/0.45 fresh4(fresh3(less_equal(divide(divide(a, d), b), divide(fresh2(less_equal(divide(divide(a, d), divide(b, d)), zero), true, zero, divide(divide(a, d), divide(b, d))), b)), true, divide(divide(a, d), b), zero), zero, divide(a, d), b)
% 0.20/0.45 = { by axiom 12 (less_equal_and_equal) }
% 0.20/0.45 fresh4(fresh3(less_equal(divide(divide(a, d), b), divide(fresh(less_equal(zero, divide(divide(a, d), divide(b, d))), true, zero, divide(divide(a, d), divide(b, d))), b)), true, divide(divide(a, d), b), zero), zero, divide(a, d), b)
% 0.20/0.45 = { by axiom 1 (zero_is_smallest) }
% 0.20/0.45 fresh4(fresh3(less_equal(divide(divide(a, d), b), divide(fresh(true, true, zero, divide(divide(a, d), divide(b, d))), b)), true, divide(divide(a, d), b), zero), zero, divide(a, d), b)
% 0.20/0.45 = { by axiom 4 (less_equal_and_equal) }
% 0.20/0.45 fresh4(fresh3(less_equal(divide(divide(a, d), b), divide(divide(divide(a, d), divide(b, d)), b)), true, divide(divide(a, d), b), zero), zero, divide(a, d), b)
% 0.20/0.45 = { by lemma 14 R->L }
% 0.20/0.45 fresh4(fresh3(less_equal(divide(divide(divide(a, d), b), zero), divide(divide(divide(a, d), divide(b, d)), b)), true, divide(divide(a, d), b), zero), zero, divide(a, d), b)
% 0.20/0.45 = { by axiom 6 (quotient_less_equal1) R->L }
% 0.20/0.45 fresh4(fresh3(less_equal(divide(divide(divide(a, d), b), fresh3(true, true, divide(b, d), b)), divide(divide(divide(a, d), divide(b, d)), b)), true, divide(divide(a, d), b), zero), zero, divide(a, d), b)
% 0.20/0.45 = { by axiom 8 (quotient_smaller_than_numerator) R->L }
% 0.20/0.45 fresh4(fresh3(less_equal(divide(divide(divide(a, d), b), fresh3(less_equal(divide(b, d), b), true, divide(b, d), b)), divide(divide(divide(a, d), divide(b, d)), b)), true, divide(divide(a, d), b), zero), zero, divide(a, d), b)
% 0.20/0.45 = { by axiom 11 (quotient_less_equal1) }
% 0.20/0.45 fresh4(fresh3(less_equal(divide(divide(divide(a, d), b), divide(divide(b, d), b)), divide(divide(divide(a, d), divide(b, d)), b)), true, divide(divide(a, d), b), zero), zero, divide(a, d), b)
% 0.20/0.45 = { by axiom 13 (quotient_property) }
% 0.20/0.45 fresh4(fresh3(true, true, divide(divide(a, d), b), zero), zero, divide(a, d), b)
% 0.20/0.45 = { by axiom 6 (quotient_less_equal1) }
% 0.20/0.45 fresh4(zero, zero, divide(a, d), b)
% 0.20/0.45 = { by axiom 5 (quotient_less_equal2) }
% 0.20/0.45 true
% 0.20/0.45 % SZS output end Proof
% 0.20/0.45
% 0.20/0.45 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------