%------------------------------------------------------------------------------
% File     : Twee---2.4.2
% Problem  : SET002+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 : n005.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 15:30:28 EDT 2023

% Result   : Theorem 0.19s 0.39s
% Output   : Proof 0.19s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12  % Problem  : SET002+3 : TPTP v8.1.2. Released v2.2.0.
% 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 : n005.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 08:54:38 EDT 2023
% 0.12/0.34  % CPUTime  : 
% 0.19/0.39  Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.19/0.39  
% 0.19/0.39  % SZS status Theorem
% 0.19/0.39  
% 0.19/0.39  % SZS output start Proof
% 0.19/0.39  Take the following subset of the input axioms:
% 0.19/0.39    fof(equal_defn, axiom, ![B, C]: (B=C <=> (subset(B, C) & subset(C, B)))).
% 0.19/0.39    fof(prove_idempotency_of_union, conjecture, ![B2]: union(B2, B2)=B2).
% 0.19/0.39    fof(subset_union, axiom, ![B2, C2]: (subset(B2, C2) => union(B2, C2)=C2)).
% 0.19/0.39  
% 0.19/0.39  Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.39  http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.39  We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.39    fresh(y, y, x1...xn) = u
% 0.19/0.39    C => fresh(s, t, x1...xn) = v
% 0.19/0.39  where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.39  variables of u and v.
% 0.19/0.39  A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.39  input problem has no model of domain size 1).
% 0.19/0.39  
% 0.19/0.39  The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.39  
% 0.19/0.39  Axiom 1 (equal_defn): subset(X, X) = true.
% 0.19/0.39  Axiom 2 (subset_union): fresh5(X, X, Y, Z) = Z.
% 0.19/0.39  Axiom 3 (subset_union): fresh5(subset(X, Y), true, X, Y) = union(X, Y).
% 0.19/0.39  
% 0.19/0.39  Goal 1 (prove_idempotency_of_union): union(b, b) = b.
% 0.19/0.39  Proof:
% 0.19/0.39    union(b, b)
% 0.19/0.39  = { by axiom 3 (subset_union) R->L }
% 0.19/0.39    fresh5(subset(b, b), true, b, b)
% 0.19/0.39  = { by axiom 1 (equal_defn) }
% 0.19/0.39    fresh5(true, true, b, b)
% 0.19/0.39  = { by axiom 2 (subset_union) }
% 0.19/0.39    b
% 0.19/0.39  % SZS output end Proof
% 0.19/0.39  
% 0.19/0.39  RESULT: Theorem (the conjecture is true).
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