TSTP Solution File: SYN356+1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : SYN356+1 : TPTP v8.1.2. Released v2.0.0.
% 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 : n016.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 : Fri Sep 1 03:34:16 EDT 2023
% Result : Theorem 0.19s 0.37s
% 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
% 0.00/0.12 % Problem : SYN356+1 : TPTP v8.1.2. Released v2.0.0.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.34 % Computer : n016.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Sat Aug 26 21:01:42 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.19/0.37 Command-line arguments: --ground-connectedness --complete-subsets
% 0.19/0.37
% 0.19/0.37 % SZS status Theorem
% 0.19/0.37
% 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(x2107, conjecture, (big_r(a, b) & (![X, Y]: (big_r(X, Y) => (big_r(Y, X) & big_q(X, Y))) & ![U, V]: (big_q(U, V) => big_q(U, U)))) => (big_q(a, a) & big_q(b, b))).
% 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 (x2107): big_r(a, b) = true.
% 0.19/0.38 Axiom 2 (x2107_3): fresh(X, X, Y) = true.
% 0.19/0.38 Axiom 3 (x2107_2): fresh3(X, X, Y, Z) = true.
% 0.19/0.38 Axiom 4 (x2107_1): fresh2(X, X, Y, Z) = true.
% 0.19/0.38 Axiom 5 (x2107_3): fresh(big_q(X, Y), true, X) = big_q(X, X).
% 0.19/0.38 Axiom 6 (x2107_2): fresh3(big_r(X, Y), true, X, Y) = big_q(X, Y).
% 0.19/0.38 Axiom 7 (x2107_1): fresh2(big_r(X, Y), true, X, Y) = big_r(Y, X).
% 0.19/0.38
% 0.19/0.38 Lemma 8: big_q(a, a) = true.
% 0.19/0.38 Proof:
% 0.19/0.38 big_q(a, a)
% 0.19/0.38 = { by axiom 5 (x2107_3) R->L }
% 0.19/0.38 fresh(big_q(a, b), true, a)
% 0.19/0.38 = { by axiom 6 (x2107_2) R->L }
% 0.19/0.38 fresh(fresh3(big_r(a, b), true, a, b), true, a)
% 0.19/0.38 = { by axiom 1 (x2107) }
% 0.19/0.38 fresh(fresh3(true, true, a, b), true, a)
% 0.19/0.38 = { by axiom 3 (x2107_2) }
% 0.19/0.38 fresh(true, true, a)
% 0.19/0.38 = { by axiom 2 (x2107_3) }
% 0.19/0.38 true
% 0.19/0.38
% 0.19/0.38 Goal 1 (x2107_4): tuple(big_q(a, a), big_q(b, b)) = tuple(true, true).
% 0.19/0.38 Proof:
% 0.19/0.38 tuple(big_q(a, a), big_q(b, b))
% 0.19/0.38 = { by axiom 5 (x2107_3) R->L }
% 0.19/0.38 tuple(big_q(a, a), fresh(big_q(b, a), true, b))
% 0.19/0.38 = { by axiom 6 (x2107_2) R->L }
% 0.19/0.38 tuple(big_q(a, a), fresh(fresh3(big_r(b, a), true, b, a), true, b))
% 0.19/0.38 = { by axiom 7 (x2107_1) R->L }
% 0.19/0.38 tuple(big_q(a, a), fresh(fresh3(fresh2(big_r(a, b), true, a, b), true, b, a), true, b))
% 0.19/0.38 = { by axiom 1 (x2107) }
% 0.19/0.38 tuple(big_q(a, a), fresh(fresh3(fresh2(true, true, a, b), true, b, a), true, b))
% 0.19/0.38 = { by axiom 4 (x2107_1) }
% 0.19/0.38 tuple(big_q(a, a), fresh(fresh3(true, true, b, a), true, b))
% 0.19/0.38 = { by lemma 8 R->L }
% 0.19/0.38 tuple(big_q(a, a), fresh(fresh3(big_q(a, a), true, b, a), true, b))
% 0.19/0.38 = { by lemma 8 R->L }
% 0.19/0.38 tuple(big_q(a, a), fresh(fresh3(big_q(a, a), big_q(a, a), b, a), true, b))
% 0.19/0.38 = { by axiom 3 (x2107_2) }
% 0.19/0.38 tuple(big_q(a, a), fresh(true, true, b))
% 0.19/0.38 = { by lemma 8 R->L }
% 0.19/0.38 tuple(big_q(a, a), fresh(big_q(a, a), true, b))
% 0.19/0.38 = { by lemma 8 R->L }
% 0.19/0.38 tuple(big_q(a, a), fresh(big_q(a, a), big_q(a, a), b))
% 0.19/0.38 = { by axiom 2 (x2107_3) }
% 0.19/0.38 tuple(big_q(a, a), true)
% 0.19/0.38 = { by lemma 8 }
% 0.19/0.38 tuple(true, true)
% 0.19/0.38 % SZS output end Proof
% 0.19/0.38
% 0.19/0.38 RESULT: Theorem (the conjecture is true).
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