TSTP Solution File: LCL056-1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL056-1 : TPTP v8.1.2. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n014.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 08:17:13 EDT 2023
% Result : Unsatisfiable 4.43s 0.96s
% Output : Proof 4.97s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LCL056-1 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.35 % Computer : n014.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Thu Aug 24 17:04:03 EDT 2023
% 0.13/0.35 % CPUTime :
% 4.43/0.96 Command-line arguments: --no-flatten-goal
% 4.43/0.96
% 4.43/0.96 % SZS status Unsatisfiable
% 4.43/0.96
% 4.43/1.00 % SZS output start Proof
% 4.43/1.00 Take the following subset of the input axioms:
% 4.43/1.00 fof(cn_1, axiom, ![X, Y, Z]: is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))))).
% 4.43/1.00 fof(cn_2, axiom, ![X2]: is_a_theorem(implies(implies(not(X2), X2), X2))).
% 4.43/1.00 fof(cn_3, axiom, ![X2, Y2]: is_a_theorem(implies(X2, implies(not(X2), Y2)))).
% 4.43/1.00 fof(condensed_detachment, axiom, ![X2, Y2]: (~is_a_theorem(implies(X2, Y2)) | (~is_a_theorem(X2) | is_a_theorem(Y2)))).
% 4.43/1.00 fof(prove_cn_39, negated_conjecture, ~is_a_theorem(implies(not(not(a)), a))).
% 4.43/1.00
% 4.43/1.00 Now clausify the problem and encode Horn clauses using encoding 3 of
% 4.43/1.00 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 4.43/1.00 We repeatedly replace C & s=t => u=v by the two clauses:
% 4.43/1.00 fresh(y, y, x1...xn) = u
% 4.43/1.00 C => fresh(s, t, x1...xn) = v
% 4.43/1.00 where fresh is a fresh function symbol and x1..xn are the free
% 4.43/1.00 variables of u and v.
% 4.43/1.00 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 4.43/1.00 input problem has no model of domain size 1).
% 4.43/1.00
% 4.43/1.00 The encoding turns the above axioms into the following unit equations and goals:
% 4.43/1.00
% 4.43/1.00 Axiom 1 (condensed_detachment): fresh2(X, X, Y) = true.
% 4.43/1.00 Axiom 2 (condensed_detachment): fresh(X, X, Y, Z) = is_a_theorem(Z).
% 4.43/1.00 Axiom 3 (cn_3): is_a_theorem(implies(X, implies(not(X), Y))) = true.
% 4.43/1.00 Axiom 4 (cn_2): is_a_theorem(implies(implies(not(X), X), X)) = true.
% 4.43/1.00 Axiom 5 (condensed_detachment): fresh(is_a_theorem(implies(X, Y)), true, X, Y) = fresh2(is_a_theorem(X), true, Y).
% 4.43/1.00 Axiom 6 (cn_1): is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))) = true.
% 4.97/1.00
% 4.97/1.00 Lemma 7: fresh2(is_a_theorem(implies(not(X), X)), true, X) = is_a_theorem(X).
% 4.97/1.00 Proof:
% 4.97/1.00 fresh2(is_a_theorem(implies(not(X), X)), true, X)
% 4.97/1.00 = { by axiom 5 (condensed_detachment) R->L }
% 4.97/1.00 fresh(is_a_theorem(implies(implies(not(X), X), X)), true, implies(not(X), X), X)
% 4.97/1.00 = { by axiom 4 (cn_2) }
% 4.97/1.00 fresh(true, true, implies(not(X), X), X)
% 4.97/1.00 = { by axiom 2 (condensed_detachment) }
% 4.97/1.00 is_a_theorem(X)
% 4.97/1.00
% 4.97/1.00 Lemma 8: fresh2(is_a_theorem(implies(X, Y)), true, implies(implies(Y, Z), implies(X, Z))) = is_a_theorem(implies(implies(Y, Z), implies(X, Z))).
% 4.97/1.00 Proof:
% 4.97/1.00 fresh2(is_a_theorem(implies(X, Y)), true, implies(implies(Y, Z), implies(X, Z)))
% 4.97/1.00 = { by axiom 5 (condensed_detachment) R->L }
% 4.97/1.00 fresh(is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))), true, implies(X, Y), implies(implies(Y, Z), implies(X, Z)))
% 4.97/1.00 = { by axiom 6 (cn_1) }
% 4.97/1.00 fresh(true, true, implies(X, Y), implies(implies(Y, Z), implies(X, Z)))
% 4.97/1.00 = { by axiom 2 (condensed_detachment) }
% 4.97/1.00 is_a_theorem(implies(implies(Y, Z), implies(X, Z)))
% 4.97/1.00
% 4.97/1.00 Lemma 9: is_a_theorem(implies(implies(X, Y), implies(implies(not(X), X), Y))) = true.
% 4.97/1.00 Proof:
% 4.97/1.00 is_a_theorem(implies(implies(X, Y), implies(implies(not(X), X), Y)))
% 4.97/1.00 = { by lemma 8 R->L }
% 4.97/1.00 fresh2(is_a_theorem(implies(implies(not(X), X), X)), true, implies(implies(X, Y), implies(implies(not(X), X), Y)))
% 4.97/1.00 = { by axiom 4 (cn_2) }
% 4.97/1.00 fresh2(true, true, implies(implies(X, Y), implies(implies(not(X), X), Y)))
% 4.97/1.00 = { by axiom 1 (condensed_detachment) }
% 4.97/1.00 true
% 4.97/1.00
% 4.97/1.00 Lemma 10: fresh2(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), true, implies(implies(Z, X), W)) = is_a_theorem(implies(implies(Z, X), W)).
% 4.97/1.00 Proof:
% 4.97/1.00 fresh2(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), true, implies(implies(Z, X), W))
% 4.97/1.00 = { by axiom 5 (condensed_detachment) R->L }
% 4.97/1.00 fresh(is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))), true, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))
% 4.97/1.00 = { by lemma 8 R->L }
% 4.97/1.00 fresh(fresh2(is_a_theorem(implies(implies(Z, X), implies(implies(X, Y), implies(Z, Y)))), true, implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))), true, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))
% 4.97/1.00 = { by axiom 6 (cn_1) }
% 4.97/1.00 fresh(fresh2(true, true, implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))), true, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))
% 4.97/1.00 = { by axiom 1 (condensed_detachment) }
% 4.97/1.00 fresh(true, true, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))
% 4.97/1.00 = { by axiom 2 (condensed_detachment) }
% 4.97/1.00 is_a_theorem(implies(implies(Z, X), W))
% 4.97/1.00
% 4.97/1.00 Lemma 11: is_a_theorem(implies(implies(implies(not(X), Y), Z), implies(X, Z))) = true.
% 4.97/1.00 Proof:
% 4.97/1.00 is_a_theorem(implies(implies(implies(not(X), Y), Z), implies(X, Z)))
% 4.97/1.01 = { by lemma 8 R->L }
% 4.97/1.01 fresh2(is_a_theorem(implies(X, implies(not(X), Y))), true, implies(implies(implies(not(X), Y), Z), implies(X, Z)))
% 4.97/1.01 = { by axiom 3 (cn_3) }
% 4.97/1.01 fresh2(true, true, implies(implies(implies(not(X), Y), Z), implies(X, Z)))
% 4.97/1.01 = { by axiom 1 (condensed_detachment) }
% 4.97/1.01 true
% 4.97/1.01
% 4.97/1.01 Lemma 12: is_a_theorem(implies(implies(X, not(Y)), implies(Y, implies(X, Z)))) = true.
% 4.97/1.01 Proof:
% 4.97/1.01 is_a_theorem(implies(implies(X, not(Y)), implies(Y, implies(X, Z))))
% 4.97/1.01 = { by lemma 10 R->L }
% 4.97/1.01 fresh2(is_a_theorem(implies(implies(implies(not(Y), Z), implies(X, Z)), implies(Y, implies(X, Z)))), true, implies(implies(X, not(Y)), implies(Y, implies(X, Z))))
% 4.97/1.01 = { by lemma 11 }
% 4.97/1.01 fresh2(true, true, implies(implies(X, not(Y)), implies(Y, implies(X, Z))))
% 4.97/1.01 = { by axiom 1 (condensed_detachment) }
% 4.97/1.01 true
% 4.97/1.01
% 4.97/1.01 Lemma 13: fresh2(is_a_theorem(implies(implies(not(X), Y), Z)), true, implies(X, Z)) = is_a_theorem(implies(X, Z)).
% 4.97/1.01 Proof:
% 4.97/1.01 fresh2(is_a_theorem(implies(implies(not(X), Y), Z)), true, implies(X, Z))
% 4.97/1.01 = { by axiom 5 (condensed_detachment) R->L }
% 4.97/1.01 fresh(is_a_theorem(implies(implies(implies(not(X), Y), Z), implies(X, Z))), true, implies(implies(not(X), Y), Z), implies(X, Z))
% 4.97/1.01 = { by lemma 11 }
% 4.97/1.01 fresh(true, true, implies(implies(not(X), Y), Z), implies(X, Z))
% 4.97/1.01 = { by axiom 2 (condensed_detachment) }
% 4.97/1.01 is_a_theorem(implies(X, Z))
% 4.97/1.01
% 4.97/1.01 Lemma 14: fresh2(is_a_theorem(implies(implies(implies(not(X), X), Y), Z)), true, implies(implies(X, Y), Z)) = is_a_theorem(implies(implies(X, Y), Z)).
% 4.97/1.01 Proof:
% 4.97/1.01 fresh2(is_a_theorem(implies(implies(implies(not(X), X), Y), Z)), true, implies(implies(X, Y), Z))
% 4.97/1.01 = { by axiom 5 (condensed_detachment) R->L }
% 4.97/1.01 fresh(is_a_theorem(implies(implies(implies(implies(not(X), X), Y), Z), implies(implies(X, Y), Z))), true, implies(implies(implies(not(X), X), Y), Z), implies(implies(X, Y), Z))
% 4.97/1.01 = { by lemma 8 R->L }
% 4.97/1.01 fresh(fresh2(is_a_theorem(implies(implies(X, Y), implies(implies(not(X), X), Y))), true, implies(implies(implies(implies(not(X), X), Y), Z), implies(implies(X, Y), Z))), true, implies(implies(implies(not(X), X), Y), Z), implies(implies(X, Y), Z))
% 4.97/1.01 = { by lemma 9 }
% 4.97/1.01 fresh(fresh2(true, true, implies(implies(implies(implies(not(X), X), Y), Z), implies(implies(X, Y), Z))), true, implies(implies(implies(not(X), X), Y), Z), implies(implies(X, Y), Z))
% 4.97/1.01 = { by axiom 1 (condensed_detachment) }
% 4.97/1.01 fresh(true, true, implies(implies(implies(not(X), X), Y), Z), implies(implies(X, Y), Z))
% 4.97/1.01 = { by axiom 2 (condensed_detachment) }
% 4.97/1.01 is_a_theorem(implies(implies(X, Y), Z))
% 4.97/1.01
% 4.97/1.01 Lemma 15: fresh2(is_a_theorem(implies(X, implies(not(Y), Y))), true, implies(Z, implies(X, Y))) = is_a_theorem(implies(Z, implies(X, Y))).
% 4.97/1.01 Proof:
% 4.97/1.01 fresh2(is_a_theorem(implies(X, implies(not(Y), Y))), true, implies(Z, implies(X, Y)))
% 4.97/1.01 = { by axiom 5 (condensed_detachment) R->L }
% 4.97/1.01 fresh(is_a_theorem(implies(implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))
% 4.97/1.01 = { by lemma 10 R->L }
% 4.97/1.01 fresh(fresh2(is_a_theorem(implies(implies(implies(implies(not(Y), Y), Y), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))
% 4.97/1.01 = { by lemma 14 R->L }
% 4.97/1.01 fresh(fresh2(fresh2(is_a_theorem(implies(implies(implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(implies(implies(not(Y), Y), Y), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))
% 4.97/1.01 = { by lemma 8 R->L }
% 4.97/1.01 fresh(fresh2(fresh2(fresh2(is_a_theorem(implies(Z, implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)))), true, implies(implies(implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(implies(implies(not(Y), Y), Y), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))
% 4.97/1.01 = { by lemma 13 R->L }
% 4.97/1.01 fresh(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(not(Z), implies(implies(not(Y), Y), Y)), implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)))), true, implies(Z, implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)))), true, implies(implies(implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(implies(implies(not(Y), Y), Y), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))
% 4.97/1.01 = { by lemma 8 R->L }
% 4.97/1.01 fresh(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(not(implies(implies(not(Y), Y), Y)), not(Z))), true, implies(implies(not(Z), implies(implies(not(Y), Y), Y)), implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)))), true, implies(Z, implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)))), true, implies(implies(implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(implies(implies(not(Y), Y), Y), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))
% 4.97/1.01 = { by axiom 2 (condensed_detachment) R->L }
% 4.97/1.01 fresh(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(true, true, implies(implies(not(Y), Y), Y), implies(not(implies(implies(not(Y), Y), Y)), not(Z))), true, implies(implies(not(Z), implies(implies(not(Y), Y), Y)), implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)))), true, implies(Z, implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)))), true, implies(implies(implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(implies(implies(not(Y), Y), Y), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))
% 4.97/1.01 = { by axiom 3 (cn_3) R->L }
% 4.97/1.01 fresh(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(is_a_theorem(implies(implies(implies(not(Y), Y), Y), implies(not(implies(implies(not(Y), Y), Y)), not(Z)))), true, implies(implies(not(Y), Y), Y), implies(not(implies(implies(not(Y), Y), Y)), not(Z))), true, implies(implies(not(Z), implies(implies(not(Y), Y), Y)), implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)))), true, implies(Z, implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)))), true, implies(implies(implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(implies(implies(not(Y), Y), Y), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))
% 4.97/1.01 = { by axiom 5 (condensed_detachment) }
% 4.97/1.01 fresh(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(not(Y), Y), Y)), true, implies(not(implies(implies(not(Y), Y), Y)), not(Z))), true, implies(implies(not(Z), implies(implies(not(Y), Y), Y)), implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)))), true, implies(Z, implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)))), true, implies(implies(implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(implies(implies(not(Y), Y), Y), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))
% 4.97/1.01 = { by axiom 4 (cn_2) }
% 4.97/1.01 fresh(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(not(implies(implies(not(Y), Y), Y)), not(Z))), true, implies(implies(not(Z), implies(implies(not(Y), Y), Y)), implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)))), true, implies(Z, implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)))), true, implies(implies(implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(implies(implies(not(Y), Y), Y), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))
% 4.97/1.01 = { by axiom 1 (condensed_detachment) }
% 4.97/1.01 fresh(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(not(Z), implies(implies(not(Y), Y), Y)), implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)))), true, implies(Z, implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)))), true, implies(implies(implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(implies(implies(not(Y), Y), Y), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))
% 4.97/1.01 = { by axiom 1 (condensed_detachment) }
% 4.97/1.01 fresh(fresh2(fresh2(fresh2(fresh2(true, true, implies(Z, implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)))), true, implies(implies(implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(implies(implies(not(Y), Y), Y), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))
% 4.97/1.01 = { by axiom 1 (condensed_detachment) }
% 4.97/1.01 fresh(fresh2(fresh2(fresh2(true, true, implies(implies(implies(not(implies(implies(not(Y), Y), Y)), implies(implies(not(Y), Y), Y)), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(implies(implies(not(Y), Y), Y), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))
% 4.97/1.01 = { by axiom 1 (condensed_detachment) }
% 4.97/1.01 fresh(fresh2(fresh2(true, true, implies(implies(implies(implies(not(Y), Y), Y), implies(X, Y)), implies(Z, implies(X, Y)))), true, implies(implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))
% 4.97/1.01 = { by axiom 1 (condensed_detachment) }
% 4.97/1.01 fresh(fresh2(true, true, implies(implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))), true, implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))
% 4.97/1.01 = { by axiom 1 (condensed_detachment) }
% 4.97/1.01 fresh(true, true, implies(X, implies(not(Y), Y)), implies(Z, implies(X, Y)))
% 4.97/1.01 = { by axiom 2 (condensed_detachment) }
% 4.97/1.01 is_a_theorem(implies(Z, implies(X, Y)))
% 4.97/1.01
% 4.97/1.01 Lemma 16: is_a_theorem(implies(implies(implies(implies(not(not(X)), not(X)), Y), Z), implies(X, Z))) = true.
% 4.97/1.01 Proof:
% 4.97/1.01 is_a_theorem(implies(implies(implies(implies(not(not(X)), not(X)), Y), Z), implies(X, Z)))
% 4.97/1.01 = { by lemma 8 R->L }
% 4.97/1.01 fresh2(is_a_theorem(implies(X, implies(implies(not(not(X)), not(X)), Y))), true, implies(implies(implies(implies(not(not(X)), not(X)), Y), Z), implies(X, Z)))
% 4.97/1.01 = { by lemma 13 R->L }
% 4.97/1.01 fresh2(fresh2(is_a_theorem(implies(implies(not(X), Y), implies(implies(not(not(X)), not(X)), Y))), true, implies(X, implies(implies(not(not(X)), not(X)), Y))), true, implies(implies(implies(implies(not(not(X)), not(X)), Y), Z), implies(X, Z)))
% 4.97/1.01 = { by lemma 9 }
% 4.97/1.01 fresh2(fresh2(true, true, implies(X, implies(implies(not(not(X)), not(X)), Y))), true, implies(implies(implies(implies(not(not(X)), not(X)), Y), Z), implies(X, Z)))
% 4.97/1.01 = { by axiom 1 (condensed_detachment) }
% 4.97/1.01 fresh2(true, true, implies(implies(implies(implies(not(not(X)), not(X)), Y), Z), implies(X, Z)))
% 4.97/1.01 = { by axiom 1 (condensed_detachment) }
% 4.97/1.02 true
% 4.97/1.02
% 4.97/1.02 Goal 1 (prove_cn_39): is_a_theorem(implies(not(not(a)), a)) = true.
% 4.97/1.02 Proof:
% 4.97/1.02 is_a_theorem(implies(not(not(a)), a))
% 4.97/1.02 = { by lemma 7 R->L }
% 4.97/1.02 fresh2(is_a_theorem(implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.02 = { by lemma 15 R->L }
% 4.97/1.02 fresh2(fresh2(is_a_theorem(implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.02 = { by lemma 13 R->L }
% 4.97/1.02 fresh2(fresh2(fresh2(is_a_theorem(implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.02 = { by axiom 2 (condensed_detachment) R->L }
% 4.97/1.02 fresh2(fresh2(fresh2(fresh(true, true, implies(not(a), not(implies(not(not(not(a))), not(not(a))))), implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.02 = { by lemma 12 R->L }
% 4.97/1.02 fresh2(fresh2(fresh2(fresh(is_a_theorem(implies(implies(not(a), not(implies(not(not(not(a))), not(not(a))))), implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a)))), true, implies(not(a), not(implies(not(not(not(a))), not(not(a))))), implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.02 = { by axiom 5 (condensed_detachment) }
% 4.97/1.02 fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(not(a), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.02 = { by axiom 2 (condensed_detachment) R->L }
% 4.97/1.02 fresh2(fresh2(fresh2(fresh2(fresh(true, true, implies(implies(implies(not(not(not(a))), not(not(a))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))), implies(not(a), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.02 = { by lemma 16 R->L }
% 4.97/1.02 fresh2(fresh2(fresh2(fresh2(fresh(is_a_theorem(implies(implies(implies(implies(not(not(not(a))), not(not(a))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))), implies(not(a), not(implies(not(not(not(a))), not(not(a))))))), true, implies(implies(implies(not(not(not(a))), not(not(a))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))), implies(not(a), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.02 = { by axiom 5 (condensed_detachment) }
% 4.97/1.02 fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(not(not(not(a))), not(not(a))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(not(a), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.02 = { by lemma 14 R->L }
% 4.97/1.02 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(not(not(a))), not(not(a))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(not(a), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.03 = { by lemma 7 R->L }
% 4.97/1.03 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(not(implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(not(not(a))), not(not(a))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(not(a), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.03 = { by lemma 15 R->L }
% 4.97/1.03 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), implies(not(not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(not(implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(not(not(a))), not(not(a))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(not(a), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.03 = { by lemma 8 R->L }
% 4.97/1.03 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(not(not(implies(not(not(not(a))), not(not(a))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), implies(not(not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(not(implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(not(not(a))), not(not(a))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(not(a), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.03 = { by axiom 2 (condensed_detachment) R->L }
% 4.97/1.03 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(true, true, implies(implies(not(not(not(a))), implies(not(not(not(implies(not(not(not(a))), not(not(a)))))), not(not(implies(not(not(not(a))), not(not(a))))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a))))), implies(not(not(implies(not(not(not(a))), not(not(a))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), implies(not(not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(not(implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(not(not(a))), not(not(a))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(not(a), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.04 = { by axiom 1 (condensed_detachment) R->L }
% 4.97/1.04 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(true, true, implies(implies(implies(not(not(not(a))), implies(not(not(not(implies(not(not(not(a))), not(not(a)))))), not(not(implies(not(not(not(a))), not(not(a))))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a))))), implies(not(not(implies(not(not(not(a))), not(not(a))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a))))))), true, implies(implies(not(not(not(a))), implies(not(not(not(implies(not(not(not(a))), not(not(a)))))), not(not(implies(not(not(not(a))), not(not(a))))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a))))), implies(not(not(implies(not(not(not(a))), not(not(a))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), implies(not(not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(not(implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(not(not(a))), not(not(a))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(not(a), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.04 = { by axiom 1 (condensed_detachment) R->L }
% 4.97/1.04 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(true, true, implies(not(not(implies(not(not(not(a))), not(not(a))))), implies(not(not(not(a))), implies(not(not(not(implies(not(not(not(a))), not(not(a)))))), not(not(implies(not(not(not(a))), not(not(a))))))))), true, implies(implies(implies(not(not(not(a))), implies(not(not(not(implies(not(not(not(a))), not(not(a)))))), not(not(implies(not(not(not(a))), not(not(a))))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a))))), implies(not(not(implies(not(not(not(a))), not(not(a))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a))))))), true, implies(implies(not(not(not(a))), implies(not(not(not(implies(not(not(not(a))), not(not(a)))))), not(not(implies(not(not(not(a))), not(not(a))))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a))))), implies(not(not(implies(not(not(not(a))), not(not(a))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), implies(not(not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(not(implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(not(not(a))), not(not(a))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(not(a), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.04 = { by lemma 12 R->L }
% 4.97/1.05 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(fresh2(is_a_theorem(implies(implies(not(not(not(implies(not(not(not(a))), not(not(a)))))), not(not(not(not(a))))), implies(not(not(not(a))), implies(not(not(not(implies(not(not(not(a))), not(not(a)))))), not(not(implies(not(not(not(a))), not(not(a))))))))), true, implies(not(not(implies(not(not(not(a))), not(not(a))))), implies(not(not(not(a))), implies(not(not(not(implies(not(not(not(a))), not(not(a)))))), not(not(implies(not(not(not(a))), not(not(a))))))))), true, implies(implies(implies(not(not(not(a))), implies(not(not(not(implies(not(not(not(a))), not(not(a)))))), not(not(implies(not(not(not(a))), not(not(a))))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a))))), implies(not(not(implies(not(not(not(a))), not(not(a))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a))))))), true, implies(implies(not(not(not(a))), implies(not(not(not(implies(not(not(not(a))), not(not(a)))))), not(not(implies(not(not(not(a))), not(not(a))))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a))))), implies(not(not(implies(not(not(not(a))), not(not(a))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), implies(not(not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(not(implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(not(not(a))), not(not(a))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(not(a), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.05 = { by lemma 13 }
% 4.97/1.05 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(is_a_theorem(implies(not(not(implies(not(not(not(a))), not(not(a))))), implies(not(not(not(a))), implies(not(not(not(implies(not(not(not(a))), not(not(a)))))), not(not(implies(not(not(not(a))), not(not(a))))))))), true, implies(implies(implies(not(not(not(a))), implies(not(not(not(implies(not(not(not(a))), not(not(a)))))), not(not(implies(not(not(not(a))), not(not(a))))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a))))), implies(not(not(implies(not(not(not(a))), not(not(a))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a))))))), true, implies(implies(not(not(not(a))), implies(not(not(not(implies(not(not(not(a))), not(not(a)))))), not(not(implies(not(not(not(a))), not(not(a))))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a))))), implies(not(not(implies(not(not(not(a))), not(not(a))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), implies(not(not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(not(implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(not(not(a))), not(not(a))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(not(a), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.05 = { by lemma 8 }
% 4.97/1.05 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(is_a_theorem(implies(implies(implies(not(not(not(a))), implies(not(not(not(implies(not(not(not(a))), not(not(a)))))), not(not(implies(not(not(not(a))), not(not(a))))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a))))), implies(not(not(implies(not(not(not(a))), not(not(a))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a))))))), true, implies(implies(not(not(not(a))), implies(not(not(not(implies(not(not(not(a))), not(not(a)))))), not(not(implies(not(not(not(a))), not(not(a))))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a))))), implies(not(not(implies(not(not(not(a))), not(not(a))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), implies(not(not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(not(implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(not(not(a))), not(not(a))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(not(a), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.05 = { by axiom 5 (condensed_detachment) }
% 4.97/1.06 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(not(not(not(a))), implies(not(not(not(implies(not(not(not(a))), not(not(a)))))), not(not(implies(not(not(not(a))), not(not(a))))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))))), true, implies(not(not(implies(not(not(not(a))), not(not(a))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), implies(not(not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(not(implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(not(not(a))), not(not(a))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(not(a), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.06 = { by lemma 10 R->L }
% 4.97/1.06 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(not(not(not(implies(not(not(not(a))), not(not(a)))))), not(not(implies(not(not(not(a))), not(not(a)))))), not(not(a))), implies(not(not(not(a))), not(not(a)))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))))), true, implies(implies(not(not(not(a))), implies(not(not(not(implies(not(not(not(a))), not(not(a)))))), not(not(implies(not(not(not(a))), not(not(a))))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))))), true, implies(not(not(implies(not(not(not(a))), not(not(a))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), implies(not(not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(not(implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(not(not(a))), not(not(a))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(not(a), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.06 = { by lemma 16 }
% 4.97/1.06 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(not(not(not(a))), implies(not(not(not(implies(not(not(not(a))), not(not(a)))))), not(not(implies(not(not(not(a))), not(not(a))))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))))), true, implies(not(not(implies(not(not(not(a))), not(not(a))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), implies(not(not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(not(implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(not(not(a))), not(not(a))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(not(a), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.06 = { by axiom 1 (condensed_detachment) }
% 4.97/1.06 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(not(not(implies(not(not(not(a))), not(not(a))))), implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), implies(not(not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(not(implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(not(not(a))), not(not(a))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(not(a), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.06 = { by axiom 1 (condensed_detachment) }
% 4.97/1.06 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), implies(not(not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(not(implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(not(not(a))), not(not(a))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(not(a), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.06 = { by axiom 1 (condensed_detachment) }
% 4.97/1.06 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(not(implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a))))))), true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(not(not(a))), not(not(a))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(not(a), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.06 = { by axiom 1 (condensed_detachment) }
% 4.97/1.06 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(implies(not(implies(not(not(not(a))), not(not(a)))), implies(not(not(not(a))), not(not(a)))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(implies(not(not(not(a))), not(not(a))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(not(a), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.06 = { by axiom 1 (condensed_detachment) }
% 4.97/1.06 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(implies(not(not(not(a))), not(not(a))), not(implies(not(not(not(a))), not(not(a))))), not(implies(not(not(not(a))), not(not(a)))))), true, implies(not(a), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.06 = { by axiom 1 (condensed_detachment) }
% 4.97/1.06 fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(not(a), not(implies(not(not(not(a))), not(not(a)))))), true, implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.06 = { by axiom 1 (condensed_detachment) }
% 4.97/1.06 fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(not(not(not(a))), not(not(a))), implies(not(a), a))), true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.06 = { by axiom 1 (condensed_detachment) }
% 4.97/1.07 fresh2(fresh2(fresh2(true, true, implies(not(not(a)), implies(not(a), a))), true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.07 = { by axiom 1 (condensed_detachment) }
% 4.97/1.07 fresh2(fresh2(true, true, implies(not(implies(not(not(a)), a)), implies(not(not(a)), a))), true, implies(not(not(a)), a))
% 4.97/1.07 = { by axiom 1 (condensed_detachment) }
% 4.97/1.07 fresh2(true, true, implies(not(not(a)), a))
% 4.97/1.07 = { by axiom 1 (condensed_detachment) }
% 4.97/1.07 true
% 4.97/1.07 % SZS output end Proof
% 4.97/1.07
% 5.49/1.07 RESULT: Unsatisfiable (the axioms are contradictory).
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