TSTP Solution File: LCL368-1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL368-1 : TPTP v8.1.2. Released v2.3.0.
% 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 : n017.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 08:18:40 EDT 2023
% Result : Unsatisfiable 69.08s 9.13s
% Output : Proof 69.24s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : LCL368-1 : TPTP v8.1.2. Released v2.3.0.
% 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 : n017.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 : Fri Aug 25 03:55:54 EDT 2023
% 0.20/0.35 % CPUTime :
% 69.08/9.13 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 69.08/9.13
% 69.08/9.13 % SZS status Unsatisfiable
% 69.08/9.13
% 69.24/9.17 % SZS output start Proof
% 69.24/9.17 Take the following subset of the input axioms:
% 69.24/9.17 fof(cn_1, axiom, ![X, Y, Z]: is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))))).
% 69.24/9.17 fof(cn_2, axiom, ![X2]: is_a_theorem(implies(implies(not(X2), X2), X2))).
% 69.24/9.17 fof(cn_3, axiom, ![X2, Y2]: is_a_theorem(implies(X2, implies(not(X2), Y2)))).
% 69.24/9.17 fof(condensed_detachment, axiom, ![X2, Y2]: (~is_a_theorem(implies(X2, Y2)) | (~is_a_theorem(X2) | is_a_theorem(Y2)))).
% 69.24/9.17 fof(prove_cn_23, negated_conjecture, ~is_a_theorem(implies(implies(implies(x, implies(y, z)), u), implies(implies(y, implies(x, z)), u)))).
% 69.24/9.17
% 69.24/9.17 Now clausify the problem and encode Horn clauses using encoding 3 of
% 69.24/9.17 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 69.24/9.17 We repeatedly replace C & s=t => u=v by the two clauses:
% 69.24/9.17 fresh(y, y, x1...xn) = u
% 69.24/9.17 C => fresh(s, t, x1...xn) = v
% 69.24/9.17 where fresh is a fresh function symbol and x1..xn are the free
% 69.24/9.17 variables of u and v.
% 69.24/9.17 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 69.24/9.17 input problem has no model of domain size 1).
% 69.24/9.17
% 69.24/9.17 The encoding turns the above axioms into the following unit equations and goals:
% 69.24/9.17
% 69.24/9.17 Axiom 1 (condensed_detachment): fresh2(X, X, Y) = true.
% 69.24/9.17 Axiom 2 (condensed_detachment): fresh(X, X, Y, Z) = is_a_theorem(Z).
% 69.24/9.17 Axiom 3 (cn_3): is_a_theorem(implies(X, implies(not(X), Y))) = true.
% 69.24/9.17 Axiom 4 (cn_2): is_a_theorem(implies(implies(not(X), X), X)) = true.
% 69.24/9.17 Axiom 5 (condensed_detachment): fresh(is_a_theorem(implies(X, Y)), true, X, Y) = fresh2(is_a_theorem(X), true, Y).
% 69.24/9.17 Axiom 6 (cn_1): is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))) = true.
% 69.24/9.17
% 69.24/9.17 Lemma 7: 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))).
% 69.24/9.17 Proof:
% 69.24/9.17 fresh2(is_a_theorem(implies(X, Y)), true, implies(implies(Y, Z), implies(X, Z)))
% 69.24/9.17 = { by axiom 5 (condensed_detachment) R->L }
% 69.24/9.17 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)))
% 69.24/9.17 = { by axiom 6 (cn_1) }
% 69.24/9.17 fresh(true, true, implies(X, Y), implies(implies(Y, Z), implies(X, Z)))
% 69.24/9.17 = { by axiom 2 (condensed_detachment) }
% 69.24/9.17 is_a_theorem(implies(implies(Y, Z), implies(X, Z)))
% 69.24/9.17
% 69.24/9.17 Lemma 8: fresh2(is_a_theorem(implies(implies(not(X), Y), Z)), true, implies(X, Z)) = is_a_theorem(implies(X, Z)).
% 69.24/9.17 Proof:
% 69.24/9.17 fresh2(is_a_theorem(implies(implies(not(X), Y), Z)), true, implies(X, Z))
% 69.24/9.17 = { by axiom 5 (condensed_detachment) R->L }
% 69.24/9.17 fresh(is_a_theorem(implies(implies(implies(not(X), Y), Z), implies(X, Z))), true, implies(implies(not(X), Y), Z), implies(X, Z))
% 69.24/9.17 = { by lemma 7 R->L }
% 69.24/9.17 fresh(fresh2(is_a_theorem(implies(X, implies(not(X), Y))), true, implies(implies(implies(not(X), Y), Z), implies(X, Z))), true, implies(implies(not(X), Y), Z), implies(X, Z))
% 69.24/9.17 = { by axiom 3 (cn_3) }
% 69.24/9.17 fresh(fresh2(true, true, implies(implies(implies(not(X), Y), Z), implies(X, Z))), true, implies(implies(not(X), Y), Z), implies(X, Z))
% 69.24/9.17 = { by axiom 1 (condensed_detachment) }
% 69.24/9.17 fresh(true, true, implies(implies(not(X), Y), Z), implies(X, Z))
% 69.24/9.17 = { by axiom 2 (condensed_detachment) }
% 69.24/9.17 is_a_theorem(implies(X, Z))
% 69.24/9.17
% 69.24/9.17 Lemma 9: is_a_theorem(implies(X, X)) = true.
% 69.24/9.17 Proof:
% 69.24/9.17 is_a_theorem(implies(X, X))
% 69.24/9.17 = { by lemma 8 R->L }
% 69.24/9.17 fresh2(is_a_theorem(implies(implies(not(X), X), X)), true, implies(X, X))
% 69.24/9.17 = { by axiom 4 (cn_2) }
% 69.24/9.17 fresh2(true, true, implies(X, X))
% 69.24/9.17 = { by axiom 1 (condensed_detachment) }
% 69.24/9.17 true
% 69.24/9.17
% 69.24/9.17 Lemma 10: fresh2(is_a_theorem(implies(not(X), X)), true, X) = is_a_theorem(X).
% 69.24/9.17 Proof:
% 69.24/9.17 fresh2(is_a_theorem(implies(not(X), X)), true, X)
% 69.24/9.17 = { by axiom 5 (condensed_detachment) R->L }
% 69.24/9.17 fresh(is_a_theorem(implies(implies(not(X), X), X)), true, implies(not(X), X), X)
% 69.24/9.17 = { by axiom 4 (cn_2) }
% 69.24/9.17 fresh(true, true, implies(not(X), X), X)
% 69.24/9.17 = { by axiom 2 (condensed_detachment) }
% 69.24/9.17 is_a_theorem(X)
% 69.24/9.17
% 69.24/9.17 Lemma 11: is_a_theorem(implies(implies(X, Y), implies(implies(not(X), X), Y))) = true.
% 69.24/9.17 Proof:
% 69.24/9.17 is_a_theorem(implies(implies(X, Y), implies(implies(not(X), X), Y)))
% 69.24/9.17 = { by lemma 7 R->L }
% 69.24/9.17 fresh2(is_a_theorem(implies(implies(not(X), X), X)), true, implies(implies(X, Y), implies(implies(not(X), X), Y)))
% 69.24/9.17 = { by axiom 4 (cn_2) }
% 69.24/9.17 fresh2(true, true, implies(implies(X, Y), implies(implies(not(X), X), Y)))
% 69.24/9.17 = { by axiom 1 (condensed_detachment) }
% 69.24/9.17 true
% 69.24/9.17
% 69.24/9.17 Lemma 12: 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)).
% 69.24/9.17 Proof:
% 69.24/9.17 fresh2(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), true, implies(implies(Z, X), W))
% 69.24/9.17 = { by axiom 5 (condensed_detachment) R->L }
% 69.24/9.17 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))
% 69.24/9.17 = { by lemma 7 R->L }
% 69.24/9.17 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))
% 69.24/9.17 = { by axiom 6 (cn_1) }
% 69.24/9.17 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))
% 69.24/9.17 = { by axiom 1 (condensed_detachment) }
% 69.24/9.17 fresh(true, true, implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))
% 69.24/9.17 = { by axiom 2 (condensed_detachment) }
% 69.24/9.17 is_a_theorem(implies(implies(Z, X), W))
% 69.24/9.17
% 69.24/9.17 Lemma 13: is_a_theorem(implies(X, implies(implies(X, Y), implies(implies(Z, X), Y)))) = true.
% 69.24/9.17 Proof:
% 69.24/9.17 is_a_theorem(implies(X, implies(implies(X, Y), implies(implies(Z, X), Y))))
% 69.24/9.17 = { by lemma 8 R->L }
% 69.24/9.17 fresh2(is_a_theorem(implies(implies(not(X), Z), implies(implies(X, Y), implies(implies(Z, X), Y)))), true, implies(X, implies(implies(X, Y), implies(implies(Z, X), Y))))
% 69.24/9.17 = { by lemma 12 R->L }
% 69.24/9.17 fresh2(fresh2(is_a_theorem(implies(implies(implies(Z, X), implies(not(X), X)), implies(implies(X, Y), implies(implies(Z, X), Y)))), true, implies(implies(not(X), Z), implies(implies(X, Y), implies(implies(Z, X), Y)))), true, implies(X, implies(implies(X, Y), implies(implies(Z, X), Y))))
% 69.24/9.18 = { by lemma 12 R->L }
% 69.24/9.18 fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(not(X), X), Y), implies(implies(Z, X), Y)), implies(implies(X, Y), implies(implies(Z, X), Y)))), true, implies(implies(implies(Z, X), implies(not(X), X)), implies(implies(X, Y), implies(implies(Z, X), Y)))), true, implies(implies(not(X), Z), implies(implies(X, Y), implies(implies(Z, X), Y)))), true, implies(X, implies(implies(X, Y), implies(implies(Z, X), Y))))
% 69.24/9.18 = { by lemma 7 R->L }
% 69.24/9.18 fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(X, Y), implies(implies(not(X), X), Y))), true, implies(implies(implies(implies(not(X), X), Y), implies(implies(Z, X), Y)), implies(implies(X, Y), implies(implies(Z, X), Y)))), true, implies(implies(implies(Z, X), implies(not(X), X)), implies(implies(X, Y), implies(implies(Z, X), Y)))), true, implies(implies(not(X), Z), implies(implies(X, Y), implies(implies(Z, X), Y)))), true, implies(X, implies(implies(X, Y), implies(implies(Z, X), Y))))
% 69.24/9.18 = { by lemma 11 }
% 69.24/9.18 fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(implies(implies(not(X), X), Y), implies(implies(Z, X), Y)), implies(implies(X, Y), implies(implies(Z, X), Y)))), true, implies(implies(implies(Z, X), implies(not(X), X)), implies(implies(X, Y), implies(implies(Z, X), Y)))), true, implies(implies(not(X), Z), implies(implies(X, Y), implies(implies(Z, X), Y)))), true, implies(X, implies(implies(X, Y), implies(implies(Z, X), Y))))
% 69.24/9.18 = { by axiom 1 (condensed_detachment) }
% 69.24/9.18 fresh2(fresh2(fresh2(true, true, implies(implies(implies(Z, X), implies(not(X), X)), implies(implies(X, Y), implies(implies(Z, X), Y)))), true, implies(implies(not(X), Z), implies(implies(X, Y), implies(implies(Z, X), Y)))), true, implies(X, implies(implies(X, Y), implies(implies(Z, X), Y))))
% 69.24/9.18 = { by axiom 1 (condensed_detachment) }
% 69.24/9.18 fresh2(fresh2(true, true, implies(implies(not(X), Z), implies(implies(X, Y), implies(implies(Z, X), Y)))), true, implies(X, implies(implies(X, Y), implies(implies(Z, X), Y))))
% 69.24/9.18 = { by axiom 1 (condensed_detachment) }
% 69.24/9.18 fresh2(true, true, implies(X, implies(implies(X, Y), implies(implies(Z, X), Y))))
% 69.24/9.18 = { by axiom 1 (condensed_detachment) }
% 69.24/9.18 true
% 69.24/9.18
% 69.24/9.18 Lemma 14: fresh2(is_a_theorem(implies(implies(X, X), Y)), true, implies(implies(Z, implies(X, X)), Y)) = is_a_theorem(implies(implies(Z, implies(X, X)), Y)).
% 69.24/9.18 Proof:
% 69.24/9.18 fresh2(is_a_theorem(implies(implies(X, X), Y)), true, implies(implies(Z, implies(X, X)), Y))
% 69.24/9.18 = { by axiom 5 (condensed_detachment) R->L }
% 69.24/9.18 fresh(is_a_theorem(implies(implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))), true, implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))
% 69.24/9.18 = { by axiom 2 (condensed_detachment) R->L }
% 69.24/9.18 fresh(fresh(true, true, implies(X, X), implies(implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))), true, implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))
% 69.24/9.18 = { by lemma 13 R->L }
% 69.24/9.18 fresh(fresh(is_a_theorem(implies(implies(X, X), implies(implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y)))), true, implies(X, X), implies(implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))), true, implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))
% 69.24/9.18 = { by axiom 5 (condensed_detachment) }
% 69.24/9.18 fresh(fresh2(is_a_theorem(implies(X, X)), true, implies(implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))), true, implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))
% 69.24/9.18 = { by lemma 9 }
% 69.24/9.18 fresh(fresh2(true, true, implies(implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))), true, implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))
% 69.24/9.18 = { by axiom 1 (condensed_detachment) }
% 69.24/9.18 fresh(true, true, implies(implies(X, X), Y), implies(implies(Z, implies(X, X)), Y))
% 69.24/9.18 = { by axiom 2 (condensed_detachment) }
% 69.24/9.18 is_a_theorem(implies(implies(Z, implies(X, X)), Y))
% 69.24/9.18
% 69.24/9.18 Lemma 15: is_a_theorem(implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z))) = true.
% 69.24/9.18 Proof:
% 69.24/9.18 is_a_theorem(implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.18 = { by lemma 7 R->L }
% 69.24/9.18 fresh2(is_a_theorem(implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.18 = { by lemma 10 R->L }
% 69.24/9.18 fresh2(fresh2(is_a_theorem(implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.18 = { by axiom 2 (condensed_detachment) R->L }
% 69.24/9.18 fresh2(fresh2(fresh(true, true, implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.18 = { by axiom 1 (condensed_detachment) R->L }
% 69.24/9.18 fresh2(fresh2(fresh(fresh2(true, true, implies(implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y))))), true, implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.18 = { by axiom 1 (condensed_detachment) R->L }
% 69.24/9.18 fresh2(fresh2(fresh(fresh2(fresh2(true, true, implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(Y), implies(X, X)))), true, implies(implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y))))), true, implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.18 = { by axiom 1 (condensed_detachment) R->L }
% 69.24/9.18 fresh2(fresh2(fresh(fresh2(fresh2(fresh2(true, true, implies(implies(not(not(implies(W, implies(implies(implies(X, X), Y), Y)))), implies(implies(X, X), implies(X, X))), implies(not(Y), implies(X, X)))), true, implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(Y), implies(X, X)))), true, implies(implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y))))), true, implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.18 = { by axiom 1 (condensed_detachment) R->L }
% 69.24/9.18 fresh2(fresh2(fresh(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(implies(X, X), implies(X, X)), implies(not(Y), implies(X, X)))), true, implies(implies(not(not(implies(W, implies(implies(implies(X, X), Y), Y)))), implies(implies(X, X), implies(X, X))), implies(not(Y), implies(X, X)))), true, implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(Y), implies(X, X)))), true, implies(implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y))))), true, implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.18 = { by axiom 1 (condensed_detachment) R->L }
% 69.24/9.18 fresh2(fresh2(fresh(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(not(Y), implies(X, X))), true, implies(implies(implies(X, X), implies(X, X)), implies(not(Y), implies(X, X)))), true, implies(implies(not(not(implies(W, implies(implies(implies(X, X), Y), Y)))), implies(implies(X, X), implies(X, X))), implies(not(Y), implies(X, X)))), true, implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(Y), implies(X, X)))), true, implies(implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y))))), true, implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.18 = { by axiom 1 (condensed_detachment) R->L }
% 69.24/9.18 fresh2(fresh2(fresh(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(not(not(Y)), implies(X, X)), implies(X, X))), true, implies(not(Y), implies(X, X))), true, implies(implies(implies(X, X), implies(X, X)), implies(not(Y), implies(X, X)))), true, implies(implies(not(not(implies(W, implies(implies(implies(X, X), Y), Y)))), implies(implies(X, X), implies(X, X))), implies(not(Y), implies(X, X)))), true, implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(Y), implies(X, X)))), true, implies(implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y))))), true, implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.18 = { by lemma 9 R->L }
% 69.24/9.18 fresh2(fresh2(fresh(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(X, X), implies(X, X))), true, implies(implies(not(not(Y)), implies(X, X)), implies(X, X))), true, implies(not(Y), implies(X, X))), true, implies(implies(implies(X, X), implies(X, X)), implies(not(Y), implies(X, X)))), true, implies(implies(not(not(implies(W, implies(implies(implies(X, X), Y), Y)))), implies(implies(X, X), implies(X, X))), implies(not(Y), implies(X, X)))), true, implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(Y), implies(X, X)))), true, implies(implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y))))), true, implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.18 = { by lemma 14 }
% 69.24/9.18 fresh2(fresh2(fresh(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(not(not(Y)), implies(X, X)), implies(X, X))), true, implies(not(Y), implies(X, X))), true, implies(implies(implies(X, X), implies(X, X)), implies(not(Y), implies(X, X)))), true, implies(implies(not(not(implies(W, implies(implies(implies(X, X), Y), Y)))), implies(implies(X, X), implies(X, X))), implies(not(Y), implies(X, X)))), true, implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(Y), implies(X, X)))), true, implies(implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y))))), true, implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.18 = { by lemma 8 }
% 69.24/9.19 fresh2(fresh2(fresh(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(not(Y), implies(X, X))), true, implies(implies(implies(X, X), implies(X, X)), implies(not(Y), implies(X, X)))), true, implies(implies(not(not(implies(W, implies(implies(implies(X, X), Y), Y)))), implies(implies(X, X), implies(X, X))), implies(not(Y), implies(X, X)))), true, implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(Y), implies(X, X)))), true, implies(implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y))))), true, implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.19 = { by lemma 7 }
% 69.24/9.19 fresh2(fresh2(fresh(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(X, X), implies(X, X)), implies(not(Y), implies(X, X)))), true, implies(implies(not(not(implies(W, implies(implies(implies(X, X), Y), Y)))), implies(implies(X, X), implies(X, X))), implies(not(Y), implies(X, X)))), true, implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(Y), implies(X, X)))), true, implies(implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y))))), true, implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.19 = { by lemma 14 }
% 69.24/9.19 fresh2(fresh2(fresh(fresh2(fresh2(is_a_theorem(implies(implies(not(not(implies(W, implies(implies(implies(X, X), Y), Y)))), implies(implies(X, X), implies(X, X))), implies(not(Y), implies(X, X)))), true, implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(Y), implies(X, X)))), true, implies(implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y))))), true, implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.19 = { by lemma 8 }
% 69.24/9.19 fresh2(fresh2(fresh(fresh2(is_a_theorem(implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(Y), implies(X, X)))), true, implies(implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y))))), true, implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.19 = { by lemma 7 }
% 69.24/9.19 fresh2(fresh2(fresh(is_a_theorem(implies(implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y))))), true, implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y))), implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.19 = { by axiom 5 (condensed_detachment) }
% 69.24/9.19 fresh2(fresh2(fresh2(is_a_theorem(implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.19 = { by lemma 12 R->L }
% 69.24/9.19 fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(X, X), Y), implies(not(Y), Y)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.19 = { by lemma 12 R->L }
% 69.24/9.19 fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(not(Y), Y), Y), implies(implies(implies(X, X), Y), Y)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(implies(implies(implies(X, X), Y), implies(not(Y), Y)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.19 = { by lemma 7 R->L }
% 69.24/9.19 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(W, implies(implies(not(Y), Y), Y))), true, implies(implies(implies(implies(not(Y), Y), Y), implies(implies(implies(X, X), Y), Y)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(implies(implies(implies(X, X), Y), implies(not(Y), Y)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.19 = { by lemma 8 R->L }
% 69.24/9.19 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(not(W), implies(Y, Y)), implies(implies(not(Y), Y), Y))), true, implies(W, implies(implies(not(Y), Y), Y))), true, implies(implies(implies(implies(not(Y), Y), Y), implies(implies(implies(X, X), Y), Y)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(implies(implies(implies(X, X), Y), implies(not(Y), Y)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.19 = { by lemma 14 R->L }
% 69.24/9.19 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(Y, Y), implies(implies(not(Y), Y), Y))), true, implies(implies(not(W), implies(Y, Y)), implies(implies(not(Y), Y), Y))), true, implies(W, implies(implies(not(Y), Y), Y))), true, implies(implies(implies(implies(not(Y), Y), Y), implies(implies(implies(X, X), Y), Y)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(implies(implies(implies(X, X), Y), implies(not(Y), Y)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.19 = { by lemma 11 }
% 69.24/9.19 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(not(W), implies(Y, Y)), implies(implies(not(Y), Y), Y))), true, implies(W, implies(implies(not(Y), Y), Y))), true, implies(implies(implies(implies(not(Y), Y), Y), implies(implies(implies(X, X), Y), Y)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(implies(implies(implies(X, X), Y), implies(not(Y), Y)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.19 = { by axiom 1 (condensed_detachment) }
% 69.24/9.19 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(W, implies(implies(not(Y), Y), Y))), true, implies(implies(implies(implies(not(Y), Y), Y), implies(implies(implies(X, X), Y), Y)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(implies(implies(implies(X, X), Y), implies(not(Y), Y)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.19 = { by axiom 1 (condensed_detachment) }
% 69.24/9.19 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(implies(implies(not(Y), Y), Y), implies(implies(implies(X, X), Y), Y)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(implies(implies(implies(X, X), Y), implies(not(Y), Y)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.19 = { by axiom 1 (condensed_detachment) }
% 69.24/9.19 fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(implies(implies(X, X), Y), implies(not(Y), Y)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.19 = { by axiom 1 (condensed_detachment) }
% 69.24/9.19 fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(not(Y), implies(X, X)), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.19 = { by axiom 1 (condensed_detachment) }
% 69.24/9.19 fresh2(fresh2(fresh2(true, true, implies(not(implies(W, implies(implies(implies(X, X), Y), Y))), implies(W, implies(implies(implies(X, X), Y), Y)))), true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.19 = { by axiom 1 (condensed_detachment) }
% 69.24/9.19 fresh2(fresh2(true, true, implies(W, implies(implies(implies(X, X), Y), Y))), true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.19 = { by axiom 1 (condensed_detachment) }
% 69.24/9.19 fresh2(true, true, implies(implies(implies(implies(implies(X, X), Y), Y), Z), implies(W, Z)))
% 69.24/9.19 = { by axiom 1 (condensed_detachment) }
% 69.24/9.19 true
% 69.24/9.19
% 69.24/9.19 Goal 1 (prove_cn_23): is_a_theorem(implies(implies(implies(x, implies(y, z)), u), implies(implies(y, implies(x, z)), u))) = true.
% 69.24/9.19 Proof:
% 69.24/9.19 is_a_theorem(implies(implies(implies(x, implies(y, z)), u), implies(implies(y, implies(x, z)), u)))
% 69.24/9.19 = { by lemma 7 R->L }
% 69.24/9.19 fresh2(is_a_theorem(implies(implies(y, implies(x, z)), implies(x, implies(y, z)))), true, implies(implies(implies(x, implies(y, z)), u), implies(implies(y, implies(x, z)), u)))
% 69.24/9.19 = { by lemma 12 R->L }
% 69.24/9.19 fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(x, z), z), implies(y, z)), implies(x, implies(y, z)))), true, implies(implies(y, implies(x, z)), implies(x, implies(y, z)))), true, implies(implies(implies(x, implies(y, z)), u), implies(implies(y, implies(x, z)), u)))
% 69.24/9.19 = { by lemma 7 R->L }
% 69.24/9.19 fresh2(fresh2(fresh2(is_a_theorem(implies(x, implies(implies(x, z), z))), true, implies(implies(implies(implies(x, z), z), implies(y, z)), implies(x, implies(y, z)))), true, implies(implies(y, implies(x, z)), implies(x, implies(y, z)))), true, implies(implies(implies(x, implies(y, z)), u), implies(implies(y, implies(x, z)), u)))
% 69.24/9.19 = { by lemma 10 R->L }
% 69.24/9.19 fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(not(implies(x, implies(implies(x, z), z))), implies(x, implies(implies(x, z), z)))), true, implies(x, implies(implies(x, z), z))), true, implies(implies(implies(implies(x, z), z), implies(y, z)), implies(x, implies(y, z)))), true, implies(implies(y, implies(x, z)), implies(x, implies(y, z)))), true, implies(implies(implies(x, implies(y, z)), u), implies(implies(y, implies(x, z)), u)))
% 69.24/9.19 = { by axiom 2 (condensed_detachment) R->L }
% 69.24/9.19 fresh2(fresh2(fresh2(fresh2(fresh(true, true, implies(implies(implies(implies(X, X), implies(implies(x, z), z)), implies(implies(x, z), z)), implies(x, implies(implies(x, z), z))), implies(not(implies(x, implies(implies(x, z), z))), implies(x, implies(implies(x, z), z)))), true, implies(x, implies(implies(x, z), z))), true, implies(implies(implies(implies(x, z), z), implies(y, z)), implies(x, implies(y, z)))), true, implies(implies(y, implies(x, z)), implies(x, implies(y, z)))), true, implies(implies(implies(x, implies(y, z)), u), implies(implies(y, implies(x, z)), u)))
% 69.24/9.19 = { by lemma 15 R->L }
% 69.24/9.20 fresh2(fresh2(fresh2(fresh2(fresh(is_a_theorem(implies(implies(implies(implies(implies(X, X), implies(implies(x, z), z)), implies(implies(x, z), z)), implies(x, implies(implies(x, z), z))), implies(not(implies(x, implies(implies(x, z), z))), implies(x, implies(implies(x, z), z))))), true, implies(implies(implies(implies(X, X), implies(implies(x, z), z)), implies(implies(x, z), z)), implies(x, implies(implies(x, z), z))), implies(not(implies(x, implies(implies(x, z), z))), implies(x, implies(implies(x, z), z)))), true, implies(x, implies(implies(x, z), z))), true, implies(implies(implies(implies(x, z), z), implies(y, z)), implies(x, implies(y, z)))), true, implies(implies(y, implies(x, z)), implies(x, implies(y, z)))), true, implies(implies(implies(x, implies(y, z)), u), implies(implies(y, implies(x, z)), u)))
% 69.24/9.20 = { by axiom 5 (condensed_detachment) }
% 69.24/9.20 fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(X, X), implies(implies(x, z), z)), implies(implies(x, z), z)), implies(x, implies(implies(x, z), z)))), true, implies(not(implies(x, implies(implies(x, z), z))), implies(x, implies(implies(x, z), z)))), true, implies(x, implies(implies(x, z), z))), true, implies(implies(implies(implies(x, z), z), implies(y, z)), implies(x, implies(y, z)))), true, implies(implies(y, implies(x, z)), implies(x, implies(y, z)))), true, implies(implies(implies(x, implies(y, z)), u), implies(implies(y, implies(x, z)), u)))
% 69.24/9.20 = { by lemma 7 R->L }
% 69.24/9.20 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(x, implies(implies(X, X), implies(implies(x, z), z)))), true, implies(implies(implies(implies(X, X), implies(implies(x, z), z)), implies(implies(x, z), z)), implies(x, implies(implies(x, z), z)))), true, implies(not(implies(x, implies(implies(x, z), z))), implies(x, implies(implies(x, z), z)))), true, implies(x, implies(implies(x, z), z))), true, implies(implies(implies(implies(x, z), z), implies(y, z)), implies(x, implies(y, z)))), true, implies(implies(y, implies(x, z)), implies(x, implies(y, z)))), true, implies(implies(implies(x, implies(y, z)), u), implies(implies(y, implies(x, z)), u)))
% 69.24/9.20 = { by axiom 2 (condensed_detachment) R->L }
% 69.24/9.20 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(true, true, implies(implies(implies(x, z), implies(implies(x, x), z)), implies(implies(X, X), implies(implies(x, z), z))), implies(x, implies(implies(X, X), implies(implies(x, z), z)))), true, implies(implies(implies(implies(X, X), implies(implies(x, z), z)), implies(implies(x, z), z)), implies(x, implies(implies(x, z), z)))), true, implies(not(implies(x, implies(implies(x, z), z))), implies(x, implies(implies(x, z), z)))), true, implies(x, implies(implies(x, z), z))), true, implies(implies(implies(implies(x, z), z), implies(y, z)), implies(x, implies(y, z)))), true, implies(implies(y, implies(x, z)), implies(x, implies(y, z)))), true, implies(implies(implies(x, implies(y, z)), u), implies(implies(y, implies(x, z)), u)))
% 69.24/9.20 = { by axiom 1 (condensed_detachment) R->L }
% 69.24/9.20 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(true, true, implies(implies(implies(implies(x, z), implies(implies(x, x), z)), implies(implies(X, X), implies(implies(x, z), z))), implies(x, implies(implies(X, X), implies(implies(x, z), z))))), true, implies(implies(implies(x, z), implies(implies(x, x), z)), implies(implies(X, X), implies(implies(x, z), z))), implies(x, implies(implies(X, X), implies(implies(x, z), z)))), true, implies(implies(implies(implies(X, X), implies(implies(x, z), z)), implies(implies(x, z), z)), implies(x, implies(implies(x, z), z)))), true, implies(not(implies(x, implies(implies(x, z), z))), implies(x, implies(implies(x, z), z)))), true, implies(x, implies(implies(x, z), z))), true, implies(implies(implies(implies(x, z), z), implies(y, z)), implies(x, implies(y, z)))), true, implies(implies(y, implies(x, z)), implies(x, implies(y, z)))), true, implies(implies(implies(x, implies(y, z)), u), implies(implies(y, implies(x, z)), u)))
% 69.24/9.20 = { by lemma 13 R->L }
% 69.24/9.20 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(fresh2(is_a_theorem(implies(x, implies(implies(x, z), implies(implies(x, x), z)))), true, implies(implies(implies(implies(x, z), implies(implies(x, x), z)), implies(implies(X, X), implies(implies(x, z), z))), implies(x, implies(implies(X, X), implies(implies(x, z), z))))), true, implies(implies(implies(x, z), implies(implies(x, x), z)), implies(implies(X, X), implies(implies(x, z), z))), implies(x, implies(implies(X, X), implies(implies(x, z), z)))), true, implies(implies(implies(implies(X, X), implies(implies(x, z), z)), implies(implies(x, z), z)), implies(x, implies(implies(x, z), z)))), true, implies(not(implies(x, implies(implies(x, z), z))), implies(x, implies(implies(x, z), z)))), true, implies(x, implies(implies(x, z), z))), true, implies(implies(implies(implies(x, z), z), implies(y, z)), implies(x, implies(y, z)))), true, implies(implies(y, implies(x, z)), implies(x, implies(y, z)))), true, implies(implies(implies(x, implies(y, z)), u), implies(implies(y, implies(x, z)), u)))
% 69.24/9.20 = { by lemma 7 }
% 69.24/9.20 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh(is_a_theorem(implies(implies(implies(implies(x, z), implies(implies(x, x), z)), implies(implies(X, X), implies(implies(x, z), z))), implies(x, implies(implies(X, X), implies(implies(x, z), z))))), true, implies(implies(implies(x, z), implies(implies(x, x), z)), implies(implies(X, X), implies(implies(x, z), z))), implies(x, implies(implies(X, X), implies(implies(x, z), z)))), true, implies(implies(implies(implies(X, X), implies(implies(x, z), z)), implies(implies(x, z), z)), implies(x, implies(implies(x, z), z)))), true, implies(not(implies(x, implies(implies(x, z), z))), implies(x, implies(implies(x, z), z)))), true, implies(x, implies(implies(x, z), z))), true, implies(implies(implies(implies(x, z), z), implies(y, z)), implies(x, implies(y, z)))), true, implies(implies(y, implies(x, z)), implies(x, implies(y, z)))), true, implies(implies(implies(x, implies(y, z)), u), implies(implies(y, implies(x, z)), u)))
% 69.24/9.20 = { by axiom 5 (condensed_detachment) }
% 69.24/9.20 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(x, z), implies(implies(x, x), z)), implies(implies(X, X), implies(implies(x, z), z)))), true, implies(x, implies(implies(X, X), implies(implies(x, z), z)))), true, implies(implies(implies(implies(X, X), implies(implies(x, z), z)), implies(implies(x, z), z)), implies(x, implies(implies(x, z), z)))), true, implies(not(implies(x, implies(implies(x, z), z))), implies(x, implies(implies(x, z), z)))), true, implies(x, implies(implies(x, z), z))), true, implies(implies(implies(implies(x, z), z), implies(y, z)), implies(x, implies(y, z)))), true, implies(implies(y, implies(x, z)), implies(x, implies(y, z)))), true, implies(implies(implies(x, implies(y, z)), u), implies(implies(y, implies(x, z)), u)))
% 69.24/9.20 = { by lemma 12 R->L }
% 69.24/9.20 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(implies(x, x), z), z), implies(implies(x, z), z)), implies(implies(X, X), implies(implies(x, z), z)))), true, implies(implies(implies(x, z), implies(implies(x, x), z)), implies(implies(X, X), implies(implies(x, z), z)))), true, implies(x, implies(implies(X, X), implies(implies(x, z), z)))), true, implies(implies(implies(implies(X, X), implies(implies(x, z), z)), implies(implies(x, z), z)), implies(x, implies(implies(x, z), z)))), true, implies(not(implies(x, implies(implies(x, z), z))), implies(x, implies(implies(x, z), z)))), true, implies(x, implies(implies(x, z), z))), true, implies(implies(implies(implies(x, z), z), implies(y, z)), implies(x, implies(y, z)))), true, implies(implies(y, implies(x, z)), implies(x, implies(y, z)))), true, implies(implies(implies(x, implies(y, z)), u), implies(implies(y, implies(x, z)), u)))
% 69.24/9.20 = { by lemma 15 }
% 69.24/9.20 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(implies(x, z), implies(implies(x, x), z)), implies(implies(X, X), implies(implies(x, z), z)))), true, implies(x, implies(implies(X, X), implies(implies(x, z), z)))), true, implies(implies(implies(implies(X, X), implies(implies(x, z), z)), implies(implies(x, z), z)), implies(x, implies(implies(x, z), z)))), true, implies(not(implies(x, implies(implies(x, z), z))), implies(x, implies(implies(x, z), z)))), true, implies(x, implies(implies(x, z), z))), true, implies(implies(implies(implies(x, z), z), implies(y, z)), implies(x, implies(y, z)))), true, implies(implies(y, implies(x, z)), implies(x, implies(y, z)))), true, implies(implies(implies(x, implies(y, z)), u), implies(implies(y, implies(x, z)), u)))
% 69.24/9.20 = { by axiom 1 (condensed_detachment) }
% 69.24/9.20 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(x, implies(implies(X, X), implies(implies(x, z), z)))), true, implies(implies(implies(implies(X, X), implies(implies(x, z), z)), implies(implies(x, z), z)), implies(x, implies(implies(x, z), z)))), true, implies(not(implies(x, implies(implies(x, z), z))), implies(x, implies(implies(x, z), z)))), true, implies(x, implies(implies(x, z), z))), true, implies(implies(implies(implies(x, z), z), implies(y, z)), implies(x, implies(y, z)))), true, implies(implies(y, implies(x, z)), implies(x, implies(y, z)))), true, implies(implies(implies(x, implies(y, z)), u), implies(implies(y, implies(x, z)), u)))
% 69.24/9.20 = { by axiom 1 (condensed_detachment) }
% 69.24/9.20 fresh2(fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(implies(implies(implies(X, X), implies(implies(x, z), z)), implies(implies(x, z), z)), implies(x, implies(implies(x, z), z)))), true, implies(not(implies(x, implies(implies(x, z), z))), implies(x, implies(implies(x, z), z)))), true, implies(x, implies(implies(x, z), z))), true, implies(implies(implies(implies(x, z), z), implies(y, z)), implies(x, implies(y, z)))), true, implies(implies(y, implies(x, z)), implies(x, implies(y, z)))), true, implies(implies(implies(x, implies(y, z)), u), implies(implies(y, implies(x, z)), u)))
% 69.24/9.20 = { by axiom 1 (condensed_detachment) }
% 69.24/9.20 fresh2(fresh2(fresh2(fresh2(fresh2(true, true, implies(not(implies(x, implies(implies(x, z), z))), implies(x, implies(implies(x, z), z)))), true, implies(x, implies(implies(x, z), z))), true, implies(implies(implies(implies(x, z), z), implies(y, z)), implies(x, implies(y, z)))), true, implies(implies(y, implies(x, z)), implies(x, implies(y, z)))), true, implies(implies(implies(x, implies(y, z)), u), implies(implies(y, implies(x, z)), u)))
% 69.24/9.20 = { by axiom 1 (condensed_detachment) }
% 69.24/9.20 fresh2(fresh2(fresh2(fresh2(true, true, implies(x, implies(implies(x, z), z))), true, implies(implies(implies(implies(x, z), z), implies(y, z)), implies(x, implies(y, z)))), true, implies(implies(y, implies(x, z)), implies(x, implies(y, z)))), true, implies(implies(implies(x, implies(y, z)), u), implies(implies(y, implies(x, z)), u)))
% 69.24/9.20 = { by axiom 1 (condensed_detachment) }
% 69.24/9.20 fresh2(fresh2(fresh2(true, true, implies(implies(implies(implies(x, z), z), implies(y, z)), implies(x, implies(y, z)))), true, implies(implies(y, implies(x, z)), implies(x, implies(y, z)))), true, implies(implies(implies(x, implies(y, z)), u), implies(implies(y, implies(x, z)), u)))
% 69.24/9.20 = { by axiom 1 (condensed_detachment) }
% 69.24/9.20 fresh2(fresh2(true, true, implies(implies(y, implies(x, z)), implies(x, implies(y, z)))), true, implies(implies(implies(x, implies(y, z)), u), implies(implies(y, implies(x, z)), u)))
% 69.24/9.20 = { by axiom 1 (condensed_detachment) }
% 69.24/9.20 fresh2(true, true, implies(implies(implies(x, implies(y, z)), u), implies(implies(y, implies(x, z)), u)))
% 69.24/9.20 = { by axiom 1 (condensed_detachment) }
% 69.24/9.20 true
% 69.24/9.20 % SZS output end Proof
% 69.24/9.20
% 69.24/9.20 RESULT: Unsatisfiable (the axioms are contradictory).
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