TSTP Solution File: LCL083-1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL083-1 : 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 : n013.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 08:17:20 EDT 2023
% Result : Unsatisfiable 3.02s 0.82s
% Output : Proof 4.37s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LCL083-1 : 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
% 0.13/0.34 % Computer : n013.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 : Fri Aug 25 01:16:32 EDT 2023
% 0.13/0.34 % CPUTime :
% 3.02/0.82 Command-line arguments: --no-flatten-goal
% 3.02/0.82
% 3.02/0.82 % SZS status Unsatisfiable
% 3.02/0.82
% 4.37/0.91 % SZS output start Proof
% 4.37/0.91 Take the following subset of the input axioms:
% 4.37/0.92 fof(condensed_detachment, axiom, ![X, Y]: (~is_a_theorem(implies(X, Y)) | (~is_a_theorem(X) | is_a_theorem(Y)))).
% 4.37/0.92 fof(ic_JLukasiewicz, axiom, ![Z, U, X2, Y2]: is_a_theorem(implies(implies(implies(X2, Y2), Z), implies(implies(Z, X2), implies(U, X2))))).
% 4.37/0.92 fof(prove_ic_3, negated_conjecture, ~is_a_theorem(implies(implies(implies(a, b), a), a))).
% 4.37/0.92
% 4.37/0.92 Now clausify the problem and encode Horn clauses using encoding 3 of
% 4.37/0.92 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 4.37/0.92 We repeatedly replace C & s=t => u=v by the two clauses:
% 4.37/0.92 fresh(y, y, x1...xn) = u
% 4.37/0.92 C => fresh(s, t, x1...xn) = v
% 4.37/0.92 where fresh is a fresh function symbol and x1..xn are the free
% 4.37/0.92 variables of u and v.
% 4.37/0.92 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 4.37/0.92 input problem has no model of domain size 1).
% 4.37/0.92
% 4.37/0.92 The encoding turns the above axioms into the following unit equations and goals:
% 4.37/0.92
% 4.37/0.92 Axiom 1 (condensed_detachment): fresh2(X, X, Y) = true.
% 4.37/0.92 Axiom 2 (condensed_detachment): fresh(X, X, Y, Z) = is_a_theorem(Z).
% 4.37/0.92 Axiom 3 (condensed_detachment): fresh(is_a_theorem(implies(X, Y)), true, X, Y) = fresh2(is_a_theorem(X), true, Y).
% 4.37/0.92 Axiom 4 (ic_JLukasiewicz): is_a_theorem(implies(implies(implies(X, Y), Z), implies(implies(Z, X), implies(W, X)))) = true.
% 4.37/0.92
% 4.37/0.92 Lemma 5: fresh2(is_a_theorem(implies(implies(X, Y), Z)), true, implies(implies(Z, X), implies(W, X))) = is_a_theorem(implies(implies(Z, X), implies(W, X))).
% 4.37/0.92 Proof:
% 4.37/0.92 fresh2(is_a_theorem(implies(implies(X, Y), Z)), true, implies(implies(Z, X), implies(W, X)))
% 4.37/0.92 = { by axiom 3 (condensed_detachment) R->L }
% 4.37/0.92 fresh(is_a_theorem(implies(implies(implies(X, Y), Z), implies(implies(Z, X), implies(W, X)))), true, implies(implies(X, Y), Z), implies(implies(Z, X), implies(W, X)))
% 4.37/0.92 = { by axiom 4 (ic_JLukasiewicz) }
% 4.37/0.92 fresh(true, true, implies(implies(X, Y), Z), implies(implies(Z, X), implies(W, X)))
% 4.37/0.92 = { by axiom 2 (condensed_detachment) }
% 4.37/0.92 is_a_theorem(implies(implies(Z, X), implies(W, X)))
% 4.37/0.92
% 4.37/0.92 Lemma 6: is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), implies(Y, W)), implies(V, implies(Y, W)))) = true.
% 4.37/0.92 Proof:
% 4.37/0.92 is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), implies(Y, W)), implies(V, implies(Y, W))))
% 4.37/0.92 = { by lemma 5 R->L }
% 4.37/0.92 fresh2(is_a_theorem(implies(implies(implies(Y, W), X), implies(implies(X, Y), implies(Z, Y)))), true, implies(implies(implies(implies(X, Y), implies(Z, Y)), implies(Y, W)), implies(V, implies(Y, W))))
% 4.37/0.92 = { by axiom 4 (ic_JLukasiewicz) }
% 4.37/0.92 fresh2(true, true, implies(implies(implies(implies(X, Y), implies(Z, Y)), implies(Y, W)), implies(V, implies(Y, W))))
% 4.37/0.92 = { by axiom 1 (condensed_detachment) }
% 4.37/0.92 true
% 4.37/0.92
% 4.37/0.92 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))).
% 4.37/0.92 Proof:
% 4.37/0.92 fresh2(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), implies(Y, W))), true, implies(V, implies(Y, W)))
% 4.37/0.92 = { by axiom 3 (condensed_detachment) R->L }
% 4.37/0.92 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)))
% 4.37/0.92 = { by lemma 6 }
% 4.37/0.92 fresh(true, true, implies(implies(implies(X, Y), implies(Z, Y)), implies(Y, W)), implies(V, implies(Y, W)))
% 4.37/0.92 = { by axiom 2 (condensed_detachment) }
% 4.37/0.92 is_a_theorem(implies(V, implies(Y, W)))
% 4.37/0.92
% 4.37/0.92 Lemma 8: is_a_theorem(implies(implies(implies(X, implies(Y, Z)), implies(implies(W, Y), implies(V, Y))), implies(U, implies(implies(W, Y), implies(V, Y))))) = true.
% 4.37/0.92 Proof:
% 4.37/0.92 is_a_theorem(implies(implies(implies(X, implies(Y, Z)), implies(implies(W, Y), implies(V, Y))), implies(U, implies(implies(W, Y), implies(V, Y)))))
% 4.37/0.92 = { by lemma 5 R->L }
% 4.37/0.92 fresh2(is_a_theorem(implies(implies(implies(implies(W, Y), implies(V, Y)), implies(Y, Z)), implies(X, implies(Y, Z)))), true, implies(implies(implies(X, implies(Y, Z)), implies(implies(W, Y), implies(V, Y))), implies(U, implies(implies(W, Y), implies(V, Y)))))
% 4.37/0.92 = { by lemma 6 }
% 4.37/0.92 fresh2(true, true, implies(implies(implies(X, implies(Y, Z)), implies(implies(W, Y), implies(V, Y))), implies(U, implies(implies(W, Y), implies(V, Y)))))
% 4.37/0.92 = { by axiom 1 (condensed_detachment) }
% 4.37/0.92 true
% 4.37/0.92
% 4.37/0.92 Lemma 9: is_a_theorem(implies(X, implies(implies(Y, Y), implies(implies(Z, Y), implies(Y, Y))))) = true.
% 4.37/0.92 Proof:
% 4.37/0.92 is_a_theorem(implies(X, implies(implies(Y, Y), implies(implies(Z, Y), implies(Y, Y)))))
% 4.37/0.92 = { by lemma 7 R->L }
% 4.37/0.92 fresh2(is_a_theorem(implies(implies(implies(W, implies(Y, Y)), implies(implies(Z, Y), implies(Y, Y))), implies(implies(Y, Y), implies(implies(Z, Y), implies(Y, Y))))), true, implies(X, implies(implies(Y, Y), implies(implies(Z, Y), implies(Y, Y)))))
% 4.37/0.92 = { by lemma 8 }
% 4.37/0.92 fresh2(true, true, implies(X, implies(implies(Y, Y), implies(implies(Z, Y), implies(Y, Y)))))
% 4.37/0.92 = { by axiom 1 (condensed_detachment) }
% 4.37/0.92 true
% 4.37/0.92
% 4.37/0.92 Lemma 10: is_a_theorem(implies(implies(X, Y), implies(Y, Y))) = true.
% 4.37/0.92 Proof:
% 4.37/0.92 is_a_theorem(implies(implies(X, Y), implies(Y, Y)))
% 4.37/0.92 = { by axiom 2 (condensed_detachment) R->L }
% 4.37/0.92 fresh(true, true, implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))
% 4.37/0.92 = { by axiom 1 (condensed_detachment) R->L }
% 4.37/0.92 fresh(fresh2(true, true, implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))), true, implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))
% 4.37/0.92 = { by axiom 4 (ic_JLukasiewicz) R->L }
% 4.37/0.92 fresh(fresh2(is_a_theorem(implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T)))), true, implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))), true, implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))
% 4.37/0.92 = { by axiom 3 (condensed_detachment) R->L }
% 4.37/0.92 fresh(fresh(is_a_theorem(implies(implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y))))), true, implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))), true, implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))
% 4.37/0.92 = { by axiom 2 (condensed_detachment) R->L }
% 4.37/0.92 fresh(fresh(fresh(true, true, implies(implies(implies(Y, Y), implies(implies(X, Y), implies(Y, Y))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))), implies(implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y))))), true, implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))), true, implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))
% 4.37/0.92 = { by axiom 1 (condensed_detachment) R->L }
% 4.37/0.92 fresh(fresh(fresh(fresh2(true, true, implies(implies(implies(implies(Y, Y), implies(implies(X, Y), implies(Y, Y))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))), implies(implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))))), true, implies(implies(implies(Y, Y), implies(implies(X, Y), implies(Y, Y))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))), implies(implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y))))), true, implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))), true, implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))
% 4.37/0.92 = { by lemma 9 R->L }
% 4.37/0.92 fresh(fresh(fresh(fresh2(is_a_theorem(implies(implies(implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y))), Z2), implies(implies(Y, Y), implies(implies(X, Y), implies(Y, Y))))), true, implies(implies(implies(implies(Y, Y), implies(implies(X, Y), implies(Y, Y))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))), implies(implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))))), true, implies(implies(implies(Y, Y), implies(implies(X, Y), implies(Y, Y))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))), implies(implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y))))), true, implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))), true, implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))
% 4.37/0.92 = { by lemma 5 }
% 4.37/0.92 fresh(fresh(fresh(is_a_theorem(implies(implies(implies(implies(Y, Y), implies(implies(X, Y), implies(Y, Y))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))), implies(implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))))), true, implies(implies(implies(Y, Y), implies(implies(X, Y), implies(Y, Y))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))), implies(implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y))))), true, implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))), true, implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))
% 4.37/0.92 = { by axiom 3 (condensed_detachment) }
% 4.37/0.92 fresh(fresh(fresh2(is_a_theorem(implies(implies(implies(Y, Y), implies(implies(X, Y), implies(Y, Y))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y))))), true, implies(implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y))))), true, implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))), true, implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))
% 4.37/0.92 = { by lemma 5 R->L }
% 4.37/0.92 fresh(fresh(fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(X, Y), implies(Y, Y)), Y), implies(Y, Y))), true, implies(implies(implies(Y, Y), implies(implies(X, Y), implies(Y, Y))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y))))), true, implies(implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y))))), true, implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))), true, implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))
% 4.37/0.92 = { by lemma 5 R->L }
% 4.37/0.92 fresh(fresh(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(Y, Y), implies(implies(X, Y), implies(Y, Y)))), true, implies(implies(implies(implies(X, Y), implies(Y, Y)), Y), implies(Y, Y))), true, implies(implies(implies(Y, Y), implies(implies(X, Y), implies(Y, Y))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y))))), true, implies(implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y))))), true, implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))), true, implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))
% 4.37/0.93 = { by axiom 2 (condensed_detachment) R->L }
% 4.37/0.93 fresh(fresh(fresh2(fresh2(fresh2(fresh(true, true, implies(implies(implies(W2, V2), U2), implies(implies(U2, W2), implies(T2, W2))), implies(implies(Y, Y), implies(implies(X, Y), implies(Y, Y)))), true, implies(implies(implies(implies(X, Y), implies(Y, Y)), Y), implies(Y, Y))), true, implies(implies(implies(Y, Y), implies(implies(X, Y), implies(Y, Y))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y))))), true, implies(implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y))))), true, implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))), true, implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))
% 4.37/0.93 = { by lemma 9 R->L }
% 4.37/0.93 fresh(fresh(fresh2(fresh2(fresh2(fresh(is_a_theorem(implies(implies(implies(implies(W2, V2), U2), implies(implies(U2, W2), implies(T2, W2))), implies(implies(Y, Y), implies(implies(X, Y), implies(Y, Y))))), true, implies(implies(implies(W2, V2), U2), implies(implies(U2, W2), implies(T2, W2))), implies(implies(Y, Y), implies(implies(X, Y), implies(Y, Y)))), true, implies(implies(implies(implies(X, Y), implies(Y, Y)), Y), implies(Y, Y))), true, implies(implies(implies(Y, Y), implies(implies(X, Y), implies(Y, Y))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y))))), true, implies(implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y))))), true, implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))), true, implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))
% 4.37/0.93 = { by axiom 3 (condensed_detachment) }
% 4.37/0.93 fresh(fresh(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(W2, V2), U2), implies(implies(U2, W2), implies(T2, W2)))), true, implies(implies(Y, Y), implies(implies(X, Y), implies(Y, Y)))), true, implies(implies(implies(implies(X, Y), implies(Y, Y)), Y), implies(Y, Y))), true, implies(implies(implies(Y, Y), implies(implies(X, Y), implies(Y, Y))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y))))), true, implies(implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y))))), true, implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))), true, implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))
% 4.37/0.93 = { by axiom 4 (ic_JLukasiewicz) }
% 4.37/0.93 fresh(fresh(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(Y, Y), implies(implies(X, Y), implies(Y, Y)))), true, implies(implies(implies(implies(X, Y), implies(Y, Y)), Y), implies(Y, Y))), true, implies(implies(implies(Y, Y), implies(implies(X, Y), implies(Y, Y))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y))))), true, implies(implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y))))), true, implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))), true, implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))
% 4.37/0.93 = { by axiom 1 (condensed_detachment) }
% 4.37/0.93 fresh(fresh(fresh2(fresh2(fresh2(true, true, implies(implies(implies(implies(X, Y), implies(Y, Y)), Y), implies(Y, Y))), true, implies(implies(implies(Y, Y), implies(implies(X, Y), implies(Y, Y))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y))))), true, implies(implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y))))), true, implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))), true, implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))
% 4.37/0.93 = { by axiom 1 (condensed_detachment) }
% 4.37/0.93 fresh(fresh(fresh2(fresh2(true, true, implies(implies(implies(Y, Y), implies(implies(X, Y), implies(Y, Y))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y))))), true, implies(implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y))))), true, implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))), true, implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))
% 4.37/0.93 = { by axiom 1 (condensed_detachment) }
% 4.37/0.93 fresh(fresh(fresh2(true, true, implies(implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y))))), true, implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))), true, implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))
% 4.37/0.93 = { by axiom 1 (condensed_detachment) }
% 4.37/0.93 fresh(fresh(true, true, implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))), true, implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))
% 4.37/0.93 = { by axiom 2 (condensed_detachment) }
% 4.37/0.93 fresh(is_a_theorem(implies(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))), true, implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z))), implies(implies(X, Y), implies(Y, Y)))
% 4.37/0.93 = { by axiom 3 (condensed_detachment) }
% 4.37/0.93 fresh2(is_a_theorem(implies(implies(implies(Z, W), V), implies(implies(V, Z), implies(U, Z)))), true, implies(implies(X, Y), implies(Y, Y)))
% 4.37/0.93 = { by axiom 4 (ic_JLukasiewicz) }
% 4.37/0.93 fresh2(true, true, implies(implies(X, Y), implies(Y, Y)))
% 4.37/0.93 = { by axiom 1 (condensed_detachment) }
% 4.37/0.93 true
% 4.37/0.93
% 4.37/0.93 Lemma 11: is_a_theorem(implies(X, implies(implies(Y, Z), implies(Z, implies(Y, Z))))) = true.
% 4.37/0.93 Proof:
% 4.37/0.93 is_a_theorem(implies(X, implies(implies(Y, Z), implies(Z, implies(Y, Z)))))
% 4.37/0.93 = { by lemma 7 R->L }
% 4.37/0.93 fresh2(is_a_theorem(implies(implies(implies(implies(W, Z), implies(Y, Z)), implies(Z, implies(Y, Z))), implies(implies(Y, Z), implies(Z, implies(Y, Z))))), true, implies(X, implies(implies(Y, Z), implies(Z, implies(Y, Z)))))
% 4.37/0.93 = { by lemma 6 }
% 4.37/0.93 fresh2(true, true, implies(X, implies(implies(Y, Z), implies(Z, implies(Y, Z)))))
% 4.37/0.93 = { by axiom 1 (condensed_detachment) }
% 4.37/0.93 true
% 4.37/0.93
% 4.37/0.93 Lemma 12: fresh2(is_a_theorem(implies(X, implies(X, Y))), true, implies(Z, implies(X, Y))) = is_a_theorem(implies(Z, implies(X, Y))).
% 4.37/0.93 Proof:
% 4.37/0.93 fresh2(is_a_theorem(implies(X, implies(X, Y))), true, implies(Z, implies(X, Y)))
% 4.37/0.93 = { by axiom 3 (condensed_detachment) R->L }
% 4.37/0.93 fresh(is_a_theorem(implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.93 = { by axiom 2 (condensed_detachment) R->L }
% 4.37/0.93 fresh(fresh(true, true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.93 = { by axiom 1 (condensed_detachment) R->L }
% 4.37/0.93 fresh(fresh(fresh2(true, true, implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.93 = { by axiom 4 (ic_JLukasiewicz) R->L }
% 4.37/0.93 fresh(fresh(fresh2(is_a_theorem(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.93 = { by axiom 3 (condensed_detachment) R->L }
% 4.37/0.93 fresh(fresh(fresh(is_a_theorem(implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.93 = { by lemma 5 R->L }
% 4.37/0.93 fresh(fresh(fresh(fresh2(is_a_theorem(implies(implies(implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))), X), implies(implies(implies(X, Y), S), X))), true, implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.94 = { by lemma 5 R->L }
% 4.37/0.94 fresh(fresh(fresh(fresh2(fresh2(is_a_theorem(implies(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))), X), implies(implies(implies(X, Y), S), X))), true, implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.94 = { by axiom 2 (condensed_detachment) R->L }
% 4.37/0.94 fresh(fresh(fresh(fresh2(fresh2(fresh(true, true, implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(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))), X), implies(implies(implies(X, Y), S), X))), true, implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.94 = { by axiom 1 (condensed_detachment) R->L }
% 4.37/0.94 fresh(fresh(fresh(fresh2(fresh2(fresh(fresh2(true, true, implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(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))), X), implies(implies(implies(X, Y), S), X))), true, implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.94 = { by axiom 4 (ic_JLukasiewicz) R->L }
% 4.37/0.94 fresh(fresh(fresh(fresh2(fresh2(fresh(fresh2(is_a_theorem(implies(implies(implies(Y2, Z2), W2), implies(implies(W2, Y2), implies(V2, Y2)))), true, implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(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))), X), implies(implies(implies(X, Y), S), X))), true, implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.94 = { by axiom 3 (condensed_detachment) R->L }
% 4.37/0.94 fresh(fresh(fresh(fresh2(fresh2(fresh(fresh(is_a_theorem(implies(implies(implies(implies(Y2, Z2), W2), implies(implies(W2, Y2), implies(V2, Y2))), implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))))), true, implies(implies(implies(Y2, Z2), W2), implies(implies(W2, Y2), implies(V2, Y2))), implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(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))), X), implies(implies(implies(X, Y), S), X))), true, implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.95 = { by axiom 2 (condensed_detachment) R->L }
% 4.37/0.95 fresh(fresh(fresh(fresh2(fresh2(fresh(fresh(fresh(true, true, implies(implies(implies(implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))), U2), implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), implies(implies(implies(implies(Y2, Z2), W2), implies(implies(W2, Y2), implies(V2, Y2))), implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))))), true, implies(implies(implies(Y2, Z2), W2), implies(implies(W2, Y2), implies(V2, Y2))), implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(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))), X), implies(implies(implies(X, Y), S), X))), true, implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.95 = { by axiom 1 (condensed_detachment) R->L }
% 4.37/0.95 fresh(fresh(fresh(fresh2(fresh2(fresh(fresh(fresh(fresh2(true, true, implies(implies(implies(implies(implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))), U2), implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), implies(implies(implies(implies(Y2, Z2), W2), implies(implies(W2, Y2), implies(V2, Y2))), implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(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))), U2), implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), implies(implies(implies(implies(Y2, Z2), W2), implies(implies(W2, Y2), implies(V2, Y2))), implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))))), true, implies(implies(implies(Y2, Z2), W2), implies(implies(W2, Y2), implies(V2, Y2))), implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(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))), X), implies(implies(implies(X, Y), S), X))), true, implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.95 = { by lemma 8 R->L }
% 4.37/0.96 fresh(fresh(fresh(fresh2(fresh2(fresh(fresh(fresh(fresh2(is_a_theorem(implies(implies(implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))), U2), implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))), U2), implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), implies(implies(implies(implies(Y2, Z2), W2), implies(implies(W2, Y2), implies(V2, Y2))), implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(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))), U2), implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), implies(implies(implies(implies(Y2, Z2), W2), implies(implies(W2, Y2), implies(V2, Y2))), implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))))), true, implies(implies(implies(Y2, Z2), W2), implies(implies(W2, Y2), implies(V2, Y2))), implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(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))), X), implies(implies(implies(X, Y), S), X))), true, implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.96 = { by lemma 5 }
% 4.37/0.96 fresh(fresh(fresh(fresh2(fresh2(fresh(fresh(fresh(is_a_theorem(implies(implies(implies(implies(implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))), U2), implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), implies(implies(implies(implies(Y2, Z2), W2), implies(implies(W2, Y2), implies(V2, Y2))), implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(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))), U2), implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), implies(implies(implies(implies(Y2, Z2), W2), implies(implies(W2, Y2), implies(V2, Y2))), implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))))), true, implies(implies(implies(Y2, Z2), W2), implies(implies(W2, Y2), implies(V2, Y2))), implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(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))), X), implies(implies(implies(X, Y), S), X))), true, implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.96 = { by axiom 3 (condensed_detachment) }
% 4.37/0.96 fresh(fresh(fresh(fresh2(fresh2(fresh(fresh(fresh2(is_a_theorem(implies(implies(implies(implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))), U2), implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))))), true, implies(implies(implies(implies(Y2, Z2), W2), implies(implies(W2, Y2), implies(V2, Y2))), implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))))), true, implies(implies(implies(Y2, Z2), W2), implies(implies(W2, Y2), implies(V2, Y2))), implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(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))), X), implies(implies(implies(X, Y), S), X))), true, implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.96 = { by axiom 4 (ic_JLukasiewicz) }
% 4.37/0.96 fresh(fresh(fresh(fresh2(fresh2(fresh(fresh(fresh2(true, true, implies(implies(implies(implies(Y2, Z2), W2), implies(implies(W2, Y2), implies(V2, Y2))), implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))))), true, implies(implies(implies(Y2, Z2), W2), implies(implies(W2, Y2), implies(V2, Y2))), implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(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))), X), implies(implies(implies(X, Y), S), X))), true, implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.96 = { by axiom 1 (condensed_detachment) }
% 4.37/0.96 fresh(fresh(fresh(fresh2(fresh2(fresh(fresh(true, true, implies(implies(implies(Y2, Z2), W2), implies(implies(W2, Y2), implies(V2, Y2))), implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(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))), X), implies(implies(implies(X, Y), S), X))), true, implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.96 = { by axiom 2 (condensed_detachment) }
% 4.37/0.96 fresh(fresh(fresh(fresh2(fresh2(fresh(is_a_theorem(implies(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(X, Y), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(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))), X), implies(implies(implies(X, Y), S), X))), true, implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.97 = { by axiom 3 (condensed_detachment) }
% 4.37/0.97 fresh(fresh(fresh(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(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))), X), implies(implies(implies(X, Y), S), X))), true, implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.97 = { by lemma 5 R->L }
% 4.37/0.97 fresh(fresh(fresh(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(Z, implies(X, Y)), T2), implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)))), true, implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(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))), X), implies(implies(implies(X, Y), S), X))), true, implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.97 = { by axiom 2 (condensed_detachment) R->L }
% 4.37/0.97 fresh(fresh(fresh(fresh2(fresh2(fresh2(fresh2(fresh(true, true, implies(implies(implies(Y, X2), implies(X2, implies(Y, X2))), implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y))), implies(implies(implies(Z, implies(X, Y)), T2), implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)))), true, implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(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))), X), implies(implies(implies(X, Y), S), X))), true, implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.97 = { by axiom 1 (condensed_detachment) R->L }
% 4.37/0.97 fresh(fresh(fresh(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(true, true, implies(implies(implies(implies(Y, X2), implies(X2, implies(Y, X2))), implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y))), implies(implies(implies(Z, implies(X, Y)), T2), implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y))))), true, implies(implies(implies(Y, X2), implies(X2, implies(Y, X2))), implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y))), implies(implies(implies(Z, implies(X, Y)), T2), implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)))), true, implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(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))), X), implies(implies(implies(X, Y), S), X))), true, implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.97 = { by lemma 11 R->L }
% 4.37/0.97 fresh(fresh(fresh(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(is_a_theorem(implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), S2), implies(implies(Y, X2), implies(X2, implies(Y, X2))))), true, implies(implies(implies(implies(Y, X2), implies(X2, implies(Y, X2))), implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y))), implies(implies(implies(Z, implies(X, Y)), T2), implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y))))), true, implies(implies(implies(Y, X2), implies(X2, implies(Y, X2))), implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y))), implies(implies(implies(Z, implies(X, Y)), T2), implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)))), true, implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(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))), X), implies(implies(implies(X, Y), S), X))), true, implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.97 = { by lemma 5 }
% 4.37/0.97 fresh(fresh(fresh(fresh2(fresh2(fresh2(fresh2(fresh(is_a_theorem(implies(implies(implies(implies(Y, X2), implies(X2, implies(Y, X2))), implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y))), implies(implies(implies(Z, implies(X, Y)), T2), implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y))))), true, implies(implies(implies(Y, X2), implies(X2, implies(Y, X2))), implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y))), implies(implies(implies(Z, implies(X, Y)), T2), implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)))), true, implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(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))), X), implies(implies(implies(X, Y), S), X))), true, implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.97 = { by axiom 3 (condensed_detachment) }
% 4.37/0.97 fresh(fresh(fresh(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(Y, X2), implies(X2, implies(Y, X2))), implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)))), true, implies(implies(implies(Z, implies(X, Y)), T2), implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)))), true, implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(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))), X), implies(implies(implies(X, Y), S), X))), true, implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.97 = { by axiom 4 (ic_JLukasiewicz) }
% 4.37/0.97 fresh(fresh(fresh(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(implies(Z, implies(X, Y)), T2), implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)))), true, implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(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))), X), implies(implies(implies(X, Y), S), X))), true, implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.97 = { by axiom 1 (condensed_detachment) }
% 4.37/0.97 fresh(fresh(fresh(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(implies(implies(implies(X2, implies(Y, X2)), Y), implies(X, Y)), implies(Z, implies(X, Y))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(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))), X), implies(implies(implies(X, Y), S), X))), true, implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.98 = { by axiom 1 (condensed_detachment) }
% 4.37/0.98 fresh(fresh(fresh(fresh2(fresh2(fresh2(true, true, implies(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))), X), implies(implies(implies(X, Y), S), X))), true, implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.98 = { by axiom 1 (condensed_detachment) }
% 4.37/0.98 fresh(fresh(fresh(fresh2(fresh2(true, true, implies(implies(implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))), X), implies(implies(implies(X, Y), S), X))), true, implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.98 = { by axiom 1 (condensed_detachment) }
% 4.37/0.98 fresh(fresh(fresh(fresh2(true, true, implies(implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))))), true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.98 = { by axiom 1 (condensed_detachment) }
% 4.37/0.98 fresh(fresh(fresh(true, true, implies(implies(implies(implies(X, Y), S), X), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.98 = { by axiom 2 (condensed_detachment) }
% 4.37/0.98 fresh(fresh(is_a_theorem(implies(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y))))), true, implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W))), implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.98 = { by axiom 3 (condensed_detachment) }
% 4.37/0.98 fresh(fresh2(is_a_theorem(implies(implies(implies(W, V), U), implies(implies(U, W), implies(T, W)))), true, implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.98 = { by axiom 4 (ic_JLukasiewicz) }
% 4.37/0.98 fresh(fresh2(true, true, implies(implies(X, implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.98 = { by axiom 1 (condensed_detachment) }
% 4.37/0.98 fresh(true, true, implies(X, implies(X, Y)), implies(Z, implies(X, Y)))
% 4.37/0.98 = { by axiom 2 (condensed_detachment) }
% 4.37/0.98 is_a_theorem(implies(Z, implies(X, Y)))
% 4.37/0.98
% 4.37/0.98 Goal 1 (prove_ic_3): is_a_theorem(implies(implies(implies(a, b), a), a)) = true.
% 4.37/0.98 Proof:
% 4.37/0.98 is_a_theorem(implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by axiom 2 (condensed_detachment) R->L }
% 4.37/0.98 fresh(true, true, implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by axiom 1 (condensed_detachment) R->L }
% 4.37/0.98 fresh(fresh2(true, true, implies(implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a))), true, implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by axiom 4 (ic_JLukasiewicz) R->L }
% 4.37/0.98 fresh(fresh2(is_a_theorem(implies(implies(implies(Y, Z), W), implies(implies(W, Y), implies(V, Y)))), true, implies(implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a))), true, implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by axiom 3 (condensed_detachment) R->L }
% 4.37/0.98 fresh(fresh(is_a_theorem(implies(implies(implies(implies(Y, Z), W), implies(implies(W, Y), implies(V, Y))), implies(implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a)))), true, implies(implies(implies(Y, Z), W), implies(implies(W, Y), implies(V, Y))), implies(implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a))), true, implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by lemma 12 R->L }
% 4.37/0.98 fresh(fresh(fresh2(is_a_theorem(implies(implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a)))), true, implies(implies(implies(implies(Y, Z), W), implies(implies(W, Y), implies(V, Y))), implies(implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a)))), true, implies(implies(implies(Y, Z), W), implies(implies(W, Y), implies(V, Y))), implies(implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a))), true, implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by lemma 5 R->L }
% 4.37/0.98 fresh(fresh(fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(implies(a, b), a), a), X), implies(X, X))), true, implies(implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a)))), true, implies(implies(implies(implies(Y, Z), W), implies(implies(W, Y), implies(V, Y))), implies(implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a)))), true, implies(implies(implies(Y, Z), W), implies(implies(W, Y), implies(V, Y))), implies(implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a))), true, implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by lemma 10 }
% 4.37/0.98 fresh(fresh(fresh2(fresh2(true, true, implies(implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a)))), true, implies(implies(implies(implies(Y, Z), W), implies(implies(W, Y), implies(V, Y))), implies(implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a)))), true, implies(implies(implies(Y, Z), W), implies(implies(W, Y), implies(V, Y))), implies(implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a))), true, implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by axiom 1 (condensed_detachment) }
% 4.37/0.98 fresh(fresh(fresh2(true, true, implies(implies(implies(implies(Y, Z), W), implies(implies(W, Y), implies(V, Y))), implies(implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a)))), true, implies(implies(implies(Y, Z), W), implies(implies(W, Y), implies(V, Y))), implies(implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a))), true, implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by axiom 1 (condensed_detachment) }
% 4.37/0.98 fresh(fresh(true, true, implies(implies(implies(Y, Z), W), implies(implies(W, Y), implies(V, Y))), implies(implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a))), true, implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by axiom 2 (condensed_detachment) }
% 4.37/0.98 fresh(is_a_theorem(implies(implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a))), true, implies(implies(X, X), implies(implies(implies(a, b), a), a)), implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by axiom 3 (condensed_detachment) }
% 4.37/0.98 fresh2(is_a_theorem(implies(implies(X, X), implies(implies(implies(a, b), a), a))), true, implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by lemma 12 R->L }
% 4.37/0.98 fresh2(fresh2(is_a_theorem(implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))), true, implies(implies(X, X), implies(implies(implies(a, b), a), a))), true, implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by lemma 5 R->L }
% 4.37/0.98 fresh2(fresh2(fresh2(is_a_theorem(implies(implies(a, b), implies(a, b))), true, implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))), true, implies(implies(X, X), implies(implies(implies(a, b), a), a))), true, implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by axiom 2 (condensed_detachment) R->L }
% 4.37/0.98 fresh2(fresh2(fresh2(fresh(true, true, implies(implies(implies(U, implies(b, U)), b), implies(a, b)), implies(implies(a, b), implies(a, b))), true, implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))), true, implies(implies(X, X), implies(implies(implies(a, b), a), a))), true, implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by lemma 10 R->L }
% 4.37/0.98 fresh2(fresh2(fresh2(fresh(is_a_theorem(implies(implies(implies(implies(U, implies(b, U)), b), implies(a, b)), implies(implies(a, b), implies(a, b)))), true, implies(implies(implies(U, implies(b, U)), b), implies(a, b)), implies(implies(a, b), implies(a, b))), true, implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))), true, implies(implies(X, X), implies(implies(implies(a, b), a), a))), true, implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by axiom 3 (condensed_detachment) }
% 4.37/0.98 fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(U, implies(b, U)), b), implies(a, b))), true, implies(implies(a, b), implies(a, b))), true, implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))), true, implies(implies(X, X), implies(implies(implies(a, b), a), a))), true, implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by lemma 5 R->L }
% 4.37/0.98 fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(b, U), implies(U, implies(b, U)))), true, implies(implies(implies(U, implies(b, U)), b), implies(a, b))), true, implies(implies(a, b), implies(a, b))), true, implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))), true, implies(implies(X, X), implies(implies(implies(a, b), a), a))), true, implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by axiom 2 (condensed_detachment) R->L }
% 4.37/0.98 fresh2(fresh2(fresh2(fresh2(fresh2(fresh(true, true, implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(b, U), implies(U, implies(b, U)))), true, implies(implies(implies(U, implies(b, U)), b), implies(a, b))), true, implies(implies(a, b), implies(a, b))), true, implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))), true, implies(implies(X, X), implies(implies(implies(a, b), a), a))), true, implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by lemma 11 R->L }
% 4.37/0.98 fresh2(fresh2(fresh2(fresh2(fresh2(fresh(is_a_theorem(implies(implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(b, U), implies(U, implies(b, U))))), true, implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T))), implies(implies(b, U), implies(U, implies(b, U)))), true, implies(implies(implies(U, implies(b, U)), b), implies(a, b))), true, implies(implies(a, b), implies(a, b))), true, implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))), true, implies(implies(X, X), implies(implies(implies(a, b), a), a))), true, implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by axiom 3 (condensed_detachment) }
% 4.37/0.98 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(T, S), X2), implies(implies(X2, T), implies(Y2, T)))), true, implies(implies(b, U), implies(U, implies(b, U)))), true, implies(implies(implies(U, implies(b, U)), b), implies(a, b))), true, implies(implies(a, b), implies(a, b))), true, implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))), true, implies(implies(X, X), implies(implies(implies(a, b), a), a))), true, implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by axiom 4 (ic_JLukasiewicz) }
% 4.37/0.98 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(b, U), implies(U, implies(b, U)))), true, implies(implies(implies(U, implies(b, U)), b), implies(a, b))), true, implies(implies(a, b), implies(a, b))), true, implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))), true, implies(implies(X, X), implies(implies(implies(a, b), a), a))), true, implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by axiom 1 (condensed_detachment) }
% 4.37/0.98 fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(implies(U, implies(b, U)), b), implies(a, b))), true, implies(implies(a, b), implies(a, b))), true, implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))), true, implies(implies(X, X), implies(implies(implies(a, b), a), a))), true, implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by axiom 1 (condensed_detachment) }
% 4.37/0.98 fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(a, b), implies(a, b))), true, implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))), true, implies(implies(X, X), implies(implies(implies(a, b), a), a))), true, implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by axiom 1 (condensed_detachment) }
% 4.37/0.98 fresh2(fresh2(fresh2(true, true, implies(implies(implies(a, b), a), implies(implies(implies(a, b), a), a))), true, implies(implies(X, X), implies(implies(implies(a, b), a), a))), true, implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by axiom 1 (condensed_detachment) }
% 4.37/0.98 fresh2(fresh2(true, true, implies(implies(X, X), implies(implies(implies(a, b), a), a))), true, implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by axiom 1 (condensed_detachment) }
% 4.37/0.98 fresh2(true, true, implies(implies(implies(a, b), a), a))
% 4.37/0.98 = { by axiom 1 (condensed_detachment) }
% 4.37/0.98 true
% 4.37/0.98 % SZS output end Proof
% 4.37/0.98
% 4.37/0.98 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------