%------------------------------------------------------------------------------
% File     : Twee---2.4.2
% Problem  : SET583+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 : n017.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 15:32:30 EDT 2023

% Result   : Theorem 0.18s 0.37s
% Output   : Proof 0.18s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12  % Problem  : SET583+3 : TPTP v8.1.2. Released v2.2.0.
% 0.06/0.12  % 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 : n017.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 07:49:10 EDT 2023
% 0.12/0.34  % CPUTime  : 
% 0.18/0.37  Command-line arguments: --no-flatten-goal
% 0.18/0.37  
% 0.18/0.37  % SZS status Theorem
% 0.18/0.37  
% 0.18/0.38  % SZS output start Proof
% 0.18/0.38  Take the following subset of the input axioms:
% 0.18/0.38    fof(equal_defn, axiom, ![B, C]: (B=C <=> (subset(B, C) & subset(C, B)))).
% 0.18/0.38    fof(prove_extensionality, conjecture, ![B2, C2]: ((subset(B2, C2) & subset(C2, B2)) => B2=C2)).
% 0.18/0.38  
% 0.18/0.38  Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.18/0.38  http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.18/0.38  We repeatedly replace C & s=t => u=v by the two clauses:
% 0.18/0.38    fresh(y, y, x1...xn) = u
% 0.18/0.38    C => fresh(s, t, x1...xn) = v
% 0.18/0.38  where fresh is a fresh function symbol and x1..xn are the free
% 0.18/0.38  variables of u and v.
% 0.18/0.38  A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.18/0.38  input problem has no model of domain size 1).
% 0.18/0.38  
% 0.18/0.38  The encoding turns the above axioms into the following unit equations and goals:
% 0.18/0.38  
% 0.18/0.38  Axiom 1 (prove_extensionality): subset(b, c) = true.
% 0.18/0.38  Axiom 2 (prove_extensionality_1): subset(c, b) = true.
% 0.18/0.38  Axiom 3 (equal_defn_1): fresh(X, X, Y, Z) = Z.
% 0.18/0.38  Axiom 4 (equal_defn_1): fresh2(X, X, Y, Z) = Y.
% 0.18/0.38  Axiom 5 (equal_defn_1): fresh2(subset(X, Y), true, Y, X) = fresh(subset(Y, X), true, Y, X).
% 0.18/0.38  
% 0.18/0.38  Goal 1 (prove_extensionality_2): b = c.
% 0.18/0.38  Proof:
% 0.18/0.38    b
% 0.18/0.38  = { by axiom 3 (equal_defn_1) R->L }
% 0.18/0.38    fresh(true, true, c, b)
% 0.18/0.38  = { by axiom 2 (prove_extensionality_1) R->L }
% 0.18/0.38    fresh(subset(c, b), true, c, b)
% 0.18/0.38  = { by axiom 5 (equal_defn_1) R->L }
% 0.18/0.38    fresh2(subset(b, c), true, c, b)
% 0.18/0.38  = { by axiom 1 (prove_extensionality) }
% 0.18/0.38    fresh2(true, true, c, b)
% 0.18/0.38  = { by axiom 4 (equal_defn_1) }
% 0.18/0.38    c
% 0.18/0.38  % SZS output end Proof
% 0.18/0.38  
% 0.18/0.38  RESULT: Theorem (the conjecture is true).
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