TSTP Solution File: BOO013-3 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : BOO013-3 : 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 : n005.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 18:11:23 EDT 2023
% Result : Unsatisfiable 0.22s 0.58s
% Output : Proof 0.22s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : BOO013-3 : TPTP v8.1.2. Released v1.0.0.
% 0.07/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.15/0.35 % Computer : n005.cluster.edu
% 0.15/0.35 % Model : x86_64 x86_64
% 0.15/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.35 % Memory : 8042.1875MB
% 0.15/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.35 % CPULimit : 300
% 0.15/0.35 % WCLimit : 300
% 0.15/0.35 % DateTime : Sun Aug 27 08:37:38 EDT 2023
% 0.15/0.35 % CPUTime :
% 0.22/0.58 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.22/0.58
% 0.22/0.58 % SZS status Unsatisfiable
% 0.22/0.58
% 0.22/0.60 % SZS output start Proof
% 0.22/0.60 Take the following subset of the input axioms:
% 0.22/0.60 fof(addition_is_well_defined, axiom, ![X, Y, U, V]: (~sum(X, Y, U) | (~sum(X, Y, V) | U=V))).
% 0.22/0.60 fof(additive_identity2, axiom, ![X2]: sum(X2, additive_identity, X2)).
% 0.22/0.60 fof(closure_of_addition, axiom, ![X2, Y2]: sum(X2, Y2, add(X2, Y2))).
% 0.22/0.60 fof(commutativity_of_addition, axiom, ![Z, X2, Y2]: (~sum(X2, Y2, Z) | sum(Y2, X2, Z))).
% 0.22/0.60 fof(commutativity_of_multiplication, axiom, ![X2, Y2, Z2]: (~product(X2, Y2, Z2) | product(Y2, X2, Z2))).
% 0.22/0.60 fof(distributivity5, axiom, ![V1, V2, V3, V4, X2, Y2, Z2]: (~sum(X2, Y2, V1) | (~sum(X2, Z2, V2) | (~product(Y2, Z2, V3) | (~sum(X2, V3, V4) | product(V1, V2, V4)))))).
% 0.22/0.60 fof(multiplication_is_well_defined, axiom, ![X2, Y2, U2, V5]: (~product(X2, Y2, U2) | (~product(X2, Y2, V5) | U2=V5))).
% 0.22/0.60 fof(multiplicative_identity2, axiom, ![X2]: product(X2, multiplicative_identity, X2)).
% 0.22/0.60 fof(product_to_additive_identity1, negated_conjecture, product(x, y, additive_identity)).
% 0.22/0.60 fof(product_to_additive_identity2, negated_conjecture, product(x, z, additive_identity)).
% 0.22/0.60 fof(prove_both_inverse_are_equal, negated_conjecture, y!=z).
% 0.22/0.60 fof(sum_to_multiplicative_identity1, negated_conjecture, sum(x, y, multiplicative_identity)).
% 0.22/0.60 fof(sum_to_multiplicative_identity2, negated_conjecture, sum(x, z, multiplicative_identity)).
% 0.22/0.60
% 0.22/0.60 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.22/0.60 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.22/0.60 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.22/0.60 fresh(y, y, x1...xn) = u
% 0.22/0.60 C => fresh(s, t, x1...xn) = v
% 0.22/0.60 where fresh is a fresh function symbol and x1..xn are the free
% 0.22/0.60 variables of u and v.
% 0.22/0.60 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.22/0.60 input problem has no model of domain size 1).
% 0.22/0.60
% 0.22/0.60 The encoding turns the above axioms into the following unit equations and goals:
% 0.22/0.60
% 0.22/0.60 Axiom 1 (additive_identity2): sum(X, additive_identity, X) = true.
% 0.22/0.60 Axiom 2 (sum_to_multiplicative_identity1): sum(x, y, multiplicative_identity) = true.
% 0.22/0.60 Axiom 3 (sum_to_multiplicative_identity2): sum(x, z, multiplicative_identity) = true.
% 0.22/0.60 Axiom 4 (multiplicative_identity2): product(X, multiplicative_identity, X) = true.
% 0.22/0.60 Axiom 5 (product_to_additive_identity1): product(x, y, additive_identity) = true.
% 0.22/0.60 Axiom 6 (product_to_additive_identity2): product(x, z, additive_identity) = true.
% 0.22/0.60 Axiom 7 (multiplication_is_well_defined): fresh(X, X, Y, Z) = Z.
% 0.22/0.60 Axiom 8 (addition_is_well_defined): fresh3(X, X, Y, Z) = Z.
% 0.22/0.60 Axiom 9 (distributivity5): fresh22(X, X, Y, Z, W) = true.
% 0.22/0.60 Axiom 10 (commutativity_of_multiplication): fresh6(X, X, Y, Z, W) = true.
% 0.22/0.60 Axiom 11 (commutativity_of_addition): fresh5(X, X, Y, Z, W) = true.
% 0.22/0.60 Axiom 12 (closure_of_addition): sum(X, Y, add(X, Y)) = true.
% 0.22/0.60 Axiom 13 (addition_is_well_defined): fresh4(X, X, Y, Z, W, V) = W.
% 0.22/0.60 Axiom 14 (multiplication_is_well_defined): fresh2(X, X, Y, Z, W, V) = W.
% 0.22/0.60 Axiom 15 (distributivity5): fresh20(X, X, Y, Z, W, V, U) = product(W, V, U).
% 0.22/0.60 Axiom 16 (distributivity5): fresh21(X, X, Y, Z, W, V, U, T) = fresh22(sum(Y, Z, W), true, W, U, T).
% 0.22/0.60 Axiom 17 (commutativity_of_multiplication): fresh6(product(X, Y, Z), true, X, Y, Z) = product(Y, X, Z).
% 0.22/0.60 Axiom 18 (commutativity_of_addition): fresh5(sum(X, Y, Z), true, X, Y, Z) = sum(Y, X, Z).
% 0.22/0.60 Axiom 19 (addition_is_well_defined): fresh4(sum(X, Y, Z), true, X, Y, W, Z) = fresh3(sum(X, Y, W), true, W, Z).
% 0.22/0.60 Axiom 20 (multiplication_is_well_defined): fresh2(product(X, Y, Z), true, X, Y, W, Z) = fresh(product(X, Y, W), true, W, Z).
% 0.22/0.60 Axiom 21 (distributivity5): fresh19(X, X, Y, Z, W, V, U, T, S) = fresh20(sum(Y, V, U), true, Y, Z, W, U, S).
% 0.22/0.60 Axiom 22 (distributivity5): fresh19(product(X, Y, Z), true, W, X, V, Y, U, Z, T) = fresh21(sum(W, Z, T), true, W, X, V, Y, U, T).
% 0.22/0.60
% 0.22/0.60 Lemma 23: fresh(product(X, multiplicative_identity, Y), true, Y, X) = Y.
% 0.22/0.60 Proof:
% 0.22/0.60 fresh(product(X, multiplicative_identity, Y), true, Y, X)
% 0.22/0.60 = { by axiom 20 (multiplication_is_well_defined) R->L }
% 0.22/0.60 fresh2(product(X, multiplicative_identity, X), true, X, multiplicative_identity, Y, X)
% 0.22/0.60 = { by axiom 4 (multiplicative_identity2) }
% 0.22/0.60 fresh2(true, true, X, multiplicative_identity, Y, X)
% 0.22/0.60 = { by axiom 14 (multiplication_is_well_defined) }
% 0.22/0.60 Y
% 0.22/0.60
% 0.22/0.60 Lemma 24: fresh21(X, X, Y, Z, add(Y, Z), W, V, U) = true.
% 0.22/0.60 Proof:
% 0.22/0.60 fresh21(X, X, Y, Z, add(Y, Z), W, V, U)
% 0.22/0.60 = { by axiom 16 (distributivity5) }
% 0.22/0.60 fresh22(sum(Y, Z, add(Y, Z)), true, add(Y, Z), V, U)
% 0.22/0.60 = { by axiom 12 (closure_of_addition) }
% 0.22/0.60 fresh22(true, true, add(Y, Z), V, U)
% 0.22/0.60 = { by axiom 9 (distributivity5) }
% 0.22/0.60 true
% 0.22/0.60
% 0.22/0.60 Lemma 25: fresh19(product(X, Y, additive_identity), true, Z, X, W, Y, V, additive_identity, Z) = fresh21(U, U, Z, X, W, T, V, Z).
% 0.22/0.60 Proof:
% 0.22/0.60 fresh19(product(X, Y, additive_identity), true, Z, X, W, Y, V, additive_identity, Z)
% 0.22/0.60 = { by axiom 22 (distributivity5) }
% 0.22/0.60 fresh21(sum(Z, additive_identity, Z), true, Z, X, W, Y, V, Z)
% 0.22/0.60 = { by axiom 1 (additive_identity2) }
% 0.22/0.60 fresh21(true, true, Z, X, W, Y, V, Z)
% 0.22/0.60 = { by axiom 16 (distributivity5) }
% 0.22/0.60 fresh22(sum(Z, X, W), true, W, V, Z)
% 0.22/0.60 = { by axiom 16 (distributivity5) R->L }
% 0.22/0.61 fresh21(U, U, Z, X, W, T, V, Z)
% 0.22/0.61
% 0.22/0.61 Goal 1 (prove_both_inverse_are_equal): y = z.
% 0.22/0.61 Proof:
% 0.22/0.61 y
% 0.22/0.61 = { by lemma 23 R->L }
% 0.22/0.61 fresh(product(z, multiplicative_identity, y), true, y, z)
% 0.22/0.61 = { by lemma 23 R->L }
% 0.22/0.61 fresh(product(fresh(product(add(y, z), multiplicative_identity, z), true, z, add(y, z)), multiplicative_identity, y), true, y, z)
% 0.22/0.61 = { by axiom 8 (addition_is_well_defined) R->L }
% 0.22/0.61 fresh(product(fresh(product(fresh3(true, true, add(z, y), add(y, z)), multiplicative_identity, z), true, z, add(y, z)), multiplicative_identity, y), true, y, z)
% 0.22/0.61 = { by axiom 11 (commutativity_of_addition) R->L }
% 0.22/0.61 fresh(product(fresh(product(fresh3(fresh5(true, true, z, y, add(z, y)), true, add(z, y), add(y, z)), multiplicative_identity, z), true, z, add(y, z)), multiplicative_identity, y), true, y, z)
% 0.22/0.61 = { by axiom 12 (closure_of_addition) R->L }
% 0.22/0.61 fresh(product(fresh(product(fresh3(fresh5(sum(z, y, add(z, y)), true, z, y, add(z, y)), true, add(z, y), add(y, z)), multiplicative_identity, z), true, z, add(y, z)), multiplicative_identity, y), true, y, z)
% 0.22/0.61 = { by axiom 18 (commutativity_of_addition) }
% 0.22/0.61 fresh(product(fresh(product(fresh3(sum(y, z, add(z, y)), true, add(z, y), add(y, z)), multiplicative_identity, z), true, z, add(y, z)), multiplicative_identity, y), true, y, z)
% 0.22/0.61 = { by axiom 19 (addition_is_well_defined) R->L }
% 0.22/0.61 fresh(product(fresh(product(fresh4(sum(y, z, add(y, z)), true, y, z, add(z, y), add(y, z)), multiplicative_identity, z), true, z, add(y, z)), multiplicative_identity, y), true, y, z)
% 0.22/0.61 = { by axiom 12 (closure_of_addition) }
% 0.22/0.61 fresh(product(fresh(product(fresh4(true, true, y, z, add(z, y), add(y, z)), multiplicative_identity, z), true, z, add(y, z)), multiplicative_identity, y), true, y, z)
% 0.22/0.61 = { by axiom 13 (addition_is_well_defined) }
% 0.22/0.61 fresh(product(fresh(product(add(z, y), multiplicative_identity, z), true, z, add(y, z)), multiplicative_identity, y), true, y, z)
% 0.22/0.61 = { by axiom 15 (distributivity5) R->L }
% 0.22/0.61 fresh(product(fresh(fresh20(true, true, z, y, add(z, y), multiplicative_identity, z), true, z, add(y, z)), multiplicative_identity, y), true, y, z)
% 0.22/0.61 = { by axiom 11 (commutativity_of_addition) R->L }
% 0.22/0.61 fresh(product(fresh(fresh20(fresh5(true, true, x, z, multiplicative_identity), true, z, y, add(z, y), multiplicative_identity, z), true, z, add(y, z)), multiplicative_identity, y), true, y, z)
% 0.22/0.61 = { by axiom 3 (sum_to_multiplicative_identity2) R->L }
% 0.22/0.61 fresh(product(fresh(fresh20(fresh5(sum(x, z, multiplicative_identity), true, x, z, multiplicative_identity), true, z, y, add(z, y), multiplicative_identity, z), true, z, add(y, z)), multiplicative_identity, y), true, y, z)
% 0.22/0.61 = { by axiom 18 (commutativity_of_addition) }
% 0.22/0.61 fresh(product(fresh(fresh20(sum(z, x, multiplicative_identity), true, z, y, add(z, y), multiplicative_identity, z), true, z, add(y, z)), multiplicative_identity, y), true, y, z)
% 0.22/0.61 = { by axiom 21 (distributivity5) R->L }
% 0.22/0.61 fresh(product(fresh(fresh19(true, true, z, y, add(z, y), x, multiplicative_identity, additive_identity, z), true, z, add(y, z)), multiplicative_identity, y), true, y, z)
% 0.22/0.61 = { by axiom 10 (commutativity_of_multiplication) R->L }
% 0.22/0.61 fresh(product(fresh(fresh19(fresh6(true, true, x, y, additive_identity), true, z, y, add(z, y), x, multiplicative_identity, additive_identity, z), true, z, add(y, z)), multiplicative_identity, y), true, y, z)
% 0.22/0.61 = { by axiom 5 (product_to_additive_identity1) R->L }
% 0.22/0.61 fresh(product(fresh(fresh19(fresh6(product(x, y, additive_identity), true, x, y, additive_identity), true, z, y, add(z, y), x, multiplicative_identity, additive_identity, z), true, z, add(y, z)), multiplicative_identity, y), true, y, z)
% 0.22/0.61 = { by axiom 17 (commutativity_of_multiplication) }
% 0.22/0.61 fresh(product(fresh(fresh19(product(y, x, additive_identity), true, z, y, add(z, y), x, multiplicative_identity, additive_identity, z), true, z, add(y, z)), multiplicative_identity, y), true, y, z)
% 0.22/0.61 = { by lemma 25 }
% 0.22/0.61 fresh(product(fresh(fresh21(X, X, z, y, add(z, y), Y, multiplicative_identity, z), true, z, add(y, z)), multiplicative_identity, y), true, y, z)
% 0.22/0.61 = { by lemma 24 }
% 0.22/0.61 fresh(product(fresh(true, true, z, add(y, z)), multiplicative_identity, y), true, y, z)
% 0.22/0.61 = { by axiom 7 (multiplication_is_well_defined) }
% 0.22/0.61 fresh(product(add(y, z), multiplicative_identity, y), true, y, z)
% 0.22/0.61 = { by axiom 15 (distributivity5) R->L }
% 0.22/0.61 fresh(fresh20(true, true, y, z, add(y, z), multiplicative_identity, y), true, y, z)
% 0.22/0.61 = { by axiom 11 (commutativity_of_addition) R->L }
% 0.22/0.61 fresh(fresh20(fresh5(true, true, x, y, multiplicative_identity), true, y, z, add(y, z), multiplicative_identity, y), true, y, z)
% 0.22/0.61 = { by axiom 2 (sum_to_multiplicative_identity1) R->L }
% 0.22/0.61 fresh(fresh20(fresh5(sum(x, y, multiplicative_identity), true, x, y, multiplicative_identity), true, y, z, add(y, z), multiplicative_identity, y), true, y, z)
% 0.22/0.61 = { by axiom 18 (commutativity_of_addition) }
% 0.22/0.61 fresh(fresh20(sum(y, x, multiplicative_identity), true, y, z, add(y, z), multiplicative_identity, y), true, y, z)
% 0.22/0.61 = { by axiom 21 (distributivity5) R->L }
% 0.22/0.61 fresh(fresh19(true, true, y, z, add(y, z), x, multiplicative_identity, additive_identity, y), true, y, z)
% 0.22/0.61 = { by axiom 10 (commutativity_of_multiplication) R->L }
% 0.22/0.61 fresh(fresh19(fresh6(true, true, x, z, additive_identity), true, y, z, add(y, z), x, multiplicative_identity, additive_identity, y), true, y, z)
% 0.22/0.61 = { by axiom 6 (product_to_additive_identity2) R->L }
% 0.22/0.61 fresh(fresh19(fresh6(product(x, z, additive_identity), true, x, z, additive_identity), true, y, z, add(y, z), x, multiplicative_identity, additive_identity, y), true, y, z)
% 0.22/0.61 = { by axiom 17 (commutativity_of_multiplication) }
% 0.22/0.61 fresh(fresh19(product(z, x, additive_identity), true, y, z, add(y, z), x, multiplicative_identity, additive_identity, y), true, y, z)
% 0.22/0.61 = { by lemma 25 }
% 0.22/0.61 fresh(fresh21(Z, Z, y, z, add(y, z), W, multiplicative_identity, y), true, y, z)
% 0.22/0.61 = { by lemma 24 }
% 0.22/0.61 fresh(true, true, y, z)
% 0.22/0.61 = { by axiom 7 (multiplication_is_well_defined) }
% 0.22/0.61 z
% 0.22/0.61 % SZS output end Proof
% 0.22/0.61
% 0.22/0.61 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------