TSTP Solution File: LCL094-1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL094-1 : 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 : n014.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 08:17:23 EDT 2023
% Result : Unsatisfiable 0.21s 0.60s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LCL094-1 : 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
% 0.13/0.35 % Computer : n014.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % 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 : Thu Aug 24 23:35:33 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.21/0.60 Command-line arguments: --no-flatten-goal
% 0.21/0.60
% 0.21/0.60 % SZS status Unsatisfiable
% 0.21/0.60
% 0.21/0.64 % SZS output start Proof
% 0.21/0.64 Take the following subset of the input axioms:
% 0.21/0.64 fof(condensed_detachment, axiom, ![X, Y]: (~is_a_theorem(implies(X, Y)) | (~is_a_theorem(X) | is_a_theorem(Y)))).
% 0.21/0.64 fof(ic_JLukasiewicz_4, axiom, ![P, Q, R, S, T]: is_a_theorem(implies(implies(implies(P, Q), implies(R, S)), implies(T, implies(implies(S, P), implies(R, P)))))).
% 0.21/0.64 fof(prove_ic_5, negated_conjecture, ~is_a_theorem(implies(a, implies(implies(a, b), b)))).
% 0.21/0.64
% 0.21/0.64 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.64 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.64 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.64 fresh(y, y, x1...xn) = u
% 0.21/0.64 C => fresh(s, t, x1...xn) = v
% 0.21/0.64 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.64 variables of u and v.
% 0.21/0.64 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.64 input problem has no model of domain size 1).
% 0.21/0.64
% 0.21/0.64 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.64
% 0.21/0.64 Axiom 1 (condensed_detachment): fresh2(X, X, Y) = true.
% 0.21/0.64 Axiom 2 (condensed_detachment): fresh(X, X, Y, Z) = is_a_theorem(Z).
% 0.21/0.64 Axiom 3 (condensed_detachment): fresh(is_a_theorem(implies(X, Y)), true, X, Y) = fresh2(is_a_theorem(X), true, Y).
% 0.21/0.64 Axiom 4 (ic_JLukasiewicz_4): is_a_theorem(implies(implies(implies(X, Y), implies(Z, W)), implies(V, implies(implies(W, X), implies(Z, X))))) = true.
% 0.21/0.64
% 0.21/0.64 Lemma 5: fresh2(is_a_theorem(implies(implies(X, Y), implies(Z, W))), true, implies(V, implies(implies(W, X), implies(Z, X)))) = is_a_theorem(implies(V, implies(implies(W, X), implies(Z, X)))).
% 0.21/0.64 Proof:
% 0.21/0.64 fresh2(is_a_theorem(implies(implies(X, Y), implies(Z, W))), true, implies(V, implies(implies(W, X), implies(Z, X))))
% 0.21/0.64 = { by axiom 3 (condensed_detachment) R->L }
% 0.21/0.64 fresh(is_a_theorem(implies(implies(implies(X, Y), implies(Z, W)), implies(V, implies(implies(W, X), implies(Z, X))))), true, implies(implies(X, Y), implies(Z, W)), implies(V, implies(implies(W, X), implies(Z, X))))
% 0.21/0.64 = { by axiom 4 (ic_JLukasiewicz_4) }
% 0.21/0.64 fresh(true, true, implies(implies(X, Y), implies(Z, W)), implies(V, implies(implies(W, X), implies(Z, X))))
% 0.21/0.64 = { by axiom 2 (condensed_detachment) }
% 0.21/0.64 is_a_theorem(implies(V, implies(implies(W, X), implies(Z, X))))
% 0.21/0.64
% 0.21/0.64 Lemma 6: is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), implies(Y, W)), implies(V, implies(Y, W)))) = true.
% 0.21/0.64 Proof:
% 0.21/0.64 is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), implies(Y, W)), implies(V, implies(Y, W))))
% 0.21/0.64 = { by axiom 2 (condensed_detachment) R->L }
% 0.21/0.64 fresh(true, true, implies(implies(implies(U, T), implies(S, X2)), implies(Y2, implies(implies(X2, U), implies(S, U)))), implies(implies(implies(implies(X, Y), implies(Z, Y)), implies(Y, W)), implies(V, implies(Y, W))))
% 0.21/0.64 = { by axiom 1 (condensed_detachment) R->L }
% 0.21/0.64 fresh(fresh2(true, true, implies(implies(implies(implies(U, T), implies(S, X2)), implies(Y2, implies(implies(X2, U), implies(S, U)))), implies(implies(implies(implies(X, Y), implies(Z, Y)), implies(Y, W)), implies(V, implies(Y, W))))), true, implies(implies(implies(U, T), implies(S, X2)), implies(Y2, implies(implies(X2, U), implies(S, U)))), implies(implies(implies(implies(X, Y), implies(Z, Y)), implies(Y, W)), implies(V, implies(Y, W))))
% 0.21/0.64 = { by axiom 4 (ic_JLukasiewicz_4) R->L }
% 0.21/0.64 fresh(fresh2(is_a_theorem(implies(implies(implies(Y, W), implies(Z, X)), implies(V, implies(implies(X, Y), implies(Z, Y))))), true, implies(implies(implies(implies(U, T), implies(S, X2)), implies(Y2, implies(implies(X2, U), implies(S, U)))), implies(implies(implies(implies(X, Y), implies(Z, Y)), implies(Y, W)), implies(V, implies(Y, W))))), true, implies(implies(implies(U, T), implies(S, X2)), implies(Y2, implies(implies(X2, U), implies(S, U)))), implies(implies(implies(implies(X, Y), implies(Z, Y)), implies(Y, W)), implies(V, implies(Y, W))))
% 0.21/0.64 = { by lemma 5 }
% 0.21/0.64 fresh(is_a_theorem(implies(implies(implies(implies(U, T), implies(S, X2)), implies(Y2, implies(implies(X2, U), implies(S, U)))), implies(implies(implies(implies(X, Y), implies(Z, Y)), implies(Y, W)), implies(V, implies(Y, W))))), true, implies(implies(implies(U, T), implies(S, X2)), implies(Y2, implies(implies(X2, U), implies(S, U)))), implies(implies(implies(implies(X, Y), implies(Z, Y)), implies(Y, W)), implies(V, implies(Y, W))))
% 0.21/0.64 = { by axiom 3 (condensed_detachment) }
% 0.21/0.64 fresh2(is_a_theorem(implies(implies(implies(U, T), implies(S, X2)), implies(Y2, implies(implies(X2, U), implies(S, U))))), true, implies(implies(implies(implies(X, Y), implies(Z, Y)), implies(Y, W)), implies(V, implies(Y, W))))
% 0.21/0.64 = { by axiom 4 (ic_JLukasiewicz_4) }
% 0.21/0.64 fresh2(true, true, implies(implies(implies(implies(X, Y), implies(Z, Y)), implies(Y, W)), implies(V, implies(Y, W))))
% 0.21/0.64 = { by axiom 1 (condensed_detachment) }
% 0.21/0.64 true
% 0.21/0.64
% 0.21/0.64 Lemma 7: fresh2(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), implies(Y, W))), true, implies(V, implies(Y, W))) = is_a_theorem(implies(V, implies(Y, W))).
% 0.21/0.64 Proof:
% 0.21/0.64 fresh2(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), implies(Y, W))), true, implies(V, implies(Y, W)))
% 0.21/0.64 = { by axiom 3 (condensed_detachment) R->L }
% 0.21/0.64 fresh(is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), implies(Y, W)), implies(V, implies(Y, W)))), true, implies(implies(implies(X, Y), implies(Z, Y)), implies(Y, W)), implies(V, implies(Y, W)))
% 0.21/0.64 = { by lemma 6 }
% 0.21/0.64 fresh(true, true, implies(implies(implies(X, Y), implies(Z, Y)), implies(Y, W)), implies(V, implies(Y, W)))
% 0.21/0.64 = { by axiom 2 (condensed_detachment) }
% 0.21/0.65 is_a_theorem(implies(V, implies(Y, W)))
% 0.21/0.65
% 0.21/0.65 Lemma 8: is_a_theorem(implies(X, implies(Y, X))) = true.
% 0.21/0.65 Proof:
% 0.21/0.65 is_a_theorem(implies(X, implies(Y, X)))
% 0.21/0.65 = { by axiom 2 (condensed_detachment) R->L }
% 0.21/0.65 fresh(true, true, implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))
% 0.21/0.65 = { by axiom 1 (condensed_detachment) R->L }
% 0.21/0.65 fresh(fresh2(true, true, implies(implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))
% 0.21/0.65 = { by axiom 1 (condensed_detachment) R->L }
% 0.21/0.65 fresh(fresh2(fresh2(true, true, implies(implies(implies(implies(Y, X), X), implies(Y, X)), implies(X, implies(Y, X)))), true, implies(implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))
% 0.21/0.65 = { by axiom 4 (ic_JLukasiewicz_4) R->L }
% 0.21/0.65 fresh(fresh2(fresh2(is_a_theorem(implies(implies(implies(S, X2), implies(Y2, Z2)), implies(W2, implies(implies(Z2, S), implies(Y2, S))))), true, implies(implies(implies(implies(Y, X), X), implies(Y, X)), implies(X, implies(Y, X)))), true, implies(implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))
% 0.21/0.65 = { by axiom 3 (condensed_detachment) R->L }
% 0.21/0.65 fresh(fresh2(fresh(is_a_theorem(implies(implies(implies(implies(S, X2), implies(Y2, Z2)), implies(W2, implies(implies(Z2, S), implies(Y2, S)))), implies(implies(implies(implies(Y, X), X), implies(Y, X)), implies(X, implies(Y, X))))), true, implies(implies(implies(S, X2), implies(Y2, Z2)), implies(W2, implies(implies(Z2, S), implies(Y2, S)))), implies(implies(implies(implies(Y, X), X), implies(Y, X)), implies(X, implies(Y, X)))), true, implies(implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))
% 0.21/0.65 = { by lemma 5 R->L }
% 0.21/0.65 fresh(fresh2(fresh(fresh2(is_a_theorem(implies(implies(implies(Y, X), X), implies(X, implies(implies(Y, X), X)))), true, implies(implies(implies(implies(S, X2), implies(Y2, Z2)), implies(W2, implies(implies(Z2, S), implies(Y2, S)))), implies(implies(implies(implies(Y, X), X), implies(Y, X)), implies(X, implies(Y, X))))), true, implies(implies(implies(S, X2), implies(Y2, Z2)), implies(W2, implies(implies(Z2, S), implies(Y2, S)))), implies(implies(implies(implies(Y, X), X), implies(Y, X)), implies(X, implies(Y, X)))), true, implies(implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))
% 0.21/0.65 = { by axiom 2 (condensed_detachment) R->L }
% 0.21/0.65 fresh(fresh2(fresh(fresh2(fresh(true, true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(X3, implies(implies(S2, V2), implies(T2, V2)))), implies(implies(implies(Y, X), X), implies(X, implies(implies(Y, X), X)))), true, implies(implies(implies(implies(S, X2), implies(Y2, Z2)), implies(W2, implies(implies(Z2, S), implies(Y2, S)))), implies(implies(implies(implies(Y, X), X), implies(Y, X)), implies(X, implies(Y, X))))), true, implies(implies(implies(S, X2), implies(Y2, Z2)), implies(W2, implies(implies(Z2, S), implies(Y2, S)))), implies(implies(implies(implies(Y, X), X), implies(Y, X)), implies(X, implies(Y, X)))), true, implies(implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))
% 0.21/0.65 = { by axiom 1 (condensed_detachment) R->L }
% 0.21/0.65 fresh(fresh2(fresh(fresh2(fresh(fresh2(true, true, implies(implies(implies(implies(V2, U2), implies(T2, S2)), implies(X3, implies(implies(S2, V2), implies(T2, V2)))), implies(implies(implies(Y, X), X), implies(X, implies(implies(Y, X), X))))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(X3, implies(implies(S2, V2), implies(T2, V2)))), implies(implies(implies(Y, X), X), implies(X, implies(implies(Y, X), X)))), true, implies(implies(implies(implies(S, X2), implies(Y2, Z2)), implies(W2, implies(implies(Z2, S), implies(Y2, S)))), implies(implies(implies(implies(Y, X), X), implies(Y, X)), implies(X, implies(Y, X))))), true, implies(implies(implies(S, X2), implies(Y2, Z2)), implies(W2, implies(implies(Z2, S), implies(Y2, S)))), implies(implies(implies(implies(Y, X), X), implies(Y, X)), implies(X, implies(Y, X)))), true, implies(implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))
% 0.21/0.65 = { by lemma 6 R->L }
% 0.21/0.65 fresh(fresh2(fresh(fresh2(fresh(fresh2(is_a_theorem(implies(implies(implies(implies(Y3, X), implies(implies(Y, X), X)), implies(X, implies(implies(Y, X), X))), implies(implies(implies(Y, X), X), implies(X, implies(implies(Y, X), X))))), true, implies(implies(implies(implies(V2, U2), implies(T2, S2)), implies(X3, implies(implies(S2, V2), implies(T2, V2)))), implies(implies(implies(Y, X), X), implies(X, implies(implies(Y, X), X))))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(X3, implies(implies(S2, V2), implies(T2, V2)))), implies(implies(implies(Y, X), X), implies(X, implies(implies(Y, X), X)))), true, implies(implies(implies(implies(S, X2), implies(Y2, Z2)), implies(W2, implies(implies(Z2, S), implies(Y2, S)))), implies(implies(implies(implies(Y, X), X), implies(Y, X)), implies(X, implies(Y, X))))), true, implies(implies(implies(S, X2), implies(Y2, Z2)), implies(W2, implies(implies(Z2, S), implies(Y2, S)))), implies(implies(implies(implies(Y, X), X), implies(Y, X)), implies(X, implies(Y, X)))), true, implies(implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))
% 0.21/0.65 = { by lemma 7 }
% 0.21/0.65 fresh(fresh2(fresh(fresh2(fresh(is_a_theorem(implies(implies(implies(implies(V2, U2), implies(T2, S2)), implies(X3, implies(implies(S2, V2), implies(T2, V2)))), implies(implies(implies(Y, X), X), implies(X, implies(implies(Y, X), X))))), true, implies(implies(implies(V2, U2), implies(T2, S2)), implies(X3, implies(implies(S2, V2), implies(T2, V2)))), implies(implies(implies(Y, X), X), implies(X, implies(implies(Y, X), X)))), true, implies(implies(implies(implies(S, X2), implies(Y2, Z2)), implies(W2, implies(implies(Z2, S), implies(Y2, S)))), implies(implies(implies(implies(Y, X), X), implies(Y, X)), implies(X, implies(Y, X))))), true, implies(implies(implies(S, X2), implies(Y2, Z2)), implies(W2, implies(implies(Z2, S), implies(Y2, S)))), implies(implies(implies(implies(Y, X), X), implies(Y, X)), implies(X, implies(Y, X)))), true, implies(implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))
% 0.21/0.65 = { by axiom 3 (condensed_detachment) }
% 0.21/0.65 fresh(fresh2(fresh(fresh2(fresh2(is_a_theorem(implies(implies(implies(V2, U2), implies(T2, S2)), implies(X3, implies(implies(S2, V2), implies(T2, V2))))), true, implies(implies(implies(Y, X), X), implies(X, implies(implies(Y, X), X)))), true, implies(implies(implies(implies(S, X2), implies(Y2, Z2)), implies(W2, implies(implies(Z2, S), implies(Y2, S)))), implies(implies(implies(implies(Y, X), X), implies(Y, X)), implies(X, implies(Y, X))))), true, implies(implies(implies(S, X2), implies(Y2, Z2)), implies(W2, implies(implies(Z2, S), implies(Y2, S)))), implies(implies(implies(implies(Y, X), X), implies(Y, X)), implies(X, implies(Y, X)))), true, implies(implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))
% 0.21/0.65 = { by axiom 4 (ic_JLukasiewicz_4) }
% 0.21/0.65 fresh(fresh2(fresh(fresh2(fresh2(true, true, implies(implies(implies(Y, X), X), implies(X, implies(implies(Y, X), X)))), true, implies(implies(implies(implies(S, X2), implies(Y2, Z2)), implies(W2, implies(implies(Z2, S), implies(Y2, S)))), implies(implies(implies(implies(Y, X), X), implies(Y, X)), implies(X, implies(Y, X))))), true, implies(implies(implies(S, X2), implies(Y2, Z2)), implies(W2, implies(implies(Z2, S), implies(Y2, S)))), implies(implies(implies(implies(Y, X), X), implies(Y, X)), implies(X, implies(Y, X)))), true, implies(implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))
% 0.21/0.65 = { by axiom 1 (condensed_detachment) }
% 0.21/0.65 fresh(fresh2(fresh(fresh2(true, true, implies(implies(implies(implies(S, X2), implies(Y2, Z2)), implies(W2, implies(implies(Z2, S), implies(Y2, S)))), implies(implies(implies(implies(Y, X), X), implies(Y, X)), implies(X, implies(Y, X))))), true, implies(implies(implies(S, X2), implies(Y2, Z2)), implies(W2, implies(implies(Z2, S), implies(Y2, S)))), implies(implies(implies(implies(Y, X), X), implies(Y, X)), implies(X, implies(Y, X)))), true, implies(implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))
% 0.21/0.65 = { by axiom 1 (condensed_detachment) }
% 0.21/0.65 fresh(fresh2(fresh(true, true, implies(implies(implies(S, X2), implies(Y2, Z2)), implies(W2, implies(implies(Z2, S), implies(Y2, S)))), implies(implies(implies(implies(Y, X), X), implies(Y, X)), implies(X, implies(Y, X)))), true, implies(implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))
% 0.21/0.65 = { by axiom 2 (condensed_detachment) }
% 0.21/0.65 fresh(fresh2(is_a_theorem(implies(implies(implies(implies(Y, X), X), implies(Y, X)), implies(X, implies(Y, X)))), true, implies(implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))
% 0.21/0.65 = { by lemma 7 }
% 0.21/0.65 fresh(is_a_theorem(implies(implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))), true, implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z)))), implies(X, implies(Y, X)))
% 0.21/0.65 = { by axiom 3 (condensed_detachment) }
% 0.21/0.65 fresh2(is_a_theorem(implies(implies(implies(Z, W), implies(V, U)), implies(T, implies(implies(U, Z), implies(V, Z))))), true, implies(X, implies(Y, X)))
% 0.21/0.65 = { by axiom 4 (ic_JLukasiewicz_4) }
% 0.21/0.65 fresh2(true, true, implies(X, implies(Y, X)))
% 0.21/0.65 = { by axiom 1 (condensed_detachment) }
% 0.21/0.65 true
% 0.21/0.65
% 0.21/0.65 Lemma 9: is_a_theorem(implies(X, implies(Y, implies(implies(Y, Z), implies(W, Z))))) = true.
% 0.21/0.65 Proof:
% 0.21/0.65 is_a_theorem(implies(X, implies(Y, implies(implies(Y, Z), implies(W, Z)))))
% 0.21/0.65 = { by lemma 7 R->L }
% 0.21/0.65 fresh2(is_a_theorem(implies(implies(implies(Z, Y), implies(W, Y)), implies(Y, implies(implies(Y, Z), implies(W, Z))))), true, implies(X, implies(Y, implies(implies(Y, Z), implies(W, Z)))))
% 0.21/0.65 = { by axiom 4 (ic_JLukasiewicz_4) }
% 0.21/0.65 fresh2(true, true, implies(X, implies(Y, implies(implies(Y, Z), implies(W, Z)))))
% 0.21/0.65 = { by axiom 1 (condensed_detachment) }
% 0.21/0.65 true
% 0.21/0.65
% 0.21/0.65 Lemma 10: is_a_theorem(implies(implies(implies(X, Y), Z), implies(implies(implies(Z, W), Y), Z))) = true.
% 0.21/0.65 Proof:
% 0.21/0.65 is_a_theorem(implies(implies(implies(X, Y), Z), implies(implies(implies(Z, W), Y), Z)))
% 0.21/0.65 = { by axiom 2 (condensed_detachment) R->L }
% 0.21/0.65 fresh(true, true, implies(implies(implies(V, U), implies(T, S)), implies(X2, implies(implies(S, V), implies(T, V)))), implies(implies(implies(X, Y), Z), implies(implies(implies(Z, W), Y), Z)))
% 0.21/0.65 = { by axiom 1 (condensed_detachment) R->L }
% 0.21/0.65 fresh(fresh2(true, true, implies(implies(implies(implies(V, U), implies(T, S)), implies(X2, implies(implies(S, V), implies(T, V)))), implies(implies(implies(X, Y), Z), implies(implies(implies(Z, W), Y), Z)))), true, implies(implies(implies(V, U), implies(T, S)), implies(X2, implies(implies(S, V), implies(T, V)))), implies(implies(implies(X, Y), Z), implies(implies(implies(Z, W), Y), Z)))
% 0.21/0.65 = { by axiom 1 (condensed_detachment) R->L }
% 0.21/0.65 fresh(fresh2(fresh2(true, true, implies(implies(Z, W), implies(implies(implies(Z, W), Y), implies(X, Y)))), true, implies(implies(implies(implies(V, U), implies(T, S)), implies(X2, implies(implies(S, V), implies(T, V)))), implies(implies(implies(X, Y), Z), implies(implies(implies(Z, W), Y), Z)))), true, implies(implies(implies(V, U), implies(T, S)), implies(X2, implies(implies(S, V), implies(T, V)))), implies(implies(implies(X, Y), Z), implies(implies(implies(Z, W), Y), Z)))
% 0.21/0.65 = { by axiom 4 (ic_JLukasiewicz_4) R->L }
% 0.21/0.65 fresh(fresh2(fresh2(is_a_theorem(implies(implies(implies(Y2, Z2), implies(W2, V2)), implies(U2, implies(implies(V2, Y2), implies(W2, Y2))))), true, implies(implies(Z, W), implies(implies(implies(Z, W), Y), implies(X, Y)))), true, implies(implies(implies(implies(V, U), implies(T, S)), implies(X2, implies(implies(S, V), implies(T, V)))), implies(implies(implies(X, Y), Z), implies(implies(implies(Z, W), Y), Z)))), true, implies(implies(implies(V, U), implies(T, S)), implies(X2, implies(implies(S, V), implies(T, V)))), implies(implies(implies(X, Y), Z), implies(implies(implies(Z, W), Y), Z)))
% 0.21/0.66 = { by axiom 3 (condensed_detachment) R->L }
% 0.21/0.66 fresh(fresh2(fresh(is_a_theorem(implies(implies(implies(implies(Y2, Z2), implies(W2, V2)), implies(U2, implies(implies(V2, Y2), implies(W2, Y2)))), implies(implies(Z, W), implies(implies(implies(Z, W), Y), implies(X, Y))))), true, implies(implies(implies(Y2, Z2), implies(W2, V2)), implies(U2, implies(implies(V2, Y2), implies(W2, Y2)))), implies(implies(Z, W), implies(implies(implies(Z, W), Y), implies(X, Y)))), true, implies(implies(implies(implies(V, U), implies(T, S)), implies(X2, implies(implies(S, V), implies(T, V)))), implies(implies(implies(X, Y), Z), implies(implies(implies(Z, W), Y), Z)))), true, implies(implies(implies(V, U), implies(T, S)), implies(X2, implies(implies(S, V), implies(T, V)))), implies(implies(implies(X, Y), Z), implies(implies(implies(Z, W), Y), Z)))
% 0.21/0.66 = { by lemma 9 }
% 0.21/0.66 fresh(fresh2(fresh(true, true, implies(implies(implies(Y2, Z2), implies(W2, V2)), implies(U2, implies(implies(V2, Y2), implies(W2, Y2)))), implies(implies(Z, W), implies(implies(implies(Z, W), Y), implies(X, Y)))), true, implies(implies(implies(implies(V, U), implies(T, S)), implies(X2, implies(implies(S, V), implies(T, V)))), implies(implies(implies(X, Y), Z), implies(implies(implies(Z, W), Y), Z)))), true, implies(implies(implies(V, U), implies(T, S)), implies(X2, implies(implies(S, V), implies(T, V)))), implies(implies(implies(X, Y), Z), implies(implies(implies(Z, W), Y), Z)))
% 0.21/0.66 = { by axiom 2 (condensed_detachment) }
% 0.21/0.66 fresh(fresh2(is_a_theorem(implies(implies(Z, W), implies(implies(implies(Z, W), Y), implies(X, Y)))), true, implies(implies(implies(implies(V, U), implies(T, S)), implies(X2, implies(implies(S, V), implies(T, V)))), implies(implies(implies(X, Y), Z), implies(implies(implies(Z, W), Y), Z)))), true, implies(implies(implies(V, U), implies(T, S)), implies(X2, implies(implies(S, V), implies(T, V)))), implies(implies(implies(X, Y), Z), implies(implies(implies(Z, W), Y), Z)))
% 0.21/0.66 = { by lemma 5 }
% 0.21/0.66 fresh(is_a_theorem(implies(implies(implies(implies(V, U), implies(T, S)), implies(X2, implies(implies(S, V), implies(T, V)))), implies(implies(implies(X, Y), Z), implies(implies(implies(Z, W), Y), Z)))), true, implies(implies(implies(V, U), implies(T, S)), implies(X2, implies(implies(S, V), implies(T, V)))), implies(implies(implies(X, Y), Z), implies(implies(implies(Z, W), Y), Z)))
% 0.21/0.66 = { by axiom 3 (condensed_detachment) }
% 0.21/0.66 fresh2(is_a_theorem(implies(implies(implies(V, U), implies(T, S)), implies(X2, implies(implies(S, V), implies(T, V))))), true, implies(implies(implies(X, Y), Z), implies(implies(implies(Z, W), Y), Z)))
% 0.21/0.66 = { by axiom 4 (ic_JLukasiewicz_4) }
% 0.21/0.66 fresh2(true, true, implies(implies(implies(X, Y), Z), implies(implies(implies(Z, W), Y), Z)))
% 0.21/0.66 = { by axiom 1 (condensed_detachment) }
% 0.21/0.66 true
% 0.21/0.66
% 0.21/0.66 Lemma 11: is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(X, W))) = true.
% 0.21/0.66 Proof:
% 0.21/0.66 is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(X, W)))
% 0.21/0.66 = { by axiom 2 (condensed_detachment) R->L }
% 0.21/0.66 fresh(true, true, implies(implies(implies(V, U), implies(T, S)), implies(X2, implies(implies(S, V), implies(T, V)))), implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(X, W)))
% 0.21/0.66 = { by axiom 1 (condensed_detachment) R->L }
% 0.21/0.66 fresh(fresh2(true, true, implies(implies(implies(implies(V, U), implies(T, S)), implies(X2, implies(implies(S, V), implies(T, V)))), implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(X, W)))), true, implies(implies(implies(V, U), implies(T, S)), implies(X2, implies(implies(S, V), implies(T, V)))), implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(X, W)))
% 0.21/0.66 = { by lemma 9 R->L }
% 0.21/0.66 fresh(fresh2(is_a_theorem(implies(implies(W, Y2), implies(X, implies(implies(X, Y), implies(Z, Y))))), true, implies(implies(implies(implies(V, U), implies(T, S)), implies(X2, implies(implies(S, V), implies(T, V)))), implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(X, W)))), true, implies(implies(implies(V, U), implies(T, S)), implies(X2, implies(implies(S, V), implies(T, V)))), implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(X, W)))
% 0.21/0.66 = { by lemma 5 }
% 0.21/0.66 fresh(is_a_theorem(implies(implies(implies(implies(V, U), implies(T, S)), implies(X2, implies(implies(S, V), implies(T, V)))), implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(X, W)))), true, implies(implies(implies(V, U), implies(T, S)), implies(X2, implies(implies(S, V), implies(T, V)))), implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(X, W)))
% 0.21/0.66 = { by axiom 3 (condensed_detachment) }
% 0.21/0.66 fresh2(is_a_theorem(implies(implies(implies(V, U), implies(T, S)), implies(X2, implies(implies(S, V), implies(T, V))))), true, implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(X, W)))
% 0.21/0.66 = { by axiom 4 (ic_JLukasiewicz_4) }
% 0.21/0.66 fresh2(true, true, implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(X, W)))
% 0.21/0.66 = { by axiom 1 (condensed_detachment) }
% 0.21/0.66 true
% 0.21/0.66
% 0.21/0.66 Lemma 12: is_a_theorem(implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))) = true.
% 0.21/0.66 Proof:
% 0.21/0.66 is_a_theorem(implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))
% 0.21/0.66 = { by axiom 2 (condensed_detachment) R->L }
% 0.21/0.66 fresh(true, true, implies(implies(implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))), Y), implies(X, Y)), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))
% 0.21/0.66 = { by axiom 1 (condensed_detachment) R->L }
% 0.21/0.66 fresh(fresh2(true, true, implies(implies(implies(implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))), Y), implies(X, Y)), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))), Y), implies(X, Y)), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))
% 0.21/0.66 = { by lemma 8 R->L }
% 0.21/0.66 fresh(fresh2(is_a_theorem(implies(implies(Z, implies(X, Y)), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))), Y), implies(X, Y)), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))), Y), implies(X, Y)), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))
% 0.21/0.66 = { by axiom 3 (condensed_detachment) R->L }
% 0.21/0.66 fresh(fresh(is_a_theorem(implies(implies(implies(Z, implies(X, Y)), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))), Y), implies(X, Y)), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(Z, implies(X, Y)), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))), Y), implies(X, Y)), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))), Y), implies(X, Y)), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))
% 0.21/0.66 = { by lemma 10 }
% 0.21/0.66 fresh(fresh(true, true, implies(implies(Z, implies(X, Y)), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))), Y), implies(X, Y)), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))), Y), implies(X, Y)), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))
% 0.21/0.66 = { by axiom 2 (condensed_detachment) }
% 0.21/0.66 fresh(is_a_theorem(implies(implies(implies(implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))), Y), implies(X, Y)), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))), Y), implies(X, Y)), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))
% 0.21/0.66 = { by axiom 3 (condensed_detachment) }
% 0.21/0.66 fresh2(is_a_theorem(implies(implies(implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))), Y), implies(X, Y))), true, implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))
% 0.21/0.66 = { by lemma 11 }
% 0.21/0.66 fresh2(true, true, implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))
% 0.21/0.66 = { by axiom 1 (condensed_detachment) }
% 0.21/0.66 true
% 0.21/0.66
% 0.21/0.66 Goal 1 (prove_ic_5): is_a_theorem(implies(a, implies(implies(a, b), b))) = true.
% 0.21/0.66 Proof:
% 0.21/0.66 is_a_theorem(implies(a, implies(implies(a, b), b)))
% 0.21/0.66 = { by axiom 2 (condensed_detachment) R->L }
% 0.21/0.66 fresh(true, true, implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))
% 0.21/0.66 = { by axiom 1 (condensed_detachment) R->L }
% 0.21/0.66 fresh(fresh2(true, true, implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), true, implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))
% 0.21/0.66 = { by axiom 4 (ic_JLukasiewicz_4) R->L }
% 0.21/0.66 fresh(fresh2(is_a_theorem(implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y))))), true, implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), true, implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))
% 0.21/0.66 = { by axiom 3 (condensed_detachment) R->L }
% 0.21/0.66 fresh(fresh(is_a_theorem(implies(implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))))), true, implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), true, implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))
% 0.21/0.66 = { by axiom 2 (condensed_detachment) R->L }
% 0.21/0.67 fresh(fresh(fresh(true, true, implies(implies(implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), implies(implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))))), true, implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), true, implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))
% 0.21/0.67 = { by axiom 1 (condensed_detachment) R->L }
% 0.21/0.67 fresh(fresh(fresh(fresh2(true, true, implies(implies(implies(implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), implies(implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))))), true, implies(implies(implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), implies(implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))))), true, implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), true, implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))
% 0.21/0.67 = { by axiom 4 (ic_JLukasiewicz_4) R->L }
% 0.21/0.67 fresh(fresh(fresh(fresh2(is_a_theorem(implies(implies(implies(T, S), implies(X2, Y2)), implies(Z2, implies(implies(Y2, T), implies(X2, T))))), true, implies(implies(implies(implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), implies(implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))))), true, implies(implies(implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), implies(implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))))), true, implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), true, implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))
% 0.21/0.67 = { by axiom 3 (condensed_detachment) R->L }
% 0.21/0.67 fresh(fresh(fresh(fresh(is_a_theorem(implies(implies(implies(implies(T, S), implies(X2, Y2)), implies(Z2, implies(implies(Y2, T), implies(X2, T)))), implies(implies(implies(implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), implies(implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))))))), true, implies(implies(implies(T, S), implies(X2, Y2)), implies(Z2, implies(implies(Y2, T), implies(X2, T)))), implies(implies(implies(implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), implies(implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))))), true, implies(implies(implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), implies(implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))))), true, implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), true, implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))
% 0.21/0.67 = { by lemma 5 R->L }
% 0.21/0.67 fresh(fresh(fresh(fresh(fresh2(is_a_theorem(implies(implies(implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))))), true, implies(implies(implies(implies(T, S), implies(X2, Y2)), implies(Z2, implies(implies(Y2, T), implies(X2, T)))), implies(implies(implies(implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), implies(implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))))))), true, implies(implies(implies(T, S), implies(X2, Y2)), implies(Z2, implies(implies(Y2, T), implies(X2, T)))), implies(implies(implies(implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), implies(implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))))), true, implies(implies(implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), implies(implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))))), true, implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), true, implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))
% 0.21/0.67 = { by lemma 8 }
% 0.21/0.67 fresh(fresh(fresh(fresh(fresh2(true, true, implies(implies(implies(implies(T, S), implies(X2, Y2)), implies(Z2, implies(implies(Y2, T), implies(X2, T)))), implies(implies(implies(implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), implies(implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))))))), true, implies(implies(implies(T, S), implies(X2, Y2)), implies(Z2, implies(implies(Y2, T), implies(X2, T)))), implies(implies(implies(implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), implies(implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))))), true, implies(implies(implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), implies(implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))))), true, implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), true, implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))
% 0.21/0.68 = { by axiom 1 (condensed_detachment) }
% 0.21/0.68 fresh(fresh(fresh(fresh(true, true, implies(implies(implies(T, S), implies(X2, Y2)), implies(Z2, implies(implies(Y2, T), implies(X2, T)))), implies(implies(implies(implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), implies(implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))))), true, implies(implies(implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), implies(implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))))), true, implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), true, implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))
% 0.21/0.68 = { by axiom 2 (condensed_detachment) }
% 0.21/0.68 fresh(fresh(fresh(is_a_theorem(implies(implies(implies(implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), implies(implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))))), true, implies(implies(implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), implies(implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))))), true, implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), true, implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))
% 0.21/0.68 = { by axiom 3 (condensed_detachment) }
% 0.21/0.68 fresh(fresh(fresh2(is_a_theorem(implies(implies(implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))))), true, implies(implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))))), true, implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), true, implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))
% 0.21/0.68 = { by lemma 10 }
% 0.21/0.68 fresh(fresh(fresh2(true, true, implies(implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b))))), true, implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), true, implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))
% 0.21/0.68 = { by axiom 1 (condensed_detachment) }
% 0.21/0.68 fresh(fresh(true, true, implies(implies(implies(Y, Z), implies(W, V)), implies(U, implies(implies(V, Y), implies(W, Y)))), implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), true, implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))
% 0.21/0.68 = { by axiom 2 (condensed_detachment) }
% 0.21/0.68 fresh(is_a_theorem(implies(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))), true, implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))), implies(a, implies(implies(a, b), b)))
% 0.21/0.68 = { by axiom 3 (condensed_detachment) }
% 0.21/0.68 fresh2(is_a_theorem(implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b)))), true, implies(a, implies(implies(a, b), b)))
% 0.21/0.68 = { by axiom 2 (condensed_detachment) R->L }
% 0.21/0.68 fresh2(fresh(true, true, implies(a, implies(a, implies(implies(a, b), b))), implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b)))), true, implies(a, implies(implies(a, b), b)))
% 0.21/0.68 = { by lemma 12 R->L }
% 0.21/0.68 fresh2(fresh(is_a_theorem(implies(implies(a, implies(a, implies(implies(a, b), b))), implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b))))), true, implies(a, implies(a, implies(implies(a, b), b))), implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b)))), true, implies(a, implies(implies(a, b), b)))
% 0.21/0.68 = { by axiom 3 (condensed_detachment) }
% 0.21/0.68 fresh2(fresh2(is_a_theorem(implies(a, implies(a, implies(implies(a, b), b)))), true, implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b)))), true, implies(a, implies(implies(a, b), b)))
% 0.21/0.68 = { by axiom 2 (condensed_detachment) R->L }
% 0.21/0.68 fresh2(fresh2(fresh(true, true, implies(implies(implies(a, b), implies(implies(a, b), b)), implies(a, implies(implies(a, b), b))), implies(a, implies(a, implies(implies(a, b), b)))), true, implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b)))), true, implies(a, implies(implies(a, b), b)))
% 0.21/0.68 = { by lemma 11 R->L }
% 0.21/0.68 fresh2(fresh2(fresh(is_a_theorem(implies(implies(implies(implies(a, b), implies(implies(a, b), b)), implies(a, implies(implies(a, b), b))), implies(a, implies(a, implies(implies(a, b), b))))), true, implies(implies(implies(a, b), implies(implies(a, b), b)), implies(a, implies(implies(a, b), b))), implies(a, implies(a, implies(implies(a, b), b)))), true, implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b)))), true, implies(a, implies(implies(a, b), b)))
% 0.21/0.68 = { by axiom 3 (condensed_detachment) }
% 0.21/0.68 fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(a, b), implies(implies(a, b), b)), implies(a, implies(implies(a, b), b)))), true, implies(a, implies(a, implies(implies(a, b), b)))), true, implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b)))), true, implies(a, implies(implies(a, b), b)))
% 0.21/0.68 = { by lemma 12 }
% 0.21/0.68 fresh2(fresh2(fresh2(true, true, implies(a, implies(a, implies(implies(a, b), b)))), true, implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b)))), true, implies(a, implies(implies(a, b), b)))
% 0.21/0.68 = { by axiom 1 (condensed_detachment) }
% 0.21/0.68 fresh2(fresh2(true, true, implies(implies(implies(a, implies(implies(a, b), b)), X), implies(a, implies(implies(a, b), b)))), true, implies(a, implies(implies(a, b), b)))
% 0.21/0.68 = { by axiom 1 (condensed_detachment) }
% 0.21/0.68 fresh2(true, true, implies(a, implies(implies(a, b), b)))
% 0.21/0.68 = { by axiom 1 (condensed_detachment) }
% 0.21/0.68 true
% 0.21/0.68 % SZS output end Proof
% 0.21/0.68
% 0.21/0.68 RESULT: Unsatisfiable (the axioms are contradictory).
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