%------------------------------------------------------------------------------
% File     : Twee---2.4.2
% Problem  : SET639+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 : n007.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 15:32:49 EDT 2023

% Result   : Theorem 0.20s 0.40s
% Output   : Proof 0.20s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12  % Problem  : SET639+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
% 0.13/0.34  % Computer : n007.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
% 0.13/0.34  % CPULimit : 300
% 0.13/0.34  % WCLimit  : 300
% 0.13/0.34  % DateTime : Sat Aug 26 09:02:26 EDT 2023
% 0.13/0.35  % CPUTime  : 
% 0.20/0.40  Command-line arguments: --no-flatten-goal
% 0.20/0.40  
% 0.20/0.40  % SZS status Theorem
% 0.20/0.40  
% 0.20/0.40  % SZS output start Proof
% 0.20/0.40  Take the following subset of the input axioms:
% 0.20/0.40    fof(commutativity_of_intersection, axiom, ![B, C]: intersection(B, C)=intersection(C, B)).
% 0.20/0.40    fof(prove_th121, conjecture, ![B2, C2]: ((subset(B2, C2) & intersection(C2, B2)=empty_set) => B2=empty_set)).
% 0.20/0.40    fof(subset_intersection, axiom, ![B2, C2]: (subset(B2, C2) => intersection(B2, C2)=B2)).
% 0.20/0.40  
% 0.20/0.40  Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.40  http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.40  We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.40    fresh(y, y, x1...xn) = u
% 0.20/0.40    C => fresh(s, t, x1...xn) = v
% 0.20/0.40  where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.40  variables of u and v.
% 0.20/0.40  A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.40  input problem has no model of domain size 1).
% 0.20/0.40  
% 0.20/0.40  The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.40  
% 0.20/0.40  Axiom 1 (commutativity_of_intersection): intersection(X, Y) = intersection(Y, X).
% 0.20/0.40  Axiom 2 (prove_th121): intersection(c, b) = empty_set.
% 0.20/0.40  Axiom 3 (prove_th121_1): subset(b, c) = true2.
% 0.20/0.40  Axiom 4 (subset_intersection): fresh5(X, X, Y, Z) = Y.
% 0.20/0.40  Axiom 5 (subset_intersection): fresh5(subset(X, Y), true2, X, Y) = intersection(X, Y).
% 0.20/0.40  
% 0.20/0.40  Goal 1 (prove_th121_2): b = empty_set.
% 0.20/0.40  Proof:
% 0.20/0.40    b
% 0.20/0.40  = { by axiom 4 (subset_intersection) R->L }
% 0.20/0.40    fresh5(true2, true2, b, c)
% 0.20/0.40  = { by axiom 3 (prove_th121_1) R->L }
% 0.20/0.40    fresh5(subset(b, c), true2, b, c)
% 0.20/0.40  = { by axiom 5 (subset_intersection) }
% 0.20/0.40    intersection(b, c)
% 0.20/0.40  = { by axiom 1 (commutativity_of_intersection) }
% 0.20/0.40    intersection(c, b)
% 0.20/0.40  = { by axiom 2 (prove_th121) }
% 0.20/0.40    empty_set
% 0.20/0.40  % SZS output end Proof
% 0.20/0.40  
% 0.20/0.40  RESULT: Theorem (the conjecture is true).
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