TSTP Solution File: HEN009-6 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : HEN009-6 : 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 : n004.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 01:57:01 EDT 2023
% Result : Unsatisfiable 0.21s 0.44s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : HEN009-6 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n004.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 : Thu Aug 24 13:26:38 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.21/0.44 Command-line arguments: --no-flatten-goal
% 0.21/0.44
% 0.21/0.44 % SZS status Unsatisfiable
% 0.21/0.44
% 0.21/0.48 % SZS output start Proof
% 0.21/0.48 Take the following subset of the input axioms:
% 0.21/0.48 fof(id_divide_a_is_b, hypothesis, divide(identity, a)=b).
% 0.21/0.48 fof(id_divide_b_is_c, hypothesis, divide(identity, b)=c).
% 0.21/0.48 fof(id_divide_c_is_d, hypothesis, divide(identity, c)=d).
% 0.21/0.48 fof(identity_is_largest, axiom, ![X]: less_equal(X, identity)).
% 0.21/0.48 fof(less_equal_and_equal, axiom, ![Y, X2]: (~less_equal(X2, Y) | (~less_equal(Y, X2) | X2=Y))).
% 0.21/0.48 fof(part_of_theorem, hypothesis, divide(identity, a)!=divide(identity, divide(identity, divide(identity, a)))).
% 0.21/0.48 fof(property_of_divide1, axiom, ![Z, X2, Y2]: (~less_equal(divide(X2, Y2), Z) | less_equal(divide(X2, Z), Y2))).
% 0.21/0.48 fof(property_of_divide3, axiom, ![X2, Y2, Z2]: (~less_equal(X2, Y2) | less_equal(divide(X2, Z2), divide(Y2, Z2)))).
% 0.21/0.48 fof(prove_b_equals_d, negated_conjecture, b!=d).
% 0.21/0.48 fof(quotient_less_equal1, axiom, ![X2, Y2]: (~less_equal(X2, Y2) | divide(X2, Y2)=zero)).
% 0.21/0.48 fof(quotient_less_equal2, axiom, ![X2, Y2]: (divide(X2, Y2)!=zero | less_equal(X2, Y2))).
% 0.21/0.48 fof(quotient_smaller_than_numerator, axiom, ![X2, Y2]: less_equal(divide(X2, Y2), X2)).
% 0.21/0.48 fof(x_divide_x_is_zero, axiom, ![X2]: divide(X2, X2)=zero).
% 0.21/0.48 fof(zero_is_smallest, axiom, ![X2]: less_equal(zero, X2)).
% 0.21/0.48
% 0.21/0.48 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.48 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.48 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.48 fresh(y, y, x1...xn) = u
% 0.21/0.48 C => fresh(s, t, x1...xn) = v
% 0.21/0.48 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.48 variables of u and v.
% 0.21/0.48 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.48 input problem has no model of domain size 1).
% 0.21/0.48
% 0.21/0.48 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.48
% 0.21/0.48 Axiom 1 (identity_is_largest): less_equal(X, identity) = true.
% 0.21/0.48 Axiom 2 (zero_is_smallest): less_equal(zero, X) = true.
% 0.21/0.48 Axiom 3 (x_divide_x_is_zero): divide(X, X) = zero.
% 0.21/0.48 Axiom 4 (id_divide_c_is_d): divide(identity, c) = d.
% 0.21/0.48 Axiom 5 (id_divide_a_is_b): divide(identity, a) = b.
% 0.21/0.48 Axiom 6 (id_divide_b_is_c): divide(identity, b) = c.
% 0.21/0.48 Axiom 7 (less_equal_and_equal): fresh(X, X, Y, Z) = Z.
% 0.21/0.48 Axiom 8 (quotient_less_equal1): fresh4(X, X, Y, Z) = zero.
% 0.21/0.48 Axiom 9 (quotient_less_equal2): fresh3(X, X, Y, Z) = true.
% 0.21/0.48 Axiom 10 (less_equal_and_equal): fresh2(X, X, Y, Z) = Y.
% 0.21/0.48 Axiom 11 (quotient_smaller_than_numerator): less_equal(divide(X, Y), X) = true.
% 0.21/0.48 Axiom 12 (property_of_divide3): fresh6(X, X, Y, Z, W) = true.
% 0.21/0.48 Axiom 13 (property_of_divide1): fresh5(X, X, Y, Z, W) = true.
% 0.21/0.48 Axiom 14 (quotient_less_equal1): fresh4(less_equal(X, Y), true, X, Y) = divide(X, Y).
% 0.21/0.48 Axiom 15 (quotient_less_equal2): fresh3(divide(X, Y), zero, X, Y) = less_equal(X, Y).
% 0.21/0.48 Axiom 16 (less_equal_and_equal): fresh2(less_equal(X, Y), true, Y, X) = fresh(less_equal(Y, X), true, Y, X).
% 0.21/0.48 Axiom 17 (property_of_divide3): fresh6(less_equal(X, Y), true, X, Y, Z) = less_equal(divide(X, Z), divide(Y, Z)).
% 0.21/0.48 Axiom 18 (property_of_divide1): fresh5(less_equal(divide(X, Y), Z), true, X, Y, Z) = less_equal(divide(X, Z), Y).
% 0.21/0.48
% 0.21/0.48 Lemma 19: b = d.
% 0.21/0.48 Proof:
% 0.21/0.48 b
% 0.21/0.48 = { by axiom 7 (less_equal_and_equal) R->L }
% 0.21/0.48 fresh(true, true, d, b)
% 0.21/0.48 = { by axiom 13 (property_of_divide1) R->L }
% 0.21/0.48 fresh(fresh5(true, true, identity, b, c), true, d, b)
% 0.21/0.48 = { by axiom 12 (property_of_divide3) R->L }
% 0.21/0.48 fresh(fresh5(fresh6(true, true, identity, identity, b), true, identity, b, c), true, d, b)
% 0.21/0.48 = { by axiom 1 (identity_is_largest) R->L }
% 0.21/0.48 fresh(fresh5(fresh6(less_equal(identity, identity), true, identity, identity, b), true, identity, b, c), true, d, b)
% 0.21/0.48 = { by axiom 17 (property_of_divide3) }
% 0.21/0.48 fresh(fresh5(less_equal(divide(identity, b), divide(identity, b)), true, identity, b, c), true, d, b)
% 0.21/0.48 = { by axiom 6 (id_divide_b_is_c) }
% 0.21/0.48 fresh(fresh5(less_equal(divide(identity, b), c), true, identity, b, c), true, d, b)
% 0.21/0.48 = { by axiom 18 (property_of_divide1) }
% 0.21/0.48 fresh(less_equal(divide(identity, c), b), true, d, b)
% 0.21/0.49 = { by axiom 4 (id_divide_c_is_d) }
% 0.21/0.49 fresh(less_equal(d, b), true, d, b)
% 0.21/0.49 = { by axiom 16 (less_equal_and_equal) R->L }
% 0.21/0.49 fresh2(less_equal(b, d), true, d, b)
% 0.21/0.49 = { by axiom 5 (id_divide_a_is_b) R->L }
% 0.21/0.49 fresh2(less_equal(divide(identity, a), d), true, d, b)
% 0.21/0.49 = { by axiom 18 (property_of_divide1) R->L }
% 0.21/0.49 fresh2(fresh5(less_equal(divide(identity, d), a), true, identity, d, a), true, d, b)
% 0.21/0.49 = { by axiom 7 (less_equal_and_equal) R->L }
% 0.21/0.49 fresh2(fresh5(less_equal(fresh(true, true, divide(divide(identity, d), zero), divide(identity, d)), a), true, identity, d, a), true, d, b)
% 0.21/0.49 = { by axiom 11 (quotient_smaller_than_numerator) R->L }
% 0.21/0.49 fresh2(fresh5(less_equal(fresh(less_equal(divide(divide(identity, d), zero), divide(identity, d)), true, divide(divide(identity, d), zero), divide(identity, d)), a), true, identity, d, a), true, d, b)
% 0.21/0.49 = { by axiom 16 (less_equal_and_equal) R->L }
% 0.21/0.49 fresh2(fresh5(less_equal(fresh2(less_equal(divide(identity, d), divide(divide(identity, d), zero)), true, divide(divide(identity, d), zero), divide(identity, d)), a), true, identity, d, a), true, d, b)
% 0.21/0.49 = { by axiom 15 (quotient_less_equal2) R->L }
% 0.21/0.49 fresh2(fresh5(less_equal(fresh2(fresh3(divide(divide(identity, d), divide(divide(identity, d), zero)), zero, divide(identity, d), divide(divide(identity, d), zero)), true, divide(divide(identity, d), zero), divide(identity, d)), a), true, identity, d, a), true, d, b)
% 0.21/0.49 = { by axiom 10 (less_equal_and_equal) R->L }
% 0.21/0.49 fresh2(fresh5(less_equal(fresh2(fresh3(fresh2(true, true, divide(divide(identity, d), divide(divide(identity, d), zero)), zero), zero, divide(identity, d), divide(divide(identity, d), zero)), true, divide(divide(identity, d), zero), divide(identity, d)), a), true, identity, d, a), true, d, b)
% 0.21/0.49 = { by axiom 2 (zero_is_smallest) R->L }
% 0.21/0.49 fresh2(fresh5(less_equal(fresh2(fresh3(fresh2(less_equal(zero, divide(divide(identity, d), divide(divide(identity, d), zero))), true, divide(divide(identity, d), divide(divide(identity, d), zero)), zero), zero, divide(identity, d), divide(divide(identity, d), zero)), true, divide(divide(identity, d), zero), divide(identity, d)), a), true, identity, d, a), true, d, b)
% 0.21/0.49 = { by axiom 16 (less_equal_and_equal) }
% 0.21/0.49 fresh2(fresh5(less_equal(fresh2(fresh3(fresh(less_equal(divide(divide(identity, d), divide(divide(identity, d), zero)), zero), true, divide(divide(identity, d), divide(divide(identity, d), zero)), zero), zero, divide(identity, d), divide(divide(identity, d), zero)), true, divide(divide(identity, d), zero), divide(identity, d)), a), true, identity, d, a), true, d, b)
% 0.21/0.49 = { by axiom 18 (property_of_divide1) R->L }
% 0.21/0.49 fresh2(fresh5(less_equal(fresh2(fresh3(fresh(fresh5(less_equal(divide(divide(identity, d), zero), divide(divide(identity, d), zero)), true, divide(identity, d), zero, divide(divide(identity, d), zero)), true, divide(divide(identity, d), divide(divide(identity, d), zero)), zero), zero, divide(identity, d), divide(divide(identity, d), zero)), true, divide(divide(identity, d), zero), divide(identity, d)), a), true, identity, d, a), true, d, b)
% 0.21/0.49 = { by axiom 15 (quotient_less_equal2) R->L }
% 0.21/0.49 fresh2(fresh5(less_equal(fresh2(fresh3(fresh(fresh5(fresh3(divide(divide(divide(identity, d), zero), divide(divide(identity, d), zero)), zero, divide(divide(identity, d), zero), divide(divide(identity, d), zero)), true, divide(identity, d), zero, divide(divide(identity, d), zero)), true, divide(divide(identity, d), divide(divide(identity, d), zero)), zero), zero, divide(identity, d), divide(divide(identity, d), zero)), true, divide(divide(identity, d), zero), divide(identity, d)), a), true, identity, d, a), true, d, b)
% 0.21/0.49 = { by axiom 3 (x_divide_x_is_zero) }
% 0.21/0.49 fresh2(fresh5(less_equal(fresh2(fresh3(fresh(fresh5(fresh3(zero, zero, divide(divide(identity, d), zero), divide(divide(identity, d), zero)), true, divide(identity, d), zero, divide(divide(identity, d), zero)), true, divide(divide(identity, d), divide(divide(identity, d), zero)), zero), zero, divide(identity, d), divide(divide(identity, d), zero)), true, divide(divide(identity, d), zero), divide(identity, d)), a), true, identity, d, a), true, d, b)
% 0.21/0.49 = { by axiom 9 (quotient_less_equal2) }
% 0.21/0.49 fresh2(fresh5(less_equal(fresh2(fresh3(fresh(fresh5(true, true, divide(identity, d), zero, divide(divide(identity, d), zero)), true, divide(divide(identity, d), divide(divide(identity, d), zero)), zero), zero, divide(identity, d), divide(divide(identity, d), zero)), true, divide(divide(identity, d), zero), divide(identity, d)), a), true, identity, d, a), true, d, b)
% 0.21/0.49 = { by axiom 13 (property_of_divide1) }
% 0.21/0.49 fresh2(fresh5(less_equal(fresh2(fresh3(fresh(true, true, divide(divide(identity, d), divide(divide(identity, d), zero)), zero), zero, divide(identity, d), divide(divide(identity, d), zero)), true, divide(divide(identity, d), zero), divide(identity, d)), a), true, identity, d, a), true, d, b)
% 0.21/0.49 = { by axiom 7 (less_equal_and_equal) }
% 0.21/0.49 fresh2(fresh5(less_equal(fresh2(fresh3(zero, zero, divide(identity, d), divide(divide(identity, d), zero)), true, divide(divide(identity, d), zero), divide(identity, d)), a), true, identity, d, a), true, d, b)
% 0.21/0.49 = { by axiom 9 (quotient_less_equal2) }
% 0.21/0.49 fresh2(fresh5(less_equal(fresh2(true, true, divide(divide(identity, d), zero), divide(identity, d)), a), true, identity, d, a), true, d, b)
% 0.21/0.49 = { by axiom 10 (less_equal_and_equal) }
% 0.21/0.49 fresh2(fresh5(less_equal(divide(divide(identity, d), zero), a), true, identity, d, a), true, d, b)
% 0.21/0.49 = { by axiom 18 (property_of_divide1) R->L }
% 0.21/0.49 fresh2(fresh5(fresh5(less_equal(divide(divide(identity, d), a), zero), true, divide(identity, d), a, zero), true, identity, d, a), true, d, b)
% 0.21/0.49 = { by axiom 8 (quotient_less_equal1) R->L }
% 0.21/0.49 fresh2(fresh5(fresh5(less_equal(divide(divide(identity, d), a), fresh4(true, true, c, a)), true, divide(identity, d), a, zero), true, identity, d, a), true, d, b)
% 0.21/0.49 = { by axiom 13 (property_of_divide1) R->L }
% 0.21/0.49 fresh2(fresh5(fresh5(less_equal(divide(divide(identity, d), a), fresh4(fresh5(true, true, identity, a, b), true, c, a)), true, divide(identity, d), a, zero), true, identity, d, a), true, d, b)
% 0.21/0.49 = { by axiom 12 (property_of_divide3) R->L }
% 0.21/0.49 fresh2(fresh5(fresh5(less_equal(divide(divide(identity, d), a), fresh4(fresh5(fresh6(true, true, identity, identity, a), true, identity, a, b), true, c, a)), true, divide(identity, d), a, zero), true, identity, d, a), true, d, b)
% 0.21/0.49 = { by axiom 1 (identity_is_largest) R->L }
% 0.21/0.49 fresh2(fresh5(fresh5(less_equal(divide(divide(identity, d), a), fresh4(fresh5(fresh6(less_equal(identity, identity), true, identity, identity, a), true, identity, a, b), true, c, a)), true, divide(identity, d), a, zero), true, identity, d, a), true, d, b)
% 0.21/0.49 = { by axiom 17 (property_of_divide3) }
% 0.21/0.49 fresh2(fresh5(fresh5(less_equal(divide(divide(identity, d), a), fresh4(fresh5(less_equal(divide(identity, a), divide(identity, a)), true, identity, a, b), true, c, a)), true, divide(identity, d), a, zero), true, identity, d, a), true, d, b)
% 0.21/0.49 = { by axiom 5 (id_divide_a_is_b) }
% 0.21/0.49 fresh2(fresh5(fresh5(less_equal(divide(divide(identity, d), a), fresh4(fresh5(less_equal(divide(identity, a), b), true, identity, a, b), true, c, a)), true, divide(identity, d), a, zero), true, identity, d, a), true, d, b)
% 0.21/0.49 = { by axiom 18 (property_of_divide1) }
% 0.21/0.49 fresh2(fresh5(fresh5(less_equal(divide(divide(identity, d), a), fresh4(less_equal(divide(identity, b), a), true, c, a)), true, divide(identity, d), a, zero), true, identity, d, a), true, d, b)
% 0.21/0.49 = { by axiom 6 (id_divide_b_is_c) }
% 0.21/0.49 fresh2(fresh5(fresh5(less_equal(divide(divide(identity, d), a), fresh4(less_equal(c, a), true, c, a)), true, divide(identity, d), a, zero), true, identity, d, a), true, d, b)
% 0.21/0.49 = { by axiom 14 (quotient_less_equal1) }
% 0.21/0.49 fresh2(fresh5(fresh5(less_equal(divide(divide(identity, d), a), divide(c, a)), true, divide(identity, d), a, zero), true, identity, d, a), true, d, b)
% 0.21/0.50 = { by axiom 17 (property_of_divide3) R->L }
% 0.21/0.50 fresh2(fresh5(fresh5(fresh6(less_equal(divide(identity, d), c), true, divide(identity, d), c, a), true, divide(identity, d), a, zero), true, identity, d, a), true, d, b)
% 0.21/0.50 = { by axiom 18 (property_of_divide1) R->L }
% 0.21/0.50 fresh2(fresh5(fresh5(fresh6(fresh5(less_equal(divide(identity, c), d), true, identity, c, d), true, divide(identity, d), c, a), true, divide(identity, d), a, zero), true, identity, d, a), true, d, b)
% 0.21/0.50 = { by axiom 4 (id_divide_c_is_d) R->L }
% 0.21/0.50 fresh2(fresh5(fresh5(fresh6(fresh5(less_equal(divide(identity, c), divide(identity, c)), true, identity, c, d), true, divide(identity, d), c, a), true, divide(identity, d), a, zero), true, identity, d, a), true, d, b)
% 0.21/0.50 = { by axiom 17 (property_of_divide3) R->L }
% 0.21/0.50 fresh2(fresh5(fresh5(fresh6(fresh5(fresh6(less_equal(identity, identity), true, identity, identity, c), true, identity, c, d), true, divide(identity, d), c, a), true, divide(identity, d), a, zero), true, identity, d, a), true, d, b)
% 0.21/0.50 = { by axiom 1 (identity_is_largest) }
% 0.21/0.50 fresh2(fresh5(fresh5(fresh6(fresh5(fresh6(true, true, identity, identity, c), true, identity, c, d), true, divide(identity, d), c, a), true, divide(identity, d), a, zero), true, identity, d, a), true, d, b)
% 0.21/0.50 = { by axiom 12 (property_of_divide3) }
% 0.21/0.50 fresh2(fresh5(fresh5(fresh6(fresh5(true, true, identity, c, d), true, divide(identity, d), c, a), true, divide(identity, d), a, zero), true, identity, d, a), true, d, b)
% 0.21/0.50 = { by axiom 13 (property_of_divide1) }
% 0.21/0.50 fresh2(fresh5(fresh5(fresh6(true, true, divide(identity, d), c, a), true, divide(identity, d), a, zero), true, identity, d, a), true, d, b)
% 0.21/0.50 = { by axiom 12 (property_of_divide3) }
% 0.21/0.50 fresh2(fresh5(fresh5(true, true, divide(identity, d), a, zero), true, identity, d, a), true, d, b)
% 0.21/0.50 = { by axiom 13 (property_of_divide1) }
% 0.21/0.50 fresh2(fresh5(true, true, identity, d, a), true, d, b)
% 0.21/0.50 = { by axiom 13 (property_of_divide1) }
% 0.21/0.50 fresh2(true, true, d, b)
% 0.21/0.50 = { by axiom 10 (less_equal_and_equal) }
% 0.21/0.50 d
% 0.21/0.50
% 0.21/0.50 Goal 1 (prove_b_equals_d): b = d.
% 0.21/0.50 Proof:
% 0.21/0.50 b
% 0.21/0.50 = { by lemma 19 }
% 0.21/0.50 d
% 0.21/0.50
% 0.21/0.50 Goal 2 (part_of_theorem): divide(identity, a) = divide(identity, divide(identity, divide(identity, a))).
% 0.21/0.50 Proof:
% 0.21/0.50 divide(identity, a)
% 0.21/0.50 = { by axiom 5 (id_divide_a_is_b) }
% 0.21/0.50 b
% 0.21/0.50 = { by lemma 19 }
% 0.21/0.50 d
% 0.21/0.50 = { by axiom 4 (id_divide_c_is_d) R->L }
% 0.21/0.50 divide(identity, c)
% 0.21/0.50 = { by axiom 6 (id_divide_b_is_c) R->L }
% 0.21/0.50 divide(identity, divide(identity, b))
% 0.21/0.50 = { by axiom 5 (id_divide_a_is_b) R->L }
% 0.21/0.50 divide(identity, divide(identity, divide(identity, a)))
% 0.21/0.50 % SZS output end Proof
% 0.21/0.50
% 0.21/0.50 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------