TSTP Solution File: GRP035-3 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP035-3 : 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 : n019.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 01:16:43 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
% 0.00/0.12 % Problem : GRP035-3 : 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
% 0.14/0.33 % Computer : n019.cluster.edu
% 0.14/0.33 % Model : x86_64 x86_64
% 0.14/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.33 % Memory : 8042.1875MB
% 0.14/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Mon Aug 28 23:55:27 EDT 2023
% 0.14/0.34 % 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
% 0.20/0.43
% 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_is_in_subgroup, hypothesis, subgroup_member(a)).
% 0.20/0.44 fof(a_times_b_is_c, hypothesis, product(a, b, c)).
% 0.20/0.44 fof(associativity2, axiom, ![X, Y, Z, W, U, V]: (~product(X, Y, U) | (~product(Y, Z, V) | (~product(X, V, W) | product(U, Z, W))))).
% 0.20/0.44 fof(b_is_in_subgroup, hypothesis, subgroup_member(b)).
% 0.20/0.44 fof(closure_of_product_and_inverse, axiom, ![B, C, A2]: (~subgroup_member(A2) | (~subgroup_member(B) | (~product(A2, inverse(B), C) | subgroup_member(C))))).
% 0.20/0.44 fof(left_identity, axiom, ![X2]: product(identity, X2, X2)).
% 0.20/0.44 fof(left_inverse, axiom, ![X2]: product(inverse(X2), X2, identity)).
% 0.20/0.44 fof(prove_c_is_in_subgroup, negated_conjecture, ~subgroup_member(c)).
% 0.20/0.44 fof(right_identity, axiom, ![X2]: product(X2, identity, X2)).
% 0.20/0.44 fof(right_inverse, axiom, ![X2]: product(X2, inverse(X2), identity)).
% 0.20/0.44 fof(total_function1, axiom, ![X2, Y2]: product(X2, Y2, multiply(X2, Y2))).
% 0.20/0.44 fof(total_function2, axiom, ![X2, Y2, Z2, W2]: (~product(X2, Y2, Z2) | (~product(X2, Y2, W2) | Z2=W2))).
% 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 (a_is_in_subgroup): subgroup_member(a) = true.
% 0.20/0.44 Axiom 2 (b_is_in_subgroup): subgroup_member(b) = true.
% 0.20/0.44 Axiom 3 (closure_of_product_and_inverse): fresh3(X, X, Y) = true.
% 0.20/0.44 Axiom 4 (right_identity): product(X, identity, X) = true.
% 0.20/0.44 Axiom 5 (a_times_b_is_c): product(a, b, c) = true.
% 0.20/0.44 Axiom 6 (left_identity): product(identity, X, X) = true.
% 0.20/0.44 Axiom 7 (total_function2): fresh(X, X, Y, Z) = Z.
% 0.20/0.44 Axiom 8 (right_inverse): product(X, inverse(X), identity) = true.
% 0.20/0.44 Axiom 9 (left_inverse): product(inverse(X), X, identity) = true.
% 0.20/0.44 Axiom 10 (associativity2): fresh9(X, X, Y, Z, W) = true.
% 0.20/0.44 Axiom 11 (closure_of_product_and_inverse): fresh7(X, X, Y, Z, W) = subgroup_member(W).
% 0.20/0.44 Axiom 12 (total_function1): product(X, Y, multiply(X, Y)) = true.
% 0.20/0.44 Axiom 13 (closure_of_product_and_inverse): fresh6(X, X, Y, Z, W) = fresh7(subgroup_member(Y), true, Y, Z, W).
% 0.20/0.44 Axiom 14 (total_function2): fresh2(X, X, Y, Z, W, V) = W.
% 0.20/0.44 Axiom 15 (associativity2): fresh4(X, X, Y, Z, W, V, U) = product(W, V, U).
% 0.20/0.44 Axiom 16 (closure_of_product_and_inverse): fresh6(subgroup_member(X), true, Y, X, Z) = fresh3(product(Y, inverse(X), Z), true, Z).
% 0.20/0.44 Axiom 17 (associativity2): fresh8(X, X, Y, Z, W, V, U, T) = fresh9(product(Y, Z, W), true, W, V, T).
% 0.20/0.44 Axiom 18 (total_function2): fresh2(product(X, Y, Z), true, X, Y, W, Z) = fresh(product(X, Y, W), true, W, Z).
% 0.20/0.44 Axiom 19 (associativity2): fresh8(product(X, Y, Z), true, W, X, V, Y, Z, U) = fresh4(product(W, Z, U), true, W, X, V, Y, U).
% 0.20/0.44
% 0.20/0.44 Lemma 20: fresh6(X, X, a, Y, Z) = subgroup_member(Z).
% 0.20/0.44 Proof:
% 0.20/0.44 fresh6(X, X, a, Y, Z)
% 0.20/0.44 = { by axiom 13 (closure_of_product_and_inverse) }
% 0.20/0.44 fresh7(subgroup_member(a), true, a, Y, Z)
% 0.20/0.44 = { by axiom 1 (a_is_in_subgroup) }
% 0.20/0.44 fresh7(true, true, a, Y, Z)
% 0.20/0.44 = { by axiom 11 (closure_of_product_and_inverse) }
% 0.20/0.44 subgroup_member(Z)
% 0.20/0.44
% 0.20/0.44 Lemma 21: multiply(multiply(X, inverse(Y)), Y) = X.
% 0.20/0.44 Proof:
% 0.20/0.44 multiply(multiply(X, inverse(Y)), Y)
% 0.20/0.44 = { by axiom 7 (total_function2) R->L }
% 0.20/0.44 fresh(true, true, X, multiply(multiply(X, inverse(Y)), Y))
% 0.20/0.44 = { by axiom 10 (associativity2) R->L }
% 0.20/0.44 fresh(fresh9(true, true, multiply(X, inverse(Y)), Y, X), true, X, multiply(multiply(X, inverse(Y)), Y))
% 0.20/0.44 = { by axiom 12 (total_function1) R->L }
% 0.20/0.44 fresh(fresh9(product(X, inverse(Y), multiply(X, inverse(Y))), true, multiply(X, inverse(Y)), Y, X), true, X, multiply(multiply(X, inverse(Y)), Y))
% 0.20/0.44 = { by axiom 17 (associativity2) R->L }
% 0.20/0.44 fresh(fresh8(true, true, X, inverse(Y), multiply(X, inverse(Y)), Y, identity, X), true, X, multiply(multiply(X, inverse(Y)), Y))
% 0.20/0.44 = { by axiom 9 (left_inverse) R->L }
% 0.20/0.44 fresh(fresh8(product(inverse(Y), Y, identity), true, X, inverse(Y), multiply(X, inverse(Y)), Y, identity, X), true, X, multiply(multiply(X, inverse(Y)), Y))
% 0.20/0.44 = { by axiom 19 (associativity2) }
% 0.20/0.44 fresh(fresh4(product(X, identity, X), true, X, inverse(Y), multiply(X, inverse(Y)), Y, X), true, X, multiply(multiply(X, inverse(Y)), Y))
% 0.20/0.44 = { by axiom 4 (right_identity) }
% 0.20/0.44 fresh(fresh4(true, true, X, inverse(Y), multiply(X, inverse(Y)), Y, X), true, X, multiply(multiply(X, inverse(Y)), Y))
% 0.20/0.44 = { by axiom 15 (associativity2) }
% 0.20/0.45 fresh(product(multiply(X, inverse(Y)), Y, X), true, X, multiply(multiply(X, inverse(Y)), Y))
% 0.20/0.45 = { by axiom 18 (total_function2) R->L }
% 0.20/0.45 fresh2(product(multiply(X, inverse(Y)), Y, multiply(multiply(X, inverse(Y)), Y)), true, multiply(X, inverse(Y)), Y, X, multiply(multiply(X, inverse(Y)), Y))
% 0.20/0.45 = { by axiom 12 (total_function1) }
% 0.20/0.45 fresh2(true, true, multiply(X, inverse(Y)), Y, X, multiply(multiply(X, inverse(Y)), Y))
% 0.20/0.45 = { by axiom 14 (total_function2) }
% 0.20/0.45 X
% 0.20/0.45
% 0.20/0.45 Lemma 22: multiply(multiply(X, Y), inverse(Y)) = X.
% 0.20/0.45 Proof:
% 0.20/0.45 multiply(multiply(X, Y), inverse(Y))
% 0.20/0.45 = { by axiom 7 (total_function2) R->L }
% 0.20/0.45 multiply(multiply(X, fresh(true, true, multiply(identity, Y), Y)), inverse(Y))
% 0.20/0.45 = { by axiom 12 (total_function1) R->L }
% 0.20/0.45 multiply(multiply(X, fresh(product(identity, Y, multiply(identity, Y)), true, multiply(identity, Y), Y)), inverse(Y))
% 0.20/0.45 = { by axiom 18 (total_function2) R->L }
% 0.20/0.45 multiply(multiply(X, fresh2(product(identity, Y, Y), true, identity, Y, multiply(identity, Y), Y)), inverse(Y))
% 0.20/0.45 = { by axiom 6 (left_identity) }
% 0.20/0.45 multiply(multiply(X, fresh2(true, true, identity, Y, multiply(identity, Y), Y)), inverse(Y))
% 0.20/0.45 = { by axiom 14 (total_function2) }
% 0.20/0.45 multiply(multiply(X, multiply(identity, Y)), inverse(Y))
% 0.20/0.45 = { by axiom 7 (total_function2) R->L }
% 0.20/0.45 multiply(multiply(X, multiply(fresh(true, true, multiply(inverse(inverse(Y)), inverse(Y)), identity), Y)), inverse(Y))
% 0.20/0.45 = { by axiom 12 (total_function1) R->L }
% 0.20/0.45 multiply(multiply(X, multiply(fresh(product(inverse(inverse(Y)), inverse(Y), multiply(inverse(inverse(Y)), inverse(Y))), true, multiply(inverse(inverse(Y)), inverse(Y)), identity), Y)), inverse(Y))
% 0.20/0.45 = { by axiom 18 (total_function2) R->L }
% 0.20/0.45 multiply(multiply(X, multiply(fresh2(product(inverse(inverse(Y)), inverse(Y), identity), true, inverse(inverse(Y)), inverse(Y), multiply(inverse(inverse(Y)), inverse(Y)), identity), Y)), inverse(Y))
% 0.20/0.45 = { by axiom 9 (left_inverse) }
% 0.20/0.45 multiply(multiply(X, multiply(fresh2(true, true, inverse(inverse(Y)), inverse(Y), multiply(inverse(inverse(Y)), inverse(Y)), identity), Y)), inverse(Y))
% 0.20/0.45 = { by axiom 14 (total_function2) }
% 0.20/0.45 multiply(multiply(X, multiply(multiply(inverse(inverse(Y)), inverse(Y)), Y)), inverse(Y))
% 0.20/0.45 = { by lemma 21 }
% 0.20/0.45 multiply(multiply(X, inverse(inverse(Y))), inverse(Y))
% 0.20/0.45 = { by lemma 21 }
% 0.20/0.45 X
% 0.20/0.45
% 0.20/0.45 Goal 1 (prove_c_is_in_subgroup): subgroup_member(c) = true.
% 0.20/0.45 Proof:
% 0.20/0.45 subgroup_member(c)
% 0.20/0.45 = { by lemma 20 R->L }
% 0.20/0.45 fresh6(true, true, a, inverse(b), c)
% 0.20/0.45 = { by axiom 3 (closure_of_product_and_inverse) R->L }
% 0.20/0.45 fresh6(fresh3(true, true, inverse(b)), true, a, inverse(b), c)
% 0.20/0.45 = { by axiom 6 (left_identity) R->L }
% 0.20/0.45 fresh6(fresh3(product(identity, inverse(b), inverse(b)), true, inverse(b)), true, a, inverse(b), c)
% 0.20/0.45 = { by axiom 16 (closure_of_product_and_inverse) R->L }
% 0.20/0.45 fresh6(fresh6(subgroup_member(b), true, identity, b, inverse(b)), true, a, inverse(b), c)
% 0.20/0.45 = { by axiom 2 (b_is_in_subgroup) }
% 0.20/0.45 fresh6(fresh6(true, true, identity, b, inverse(b)), true, a, inverse(b), c)
% 0.20/0.45 = { by axiom 13 (closure_of_product_and_inverse) }
% 0.20/0.45 fresh6(fresh7(subgroup_member(identity), true, identity, b, inverse(b)), true, a, inverse(b), c)
% 0.20/0.45 = { by lemma 20 R->L }
% 0.20/0.45 fresh6(fresh7(fresh6(true, true, a, a, identity), true, identity, b, inverse(b)), true, a, inverse(b), c)
% 0.20/0.45 = { by axiom 1 (a_is_in_subgroup) R->L }
% 0.20/0.45 fresh6(fresh7(fresh6(subgroup_member(a), true, a, a, identity), true, identity, b, inverse(b)), true, a, inverse(b), c)
% 0.20/0.45 = { by axiom 16 (closure_of_product_and_inverse) }
% 0.20/0.45 fresh6(fresh7(fresh3(product(a, inverse(a), identity), true, identity), true, identity, b, inverse(b)), true, a, inverse(b), c)
% 0.20/0.45 = { by axiom 8 (right_inverse) }
% 0.20/0.45 fresh6(fresh7(fresh3(true, true, identity), true, identity, b, inverse(b)), true, a, inverse(b), c)
% 0.20/0.45 = { by axiom 3 (closure_of_product_and_inverse) }
% 0.20/0.45 fresh6(fresh7(true, true, identity, b, inverse(b)), true, a, inverse(b), c)
% 0.20/0.45 = { by axiom 11 (closure_of_product_and_inverse) }
% 0.20/0.45 fresh6(subgroup_member(inverse(b)), true, a, inverse(b), c)
% 0.20/0.45 = { by lemma 22 R->L }
% 0.20/0.45 fresh6(subgroup_member(inverse(b)), true, multiply(multiply(a, b), inverse(b)), inverse(b), c)
% 0.20/0.45 = { by axiom 14 (total_function2) R->L }
% 0.20/0.45 fresh6(subgroup_member(inverse(b)), true, multiply(fresh2(true, true, a, b, multiply(a, b), c), inverse(b)), inverse(b), c)
% 0.20/0.45 = { by axiom 5 (a_times_b_is_c) R->L }
% 0.20/0.45 fresh6(subgroup_member(inverse(b)), true, multiply(fresh2(product(a, b, c), true, a, b, multiply(a, b), c), inverse(b)), inverse(b), c)
% 0.20/0.45 = { by axiom 18 (total_function2) }
% 0.20/0.45 fresh6(subgroup_member(inverse(b)), true, multiply(fresh(product(a, b, multiply(a, b)), true, multiply(a, b), c), inverse(b)), inverse(b), c)
% 0.20/0.45 = { by axiom 12 (total_function1) }
% 0.20/0.45 fresh6(subgroup_member(inverse(b)), true, multiply(fresh(true, true, multiply(a, b), c), inverse(b)), inverse(b), c)
% 0.20/0.45 = { by axiom 7 (total_function2) }
% 0.20/0.45 fresh6(subgroup_member(inverse(b)), true, multiply(c, inverse(b)), inverse(b), c)
% 0.20/0.45 = { by lemma 22 R->L }
% 0.20/0.45 fresh6(subgroup_member(inverse(b)), true, multiply(c, inverse(b)), inverse(b), multiply(multiply(c, inverse(b)), inverse(inverse(b))))
% 0.20/0.45 = { by axiom 16 (closure_of_product_and_inverse) }
% 0.20/0.45 fresh3(product(multiply(c, inverse(b)), inverse(inverse(b)), multiply(multiply(c, inverse(b)), inverse(inverse(b)))), true, multiply(multiply(c, inverse(b)), inverse(inverse(b))))
% 0.20/0.45 = { by axiom 12 (total_function1) }
% 0.20/0.45 fresh3(true, true, multiply(multiply(c, inverse(b)), inverse(inverse(b))))
% 0.20/0.45 = { by axiom 3 (closure_of_product_and_inverse) }
% 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).
%------------------------------------------------------------------------------