TSTP Solution File: COL003-5 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : COL003-5 : TPTP v8.1.2. Released v1.0.0.
% 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 : n015.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 : Wed Aug 30 18:31:34 EDT 2023
% Result : Unsatisfiable 0.12s 0.41s
% Output : Proof 0.12s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : COL003-5 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.34 % Computer : n015.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Sun Aug 27 04:37:38 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.12/0.41 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.12/0.41
% 0.12/0.41 % SZS status Unsatisfiable
% 0.12/0.41
% 0.12/0.42 % SZS output start Proof
% 0.12/0.42 Take the following subset of the input axioms:
% 0.12/0.42 fof(b_definition, axiom, ![X, Y, Z]: apply(apply(apply(b, X), Y), Z)=apply(X, apply(Y, Z))).
% 0.12/0.42 fof(prove_strong_fixed_point, negated_conjecture, ~fixed_point(apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b)))).
% 0.12/0.42 fof(strong_fixed_point, axiom, ![Strong_fixed_point]: (apply(Strong_fixed_point, fixed_pt)!=apply(fixed_pt, apply(Strong_fixed_point, fixed_pt)) | fixed_point(Strong_fixed_point))).
% 0.12/0.42 fof(w_definition, axiom, ![X2, Y2]: apply(apply(w, X2), Y2)=apply(apply(X2, Y2), Y2)).
% 0.12/0.42
% 0.12/0.42 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.12/0.42 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.12/0.42 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.12/0.42 fresh(y, y, x1...xn) = u
% 0.12/0.42 C => fresh(s, t, x1...xn) = v
% 0.12/0.42 where fresh is a fresh function symbol and x1..xn are the free
% 0.12/0.42 variables of u and v.
% 0.12/0.42 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.12/0.42 input problem has no model of domain size 1).
% 0.12/0.42
% 0.12/0.42 The encoding turns the above axioms into the following unit equations and goals:
% 0.12/0.42
% 0.12/0.42 Axiom 1 (strong_fixed_point): fresh(X, X, Y) = true.
% 0.12/0.42 Axiom 2 (w_definition): apply(apply(w, X), Y) = apply(apply(X, Y), Y).
% 0.12/0.42 Axiom 3 (b_definition): apply(apply(apply(b, X), Y), Z) = apply(X, apply(Y, Z)).
% 0.12/0.42 Axiom 4 (strong_fixed_point): fresh(apply(X, fixed_pt), apply(fixed_pt, apply(X, fixed_pt)), X) = fixed_point(X).
% 0.12/0.42
% 0.12/0.42 Lemma 5: apply(apply(w, w), X) = apply(apply(X, X), X).
% 0.12/0.42 Proof:
% 0.12/0.42 apply(apply(w, w), X)
% 0.12/0.42 = { by axiom 2 (w_definition) }
% 0.12/0.42 apply(apply(w, X), X)
% 0.12/0.42 = { by axiom 2 (w_definition) }
% 0.12/0.42 apply(apply(X, X), X)
% 0.12/0.42
% 0.12/0.42 Goal 1 (prove_strong_fixed_point): fixed_point(apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b))) = true.
% 0.12/0.42 Proof:
% 0.12/0.42 fixed_point(apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b)))
% 0.12/0.42 = { by axiom 4 (strong_fixed_point) R->L }
% 0.12/0.42 fresh(apply(apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b)), fixed_pt), apply(fixed_pt, apply(apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b)), fixed_pt)), apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b)))
% 0.12/0.42 = { by axiom 3 (b_definition) }
% 0.12/0.42 fresh(apply(apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b)), fixed_pt), apply(fixed_pt, apply(apply(apply(b, apply(w, w)), w), apply(apply(apply(b, b), b), fixed_pt))), apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b)))
% 0.12/0.42 = { by axiom 3 (b_definition) }
% 0.12/0.42 fresh(apply(apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b)), fixed_pt), apply(fixed_pt, apply(apply(w, w), apply(w, apply(apply(apply(b, b), b), fixed_pt)))), apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b)))
% 0.12/0.42 = { by lemma 5 }
% 0.12/0.42 fresh(apply(apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b)), fixed_pt), apply(fixed_pt, apply(apply(apply(w, apply(apply(apply(b, b), b), fixed_pt)), apply(w, apply(apply(apply(b, b), b), fixed_pt))), apply(w, apply(apply(apply(b, b), b), fixed_pt)))), apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b)))
% 0.12/0.42 = { by axiom 3 (b_definition) R->L }
% 0.12/0.42 fresh(apply(apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b)), fixed_pt), apply(apply(apply(b, fixed_pt), apply(apply(w, apply(apply(apply(b, b), b), fixed_pt)), apply(w, apply(apply(apply(b, b), b), fixed_pt)))), apply(w, apply(apply(apply(b, b), b), fixed_pt))), apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b)))
% 0.12/0.42 = { by axiom 3 (b_definition) R->L }
% 0.12/0.42 fresh(apply(apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b)), fixed_pt), apply(apply(apply(apply(b, apply(b, fixed_pt)), apply(w, apply(apply(apply(b, b), b), fixed_pt))), apply(w, apply(apply(apply(b, b), b), fixed_pt))), apply(w, apply(apply(apply(b, b), b), fixed_pt))), apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b)))
% 0.12/0.42 = { by axiom 2 (w_definition) R->L }
% 0.12/0.42 fresh(apply(apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b)), fixed_pt), apply(apply(apply(w, apply(b, apply(b, fixed_pt))), apply(w, apply(apply(apply(b, b), b), fixed_pt))), apply(w, apply(apply(apply(b, b), b), fixed_pt))), apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b)))
% 0.12/0.42 = { by axiom 3 (b_definition) R->L }
% 0.12/0.43 fresh(apply(apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b)), fixed_pt), apply(apply(apply(w, apply(apply(apply(b, b), b), fixed_pt)), apply(w, apply(apply(apply(b, b), b), fixed_pt))), apply(w, apply(apply(apply(b, b), b), fixed_pt))), apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b)))
% 0.12/0.43 = { by lemma 5 R->L }
% 0.12/0.43 fresh(apply(apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b)), fixed_pt), apply(apply(w, w), apply(w, apply(apply(apply(b, b), b), fixed_pt))), apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b)))
% 0.12/0.43 = { by axiom 3 (b_definition) R->L }
% 0.12/0.43 fresh(apply(apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b)), fixed_pt), apply(apply(apply(b, apply(w, w)), w), apply(apply(apply(b, b), b), fixed_pt)), apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b)))
% 0.12/0.43 = { by axiom 3 (b_definition) R->L }
% 0.12/0.43 fresh(apply(apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b)), fixed_pt), apply(apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b)), fixed_pt), apply(apply(b, apply(apply(b, apply(w, w)), w)), apply(apply(b, b), b)))
% 0.12/0.43 = { by axiom 1 (strong_fixed_point) }
% 0.12/0.43 true
% 0.12/0.43 % SZS output end Proof
% 0.12/0.43
% 0.12/0.43 RESULT: Unsatisfiable (the axioms are contradictory).
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