TSTP Solution File: LCL018-1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL018-1 : 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 : n009.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 08:17:03 EDT 2023
% Result : Unsatisfiable 11.43s 1.84s
% Output : Proof 11.89s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : LCL018-1 : TPTP v8.1.2. Released v1.0.0.
% 0.07/0.13 % 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 : n009.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 18:15:51 EDT 2023
% 0.13/0.35 % CPUTime :
% 11.43/1.84 Command-line arguments: --no-flatten-goal
% 11.43/1.84
% 11.43/1.84 % SZS status Unsatisfiable
% 11.43/1.84
% 11.89/1.91 % SZS output start Proof
% 11.89/1.91 Take the following subset of the input axioms:
% 11.89/1.91 fof(condensed_detachment, axiom, ![X, Y]: (~is_a_theorem(equivalent(X, Y)) | (~is_a_theorem(X) | is_a_theorem(Y)))).
% 11.89/1.91 fof(prove_pyo, negated_conjecture, ~is_a_theorem(equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))).
% 11.89/1.91 fof(pym, axiom, ![Z, X2, Y2]: is_a_theorem(equivalent(equivalent(equivalent(X2, equivalent(Y2, Z)), Y2), equivalent(Z, X2)))).
% 11.89/1.91
% 11.89/1.91 Now clausify the problem and encode Horn clauses using encoding 3 of
% 11.89/1.91 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 11.89/1.91 We repeatedly replace C & s=t => u=v by the two clauses:
% 11.89/1.91 fresh(y, y, x1...xn) = u
% 11.89/1.91 C => fresh(s, t, x1...xn) = v
% 11.89/1.91 where fresh is a fresh function symbol and x1..xn are the free
% 11.89/1.91 variables of u and v.
% 11.89/1.91 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 11.89/1.91 input problem has no model of domain size 1).
% 11.89/1.91
% 11.89/1.91 The encoding turns the above axioms into the following unit equations and goals:
% 11.89/1.92
% 11.89/1.92 Axiom 1 (condensed_detachment): fresh2(X, X, Y) = true.
% 11.89/1.92 Axiom 2 (condensed_detachment): fresh(X, X, Y, Z) = is_a_theorem(Z).
% 11.89/1.92 Axiom 3 (condensed_detachment): fresh(is_a_theorem(equivalent(X, Y)), true, X, Y) = fresh2(is_a_theorem(X), true, Y).
% 11.89/1.92 Axiom 4 (pym): is_a_theorem(equivalent(equivalent(equivalent(X, equivalent(Y, Z)), Y), equivalent(Z, X))) = true.
% 11.89/1.92
% 11.89/1.92 Lemma 5: fresh2(is_a_theorem(equivalent(equivalent(X, equivalent(Y, Z)), Y)), true, equivalent(Z, X)) = is_a_theorem(equivalent(Z, X)).
% 11.89/1.92 Proof:
% 11.89/1.92 fresh2(is_a_theorem(equivalent(equivalent(X, equivalent(Y, Z)), Y)), true, equivalent(Z, X))
% 11.89/1.92 = { by axiom 3 (condensed_detachment) R->L }
% 11.89/1.92 fresh(is_a_theorem(equivalent(equivalent(equivalent(X, equivalent(Y, Z)), Y), equivalent(Z, X))), true, equivalent(equivalent(X, equivalent(Y, Z)), Y), equivalent(Z, X))
% 11.89/1.92 = { by axiom 4 (pym) }
% 11.89/1.92 fresh(true, true, equivalent(equivalent(X, equivalent(Y, Z)), Y), equivalent(Z, X))
% 11.89/1.92 = { by axiom 2 (condensed_detachment) }
% 11.89/1.92 is_a_theorem(equivalent(Z, X))
% 11.89/1.92
% 11.89/1.92 Lemma 6: is_a_theorem(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z)))) = true.
% 11.89/1.92 Proof:
% 11.89/1.92 is_a_theorem(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z))))
% 11.89/1.92 = { by lemma 5 R->L }
% 11.89/1.92 fresh2(is_a_theorem(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z)), equivalent(equivalent(Z, Y), X)), equivalent(Z, Y))), true, equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z))))
% 11.89/1.92 = { by axiom 4 (pym) }
% 11.89/1.92 fresh2(true, true, equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z))))
% 11.89/1.92 = { by axiom 1 (condensed_detachment) }
% 11.89/1.92 true
% 11.89/1.92
% 11.89/1.92 Lemma 7: fresh2(is_a_theorem(X), true, equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z))) = is_a_theorem(equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z))).
% 11.89/1.92 Proof:
% 11.89/1.92 fresh2(is_a_theorem(X), true, equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z)))
% 11.89/1.92 = { by axiom 3 (condensed_detachment) R->L }
% 11.89/1.92 fresh(is_a_theorem(equivalent(X, equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z)))), true, X, equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z)))
% 11.89/1.92 = { by lemma 6 }
% 11.89/1.92 fresh(true, true, X, equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z)))
% 11.89/1.92 = { by axiom 2 (condensed_detachment) }
% 11.89/1.92 is_a_theorem(equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z)))
% 11.89/1.92
% 11.89/1.92 Lemma 8: fresh2(is_a_theorem(X), true, equivalent(equivalent(equivalent(Y, X), equivalent(equivalent(equivalent(Z, equivalent(W, V)), W), equivalent(V, Z))), Y)) = is_a_theorem(equivalent(equivalent(equivalent(Y, X), equivalent(equivalent(equivalent(Z, equivalent(W, V)), W), equivalent(V, Z))), Y)).
% 11.89/1.92 Proof:
% 11.89/1.92 fresh2(is_a_theorem(X), true, equivalent(equivalent(equivalent(Y, X), equivalent(equivalent(equivalent(Z, equivalent(W, V)), W), equivalent(V, Z))), Y))
% 11.89/1.92 = { by axiom 3 (condensed_detachment) R->L }
% 11.89/1.92 fresh(is_a_theorem(equivalent(X, equivalent(equivalent(equivalent(Y, X), equivalent(equivalent(equivalent(Z, equivalent(W, V)), W), equivalent(V, Z))), Y))), true, X, equivalent(equivalent(equivalent(Y, X), equivalent(equivalent(equivalent(Z, equivalent(W, V)), W), equivalent(V, Z))), Y))
% 11.89/1.92 = { by lemma 7 R->L }
% 11.89/1.92 fresh(fresh2(is_a_theorem(equivalent(equivalent(equivalent(Z, equivalent(W, V)), W), equivalent(V, Z))), true, equivalent(X, equivalent(equivalent(equivalent(Y, X), equivalent(equivalent(equivalent(Z, equivalent(W, V)), W), equivalent(V, Z))), Y))), true, X, equivalent(equivalent(equivalent(Y, X), equivalent(equivalent(equivalent(Z, equivalent(W, V)), W), equivalent(V, Z))), Y))
% 11.89/1.92 = { by axiom 4 (pym) }
% 11.89/1.92 fresh(fresh2(true, true, equivalent(X, equivalent(equivalent(equivalent(Y, X), equivalent(equivalent(equivalent(Z, equivalent(W, V)), W), equivalent(V, Z))), Y))), true, X, equivalent(equivalent(equivalent(Y, X), equivalent(equivalent(equivalent(Z, equivalent(W, V)), W), equivalent(V, Z))), Y))
% 11.89/1.92 = { by axiom 1 (condensed_detachment) }
% 11.89/1.92 fresh(true, true, X, equivalent(equivalent(equivalent(Y, X), equivalent(equivalent(equivalent(Z, equivalent(W, V)), W), equivalent(V, Z))), Y))
% 11.89/1.92 = { by axiom 2 (condensed_detachment) }
% 11.89/1.92 is_a_theorem(equivalent(equivalent(equivalent(Y, X), equivalent(equivalent(equivalent(Z, equivalent(W, V)), W), equivalent(V, Z))), Y))
% 11.89/1.92
% 11.89/1.92 Lemma 9: is_a_theorem(equivalent(X, equivalent(equivalent(equivalent(Y, X), equivalent(Z, equivalent(W, equivalent(equivalent(equivalent(V, W), Z), V)))), Y))) = true.
% 11.89/1.92 Proof:
% 11.89/1.92 is_a_theorem(equivalent(X, equivalent(equivalent(equivalent(Y, X), equivalent(Z, equivalent(W, equivalent(equivalent(equivalent(V, W), Z), V)))), Y)))
% 11.89/1.92 = { by lemma 7 R->L }
% 11.89/1.92 fresh2(is_a_theorem(equivalent(Z, equivalent(W, equivalent(equivalent(equivalent(V, W), Z), V)))), true, equivalent(X, equivalent(equivalent(equivalent(Y, X), equivalent(Z, equivalent(W, equivalent(equivalent(equivalent(V, W), Z), V)))), Y)))
% 11.89/1.92 = { by lemma 6 }
% 11.89/1.92 fresh2(true, true, equivalent(X, equivalent(equivalent(equivalent(Y, X), equivalent(Z, equivalent(W, equivalent(equivalent(equivalent(V, W), Z), V)))), Y)))
% 11.89/1.92 = { by axiom 1 (condensed_detachment) }
% 11.89/1.92 true
% 11.89/1.92
% 11.89/1.92 Lemma 10: is_a_theorem(equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X), V)) = true.
% 11.89/1.92 Proof:
% 11.89/1.92 is_a_theorem(equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X), V))
% 11.89/1.92 = { by lemma 5 R->L }
% 11.89/1.92 fresh2(is_a_theorem(equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)))), true, equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X), V))
% 11.89/1.92 = { by axiom 2 (condensed_detachment) R->L }
% 11.89/1.92 fresh2(fresh(true, true, equivalent(U, equivalent(equivalent(equivalent(T, U), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X)))), T)), equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)))), true, equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X), V))
% 11.89/1.92 = { by axiom 1 (condensed_detachment) R->L }
% 11.89/1.92 fresh2(fresh(fresh2(true, true, equivalent(equivalent(U, equivalent(equivalent(equivalent(T, U), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X)))), T)), equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))))), true, equivalent(U, equivalent(equivalent(equivalent(T, U), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X)))), T)), equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)))), true, equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X), V))
% 11.89/1.92 = { by axiom 1 (condensed_detachment) R->L }
% 11.89/1.92 fresh2(fresh(fresh2(fresh2(true, true, equivalent(equivalent(equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(U, equivalent(equivalent(equivalent(T, U), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X)))), T)))), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))))), true, equivalent(equivalent(U, equivalent(equivalent(equivalent(T, U), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X)))), T)), equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))))), true, equivalent(U, equivalent(equivalent(equivalent(T, U), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X)))), T)), equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)))), true, equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X), V))
% 11.89/1.92 = { by axiom 4 (pym) R->L }
% 11.89/1.93 fresh2(fresh(fresh2(fresh2(is_a_theorem(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), true, equivalent(equivalent(equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(U, equivalent(equivalent(equivalent(T, U), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X)))), T)))), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))))), true, equivalent(equivalent(U, equivalent(equivalent(equivalent(T, U), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X)))), T)), equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))))), true, equivalent(U, equivalent(equivalent(equivalent(T, U), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X)))), T)), equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)))), true, equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X), V))
% 11.89/1.93 = { by axiom 3 (condensed_detachment) R->L }
% 11.89/1.93 fresh2(fresh(fresh2(fresh(is_a_theorem(equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(U, equivalent(equivalent(equivalent(T, U), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X)))), T)))), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X)))))), true, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(U, equivalent(equivalent(equivalent(T, U), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X)))), T)))), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))))), true, equivalent(equivalent(U, equivalent(equivalent(equivalent(T, U), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X)))), T)), equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))))), true, equivalent(U, equivalent(equivalent(equivalent(T, U), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X)))), T)), equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)))), true, equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X), V))
% 11.89/1.93 = { by lemma 9 }
% 11.89/1.93 fresh2(fresh(fresh2(fresh(true, true, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(U, equivalent(equivalent(equivalent(T, U), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X)))), T)))), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))))), true, equivalent(equivalent(U, equivalent(equivalent(equivalent(T, U), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X)))), T)), equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))))), true, equivalent(U, equivalent(equivalent(equivalent(T, U), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X)))), T)), equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)))), true, equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X), V))
% 11.89/1.93 = { by axiom 2 (condensed_detachment) }
% 11.89/1.93 fresh2(fresh(fresh2(is_a_theorem(equivalent(equivalent(equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(U, equivalent(equivalent(equivalent(T, U), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X)))), T)))), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))))), true, equivalent(equivalent(U, equivalent(equivalent(equivalent(T, U), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X)))), T)), equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))))), true, equivalent(U, equivalent(equivalent(equivalent(T, U), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X)))), T)), equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)))), true, equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X), V))
% 11.89/1.93 = { by lemma 5 }
% 11.89/1.93 fresh2(fresh(is_a_theorem(equivalent(equivalent(U, equivalent(equivalent(equivalent(T, U), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X)))), T)), equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))))), true, equivalent(U, equivalent(equivalent(equivalent(T, U), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X)))), T)), equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)))), true, equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X), V))
% 11.89/1.93 = { by axiom 3 (condensed_detachment) }
% 11.89/1.93 fresh2(fresh2(is_a_theorem(equivalent(U, equivalent(equivalent(equivalent(T, U), equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X)))), T))), true, equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)))), true, equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X), V))
% 11.89/1.93 = { by lemma 9 }
% 11.89/1.93 fresh2(fresh2(true, true, equivalent(equivalent(V, equivalent(equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X))), equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y)))), true, equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X), V))
% 11.89/1.93 = { by axiom 1 (condensed_detachment) }
% 11.89/1.93 fresh2(true, true, equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, equivalent(Z, W)), Z), equivalent(W, Y))), V), X), V))
% 11.89/1.93 = { by axiom 1 (condensed_detachment) }
% 11.89/1.93 true
% 11.89/1.93
% 11.89/1.93 Lemma 11: fresh2(is_a_theorem(equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W)))), true, Z) = is_a_theorem(Z).
% 11.89/1.93 Proof:
% 11.89/1.93 fresh2(is_a_theorem(equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W)))), true, Z)
% 11.89/1.93 = { by axiom 3 (condensed_detachment) R->L }
% 11.89/1.93 fresh(is_a_theorem(equivalent(equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z)), true, equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z)
% 11.89/1.93 = { by axiom 2 (condensed_detachment) R->L }
% 11.89/1.94 fresh(fresh(true, true, equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z), equivalent(equivalent(equivalent(Y, equivalent(W, X)), W), equivalent(X, Y))), equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z), equivalent(equivalent(equivalent(Y, equivalent(W, X)), W), equivalent(X, Y)))), equivalent(equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z)), equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z), equivalent(equivalent(equivalent(Y, equivalent(W, X)), W), equivalent(X, Y)))), equivalent(equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z)), true, equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z)
% 11.89/1.94 = { by lemma 10 R->L }
% 11.89/1.94 fresh(fresh(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z), equivalent(equivalent(equivalent(Y, equivalent(W, X)), W), equivalent(X, Y))), equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z), equivalent(equivalent(equivalent(Y, equivalent(W, X)), W), equivalent(X, Y)))), equivalent(equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z)), equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z), equivalent(equivalent(equivalent(Y, equivalent(W, X)), W), equivalent(X, Y)))), equivalent(equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z))), true, equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z), equivalent(equivalent(equivalent(Y, equivalent(W, X)), W), equivalent(X, Y))), equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z), equivalent(equivalent(equivalent(Y, equivalent(W, X)), W), equivalent(X, Y)))), equivalent(equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z)), equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z), equivalent(equivalent(equivalent(Y, equivalent(W, X)), W), equivalent(X, Y)))), equivalent(equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z)), true, equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z)
% 11.89/1.94 = { by axiom 3 (condensed_detachment) }
% 11.89/1.94 fresh(fresh2(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z), equivalent(equivalent(equivalent(Y, equivalent(W, X)), W), equivalent(X, Y))), equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z), equivalent(equivalent(equivalent(Y, equivalent(W, X)), W), equivalent(X, Y)))), equivalent(equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z)), equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z), equivalent(equivalent(equivalent(Y, equivalent(W, X)), W), equivalent(X, Y))))), true, equivalent(equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z)), true, equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z)
% 11.89/1.94 = { by lemma 10 }
% 11.89/1.94 fresh(fresh2(true, true, equivalent(equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z)), true, equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z)
% 11.89/1.94 = { by axiom 1 (condensed_detachment) }
% 11.89/1.94 fresh(true, true, equivalent(equivalent(X, Y), equivalent(Z, equivalent(equivalent(Y, equivalent(W, X)), W))), Z)
% 11.89/1.94 = { by axiom 2 (condensed_detachment) }
% 11.89/1.94 is_a_theorem(Z)
% 11.89/1.94
% 11.89/1.94 Lemma 12: is_a_theorem(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(W, Z)), W)) = true.
% 11.89/1.94 Proof:
% 11.89/1.94 is_a_theorem(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(W, Z)), W))
% 11.89/1.94 = { by lemma 11 R->L }
% 11.89/1.94 fresh2(is_a_theorem(equivalent(equivalent(Z, equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y))), equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(W, Z)), W), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(equivalent(Y, X), Z)), equivalent(Y, X))))), true, equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(W, Z)), W))
% 11.89/1.94 = { by lemma 5 R->L }
% 11.89/1.94 fresh2(fresh2(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(W, Z)), W), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(equivalent(Y, X), Z)), equivalent(Y, X))), equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(W, Z)), W), equivalent(Z, equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y))))), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(W, Z)), W))), true, equivalent(equivalent(Z, equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y))), equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(W, Z)), W), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(equivalent(Y, X), Z)), equivalent(Y, X))))), true, equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(W, Z)), W))
% 11.89/1.94 = { by lemma 8 R->L }
% 11.89/1.94 fresh2(fresh2(fresh2(is_a_theorem(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(equivalent(Y, X), Z)), equivalent(Y, X))), true, equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(W, Z)), W), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(equivalent(Y, X), Z)), equivalent(Y, X))), equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(W, Z)), W), equivalent(Z, equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y))))), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(W, Z)), W))), true, equivalent(equivalent(Z, equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y))), equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(W, Z)), W), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(equivalent(Y, X), Z)), equivalent(Y, X))))), true, equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(W, Z)), W))
% 11.89/1.94 = { by axiom 4 (pym) }
% 11.89/1.94 fresh2(fresh2(fresh2(true, true, equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(W, Z)), W), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(equivalent(Y, X), Z)), equivalent(Y, X))), equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(W, Z)), W), equivalent(Z, equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y))))), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(W, Z)), W))), true, equivalent(equivalent(Z, equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y))), equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(W, Z)), W), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(equivalent(Y, X), Z)), equivalent(Y, X))))), true, equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(W, Z)), W))
% 11.89/1.94 = { by axiom 1 (condensed_detachment) }
% 11.89/1.94 fresh2(fresh2(true, true, equivalent(equivalent(Z, equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y))), equivalent(equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(W, Z)), W), equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(equivalent(Y, X), Z)), equivalent(Y, X))))), true, equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(W, Z)), W))
% 11.89/1.94 = { by axiom 1 (condensed_detachment) }
% 11.89/1.94 fresh2(true, true, equivalent(equivalent(equivalent(X, equivalent(equivalent(equivalent(Y, X), Z), Y)), equivalent(W, Z)), W))
% 11.89/1.94 = { by axiom 1 (condensed_detachment) }
% 11.89/1.94 true
% 11.89/1.94
% 11.89/1.94 Lemma 13: is_a_theorem(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z))) = true.
% 11.89/1.94 Proof:
% 11.89/1.94 is_a_theorem(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z)))
% 11.89/1.94 = { by axiom 2 (condensed_detachment) R->L }
% 11.89/1.94 fresh(true, true, equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z))), equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z)))
% 11.89/1.94 = { by axiom 1 (condensed_detachment) R->L }
% 11.89/1.94 fresh(fresh2(true, true, equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z))), equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z)))), true, equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z))), equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z)))
% 11.89/1.94 = { by axiom 1 (condensed_detachment) R->L }
% 11.89/1.94 fresh(fresh2(fresh2(true, true, equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z))), equivalent(equivalent(equivalent(X, Y), Z), X)))), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y))), true, equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z))), equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z)))), true, equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z))), equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z)))
% 11.89/1.95 = { by lemma 6 R->L }
% 11.89/1.95 fresh(fresh2(fresh2(is_a_theorem(equivalent(Z, equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)))), true, equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z))), equivalent(equivalent(equivalent(X, Y), Z), X)))), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y))), true, equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z))), equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z)))), true, equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z))), equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z)))
% 11.89/1.95 = { by axiom 3 (condensed_detachment) R->L }
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% 11.89/1.95 = { by axiom 2 (condensed_detachment) R->L }
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% 11.89/1.97 = { by axiom 4 (pym) }
% 11.89/1.97 fresh(fresh2(fresh(fresh2(true, true, equivalent(equivalent(Z, equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z))), equivalent(equivalent(equivalent(X, Y), Z), X)))), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y)))), true, equivalent(Z, equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z))), equivalent(equivalent(equivalent(X, Y), Z), X)))), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y))), true, equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z))), equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z)))), true, equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z))), equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z)))
% 11.89/1.97 = { by axiom 1 (condensed_detachment) }
% 11.89/1.97 fresh(fresh2(fresh(true, true, equivalent(Z, equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z))), equivalent(equivalent(equivalent(X, Y), Z), X)))), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y))), true, equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z))), equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z)))), true, equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z))), equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z)))
% 11.89/1.97 = { by axiom 2 (condensed_detachment) }
% 11.89/1.97 fresh(fresh2(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z)), equivalent(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z))), equivalent(equivalent(equivalent(X, Y), Z), X)))), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y))), true, equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z))), equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z)))), true, equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z))), equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z)))
% 11.89/1.97 = { by lemma 5 }
% 11.89/1.97 fresh(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z))), equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z)))), true, equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z))), equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z)))
% 11.89/1.97 = { by axiom 3 (condensed_detachment) }
% 11.89/1.98 fresh2(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(X, Y), Z), X)), equivalent(Y, Z)), Y)), equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z))), equivalent(equivalent(equivalent(X, Y), Z), X))), true, equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z)))
% 11.89/1.98 = { by lemma 12 }
% 11.89/1.98 fresh2(true, true, equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z)))
% 11.89/1.98 = { by axiom 1 (condensed_detachment) }
% 11.89/1.98 true
% 11.89/1.98
% 11.89/1.98 Lemma 14: is_a_theorem(equivalent(X, equivalent(equivalent(equivalent(equivalent(Y, Z), X), Y), Z))) = true.
% 11.89/1.98 Proof:
% 11.89/1.98 is_a_theorem(equivalent(X, equivalent(equivalent(equivalent(equivalent(Y, Z), X), Y), Z)))
% 11.89/1.98 = { by lemma 5 R->L }
% 11.89/1.98 fresh2(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(Y, Z), X), Y), Z), equivalent(equivalent(Y, Z), X)), equivalent(Y, Z))), true, equivalent(X, equivalent(equivalent(equivalent(equivalent(Y, Z), X), Y), Z)))
% 11.89/1.98 = { by lemma 13 }
% 11.89/1.98 fresh2(true, true, equivalent(X, equivalent(equivalent(equivalent(equivalent(Y, Z), X), Y), Z)))
% 11.89/1.98 = { by axiom 1 (condensed_detachment) }
% 11.89/1.98 true
% 11.89/1.98
% 11.89/1.98 Lemma 15: fresh2(is_a_theorem(equivalent(X, Y)), true, equivalent(equivalent(equivalent(Z, Y), X), Z)) = is_a_theorem(equivalent(equivalent(equivalent(Z, Y), X), Z)).
% 11.89/1.98 Proof:
% 11.89/1.98 fresh2(is_a_theorem(equivalent(X, Y)), true, equivalent(equivalent(equivalent(Z, Y), X), Z))
% 11.89/1.98 = { by axiom 3 (condensed_detachment) R->L }
% 11.89/1.98 fresh(is_a_theorem(equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z))), true, equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z))
% 11.89/1.98 = { by axiom 2 (condensed_detachment) R->L }
% 11.89/1.98 fresh(fresh(true, true, equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z)), X), equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z))), equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z))), true, equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z))
% 11.89/1.98 = { by axiom 1 (condensed_detachment) R->L }
% 11.89/1.98 fresh(fresh(fresh2(true, true, equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z)), X), equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z))), equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z)))), true, equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z)), X), equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z))), equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z))), true, equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z))
% 11.89/1.98 = { by lemma 14 R->L }
% 11.89/1.98 fresh(fresh(fresh2(is_a_theorem(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z)), X), equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z))), equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z))), X))), true, equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z)), X), equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z))), equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z)))), true, equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z)), X), equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z))), equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z))), true, equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z))
% 11.89/1.98 = { by axiom 3 (condensed_detachment) R->L }
% 11.89/1.98 fresh(fresh(fresh(is_a_theorem(equivalent(equivalent(equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z)), X), equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z))), equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z))), X)), equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z)), X), equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z))), equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z))))), true, equivalent(equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z)), X), equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z))), equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z))), X)), equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z)), X), equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z))), equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z)))), true, equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z)), X), equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z))), equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z))), true, equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z))
% 11.89/1.98 = { by lemma 12 }
% 11.89/1.98 fresh(fresh(fresh(true, true, equivalent(equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z)), X), equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z))), equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z))), X)), equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z)), X), equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z))), equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z)))), true, equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z)), X), equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z))), equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z))), true, equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z))
% 11.89/1.98 = { by axiom 2 (condensed_detachment) }
% 11.89/1.98 fresh(fresh(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z)), X), equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z))), equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z)))), true, equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z)), X), equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z))), equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z))), true, equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z))
% 11.89/1.98 = { by axiom 3 (condensed_detachment) }
% 11.89/1.98 fresh(fresh2(is_a_theorem(equivalent(equivalent(equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z)), X), equivalent(Y, equivalent(equivalent(equivalent(Z, Y), X), Z)))), true, equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z))), true, equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z))
% 11.89/1.98 = { by lemma 13 }
% 11.89/1.98 fresh(fresh2(true, true, equivalent(equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z))), true, equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z))
% 11.89/1.98 = { by axiom 1 (condensed_detachment) }
% 11.89/1.98 fresh(true, true, equivalent(X, Y), equivalent(equivalent(equivalent(Z, Y), X), Z))
% 11.89/1.98 = { by axiom 2 (condensed_detachment) }
% 11.89/1.98 is_a_theorem(equivalent(equivalent(equivalent(Z, Y), X), Z))
% 11.89/1.98
% 11.89/1.98 Lemma 16: fresh2(is_a_theorem(equivalent(equivalent(equivalent(X, Y), Z), X)), true, equivalent(Y, Z)) = is_a_theorem(equivalent(Y, Z)).
% 11.89/1.98 Proof:
% 11.89/1.98 fresh2(is_a_theorem(equivalent(equivalent(equivalent(X, Y), Z), X)), true, equivalent(Y, Z))
% 11.89/1.98 = { by axiom 3 (condensed_detachment) R->L }
% 11.89/1.98 fresh(is_a_theorem(equivalent(equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z))), true, equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z))
% 11.89/1.98 = { by lemma 13 }
% 11.89/1.98 fresh(true, true, equivalent(equivalent(equivalent(X, Y), Z), X), equivalent(Y, Z))
% 11.89/1.98 = { by axiom 2 (condensed_detachment) }
% 11.89/1.98 is_a_theorem(equivalent(Y, Z))
% 11.89/1.98
% 11.89/1.98 Lemma 17: is_a_theorem(equivalent(equivalent(equivalent(X, equivalent(Y, Z)), equivalent(equivalent(equivalent(W, Y), Z), W)), X)) = true.
% 11.89/1.98 Proof:
% 11.89/1.98 is_a_theorem(equivalent(equivalent(equivalent(X, equivalent(Y, Z)), equivalent(equivalent(equivalent(W, Y), Z), W)), X))
% 11.89/1.98 = { by lemma 15 R->L }
% 11.89/1.98 fresh2(is_a_theorem(equivalent(equivalent(equivalent(equivalent(W, Y), Z), W), equivalent(Y, Z))), true, equivalent(equivalent(equivalent(X, equivalent(Y, Z)), equivalent(equivalent(equivalent(W, Y), Z), W)), X))
% 11.89/1.98 = { by lemma 13 }
% 11.89/1.98 fresh2(true, true, equivalent(equivalent(equivalent(X, equivalent(Y, Z)), equivalent(equivalent(equivalent(W, Y), Z), W)), X))
% 11.89/1.98 = { by axiom 1 (condensed_detachment) }
% 11.89/1.98 true
% 11.89/1.98
% 11.89/1.98 Goal 1 (prove_pyo): is_a_theorem(equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a))) = true.
% 11.89/1.98 Proof:
% 11.89/1.98 is_a_theorem(equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/1.98 = { by axiom 2 (condensed_detachment) R->L }
% 11.89/1.98 fresh(true, true, equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/1.98 = { by axiom 1 (condensed_detachment) R->L }
% 11.89/1.98 fresh(fresh2(true, true, equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))), true, equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/1.98 = { by lemma 17 R->L }
% 11.89/1.98 fresh(fresh2(is_a_theorem(equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a))), equivalent(equivalent(equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)), equivalent(b, a)), b)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))))), true, equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))), true, equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/1.98 = { by axiom 3 (condensed_detachment) R->L }
% 11.89/1.99 fresh(fresh(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a))), equivalent(equivalent(equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)), equivalent(b, a)), b)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a))))), true, equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a))), equivalent(equivalent(equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)), equivalent(b, a)), b)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))), true, equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/1.99 = { by lemma 11 R->L }
% 11.89/1.99 fresh(fresh(fresh2(is_a_theorem(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a))), equivalent(equivalent(equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)), equivalent(b, a)), b)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))), equivalent(equivalent(equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)), equivalent(b, a)), b)))), true, equivalent(equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a))), equivalent(equivalent(equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)), equivalent(b, a)), b)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a))))), true, equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a))), equivalent(equivalent(equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)), equivalent(b, a)), b)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))), true, equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/1.99 = { by lemma 14 }
% 11.89/1.99 fresh(fresh(fresh2(true, true, equivalent(equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a))), equivalent(equivalent(equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)), equivalent(b, a)), b)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a))))), true, equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a))), equivalent(equivalent(equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)), equivalent(b, a)), b)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))), true, equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/1.99 = { by axiom 1 (condensed_detachment) }
% 11.89/1.99 fresh(fresh(true, true, equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a))), equivalent(equivalent(equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)), equivalent(b, a)), b)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))), true, equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/1.99 = { by axiom 2 (condensed_detachment) }
% 11.89/1.99 fresh(is_a_theorem(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))), true, equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/1.99 = { by axiom 3 (condensed_detachment) }
% 11.89/1.99 fresh2(is_a_theorem(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), true, equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/1.99 = { by axiom 2 (condensed_detachment) R->L }
% 11.89/1.99 fresh2(fresh(true, true, equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), true, equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/1.99 = { by axiom 1 (condensed_detachment) R->L }
% 11.89/1.99 fresh2(fresh(fresh2(true, true, equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))))), true, equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), true, equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/1.99 = { by axiom 1 (condensed_detachment) R->L }
% 11.89/1.99 fresh2(fresh(fresh2(fresh2(true, true, equivalent(equivalent(equivalent(b, c), c), b)), true, equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))))), true, equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), true, equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/1.99 = { by lemma 12 R->L }
% 11.89/1.99 fresh2(fresh(fresh2(fresh2(is_a_theorem(equivalent(equivalent(equivalent(c, equivalent(equivalent(equivalent(b, c), c), b)), equivalent(c, c)), c)), true, equivalent(equivalent(equivalent(b, c), c), b)), true, equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))))), true, equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), true, equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/1.99 = { by axiom 3 (condensed_detachment) R->L }
% 11.89/1.99 fresh2(fresh(fresh2(fresh(is_a_theorem(equivalent(equivalent(equivalent(equivalent(c, equivalent(equivalent(equivalent(b, c), c), b)), equivalent(c, c)), c), equivalent(equivalent(equivalent(b, c), c), b))), true, equivalent(equivalent(equivalent(c, equivalent(equivalent(equivalent(b, c), c), b)), equivalent(c, c)), c), equivalent(equivalent(equivalent(b, c), c), b)), true, equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))))), true, equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), true, equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/1.99 = { by axiom 2 (condensed_detachment) R->L }
% 11.89/1.99 fresh2(fresh(fresh2(fresh(fresh(true, true, equivalent(c, c), equivalent(equivalent(equivalent(equivalent(c, equivalent(equivalent(equivalent(b, c), c), b)), equivalent(c, c)), c), equivalent(equivalent(equivalent(b, c), c), b))), true, equivalent(equivalent(equivalent(c, equivalent(equivalent(equivalent(b, c), c), b)), equivalent(c, c)), c), equivalent(equivalent(equivalent(b, c), c), b)), true, equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))))), true, equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), true, equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/1.99 = { by lemma 14 R->L }
% 11.89/1.99 fresh2(fresh(fresh2(fresh(fresh(is_a_theorem(equivalent(equivalent(c, c), equivalent(equivalent(equivalent(equivalent(c, equivalent(equivalent(equivalent(b, c), c), b)), equivalent(c, c)), c), equivalent(equivalent(equivalent(b, c), c), b)))), true, equivalent(c, c), equivalent(equivalent(equivalent(equivalent(c, equivalent(equivalent(equivalent(b, c), c), b)), equivalent(c, c)), c), equivalent(equivalent(equivalent(b, c), c), b))), true, equivalent(equivalent(equivalent(c, equivalent(equivalent(equivalent(b, c), c), b)), equivalent(c, c)), c), equivalent(equivalent(equivalent(b, c), c), b)), true, equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))))), true, equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), true, equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/1.99 = { by axiom 3 (condensed_detachment) }
% 11.89/1.99 fresh2(fresh(fresh2(fresh(fresh2(is_a_theorem(equivalent(c, c)), true, equivalent(equivalent(equivalent(equivalent(c, equivalent(equivalent(equivalent(b, c), c), b)), equivalent(c, c)), c), equivalent(equivalent(equivalent(b, c), c), b))), true, equivalent(equivalent(equivalent(c, equivalent(equivalent(equivalent(b, c), c), b)), equivalent(c, c)), c), equivalent(equivalent(equivalent(b, c), c), b)), true, equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))))), true, equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), true, equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/1.99 = { by lemma 16 R->L }
% 11.89/1.99 fresh2(fresh(fresh2(fresh(fresh2(fresh2(is_a_theorem(equivalent(equivalent(equivalent(equivalent(c, c), c), c), equivalent(c, c))), true, equivalent(c, c)), true, equivalent(equivalent(equivalent(equivalent(c, equivalent(equivalent(equivalent(b, c), c), b)), equivalent(c, c)), c), equivalent(equivalent(equivalent(b, c), c), b))), true, equivalent(equivalent(equivalent(c, equivalent(equivalent(equivalent(b, c), c), b)), equivalent(c, c)), c), equivalent(equivalent(equivalent(b, c), c), b)), true, equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))))), true, equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), true, equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/1.99 = { by lemma 13 }
% 11.89/1.99 fresh2(fresh(fresh2(fresh(fresh2(fresh2(true, true, equivalent(c, c)), true, equivalent(equivalent(equivalent(equivalent(c, equivalent(equivalent(equivalent(b, c), c), b)), equivalent(c, c)), c), equivalent(equivalent(equivalent(b, c), c), b))), true, equivalent(equivalent(equivalent(c, equivalent(equivalent(equivalent(b, c), c), b)), equivalent(c, c)), c), equivalent(equivalent(equivalent(b, c), c), b)), true, equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))))), true, equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), true, equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/1.99 = { by axiom 1 (condensed_detachment) }
% 11.89/1.99 fresh2(fresh(fresh2(fresh(fresh2(true, true, equivalent(equivalent(equivalent(equivalent(c, equivalent(equivalent(equivalent(b, c), c), b)), equivalent(c, c)), c), equivalent(equivalent(equivalent(b, c), c), b))), true, equivalent(equivalent(equivalent(c, equivalent(equivalent(equivalent(b, c), c), b)), equivalent(c, c)), c), equivalent(equivalent(equivalent(b, c), c), b)), true, equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))))), true, equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), true, equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/1.99 = { by axiom 1 (condensed_detachment) }
% 11.89/1.99 fresh2(fresh(fresh2(fresh(true, true, equivalent(equivalent(equivalent(c, equivalent(equivalent(equivalent(b, c), c), b)), equivalent(c, c)), c), equivalent(equivalent(equivalent(b, c), c), b)), true, equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))))), true, equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), true, equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/1.99 = { by axiom 2 (condensed_detachment) }
% 11.89/1.99 fresh2(fresh(fresh2(is_a_theorem(equivalent(equivalent(equivalent(b, c), c), b)), true, equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))))), true, equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), true, equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/1.99 = { by lemma 15 }
% 11.89/1.99 fresh2(fresh(is_a_theorem(equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))))), true, equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), true, equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/1.99 = { by axiom 3 (condensed_detachment) }
% 11.89/1.99 fresh2(fresh2(is_a_theorem(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c))), true, equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), true, equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/1.99 = { by axiom 2 (condensed_detachment) R->L }
% 11.89/1.99 fresh2(fresh2(fresh(true, true, equivalent(equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), a)), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b)), equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c))), true, equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), true, equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/2.00 = { by axiom 1 (condensed_detachment) R->L }
% 11.89/2.00 fresh2(fresh2(fresh(fresh2(true, true, equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), a)), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b)), equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)))), true, equivalent(equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), a)), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b)), equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c))), true, equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), true, equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/2.00 = { by axiom 1 (condensed_detachment) R->L }
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% 11.89/2.00 = { by axiom 1 (condensed_detachment) R->L }
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% 11.89/2.00 = { by lemma 9 R->L }
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c)), c))), b)))), true, equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), a)), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b)), equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)))), true, equivalent(equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), a)), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b)), equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c))), true, equivalent(a, equivalent(b, 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% 11.89/2.01 = { by lemma 11 }
% 11.89/2.01 fresh2(fresh2(fresh(fresh2(fresh2(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(equivalent(equivalent(a, equivalent(b, c)), c), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), a)), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b)), equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c))))), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), a)), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), a)), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b)), equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c))), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b))), equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(equivalent(equivalent(a, equivalent(b, c)), c), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), a)), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b)), equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c))))))), true, equivalent(equivalent(equivalent(equivalent(a, equivalent(b, c)), c), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), a)), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b)), equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c))), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b)))), true, equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), a)), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b)), equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)))), true, equivalent(equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), a)), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b)), equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c))), true, equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), true, equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/2.01 = { by lemma 16 }
% 11.89/2.01 fresh2(fresh2(fresh(fresh2(is_a_theorem(equivalent(equivalent(equivalent(equivalent(a, equivalent(b, c)), c), a), equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), a)), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b)), equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c))), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b)))), true, equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), a)), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b)), equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)))), true, equivalent(equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), a)), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b)), equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c))), true, equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), true, equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/2.01 = { by lemma 11 }
% 11.89/2.01 fresh2(fresh2(fresh(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), a)), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b)), equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)))), true, equivalent(equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), a)), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b)), equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c))), true, equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), true, equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/2.01 = { by axiom 3 (condensed_detachment) }
% 11.89/2.01 fresh2(fresh2(fresh2(is_a_theorem(equivalent(equivalent(equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c)), equivalent(equivalent(equivalent(a, equivalent(b, c)), c), a)), equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b))), true, equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c))), true, equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), true, equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/2.01 = { by lemma 17 }
% 11.89/2.01 fresh2(fresh2(fresh2(true, true, equivalent(equivalent(equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c))), b), equivalent(equivalent(b, c), c))), true, equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), true, equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/2.01 = { by axiom 1 (condensed_detachment) }
% 11.89/2.01 fresh2(fresh2(true, true, equivalent(a, equivalent(b, equivalent(equivalent(a, equivalent(b, c)), c)))), true, equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/2.01 = { by axiom 1 (condensed_detachment) }
% 11.89/2.01 fresh2(true, true, equivalent(equivalent(equivalent(a, equivalent(b, c)), c), equivalent(b, a)))
% 11.89/2.01 = { by axiom 1 (condensed_detachment) }
% 11.89/2.01 true
% 11.89/2.01 % SZS output end Proof
% 11.89/2.01
% 11.89/2.01 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------