%------------------------------------------------------------------------------
% File     : Twee---2.4.2
% Problem  : SET621+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 : n012.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 15:32:43 EDT 2023

% Result   : Theorem 0.18s 0.44s
% Output   : Proof 0.18s
% Verified : 
% SZS Type : -

% Comments : 
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12  % Problem  : SET621+3 : TPTP v8.1.2. Released v2.2.0.
% 0.11/0.13  % Command  : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33  % Computer : n012.cluster.edu
% 0.12/0.33  % Model    : x86_64 x86_64
% 0.12/0.33  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33  % Memory   : 8042.1875MB
% 0.12/0.33  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33  % CPULimit : 300
% 0.12/0.33  % WCLimit  : 300
% 0.12/0.33  % DateTime : Sat Aug 26 08:43:41 EDT 2023
% 0.12/0.34  % CPUTime  : 
% 0.18/0.44  Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.18/0.44  
% 0.18/0.44  % SZS status Theorem
% 0.18/0.44  
% 0.18/0.44  % SZS output start Proof
% 0.18/0.44  Take the following subset of the input axioms:
% 0.18/0.44    fof(difference_difference_union, axiom, ![B, C, D]: difference(difference(B, C), D)=difference(B, union(C, D))).
% 0.18/0.44    fof(difference_distributes_over_union, axiom, ![B2, C2, D2]: difference(union(B2, C2), D2)=union(difference(B2, D2), difference(C2, D2))).
% 0.18/0.44    fof(prove_th97, conjecture, ![B2, C2, D2]: difference(symmetric_difference(B2, C2), D2)=union(difference(B2, union(C2, D2)), difference(C2, union(B2, D2)))).
% 0.18/0.44    fof(symmetric_difference_defn, axiom, ![B2, C2]: symmetric_difference(B2, C2)=union(difference(B2, C2), difference(C2, B2))).
% 0.18/0.44  
% 0.18/0.44  Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.18/0.44  http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.18/0.44  We repeatedly replace C & s=t => u=v by the two clauses:
% 0.18/0.44    fresh(y, y, x1...xn) = u
% 0.18/0.44    C => fresh(s, t, x1...xn) = v
% 0.18/0.44  where fresh is a fresh function symbol and x1..xn are the free
% 0.18/0.44  variables of u and v.
% 0.18/0.44  A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.18/0.44  input problem has no model of domain size 1).
% 0.18/0.44  
% 0.18/0.44  The encoding turns the above axioms into the following unit equations and goals:
% 0.18/0.45  
% 0.18/0.45  Axiom 1 (difference_difference_union): difference(difference(X, Y), Z) = difference(X, union(Y, Z)).
% 0.18/0.45  Axiom 2 (symmetric_difference_defn): symmetric_difference(X, Y) = union(difference(X, Y), difference(Y, X)).
% 0.18/0.45  Axiom 3 (difference_distributes_over_union): difference(union(X, Y), Z) = union(difference(X, Z), difference(Y, Z)).
% 0.18/0.45  
% 0.18/0.45  Goal 1 (prove_th97): difference(symmetric_difference(b, c), d) = union(difference(b, union(c, d)), difference(c, union(b, d))).
% 0.18/0.45  Proof:
% 0.18/0.45    difference(symmetric_difference(b, c), d)
% 0.18/0.45  = { by axiom 2 (symmetric_difference_defn) }
% 0.18/0.45    difference(union(difference(b, c), difference(c, b)), d)
% 0.18/0.45  = { by axiom 3 (difference_distributes_over_union) }
% 0.18/0.45    union(difference(difference(b, c), d), difference(difference(c, b), d))
% 0.18/0.45  = { by axiom 1 (difference_difference_union) }
% 0.18/0.45    union(difference(b, union(c, d)), difference(difference(c, b), d))
% 0.18/0.45  = { by axiom 1 (difference_difference_union) }
% 0.18/0.45    union(difference(b, union(c, d)), difference(c, union(b, d)))
% 0.18/0.45  % SZS output end Proof
% 0.18/0.45  
% 0.18/0.45  RESULT: Theorem (the conjecture is true).
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