%------------------------------------------------------------------------------
% File     : Twee---2.4.2
% Problem  : SET813+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 : n029.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 0.21s 0.51s
% Output   : Proof 0.21s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13  % Problem  : SET813+4 : TPTP v8.1.2. Released v3.2.0.
% 0.00/0.14  % 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 : n029.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 13:20:39 EDT 2023
% 0.13/0.35  % CPUTime  : 
% 0.21/0.51  Command-line arguments: --no-flatten-goal
% 0.21/0.51  
% 0.21/0.51  % SZS status Theorem
% 0.21/0.51  
% 0.21/0.51  % SZS output start Proof
% 0.21/0.51  Take the following subset of the input axioms:
% 0.21/0.51    fof(singleton, axiom, ![X, A2]: (member(X, singleton(A2)) <=> X=A2)).
% 0.21/0.51    fof(successor, axiom, ![X2, A2_2]: (member(X2, suc(A2_2)) <=> member(X2, union(A2_2, singleton(A2_2))))).
% 0.21/0.51    fof(thV12, conjecture, ![A]: (member(A, on) => member(A, suc(A)))).
% 0.21/0.51    fof(union, axiom, ![B, X2, A2_2]: (member(X2, union(A2_2, B)) <=> (member(X2, A2_2) | member(X2, B)))).
% 0.21/0.51  
% 0.21/0.51  Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.51  http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.51  We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.51    fresh(y, y, x1...xn) = u
% 0.21/0.51    C => fresh(s, t, x1...xn) = v
% 0.21/0.51  where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.51  variables of u and v.
% 0.21/0.51  A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.51  input problem has no model of domain size 1).
% 0.21/0.51  
% 0.21/0.51  The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.51  
% 0.21/0.51  Axiom 1 (singleton): member(X, singleton(X)) = true2.
% 0.21/0.51  Axiom 2 (successor): fresh9(X, X, Y, Z) = true2.
% 0.21/0.51  Axiom 3 (union_1): fresh2(X, X, Y, Z, W) = true2.
% 0.21/0.51  Axiom 4 (union_1): fresh2(member(X, Y), true2, X, Z, Y) = member(X, union(Z, Y)).
% 0.21/0.51  Axiom 5 (successor): fresh9(member(X, union(Y, singleton(Y))), true2, Y, X) = member(X, suc(Y)).
% 0.21/0.51  
% 0.21/0.51  Goal 1 (thV12_1): member(a, suc(a)) = true2.
% 0.21/0.51  Proof:
% 0.21/0.51    member(a, suc(a))
% 0.21/0.51  = { by axiom 5 (successor) R->L }
% 0.21/0.51    fresh9(member(a, union(a, singleton(a))), true2, a, a)
% 0.21/0.51  = { by axiom 4 (union_1) R->L }
% 0.21/0.51    fresh9(fresh2(member(a, singleton(a)), true2, a, a, singleton(a)), true2, a, a)
% 0.21/0.51  = { by axiom 1 (singleton) }
% 0.21/0.51    fresh9(fresh2(true2, true2, a, a, singleton(a)), true2, a, a)
% 0.21/0.51  = { by axiom 3 (union_1) }
% 0.21/0.51    fresh9(true2, true2, a, a)
% 0.21/0.51  = { by axiom 2 (successor) }
% 0.21/0.51    true2
% 0.21/0.51  % SZS output end Proof
% 0.21/0.51  
% 0.21/0.51  RESULT: Theorem (the conjecture is true).
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