TSTP Solution File: SET664+3 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SET664+3 : TPTP v8.1.2. Released v2.2.0.
% 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 : 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 15:32:54 EDT 2023
% Result : Theorem 0.17s 0.50s
% Output : Proof 0.17s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SET664+3 : TPTP v8.1.2. Released v2.2.0.
% 0.00/0.11 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.11/0.32 % Computer : n019.cluster.edu
% 0.11/0.32 % Model : x86_64 x86_64
% 0.11/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.32 % Memory : 8042.1875MB
% 0.11/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.32 % CPULimit : 300
% 0.11/0.32 % WCLimit : 300
% 0.11/0.32 % DateTime : Sat Aug 26 12:00:58 EDT 2023
% 0.11/0.32 % CPUTime :
% 0.17/0.50 Command-line arguments: --no-flatten-goal
% 0.17/0.50
% 0.17/0.50 % SZS status Theorem
% 0.17/0.50
% 0.17/0.51 % SZS output start Proof
% 0.17/0.51 Take the following subset of the input axioms:
% 0.17/0.51 fof(p1, axiom, ![B]: (ilf_type(B, set_type) => (subset(B, empty_set) => B=empty_set))).
% 0.17/0.51 fof(p13, axiom, ![B2]: (ilf_type(B2, set_type) => (ilf_type(B2, binary_relation_type) <=> (relation_like(B2) & ilf_type(B2, set_type))))).
% 0.17/0.51 fof(p2, axiom, ![B2]: (ilf_type(B2, binary_relation_type) => ((domain_of(B2)=empty_set | range_of(B2)=empty_set) => B2=empty_set))).
% 0.17/0.51 fof(p28, axiom, ![B2]: (ilf_type(B2, set_type) => ![C]: (ilf_type(C, set_type) => ![D]: (ilf_type(D, subset_type(cross_product(B2, C))) => relation_like(D))))).
% 0.17/0.51 fof(p3, axiom, ![B2]: (ilf_type(B2, set_type) => ![C2]: (ilf_type(C2, set_type) => ![D2]: (ilf_type(D2, relation_type(B2, C2)) => (subset(domain_of(D2), B2) & subset(range_of(D2), C2)))))).
% 0.17/0.51 fof(p33, axiom, ![B2]: ilf_type(B2, set_type)).
% 0.17/0.51 fof(p6, axiom, ![B2]: (ilf_type(B2, set_type) => ![C2]: (ilf_type(C2, set_type) => (![D2]: (ilf_type(D2, subset_type(cross_product(B2, C2))) => ilf_type(D2, relation_type(B2, C2))) & ![E]: (ilf_type(E, relation_type(B2, C2)) => ilf_type(E, subset_type(cross_product(B2, C2)))))))).
% 0.17/0.51 fof(prove_relset_1_27, conjecture, ![B2]: (ilf_type(B2, set_type) => ![C2]: (ilf_type(C2, set_type) => ![D2]: (ilf_type(D2, relation_type(C2, B2)) => (ilf_type(D2, relation_type(C2, empty_set)) => D2=empty_set))))).
% 0.17/0.51
% 0.17/0.51 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.17/0.51 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.17/0.51 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.17/0.51 fresh(y, y, x1...xn) = u
% 0.17/0.51 C => fresh(s, t, x1...xn) = v
% 0.17/0.51 where fresh is a fresh function symbol and x1..xn are the free
% 0.17/0.51 variables of u and v.
% 0.17/0.51 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.17/0.51 input problem has no model of domain size 1).
% 0.17/0.51
% 0.17/0.51 The encoding turns the above axioms into the following unit equations and goals:
% 0.17/0.51
% 0.17/0.51 Axiom 1 (p33): ilf_type(X, set_type) = true2.
% 0.17/0.51 Axiom 2 (p28): fresh46(X, X, Y) = true2.
% 0.17/0.51 Axiom 3 (p1): fresh35(X, X, Y) = empty_set.
% 0.17/0.51 Axiom 4 (p13_1): fresh28(X, X, Y) = ilf_type(Y, binary_relation_type).
% 0.17/0.51 Axiom 5 (p13_1): fresh27(X, X, Y) = true2.
% 0.17/0.51 Axiom 6 (p2_1): fresh14(X, X, Y) = empty_set.
% 0.17/0.51 Axiom 7 (p1): fresh3(X, X, Y) = Y.
% 0.17/0.51 Axiom 8 (p2_1): fresh2(X, X, Y) = Y.
% 0.17/0.51 Axiom 9 (p3_1): fresh92(X, X, Y, Z) = true2.
% 0.17/0.51 Axiom 10 (p28): fresh16(X, X, Y, Z) = relation_like(Z).
% 0.17/0.51 Axiom 11 (prove_relset_1_27_3): ilf_type(d, relation_type(c, b)) = true2.
% 0.17/0.51 Axiom 12 (prove_relset_1_27_2): ilf_type(d, relation_type(c, empty_set)) = true2.
% 0.17/0.51 Axiom 13 (p6_1): fresh88(X, X, Y, Z, W) = true2.
% 0.17/0.51 Axiom 14 (p28): fresh45(X, X, Y, Z, W) = fresh46(ilf_type(Y, set_type), true2, W).
% 0.17/0.51 Axiom 15 (p13_1): fresh28(relation_like(X), true2, X) = fresh27(ilf_type(X, set_type), true2, X).
% 0.17/0.52 Axiom 16 (p3_1): fresh9(X, X, Y, Z, W) = subset(range_of(W), Z).
% 0.17/0.52 Axiom 17 (p1): fresh3(subset(X, empty_set), true2, X) = fresh35(ilf_type(X, set_type), true2, X).
% 0.17/0.52 Axiom 18 (p2_1): fresh2(ilf_type(X, binary_relation_type), true2, X) = fresh14(range_of(X), empty_set, X).
% 0.17/0.52 Axiom 19 (p6_1): fresh7(X, X, Y, Z, W) = ilf_type(W, subset_type(cross_product(Y, Z))).
% 0.17/0.52 Axiom 20 (p3_1): fresh91(X, X, Y, Z, W) = fresh92(ilf_type(Y, set_type), true2, Z, W).
% 0.17/0.52 Axiom 21 (p6_1): fresh87(X, X, Y, Z, W) = fresh88(ilf_type(Y, set_type), true2, Y, Z, W).
% 0.17/0.52 Axiom 22 (p3_1): fresh91(ilf_type(X, relation_type(Y, Z)), true2, Y, Z, X) = fresh9(ilf_type(Z, set_type), true2, Y, Z, X).
% 0.17/0.52 Axiom 23 (p6_1): fresh87(ilf_type(X, relation_type(Y, Z)), true2, Y, Z, X) = fresh7(ilf_type(Z, set_type), true2, Y, Z, X).
% 0.17/0.52 Axiom 24 (p28): fresh45(ilf_type(X, subset_type(cross_product(Y, Z))), true2, Y, Z, X) = fresh16(ilf_type(Z, set_type), true2, Y, X).
% 0.17/0.52
% 0.17/0.52 Goal 1 (prove_relset_1_27_4): d = empty_set.
% 0.17/0.52 Proof:
% 0.17/0.52 d
% 0.17/0.52 = { by axiom 8 (p2_1) R->L }
% 0.17/0.52 fresh2(true2, true2, d)
% 0.17/0.52 = { by axiom 5 (p13_1) R->L }
% 0.17/0.52 fresh2(fresh27(true2, true2, d), true2, d)
% 0.17/0.52 = { by axiom 1 (p33) R->L }
% 0.17/0.52 fresh2(fresh27(ilf_type(d, set_type), true2, d), true2, d)
% 0.17/0.52 = { by axiom 15 (p13_1) R->L }
% 0.17/0.52 fresh2(fresh28(relation_like(d), true2, d), true2, d)
% 0.17/0.52 = { by axiom 10 (p28) R->L }
% 0.17/0.52 fresh2(fresh28(fresh16(true2, true2, c, d), true2, d), true2, d)
% 0.17/0.52 = { by axiom 1 (p33) R->L }
% 0.17/0.52 fresh2(fresh28(fresh16(ilf_type(b, set_type), true2, c, d), true2, d), true2, d)
% 0.17/0.52 = { by axiom 24 (p28) R->L }
% 0.17/0.52 fresh2(fresh28(fresh45(ilf_type(d, subset_type(cross_product(c, b))), true2, c, b, d), true2, d), true2, d)
% 0.17/0.52 = { by axiom 19 (p6_1) R->L }
% 0.17/0.52 fresh2(fresh28(fresh45(fresh7(true2, true2, c, b, d), true2, c, b, d), true2, d), true2, d)
% 0.17/0.52 = { by axiom 1 (p33) R->L }
% 0.17/0.52 fresh2(fresh28(fresh45(fresh7(ilf_type(b, set_type), true2, c, b, d), true2, c, b, d), true2, d), true2, d)
% 0.17/0.52 = { by axiom 23 (p6_1) R->L }
% 0.17/0.52 fresh2(fresh28(fresh45(fresh87(ilf_type(d, relation_type(c, b)), true2, c, b, d), true2, c, b, d), true2, d), true2, d)
% 0.17/0.52 = { by axiom 11 (prove_relset_1_27_3) }
% 0.17/0.52 fresh2(fresh28(fresh45(fresh87(true2, true2, c, b, d), true2, c, b, d), true2, d), true2, d)
% 0.17/0.52 = { by axiom 21 (p6_1) }
% 0.17/0.52 fresh2(fresh28(fresh45(fresh88(ilf_type(c, set_type), true2, c, b, d), true2, c, b, d), true2, d), true2, d)
% 0.17/0.52 = { by axiom 1 (p33) }
% 0.17/0.52 fresh2(fresh28(fresh45(fresh88(true2, true2, c, b, d), true2, c, b, d), true2, d), true2, d)
% 0.17/0.52 = { by axiom 13 (p6_1) }
% 0.17/0.52 fresh2(fresh28(fresh45(true2, true2, c, b, d), true2, d), true2, d)
% 0.17/0.52 = { by axiom 14 (p28) }
% 0.17/0.52 fresh2(fresh28(fresh46(ilf_type(c, set_type), true2, d), true2, d), true2, d)
% 0.17/0.52 = { by axiom 1 (p33) }
% 0.17/0.52 fresh2(fresh28(fresh46(true2, true2, d), true2, d), true2, d)
% 0.17/0.52 = { by axiom 2 (p28) }
% 0.17/0.52 fresh2(fresh28(true2, true2, d), true2, d)
% 0.17/0.52 = { by axiom 4 (p13_1) }
% 0.17/0.52 fresh2(ilf_type(d, binary_relation_type), true2, d)
% 0.17/0.52 = { by axiom 18 (p2_1) }
% 0.17/0.52 fresh14(range_of(d), empty_set, d)
% 0.17/0.52 = { by axiom 7 (p1) R->L }
% 0.17/0.52 fresh14(fresh3(true2, true2, range_of(d)), empty_set, d)
% 0.17/0.52 = { by axiom 9 (p3_1) R->L }
% 0.17/0.52 fresh14(fresh3(fresh92(true2, true2, empty_set, d), true2, range_of(d)), empty_set, d)
% 0.17/0.52 = { by axiom 1 (p33) R->L }
% 0.17/0.52 fresh14(fresh3(fresh92(ilf_type(c, set_type), true2, empty_set, d), true2, range_of(d)), empty_set, d)
% 0.17/0.52 = { by axiom 20 (p3_1) R->L }
% 0.17/0.52 fresh14(fresh3(fresh91(true2, true2, c, empty_set, d), true2, range_of(d)), empty_set, d)
% 0.17/0.52 = { by axiom 12 (prove_relset_1_27_2) R->L }
% 0.17/0.52 fresh14(fresh3(fresh91(ilf_type(d, relation_type(c, empty_set)), true2, c, empty_set, d), true2, range_of(d)), empty_set, d)
% 0.17/0.52 = { by axiom 22 (p3_1) }
% 0.17/0.52 fresh14(fresh3(fresh9(ilf_type(empty_set, set_type), true2, c, empty_set, d), true2, range_of(d)), empty_set, d)
% 0.17/0.52 = { by axiom 1 (p33) }
% 0.17/0.52 fresh14(fresh3(fresh9(true2, true2, c, empty_set, d), true2, range_of(d)), empty_set, d)
% 0.17/0.52 = { by axiom 16 (p3_1) }
% 0.17/0.52 fresh14(fresh3(subset(range_of(d), empty_set), true2, range_of(d)), empty_set, d)
% 0.17/0.52 = { by axiom 17 (p1) }
% 0.17/0.52 fresh14(fresh35(ilf_type(range_of(d), set_type), true2, range_of(d)), empty_set, d)
% 0.17/0.52 = { by axiom 1 (p33) }
% 0.17/0.52 fresh14(fresh35(true2, true2, range_of(d)), empty_set, d)
% 0.17/0.52 = { by axiom 3 (p1) }
% 0.17/0.52 fresh14(empty_set, empty_set, d)
% 0.17/0.52 = { by axiom 6 (p2_1) }
% 0.17/0.52 empty_set
% 0.17/0.52 % SZS output end Proof
% 0.17/0.52
% 0.17/0.52 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------