%------------------------------------------------------------------------------
% File     : Twee---2.4.2
% Problem  : SET618+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 : n027.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 15:32:42 EDT 2023

% Result   : Theorem 0.20s 0.39s
% Output   : Proof 0.20s
% Verified : 
% SZS Type : -

% Comments : 
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12  % Problem  : SET618+3 : TPTP v8.1.2. Released v2.2.0.
% 0.00/0.13  % 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 : n027.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 11:31:04 EDT 2023
% 0.12/0.34  % CPUTime  : 
% 0.20/0.39  Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.20/0.39  
% 0.20/0.39  % SZS status Theorem
% 0.20/0.39  
% 0.20/0.39  % SZS output start Proof
% 0.20/0.39  Take the following subset of the input axioms:
% 0.20/0.39    fof(idempotency_of_union, axiom, ![B]: union(B, B)=B).
% 0.20/0.39    fof(prove_th93, conjecture, ![B2]: symmetric_difference(B2, B2)=empty_set).
% 0.20/0.39    fof(self_difference_is_empty_set, axiom, ![B2]: difference(B2, B2)=empty_set).
% 0.20/0.39    fof(symmetric_difference_defn, axiom, ![C, B2]: symmetric_difference(B2, C)=union(difference(B2, C), difference(C, B2))).
% 0.20/0.39  
% 0.20/0.39  Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.39  http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.39  We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.39    fresh(y, y, x1...xn) = u
% 0.20/0.39    C => fresh(s, t, x1...xn) = v
% 0.20/0.39  where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.39  variables of u and v.
% 0.20/0.39  A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.39  input problem has no model of domain size 1).
% 0.20/0.39  
% 0.20/0.39  The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.39  
% 0.20/0.39  Axiom 1 (self_difference_is_empty_set): difference(X, X) = empty_set.
% 0.20/0.39  Axiom 2 (idempotency_of_union): union(X, X) = X.
% 0.20/0.39  Axiom 3 (symmetric_difference_defn): symmetric_difference(X, Y) = union(difference(X, Y), difference(Y, X)).
% 0.20/0.39  
% 0.20/0.39  Goal 1 (prove_th93): symmetric_difference(b, b) = empty_set.
% 0.20/0.39  Proof:
% 0.20/0.39    symmetric_difference(b, b)
% 0.20/0.39  = { by axiom 3 (symmetric_difference_defn) }
% 0.20/0.39    union(difference(b, b), difference(b, b))
% 0.20/0.39  = { by axiom 1 (self_difference_is_empty_set) }
% 0.20/0.39    union(empty_set, difference(b, b))
% 0.20/0.39  = { by axiom 1 (self_difference_is_empty_set) }
% 0.20/0.39    union(empty_set, empty_set)
% 0.20/0.39  = { by axiom 2 (idempotency_of_union) }
% 0.20/0.39    empty_set
% 0.20/0.39  % SZS output end Proof
% 0.20/0.39  
% 0.20/0.39  RESULT: Theorem (the conjecture is true).
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