%------------------------------------------------------------------------------
% File     : Twee---2.4.2
% Problem  : SET604+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 15:32:38 EDT 2023

% Result   : Theorem 0.13s 0.40s
% Output   : Proof 0.13s
% Verified : 
% SZS Type : -

% Comments : 
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%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12  % Problem  : SET604+3 : TPTP v8.1.2. Released v2.2.0.
% 0.07/0.13  % Command  : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34  % Computer : n005.cluster.edu
% 0.13/0.34  % Model    : x86_64 x86_64
% 0.13/0.34  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34  % Memory   : 8042.1875MB
% 0.13/0.34  % 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 12:32:53 EDT 2023
% 0.13/0.35  % CPUTime  : 
% 0.13/0.40  Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.13/0.40  
% 0.13/0.40  % SZS status Theorem
% 0.13/0.40  
% 0.13/0.40  % SZS output start Proof
% 0.13/0.40  Take the following subset of the input axioms:
% 0.13/0.40    fof(difference_empty_set, axiom, ![B, C]: (difference(B, C)=empty_set <=> subset(B, C))).
% 0.13/0.40    fof(empty_set_subset, axiom, ![B2]: subset(empty_set, B2)).
% 0.13/0.40    fof(prove_no_difference_with_empty_set, conjecture, ![B2]: difference(empty_set, B2)=empty_set).
% 0.13/0.40  
% 0.13/0.40  Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.13/0.40  http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.13/0.40  We repeatedly replace C & s=t => u=v by the two clauses:
% 0.13/0.40    fresh(y, y, x1...xn) = u
% 0.13/0.40    C => fresh(s, t, x1...xn) = v
% 0.13/0.40  where fresh is a fresh function symbol and x1..xn are the free
% 0.13/0.40  variables of u and v.
% 0.13/0.40  A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.13/0.40  input problem has no model of domain size 1).
% 0.13/0.40  
% 0.13/0.40  The encoding turns the above axioms into the following unit equations and goals:
% 0.13/0.40  
% 0.13/0.40  Axiom 1 (empty_set_subset): subset(empty_set, X) = true2.
% 0.13/0.40  Axiom 2 (difference_empty_set_1): fresh6(X, X, Y, Z) = empty_set.
% 0.13/0.40  Axiom 3 (difference_empty_set_1): fresh6(subset(X, Y), true2, X, Y) = difference(X, Y).
% 0.13/0.40  
% 0.13/0.40  Goal 1 (prove_no_difference_with_empty_set): difference(empty_set, b) = empty_set.
% 0.13/0.40  Proof:
% 0.13/0.40    difference(empty_set, b)
% 0.13/0.40  = { by axiom 3 (difference_empty_set_1) R->L }
% 0.13/0.40    fresh6(subset(empty_set, b), true2, empty_set, b)
% 0.13/0.40  = { by axiom 1 (empty_set_subset) }
% 0.13/0.40    fresh6(true2, true2, empty_set, b)
% 0.13/0.40  = { by axiom 2 (difference_empty_set_1) }
% 0.13/0.40    empty_set
% 0.13/0.40  % SZS output end Proof
% 0.13/0.40  
% 0.13/0.40  RESULT: Theorem (the conjecture is true).
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