TSTP Solution File: REL048-1 by Twee---2.4.2

View Problem - Process Solution

%------------------------------------------------------------------------------
% File     : Twee---2.4.2
% Problem  : REL048-1 : TPTP v8.1.2. Released v4.0.0.
% 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 13:44:32 EDT 2023

% Result   : Unsatisfiable 0.21s 0.45s
% Output   : Proof 0.21s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12  % Problem  : REL048-1 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.13  % Command  : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.35  % Computer : n027.cluster.edu
% 0.14/0.35  % Model    : x86_64 x86_64
% 0.14/0.35  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35  % Memory   : 8042.1875MB
% 0.14/0.35  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35  % CPULimit : 300
% 0.14/0.35  % WCLimit  : 300
% 0.14/0.35  % DateTime : Fri Aug 25 19:36:04 EDT 2023
% 0.14/0.35  % CPUTime  : 
% 0.21/0.45  Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.21/0.45  
% 0.21/0.45  % SZS status Unsatisfiable
% 0.21/0.45  
% 0.21/0.46  % SZS output start Proof
% 0.21/0.46  Take the following subset of the input axioms:
% 0.21/0.46    fof(composition_associativity_5, axiom, ![A, B, C]: composition(A, composition(B, C))=composition(composition(A, B), C)).
% 0.21/0.46    fof(composition_identity_6, axiom, ![A2]: composition(A2, one)=A2).
% 0.21/0.46    fof(converse_cancellativity_11, axiom, ![A2, B2]: join(composition(converse(A2), complement(composition(A2, B2))), complement(B2))=complement(B2)).
% 0.21/0.46    fof(converse_idempotence_8, axiom, ![A2]: converse(converse(A2))=A2).
% 0.21/0.46    fof(converse_multiplicativity_10, axiom, ![A2, B2]: converse(composition(A2, B2))=composition(converse(B2), converse(A2))).
% 0.21/0.46    fof(def_top_12, axiom, ![A2]: top=join(A2, complement(A2))).
% 0.21/0.46    fof(def_zero_13, axiom, ![A2]: zero=meet(A2, complement(A2))).
% 0.21/0.46    fof(goals_14, negated_conjecture, join(join(sk1, sk2), sk3)=sk3).
% 0.21/0.46    fof(goals_15, negated_conjecture, join(sk1, sk3)!=sk3 | join(sk2, sk3)!=sk3).
% 0.21/0.46    fof(maddux1_join_commutativity_1, axiom, ![A2, B2]: join(A2, B2)=join(B2, A2)).
% 0.21/0.46    fof(maddux2_join_associativity_2, axiom, ![A2, B2, C2]: join(A2, join(B2, C2))=join(join(A2, B2), C2)).
% 0.21/0.46    fof(maddux3_a_kind_of_de_Morgan_3, axiom, ![A2, B2]: A2=join(complement(join(complement(A2), complement(B2))), complement(join(complement(A2), B2)))).
% 0.21/0.46    fof(maddux4_definiton_of_meet_4, axiom, ![A2, B2]: meet(A2, B2)=complement(join(complement(A2), complement(B2)))).
% 0.21/0.46  
% 0.21/0.46  Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.46  http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.46  We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.46    fresh(y, y, x1...xn) = u
% 0.21/0.46    C => fresh(s, t, x1...xn) = v
% 0.21/0.46  where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.46  variables of u and v.
% 0.21/0.46  A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.46  input problem has no model of domain size 1).
% 0.21/0.46  
% 0.21/0.46  The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.46  
% 0.21/0.46  Axiom 1 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 0.21/0.46  Axiom 2 (composition_identity_6): composition(X, one) = X.
% 0.21/0.46  Axiom 3 (converse_idempotence_8): converse(converse(X)) = X.
% 0.21/0.46  Axiom 4 (def_top_12): top = join(X, complement(X)).
% 0.21/0.46  Axiom 5 (def_zero_13): zero = meet(X, complement(X)).
% 0.21/0.46  Axiom 6 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 0.21/0.46  Axiom 7 (goals_14): join(join(sk1, sk2), sk3) = sk3.
% 0.21/0.46  Axiom 8 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 0.21/0.46  Axiom 9 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 0.21/0.46  Axiom 10 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 0.21/0.46  Axiom 11 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 0.21/0.46  Axiom 12 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 0.21/0.46  
% 0.21/0.46  Lemma 13: complement(top) = zero.
% 0.21/0.46  Proof:
% 0.21/0.46    complement(top)
% 0.21/0.46  = { by axiom 4 (def_top_12) }
% 0.21/0.46    complement(join(complement(X), complement(complement(X))))
% 0.21/0.46  = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 0.21/0.46    meet(X, complement(X))
% 0.21/0.46  = { by axiom 5 (def_zero_13) R->L }
% 0.21/0.46    zero
% 0.21/0.46  
% 0.21/0.46  Lemma 14: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 0.21/0.46  Proof:
% 0.21/0.46    join(meet(X, Y), complement(join(complement(X), Y)))
% 0.21/0.46  = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 0.21/0.46    join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 0.21/0.46  = { by axiom 12 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 0.21/0.46    X
% 0.21/0.46  
% 0.21/0.46  Lemma 15: composition(converse(one), X) = X.
% 0.21/0.46  Proof:
% 0.21/0.46    composition(converse(one), X)
% 0.21/0.46  = { by axiom 3 (converse_idempotence_8) R->L }
% 0.21/0.46    composition(converse(one), converse(converse(X)))
% 0.21/0.46  = { by axiom 8 (converse_multiplicativity_10) R->L }
% 0.21/0.46    converse(composition(converse(X), one))
% 0.21/0.46  = { by axiom 2 (composition_identity_6) }
% 0.21/0.46    converse(converse(X))
% 0.21/0.46  = { by axiom 3 (converse_idempotence_8) }
% 0.21/0.46    X
% 0.21/0.46  
% 0.21/0.46  Lemma 16: join(complement(X), complement(X)) = complement(X).
% 0.21/0.46  Proof:
% 0.21/0.46    join(complement(X), complement(X))
% 0.21/0.46  = { by lemma 15 R->L }
% 0.21/0.46    join(complement(X), composition(converse(one), complement(X)))
% 0.21/0.46  = { by lemma 15 R->L }
% 0.21/0.46    join(complement(X), composition(converse(one), complement(composition(converse(one), X))))
% 0.21/0.46  = { by axiom 2 (composition_identity_6) R->L }
% 0.21/0.46    join(complement(X), composition(converse(one), complement(composition(composition(converse(one), one), X))))
% 0.21/0.46  = { by axiom 9 (composition_associativity_5) R->L }
% 0.21/0.46    join(complement(X), composition(converse(one), complement(composition(converse(one), composition(one, X)))))
% 0.21/0.46  = { by lemma 15 }
% 0.21/0.46    join(complement(X), composition(converse(one), complement(composition(one, X))))
% 0.21/0.46  = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 0.21/0.46    join(composition(converse(one), complement(composition(one, X))), complement(X))
% 0.21/0.46  = { by axiom 11 (converse_cancellativity_11) }
% 0.21/0.46    complement(X)
% 0.21/0.46  
% 0.21/0.46  Lemma 17: join(zero, meet(X, X)) = X.
% 0.21/0.46  Proof:
% 0.21/0.46    join(zero, meet(X, X))
% 0.21/0.46  = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 0.21/0.46    join(zero, complement(join(complement(X), complement(X))))
% 0.21/0.46  = { by axiom 5 (def_zero_13) }
% 0.21/0.46    join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 0.21/0.46  = { by lemma 14 }
% 0.21/0.46    X
% 0.21/0.46  
% 0.21/0.46  Lemma 18: meet(X, X) = X.
% 0.21/0.46  Proof:
% 0.21/0.46    meet(X, X)
% 0.21/0.46  = { by lemma 14 R->L }
% 0.21/0.46    join(meet(meet(X, X), complement(meet(X, X))), complement(join(complement(meet(X, X)), complement(meet(X, X)))))
% 0.21/0.46  = { by lemma 16 }
% 0.21/0.46    join(meet(meet(X, X), complement(meet(X, X))), complement(complement(meet(X, X))))
% 0.21/0.46  = { by axiom 5 (def_zero_13) R->L }
% 0.21/0.46    join(zero, complement(complement(meet(X, X))))
% 0.21/0.46  = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 0.21/0.46    join(complement(complement(meet(X, X))), zero)
% 0.21/0.46  = { by lemma 16 R->L }
% 0.21/0.46    join(complement(join(complement(meet(X, X)), complement(meet(X, X)))), zero)
% 0.21/0.46  = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 0.21/0.46    join(meet(meet(X, X), meet(X, X)), zero)
% 0.21/0.46  = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 0.21/0.46    join(zero, meet(meet(X, X), meet(X, X)))
% 0.21/0.46  = { by lemma 13 R->L }
% 0.21/0.46    join(complement(top), meet(meet(X, X), meet(X, X)))
% 0.21/0.46  = { by lemma 16 R->L }
% 0.21/0.46    join(join(complement(top), complement(top)), meet(meet(X, X), meet(X, X)))
% 0.21/0.46  = { by lemma 13 }
% 0.21/0.46    join(join(zero, complement(top)), meet(meet(X, X), meet(X, X)))
% 0.21/0.46  = { by lemma 13 }
% 0.21/0.46    join(join(zero, zero), meet(meet(X, X), meet(X, X)))
% 0.21/0.46  = { by axiom 6 (maddux2_join_associativity_2) R->L }
% 0.21/0.46    join(zero, join(zero, meet(meet(X, X), meet(X, X))))
% 0.21/0.46  = { by lemma 17 }
% 0.21/0.46    join(zero, meet(X, X))
% 0.21/0.46  = { by lemma 17 }
% 0.21/0.47    X
% 0.21/0.47  
% 0.21/0.47  Lemma 19: join(X, X) = X.
% 0.21/0.47  Proof:
% 0.21/0.47    join(X, X)
% 0.21/0.47  = { by lemma 18 R->L }
% 0.21/0.47    join(X, meet(X, X))
% 0.21/0.47  = { by lemma 18 R->L }
% 0.21/0.47    join(meet(X, X), meet(X, X))
% 0.21/0.47  = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 0.21/0.47    join(meet(X, X), complement(join(complement(X), complement(X))))
% 0.21/0.47  = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 0.21/0.47    join(complement(join(complement(X), complement(X))), complement(join(complement(X), complement(X))))
% 0.21/0.47  = { by lemma 16 }
% 0.21/0.47    complement(join(complement(X), complement(X)))
% 0.21/0.47  = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 0.21/0.47    meet(X, X)
% 0.21/0.47  = { by lemma 18 }
% 0.21/0.47    X
% 0.21/0.47  
% 0.21/0.47  Lemma 20: join(Y, join(Z, X)) = join(X, join(Y, Z)).
% 0.21/0.47  Proof:
% 0.21/0.47    join(Y, join(Z, X))
% 0.21/0.47  = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 0.21/0.47    join(join(Z, X), Y)
% 0.21/0.47  = { by axiom 1 (maddux1_join_commutativity_1) }
% 0.21/0.47    join(join(X, Z), Y)
% 0.21/0.47  = { by axiom 6 (maddux2_join_associativity_2) R->L }
% 0.21/0.47    join(X, join(Z, Y))
% 0.21/0.47  = { by axiom 1 (maddux1_join_commutativity_1) }
% 0.21/0.47    join(X, join(Y, Z))
% 0.21/0.47  
% 0.21/0.47  Lemma 21: join(X, join(X, Y)) = join(X, Y).
% 0.21/0.47  Proof:
% 0.21/0.47    join(X, join(X, Y))
% 0.21/0.47  = { by lemma 20 }
% 0.21/0.47    join(Y, join(X, X))
% 0.21/0.47  = { by lemma 19 }
% 0.21/0.47    join(Y, X)
% 0.21/0.47  = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 0.21/0.47    join(X, Y)
% 0.21/0.47  
% 0.21/0.47  Lemma 22: join(Z, join(Y, X)) = join(X, join(Y, Z)).
% 0.21/0.47  Proof:
% 0.21/0.47    join(Z, join(Y, X))
% 0.21/0.47  = { by lemma 20 }
% 0.21/0.47    join(X, join(Z, Y))
% 0.21/0.47  = { by axiom 1 (maddux1_join_commutativity_1) }
% 0.21/0.47    join(X, join(Y, Z))
% 0.21/0.47  
% 0.21/0.47  Lemma 23: join(sk1, join(sk2, sk3)) = sk3.
% 0.21/0.47  Proof:
% 0.21/0.47    join(sk1, join(sk2, sk3))
% 0.21/0.47  = { by axiom 6 (maddux2_join_associativity_2) }
% 0.21/0.47    join(join(sk1, sk2), sk3)
% 0.21/0.47  = { by axiom 7 (goals_14) }
% 0.21/0.47    sk3
% 0.21/0.47  
% 0.21/0.47  Lemma 24: join(sk2, join(sk1, sk3)) = sk3.
% 0.21/0.47  Proof:
% 0.21/0.47    join(sk2, join(sk1, sk3))
% 0.21/0.47  = { by lemma 22 }
% 0.21/0.47    join(sk3, join(sk1, sk2))
% 0.21/0.47  = { by lemma 20 R->L }
% 0.21/0.47    join(sk1, join(sk2, sk3))
% 0.21/0.47  = { by lemma 23 }
% 0.21/0.47    sk3
% 0.21/0.47  
% 0.21/0.47  Goal 1 (goals_15): tuple(join(sk1, sk3), join(sk2, sk3)) = tuple(sk3, sk3).
% 0.21/0.47  Proof:
% 0.21/0.47    tuple(join(sk1, sk3), join(sk2, sk3))
% 0.21/0.47  = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 0.21/0.47    tuple(join(sk1, sk3), join(sk3, sk2))
% 0.21/0.47  = { by lemma 23 R->L }
% 0.21/0.47    tuple(join(sk1, sk3), join(join(sk1, join(sk2, sk3)), sk2))
% 0.21/0.47  = { by axiom 6 (maddux2_join_associativity_2) R->L }
% 0.21/0.47    tuple(join(sk1, sk3), join(sk1, join(join(sk2, sk3), sk2)))
% 0.21/0.47  = { by axiom 1 (maddux1_join_commutativity_1) }
% 0.21/0.47    tuple(join(sk1, sk3), join(sk1, join(sk2, join(sk2, sk3))))
% 0.21/0.47  = { by lemma 21 }
% 0.21/0.47    tuple(join(sk1, sk3), join(sk1, join(sk2, sk3)))
% 0.21/0.47  = { by lemma 23 }
% 0.21/0.47    tuple(join(sk1, sk3), sk3)
% 0.21/0.47  = { by lemma 19 R->L }
% 0.21/0.47    tuple(join(join(sk1, sk3), join(sk1, sk3)), sk3)
% 0.21/0.47  = { by lemma 22 }
% 0.21/0.47    tuple(join(sk3, join(sk1, join(sk1, sk3))), sk3)
% 0.21/0.47  = { by lemma 21 }
% 0.21/0.47    tuple(join(sk3, join(sk1, sk3)), sk3)
% 0.21/0.47  = { by lemma 24 R->L }
% 0.21/0.47    tuple(join(join(sk2, join(sk1, sk3)), join(sk1, sk3)), sk3)
% 0.21/0.47  = { by axiom 6 (maddux2_join_associativity_2) R->L }
% 0.21/0.47    tuple(join(sk2, join(join(sk1, sk3), join(sk1, sk3))), sk3)
% 0.21/0.47  = { by lemma 19 }
% 0.21/0.47    tuple(join(sk2, join(sk1, sk3)), sk3)
% 0.21/0.47  = { by lemma 24 }
% 0.21/0.47    tuple(sk3, sk3)
% 0.21/0.47  % SZS output end Proof
% 0.21/0.47  
% 0.21/0.47  RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------