TSTP Solution File: SET979+1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SET979+1 : TPTP v8.1.2. Released v3.2.0.
% 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 : n002.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 15:33:53 EDT 2023
% Result : Theorem 0.19s 0.38s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : SET979+1 : TPTP v8.1.2. Released v3.2.0.
% 0.13/0.13 % 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 : n002.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 : Sat Aug 26 09:30:32 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.19/0.38 Command-line arguments: --no-flatten-goal
% 0.19/0.38
% 0.19/0.38 % SZS status Theorem
% 0.19/0.38
% 0.19/0.38 % SZS output start Proof
% 0.19/0.38 Take the following subset of the input axioms:
% 0.19/0.38 fof(t120_zfmisc_1, axiom, ![A, B, C]: (cartesian_product2(set_union2(A, B), C)=set_union2(cartesian_product2(A, C), cartesian_product2(B, C)) & cartesian_product2(C, set_union2(A, B))=set_union2(cartesian_product2(C, A), cartesian_product2(C, B)))).
% 0.19/0.38 fof(t132_zfmisc_1, conjecture, ![A2, B2, C2]: (cartesian_product2(unordered_pair(A2, B2), C2)=set_union2(cartesian_product2(singleton(A2), C2), cartesian_product2(singleton(B2), C2)) & cartesian_product2(C2, unordered_pair(A2, B2))=set_union2(cartesian_product2(C2, singleton(A2)), cartesian_product2(C2, singleton(B2))))).
% 0.19/0.38 fof(t41_enumset1, axiom, ![A2, B2]: unordered_pair(A2, B2)=set_union2(singleton(A2), singleton(B2))).
% 0.19/0.38
% 0.19/0.38 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.38 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.38 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.38 fresh(y, y, x1...xn) = u
% 0.19/0.38 C => fresh(s, t, x1...xn) = v
% 0.19/0.38 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.38 variables of u and v.
% 0.19/0.38 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.38 input problem has no model of domain size 1).
% 0.19/0.38
% 0.19/0.38 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.38
% 0.19/0.38 Axiom 1 (t41_enumset1): unordered_pair(X, Y) = set_union2(singleton(X), singleton(Y)).
% 0.19/0.38 Axiom 2 (t120_zfmisc_1_1): cartesian_product2(X, set_union2(Y, Z)) = set_union2(cartesian_product2(X, Y), cartesian_product2(X, Z)).
% 0.19/0.38 Axiom 3 (t120_zfmisc_1): cartesian_product2(set_union2(X, Y), Z) = set_union2(cartesian_product2(X, Z), cartesian_product2(Y, Z)).
% 0.19/0.38
% 0.19/0.38 Goal 1 (t132_zfmisc_1): tuple(cartesian_product2(unordered_pair(a2, b2), c2), cartesian_product2(c, unordered_pair(a, b))) = tuple(set_union2(cartesian_product2(singleton(a2), c2), cartesian_product2(singleton(b2), c2)), set_union2(cartesian_product2(c, singleton(a)), cartesian_product2(c, singleton(b)))).
% 0.19/0.38 Proof:
% 0.19/0.38 tuple(cartesian_product2(unordered_pair(a2, b2), c2), cartesian_product2(c, unordered_pair(a, b)))
% 0.19/0.39 = { by axiom 1 (t41_enumset1) }
% 0.19/0.39 tuple(cartesian_product2(set_union2(singleton(a2), singleton(b2)), c2), cartesian_product2(c, unordered_pair(a, b)))
% 0.19/0.39 = { by axiom 3 (t120_zfmisc_1) }
% 0.19/0.39 tuple(set_union2(cartesian_product2(singleton(a2), c2), cartesian_product2(singleton(b2), c2)), cartesian_product2(c, unordered_pair(a, b)))
% 0.19/0.39 = { by axiom 1 (t41_enumset1) }
% 0.19/0.39 tuple(set_union2(cartesian_product2(singleton(a2), c2), cartesian_product2(singleton(b2), c2)), cartesian_product2(c, set_union2(singleton(a), singleton(b))))
% 0.19/0.39 = { by axiom 2 (t120_zfmisc_1_1) }
% 0.19/0.39 tuple(set_union2(cartesian_product2(singleton(a2), c2), cartesian_product2(singleton(b2), c2)), set_union2(cartesian_product2(c, singleton(a)), cartesian_product2(c, singleton(b))))
% 0.19/0.39 % SZS output end Proof
% 0.19/0.39
% 0.19/0.39 RESULT: Theorem (the conjecture is true).
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