TSTP Solution File: SET799+4 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SET799+4 : 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 : n007.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 15:33:16 EDT 2023
% Result : Theorem 0.20s 0.66s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : SET799+4 : TPTP v8.1.2. Released v3.2.0.
% 0.07/0.13 % 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 : n007.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 15:47:11 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.20/0.66 Command-line arguments: --flatten
% 0.20/0.66
% 0.20/0.66 % SZS status Theorem
% 0.20/0.66
% 0.20/0.66 % SZS output start Proof
% 0.20/0.66 Take the following subset of the input axioms:
% 0.20/0.67 fof(least_upper_bound, axiom, ![X, E, R, A2]: (least_upper_bound(A2, X, R, E) <=> (member(A2, X) & (upper_bound(A2, R, X) & ![M]: ((member(M, E) & upper_bound(M, R, X)) => apply(R, A2, M)))))).
% 0.20/0.67 fof(subset, axiom, ![B, A2_2]: (subset(A2_2, B) <=> ![X3]: (member(X3, A2_2) => member(X3, B)))).
% 0.20/0.67 fof(thIV11, conjecture, ![E2, R2]: (order(R2, E2) => ![X1, X2]: ((subset(X1, E2) & (subset(X2, E2) & subset(X1, X2))) => ![M1, M2]: ((least_upper_bound(M1, X1, R2, E2) & least_upper_bound(M2, X2, R2, E2)) => apply(R2, M1, M2))))).
% 0.20/0.67 fof(upper_bound, axiom, ![E2, R2, M3]: (upper_bound(M3, R2, E2) <=> ![X3]: (member(X3, E2) => apply(R2, X3, M3)))).
% 0.20/0.67
% 0.20/0.67 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.67 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.67 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.67 fresh(y, y, x1...xn) = u
% 0.20/0.67 C => fresh(s, t, x1...xn) = v
% 0.20/0.67 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.67 variables of u and v.
% 0.20/0.67 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.67 input problem has no model of domain size 1).
% 0.20/0.67
% 0.20/0.67 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.67
% 0.20/0.67 Axiom 1 (thIV11_3): order(r, e) = true2.
% 0.20/0.67 Axiom 2 (thIV11): subset(x2, e) = true2.
% 0.20/0.67 Axiom 3 (thIV11_2): subset(x1, x2) = true2.
% 0.20/0.67 Axiom 4 (least_upper_bound_4): fresh43(X, X, Y, Z) = true2.
% 0.20/0.67 Axiom 5 (subset_1): fresh19(X, X, Y, Z) = true2.
% 0.20/0.67 Axiom 6 (thIV11_4): least_upper_bound(m2, x2, r, e) = true2.
% 0.20/0.67 Axiom 7 (thIV11_5): least_upper_bound(m1, x1, r, e) = true2.
% 0.20/0.67 Axiom 8 (least_upper_bound_5): fresh42(X, X, Y, Z, W) = true2.
% 0.20/0.67 Axiom 9 (subset_1): fresh20(X, X, Y, Z, W) = member(W, Z).
% 0.20/0.67 Axiom 10 (upper_bound_1): fresh5(X, X, Y, Z, W) = true2.
% 0.20/0.67 Axiom 11 (upper_bound_1): fresh6(X, X, Y, Z, W, V) = apply(Y, V, W).
% 0.20/0.67 Axiom 12 (subset_1): fresh20(member(X, Y), true2, Y, Z, X) = fresh19(subset(Y, Z), true2, Z, X).
% 0.20/0.67 Axiom 13 (least_upper_bound_4): fresh43(least_upper_bound(X, Y, Z, W), true2, X, Y) = member(X, Y).
% 0.20/0.67 Axiom 14 (least_upper_bound_5): fresh42(least_upper_bound(X, Y, Z, W), true2, X, Y, Z) = upper_bound(X, Z, Y).
% 0.20/0.67 Axiom 15 (upper_bound_1): fresh6(upper_bound(X, Y, Z), true2, Y, Z, X, W) = fresh5(member(W, Z), true2, Y, X, W).
% 0.20/0.67
% 0.20/0.67 Lemma 16: subset(x2, e) = order(r, e).
% 0.20/0.67 Proof:
% 0.20/0.67 subset(x2, e)
% 0.20/0.67 = { by axiom 2 (thIV11) }
% 0.20/0.67 true2
% 0.20/0.67 = { by axiom 1 (thIV11_3) R->L }
% 0.20/0.67 order(r, e)
% 0.20/0.67
% 0.20/0.67 Goal 1 (thIV11_6): apply(r, m1, m2) = true2.
% 0.20/0.67 Proof:
% 0.20/0.67 apply(r, m1, m2)
% 0.20/0.67 = { by axiom 11 (upper_bound_1) R->L }
% 0.20/0.67 fresh6(subset(x2, e), subset(x2, e), r, x2, m2, m1)
% 0.20/0.67 = { by lemma 16 }
% 0.20/0.67 fresh6(order(r, e), subset(x2, e), r, x2, m2, m1)
% 0.20/0.67 = { by axiom 1 (thIV11_3) }
% 0.20/0.67 fresh6(true2, subset(x2, e), r, x2, m2, m1)
% 0.20/0.67 = { by axiom 8 (least_upper_bound_5) R->L }
% 0.20/0.67 fresh6(fresh42(subset(x2, e), subset(x2, e), m2, x2, r), subset(x2, e), r, x2, m2, m1)
% 0.20/0.67 = { by lemma 16 }
% 0.20/0.67 fresh6(fresh42(order(r, e), subset(x2, e), m2, x2, r), subset(x2, e), r, x2, m2, m1)
% 0.20/0.67 = { by axiom 1 (thIV11_3) }
% 0.20/0.67 fresh6(fresh42(true2, subset(x2, e), m2, x2, r), subset(x2, e), r, x2, m2, m1)
% 0.20/0.67 = { by axiom 6 (thIV11_4) R->L }
% 0.20/0.67 fresh6(fresh42(least_upper_bound(m2, x2, r, e), subset(x2, e), m2, x2, r), subset(x2, e), r, x2, m2, m1)
% 0.20/0.67 = { by lemma 16 }
% 0.20/0.67 fresh6(fresh42(least_upper_bound(m2, x2, r, e), order(r, e), m2, x2, r), subset(x2, e), r, x2, m2, m1)
% 0.20/0.67 = { by axiom 1 (thIV11_3) }
% 0.20/0.67 fresh6(fresh42(least_upper_bound(m2, x2, r, e), true2, m2, x2, r), subset(x2, e), r, x2, m2, m1)
% 0.20/0.67 = { by axiom 14 (least_upper_bound_5) }
% 0.20/0.67 fresh6(upper_bound(m2, r, x2), subset(x2, e), r, x2, m2, m1)
% 0.20/0.67 = { by lemma 16 }
% 0.20/0.67 fresh6(upper_bound(m2, r, x2), order(r, e), r, x2, m2, m1)
% 0.20/0.67 = { by axiom 1 (thIV11_3) }
% 0.20/0.67 fresh6(upper_bound(m2, r, x2), true2, r, x2, m2, m1)
% 0.20/0.67 = { by axiom 15 (upper_bound_1) }
% 0.20/0.67 fresh5(member(m1, x2), true2, r, m2, m1)
% 0.20/0.67 = { by axiom 1 (thIV11_3) R->L }
% 0.20/0.67 fresh5(member(m1, x2), order(r, e), r, m2, m1)
% 0.20/0.67 = { by lemma 16 R->L }
% 0.20/0.67 fresh5(member(m1, x2), subset(x2, e), r, m2, m1)
% 0.20/0.67 = { by axiom 9 (subset_1) R->L }
% 0.20/0.67 fresh5(fresh20(subset(x2, e), subset(x2, e), x1, x2, m1), subset(x2, e), r, m2, m1)
% 0.20/0.67 = { by lemma 16 }
% 0.20/0.67 fresh5(fresh20(order(r, e), subset(x2, e), x1, x2, m1), subset(x2, e), r, m2, m1)
% 0.20/0.67 = { by axiom 1 (thIV11_3) }
% 0.20/0.67 fresh5(fresh20(true2, subset(x2, e), x1, x2, m1), subset(x2, e), r, m2, m1)
% 0.20/0.67 = { by axiom 4 (least_upper_bound_4) R->L }
% 0.20/0.67 fresh5(fresh20(fresh43(subset(x2, e), subset(x2, e), m1, x1), subset(x2, e), x1, x2, m1), subset(x2, e), r, m2, m1)
% 0.20/0.67 = { by lemma 16 }
% 0.20/0.67 fresh5(fresh20(fresh43(order(r, e), subset(x2, e), m1, x1), subset(x2, e), x1, x2, m1), subset(x2, e), r, m2, m1)
% 0.20/0.67 = { by axiom 1 (thIV11_3) }
% 0.20/0.67 fresh5(fresh20(fresh43(true2, subset(x2, e), m1, x1), subset(x2, e), x1, x2, m1), subset(x2, e), r, m2, m1)
% 0.20/0.67 = { by axiom 7 (thIV11_5) R->L }
% 0.20/0.67 fresh5(fresh20(fresh43(least_upper_bound(m1, x1, r, e), subset(x2, e), m1, x1), subset(x2, e), x1, x2, m1), subset(x2, e), r, m2, m1)
% 0.20/0.67 = { by lemma 16 }
% 0.20/0.67 fresh5(fresh20(fresh43(least_upper_bound(m1, x1, r, e), order(r, e), m1, x1), subset(x2, e), x1, x2, m1), subset(x2, e), r, m2, m1)
% 0.20/0.67 = { by axiom 1 (thIV11_3) }
% 0.20/0.67 fresh5(fresh20(fresh43(least_upper_bound(m1, x1, r, e), true2, m1, x1), subset(x2, e), x1, x2, m1), subset(x2, e), r, m2, m1)
% 0.20/0.67 = { by axiom 13 (least_upper_bound_4) }
% 0.20/0.67 fresh5(fresh20(member(m1, x1), subset(x2, e), x1, x2, m1), subset(x2, e), r, m2, m1)
% 0.20/0.67 = { by lemma 16 }
% 0.20/0.67 fresh5(fresh20(member(m1, x1), order(r, e), x1, x2, m1), subset(x2, e), r, m2, m1)
% 0.20/0.67 = { by axiom 1 (thIV11_3) }
% 0.20/0.67 fresh5(fresh20(member(m1, x1), true2, x1, x2, m1), subset(x2, e), r, m2, m1)
% 0.20/0.67 = { by axiom 12 (subset_1) }
% 0.20/0.67 fresh5(fresh19(subset(x1, x2), true2, x2, m1), subset(x2, e), r, m2, m1)
% 0.20/0.67 = { by axiom 1 (thIV11_3) R->L }
% 0.20/0.67 fresh5(fresh19(subset(x1, x2), order(r, e), x2, m1), subset(x2, e), r, m2, m1)
% 0.20/0.67 = { by lemma 16 R->L }
% 0.20/0.67 fresh5(fresh19(subset(x1, x2), subset(x2, e), x2, m1), subset(x2, e), r, m2, m1)
% 0.20/0.67 = { by axiom 3 (thIV11_2) }
% 0.20/0.67 fresh5(fresh19(true2, subset(x2, e), x2, m1), subset(x2, e), r, m2, m1)
% 0.20/0.67 = { by axiom 1 (thIV11_3) R->L }
% 0.20/0.67 fresh5(fresh19(order(r, e), subset(x2, e), x2, m1), subset(x2, e), r, m2, m1)
% 0.20/0.67 = { by lemma 16 R->L }
% 0.20/0.67 fresh5(fresh19(subset(x2, e), subset(x2, e), x2, m1), subset(x2, e), r, m2, m1)
% 0.20/0.67 = { by axiom 5 (subset_1) }
% 0.20/0.67 fresh5(true2, subset(x2, e), r, m2, m1)
% 0.20/0.67 = { by axiom 1 (thIV11_3) R->L }
% 0.20/0.67 fresh5(order(r, e), subset(x2, e), r, m2, m1)
% 0.20/0.67 = { by lemma 16 R->L }
% 0.20/0.67 fresh5(subset(x2, e), subset(x2, e), r, m2, m1)
% 0.20/0.67 = { by axiom 10 (upper_bound_1) }
% 0.20/0.67 true2
% 0.20/0.67 % SZS output end Proof
% 0.20/0.67
% 0.20/0.67 RESULT: Theorem (the conjecture is true).
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