TSTP Solution File: ITP209_1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : ITP209_1 : TPTP v8.1.2. Released v8.0.0.
% 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 04:17:20 EDT 2023
% Result : Theorem 0.21s 0.41s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP209_1 : TPTP v8.1.2. Released v8.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.16/0.34 % Computer : n021.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.20/0.34 % WCLimit : 300
% 0.20/0.34 % DateTime : Sun Aug 27 13:38:42 EDT 2023
% 0.20/0.35 % CPUTime :
% 0.21/0.41 Command-line arguments: --no-flatten-goal
% 0.21/0.41
% 0.21/0.41 % SZS status Theorem
% 0.21/0.41
% 0.21/0.41 % SZS output start Proof
% 0.21/0.41 Take the following subset of the input axioms:
% 0.21/0.42 fof(conj_0, conjecture, aa_a_a(aa_a_fun_a_a(f, aa_a_a(aa_a_fun_a_a(f, a2), b)), c)=aa_a_a(aa_a_fun_a_a(f, aa_a_a(aa_a_fun_a_a(f, a2), c)), b)).
% 0.21/0.42 fof(fact_0_ac__operator__axioms, axiom, syntax_ac_operator_a(f)).
% 0.21/0.42 fof(fact_1_commute, axiom, ![A, B]: aa_a_a(aa_a_fun_a_a(f, A), B)=aa_a_a(aa_a_fun_a_a(f, B), A)).
% 0.21/0.42 fof(fact_3_right__assoc, axiom, ![C, B2, A3]: aa_a_a(aa_a_fun_a_a(f, aa_a_a(aa_a_fun_a_a(f, A3), B2)), C)=aa_a_a(aa_a_fun_a_a(f, A3), aa_a_a(aa_a_fun_a_a(f, B2), C))).
% 0.21/0.42 fof(fact_6_ac__operator_Oleft__assoc, axiom, ![F, A2, B2, C2]: (syntax_ac_operator_a(F) => aa_a_a(aa_a_fun_a_a(F, A2), aa_a_a(aa_a_fun_a_a(F, B2), C2))=aa_a_a(aa_a_fun_a_a(F, aa_a_a(aa_a_fun_a_a(F, A2), B2)), C2))).
% 0.21/0.42 fof(fact_8_ac__operator_Oleft__commute, axiom, ![B2, C2, F2, A2_2]: (syntax_ac_operator_a(F2) => aa_a_a(aa_a_fun_a_a(F2, A2_2), aa_a_a(aa_a_fun_a_a(F2, B2), C2))=aa_a_a(aa_a_fun_a_a(F2, B2), aa_a_a(aa_a_fun_a_a(F2, A2_2), C2)))).
% 0.21/0.42
% 0.21/0.42 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.42 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.42 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.42 fresh(y, y, x1...xn) = u
% 0.21/0.42 C => fresh(s, t, x1...xn) = v
% 0.21/0.42 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.42 variables of u and v.
% 0.21/0.42 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.42 input problem has no model of domain size 1).
% 0.21/0.42
% 0.21/0.42 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.42
% 0.21/0.42 Axiom 1 (fact_0_ac__operator__axioms): syntax_ac_operator_a(f) = true.
% 0.21/0.42 Axiom 2 (fact_1_commute): aa_a_a(aa_a_fun_a_a(f, X), Y) = aa_a_a(aa_a_fun_a_a(f, Y), X).
% 0.21/0.42 Axiom 3 (fact_6_ac__operator_Oleft__assoc): fresh5(syntax_ac_operator_a(X), true, X, Y, Z, W) = aa_a_a(aa_a_fun_a_a(X, Y), aa_a_a(aa_a_fun_a_a(X, Z), W)).
% 0.21/0.42 Axiom 4 (fact_8_ac__operator_Oleft__commute): fresh3(syntax_ac_operator_a(X), true, X, Y, Z, W) = aa_a_a(aa_a_fun_a_a(X, Y), aa_a_a(aa_a_fun_a_a(X, Z), W)).
% 0.21/0.42 Axiom 5 (fact_7_ac__operator_Oright__assoc): fresh4(X, X, Y, Z, W, V) = aa_a_a(aa_a_fun_a_a(Y, Z), aa_a_a(aa_a_fun_a_a(Y, W), V)).
% 0.21/0.42 Axiom 6 (fact_8_ac__operator_Oleft__commute): fresh3(X, X, Y, Z, W, V) = aa_a_a(aa_a_fun_a_a(Y, W), aa_a_a(aa_a_fun_a_a(Y, Z), V)).
% 0.21/0.42 Axiom 7 (fact_6_ac__operator_Oleft__assoc): fresh5(X, X, Y, Z, W, V) = aa_a_a(aa_a_fun_a_a(Y, aa_a_a(aa_a_fun_a_a(Y, Z), W)), V).
% 0.21/0.42 Axiom 8 (fact_3_right__assoc): aa_a_a(aa_a_fun_a_a(f, aa_a_a(aa_a_fun_a_a(f, X), Y)), Z) = aa_a_a(aa_a_fun_a_a(f, X), aa_a_a(aa_a_fun_a_a(f, Y), Z)).
% 0.21/0.42
% 0.21/0.42 Lemma 9: fresh5(syntax_ac_operator_a(X), true, X, Y, Z, W) = fresh4(V, V, X, Y, Z, W).
% 0.21/0.42 Proof:
% 0.21/0.42 fresh5(syntax_ac_operator_a(X), true, X, Y, Z, W)
% 0.21/0.42 = { by axiom 3 (fact_6_ac__operator_Oleft__assoc) }
% 0.21/0.42 aa_a_a(aa_a_fun_a_a(X, Y), aa_a_a(aa_a_fun_a_a(X, Z), W))
% 0.21/0.42 = { by axiom 5 (fact_7_ac__operator_Oright__assoc) R->L }
% 0.21/0.42 fresh4(V, V, X, Y, Z, W)
% 0.21/0.42
% 0.21/0.42 Lemma 10: fresh4(X, X, f, Y, Z, W) = fresh5(V, V, f, Y, Z, W).
% 0.21/0.42 Proof:
% 0.21/0.42 fresh4(X, X, f, Y, Z, W)
% 0.21/0.42 = { by lemma 9 R->L }
% 0.21/0.42 fresh5(syntax_ac_operator_a(f), true, f, Y, Z, W)
% 0.21/0.42 = { by axiom 3 (fact_6_ac__operator_Oleft__assoc) }
% 0.21/0.42 aa_a_a(aa_a_fun_a_a(f, Y), aa_a_a(aa_a_fun_a_a(f, Z), W))
% 0.21/0.42 = { by axiom 8 (fact_3_right__assoc) R->L }
% 0.21/0.42 aa_a_a(aa_a_fun_a_a(f, aa_a_a(aa_a_fun_a_a(f, Y), Z)), W)
% 0.21/0.42 = { by axiom 7 (fact_6_ac__operator_Oleft__assoc) R->L }
% 0.21/0.42 fresh5(V, V, f, Y, Z, W)
% 0.21/0.42
% 0.21/0.42 Lemma 11: fresh5(V, V, f, Z, Y, W) = fresh5(X, X, f, Y, Z, W).
% 0.21/0.42 Proof:
% 0.21/0.42 fresh5(V, V, f, Z, Y, W)
% 0.21/0.42 = { by lemma 10 R->L }
% 0.21/0.42 fresh4(T, T, f, Z, Y, W)
% 0.21/0.42 = { by lemma 9 R->L }
% 0.21/0.42 fresh5(syntax_ac_operator_a(f), true, f, Z, Y, W)
% 0.21/0.42 = { by axiom 3 (fact_6_ac__operator_Oleft__assoc) }
% 0.21/0.42 aa_a_a(aa_a_fun_a_a(f, Z), aa_a_a(aa_a_fun_a_a(f, Y), W))
% 0.21/0.42 = { by axiom 6 (fact_8_ac__operator_Oleft__commute) R->L }
% 0.21/0.42 fresh3(true, true, f, Y, Z, W)
% 0.21/0.42 = { by axiom 1 (fact_0_ac__operator__axioms) R->L }
% 0.21/0.42 fresh3(syntax_ac_operator_a(f), true, f, Y, Z, W)
% 0.21/0.42 = { by axiom 4 (fact_8_ac__operator_Oleft__commute) }
% 0.21/0.42 aa_a_a(aa_a_fun_a_a(f, Y), aa_a_a(aa_a_fun_a_a(f, Z), W))
% 0.21/0.42 = { by axiom 3 (fact_6_ac__operator_Oleft__assoc) R->L }
% 0.21/0.42 fresh5(syntax_ac_operator_a(f), true, f, Y, Z, W)
% 0.21/0.42 = { by lemma 9 }
% 0.21/0.42 fresh4(U, U, f, Y, Z, W)
% 0.21/0.42 = { by lemma 10 }
% 0.21/0.42 fresh5(X, X, f, Y, Z, W)
% 0.21/0.42
% 0.21/0.42 Goal 1 (conj_0): aa_a_a(aa_a_fun_a_a(f, aa_a_a(aa_a_fun_a_a(f, a2), b)), c) = aa_a_a(aa_a_fun_a_a(f, aa_a_a(aa_a_fun_a_a(f, a2), c)), b).
% 0.21/0.42 Proof:
% 0.21/0.42 aa_a_a(aa_a_fun_a_a(f, aa_a_a(aa_a_fun_a_a(f, a2), b)), c)
% 0.21/0.42 = { by axiom 2 (fact_1_commute) }
% 0.21/0.42 aa_a_a(aa_a_fun_a_a(f, c), aa_a_a(aa_a_fun_a_a(f, a2), b))
% 0.21/0.42 = { by axiom 3 (fact_6_ac__operator_Oleft__assoc) R->L }
% 0.21/0.42 fresh5(syntax_ac_operator_a(f), true, f, c, a2, b)
% 0.21/0.42 = { by axiom 1 (fact_0_ac__operator__axioms) }
% 0.21/0.42 fresh5(true, true, f, c, a2, b)
% 0.21/0.42 = { by lemma 11 R->L }
% 0.21/0.42 fresh5(X, X, f, a2, c, b)
% 0.21/0.42 = { by lemma 10 R->L }
% 0.21/0.42 fresh4(Y, Y, f, a2, c, b)
% 0.21/0.42 = { by lemma 9 R->L }
% 0.21/0.42 fresh5(syntax_ac_operator_a(f), true, f, a2, c, b)
% 0.21/0.42 = { by axiom 3 (fact_6_ac__operator_Oleft__assoc) }
% 0.21/0.42 aa_a_a(aa_a_fun_a_a(f, a2), aa_a_a(aa_a_fun_a_a(f, c), b))
% 0.21/0.42 = { by axiom 2 (fact_1_commute) }
% 0.21/0.42 aa_a_a(aa_a_fun_a_a(f, a2), aa_a_a(aa_a_fun_a_a(f, b), c))
% 0.21/0.42 = { by axiom 3 (fact_6_ac__operator_Oleft__assoc) R->L }
% 0.21/0.42 fresh5(syntax_ac_operator_a(f), true, f, a2, b, c)
% 0.21/0.42 = { by lemma 9 }
% 0.21/0.42 fresh4(Z, Z, f, a2, b, c)
% 0.21/0.42 = { by lemma 10 }
% 0.21/0.42 fresh5(W, W, f, a2, b, c)
% 0.21/0.42 = { by lemma 11 }
% 0.21/0.42 fresh5(true, true, f, b, a2, c)
% 0.21/0.42 = { by axiom 1 (fact_0_ac__operator__axioms) R->L }
% 0.21/0.42 fresh5(syntax_ac_operator_a(f), true, f, b, a2, c)
% 0.21/0.42 = { by axiom 3 (fact_6_ac__operator_Oleft__assoc) }
% 0.21/0.42 aa_a_a(aa_a_fun_a_a(f, b), aa_a_a(aa_a_fun_a_a(f, a2), c))
% 0.21/0.42 = { by axiom 2 (fact_1_commute) R->L }
% 0.21/0.42 aa_a_a(aa_a_fun_a_a(f, aa_a_a(aa_a_fun_a_a(f, a2), c)), b)
% 0.21/0.42 % SZS output end Proof
% 0.21/0.42
% 0.21/0.42 RESULT: Theorem (the conjecture is true).
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