TSTP Solution File: SET810+4 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SET810+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 : n011.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 15:33:18 EDT 2023
% Result : Theorem 0.20s 0.68s
% Output : Proof 2.56s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : SET810+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
% 0.13/0.35 % Computer : n011.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Sat Aug 26 11:15:24 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.20/0.68 Command-line arguments: --flip-ordering --lhs-weight 1 --depth-weight 60 --distributivity-heuristic
% 0.20/0.68
% 0.20/0.68 % SZS status Theorem
% 0.20/0.68
% 2.33/0.69 % SZS output start Proof
% 2.33/0.69 Take the following subset of the input axioms:
% 2.33/0.69 fof(difference, axiom, ![A, B, E]: (member(B, difference(E, A)) <=> (member(B, E) & ~member(B, A)))).
% 2.33/0.69 fof(empty_set, axiom, ![X]: ~member(X, empty_set)).
% 2.33/0.69 fof(ordinal_number, axiom, ![A2]: (member(A2, on) <=> (set(A2) & (strict_well_order(member_predicate, A2) & ![X2]: (member(X2, A2) => subset(X2, A2)))))).
% 2.33/0.69 fof(rel_member, axiom, ![Y, X2]: (apply(member_predicate, X2, Y) <=> member(X2, Y))).
% 2.33/0.69 fof(strict_order, axiom, ![R, E2]: (strict_order(R, E2) <=> (![X2, Y2]: ((member(X2, E2) & member(Y2, E2)) => ~(apply(R, X2, Y2) & apply(R, Y2, X2))) & ![Z, X2, Y2]: ((member(X2, E2) & (member(Y2, E2) & member(Z, E2))) => ((apply(R, X2, Y2) & apply(R, Y2, Z)) => apply(R, X2, Z)))))).
% 2.33/0.69 fof(strict_well_order, axiom, ![E2, R2]: (strict_well_order(R2, E2) <=> (strict_order(R2, E2) & ![A2_2]: ((subset(A2_2, E2) & ?[X2]: member(X2, A2_2)) => ?[Y2]: least(Y2, R2, A2_2))))).
% 2.33/0.69 fof(subset, axiom, ![B2, A2_2]: (subset(A2_2, B2) <=> ![X2]: (member(X2, A2_2) => member(X2, B2)))).
% 2.33/0.69 fof(thV3, conjecture, ![A3, B2]: ((member(A3, on) & member(B2, on)) => ~(member(A3, B2) & member(B2, A3)))).
% 2.33/0.69
% 2.33/0.69 Now clausify the problem and encode Horn clauses using encoding 3 of
% 2.33/0.69 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 2.33/0.69 We repeatedly replace C & s=t => u=v by the two clauses:
% 2.33/0.69 fresh(y, y, x1...xn) = u
% 2.33/0.69 C => fresh(s, t, x1...xn) = v
% 2.33/0.69 where fresh is a fresh function symbol and x1..xn are the free
% 2.33/0.69 variables of u and v.
% 2.33/0.69 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 2.33/0.69 input problem has no model of domain size 1).
% 2.33/0.69
% 2.56/0.69 The encoding turns the above axioms into the following unit equations and goals:
% 2.56/0.69
% 2.56/0.69 Axiom 1 (thV3_1): member(b, a) = true2.
% 2.56/0.69 Axiom 2 (thV3_2): member(a, on) = true2.
% 2.56/0.69 Axiom 3 (thV3_3): member(a, b) = true2.
% 2.56/0.69 Axiom 4 (ordinal_number_2): fresh33(X, X, Y) = true2.
% 2.56/0.69 Axiom 5 (ordinal_number_3): fresh32(X, X, Y, Z) = subset(Z, Y).
% 2.56/0.69 Axiom 6 (ordinal_number_3): fresh31(X, X, Y, Z) = true2.
% 2.56/0.69 Axiom 7 (rel_member): fresh25(X, X, Y, Z) = true2.
% 2.56/0.69 Axiom 8 (strict_well_order_1): fresh15(X, X, Y, Z) = true2.
% 2.56/0.69 Axiom 9 (subset_1): fresh11(X, X, Y, Z) = true2.
% 2.56/0.69 Axiom 10 (ordinal_number_2): fresh33(member(X, on), true2, X) = strict_well_order(member_predicate, X).
% 2.56/0.69 Axiom 11 (subset_1): fresh12(X, X, Y, Z, W) = member(W, Z).
% 2.56/0.69 Axiom 12 (ordinal_number_3): fresh32(member(X, Y), true2, Y, X) = fresh31(member(Y, on), true2, Y, X).
% 2.56/0.69 Axiom 13 (rel_member): fresh25(member(X, Y), true2, X, Y) = apply(member_predicate, X, Y).
% 2.56/0.69 Axiom 14 (strict_well_order_1): fresh15(strict_well_order(X, Y), true2, X, Y) = strict_order(X, Y).
% 2.56/0.69 Axiom 15 (subset_1): fresh12(member(X, Y), true2, Y, Z, X) = fresh11(subset(Y, Z), true2, Z, X).
% 2.56/0.69
% 2.56/0.69 Goal 1 (strict_order_4): tuple(member(X, Y), member(Z, Y), strict_order(W, Y), apply(W, X, Z), apply(W, Z, X)) = tuple(true2, true2, true2, true2, true2).
% 2.56/0.69 The goal is true when:
% 2.56/0.69 X = b
% 2.56/0.69 Y = a
% 2.56/0.69 Z = a
% 2.56/0.69 W = member_predicate
% 2.56/0.69
% 2.56/0.69 Proof:
% 2.56/0.69 tuple(member(b, a), member(a, a), strict_order(member_predicate, a), apply(member_predicate, b, a), apply(member_predicate, a, b))
% 2.56/0.69 = { by axiom 1 (thV3_1) }
% 2.56/0.69 tuple(true2, member(a, a), strict_order(member_predicate, a), apply(member_predicate, b, a), apply(member_predicate, a, b))
% 2.56/0.69 = { by axiom 13 (rel_member) R->L }
% 2.56/0.69 tuple(true2, member(a, a), strict_order(member_predicate, a), fresh25(member(b, a), true2, b, a), apply(member_predicate, a, b))
% 2.56/0.69 = { by axiom 1 (thV3_1) }
% 2.56/0.69 tuple(true2, member(a, a), strict_order(member_predicate, a), fresh25(true2, true2, b, a), apply(member_predicate, a, b))
% 2.56/0.69 = { by axiom 7 (rel_member) }
% 2.56/0.69 tuple(true2, member(a, a), strict_order(member_predicate, a), true2, apply(member_predicate, a, b))
% 2.56/0.69 = { by axiom 11 (subset_1) R->L }
% 2.56/0.69 tuple(true2, fresh12(true2, true2, b, a, a), strict_order(member_predicate, a), true2, apply(member_predicate, a, b))
% 2.56/0.69 = { by axiom 3 (thV3_3) R->L }
% 2.56/0.69 tuple(true2, fresh12(member(a, b), true2, b, a, a), strict_order(member_predicate, a), true2, apply(member_predicate, a, b))
% 2.56/0.69 = { by axiom 15 (subset_1) }
% 2.56/0.69 tuple(true2, fresh11(subset(b, a), true2, a, a), strict_order(member_predicate, a), true2, apply(member_predicate, a, b))
% 2.56/0.69 = { by axiom 5 (ordinal_number_3) R->L }
% 2.56/0.69 tuple(true2, fresh11(fresh32(true2, true2, a, b), true2, a, a), strict_order(member_predicate, a), true2, apply(member_predicate, a, b))
% 2.56/0.69 = { by axiom 1 (thV3_1) R->L }
% 2.56/0.69 tuple(true2, fresh11(fresh32(member(b, a), true2, a, b), true2, a, a), strict_order(member_predicate, a), true2, apply(member_predicate, a, b))
% 2.56/0.69 = { by axiom 12 (ordinal_number_3) }
% 2.56/0.69 tuple(true2, fresh11(fresh31(member(a, on), true2, a, b), true2, a, a), strict_order(member_predicate, a), true2, apply(member_predicate, a, b))
% 2.56/0.69 = { by axiom 2 (thV3_2) }
% 2.56/0.69 tuple(true2, fresh11(fresh31(true2, true2, a, b), true2, a, a), strict_order(member_predicate, a), true2, apply(member_predicate, a, b))
% 2.56/0.69 = { by axiom 6 (ordinal_number_3) }
% 2.56/0.69 tuple(true2, fresh11(true2, true2, a, a), strict_order(member_predicate, a), true2, apply(member_predicate, a, b))
% 2.56/0.69 = { by axiom 9 (subset_1) }
% 2.56/0.69 tuple(true2, true2, strict_order(member_predicate, a), true2, apply(member_predicate, a, b))
% 2.56/0.69 = { by axiom 14 (strict_well_order_1) R->L }
% 2.56/0.69 tuple(true2, true2, fresh15(strict_well_order(member_predicate, a), true2, member_predicate, a), true2, apply(member_predicate, a, b))
% 2.56/0.69 = { by axiom 10 (ordinal_number_2) R->L }
% 2.56/0.69 tuple(true2, true2, fresh15(fresh33(member(a, on), true2, a), true2, member_predicate, a), true2, apply(member_predicate, a, b))
% 2.56/0.69 = { by axiom 2 (thV3_2) }
% 2.56/0.69 tuple(true2, true2, fresh15(fresh33(true2, true2, a), true2, member_predicate, a), true2, apply(member_predicate, a, b))
% 2.56/0.69 = { by axiom 4 (ordinal_number_2) }
% 2.56/0.69 tuple(true2, true2, fresh15(true2, true2, member_predicate, a), true2, apply(member_predicate, a, b))
% 2.56/0.69 = { by axiom 8 (strict_well_order_1) }
% 2.56/0.69 tuple(true2, true2, true2, true2, apply(member_predicate, a, b))
% 2.56/0.69 = { by axiom 13 (rel_member) R->L }
% 2.56/0.69 tuple(true2, true2, true2, true2, fresh25(member(a, b), true2, a, b))
% 2.56/0.69 = { by axiom 3 (thV3_3) }
% 2.56/0.69 tuple(true2, true2, true2, true2, fresh25(true2, true2, a, b))
% 2.56/0.69 = { by axiom 7 (rel_member) }
% 2.56/0.69 tuple(true2, true2, true2, true2, true2)
% 2.56/0.69 % SZS output end Proof
% 2.56/0.69
% 2.56/0.69 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------