TSTP Solution File: LCL036-1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL036-1 : 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 : n020.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 08:17:07 EDT 2023
% Result : Unsatisfiable 0.20s 0.64s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LCL036-1 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n020.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Aug 25 05:25:00 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.20/0.64 Command-line arguments: --no-flatten-goal
% 0.20/0.64
% 0.20/0.64 % SZS status Unsatisfiable
% 0.20/0.64
% 0.20/0.66 % SZS output start Proof
% 0.20/0.66 Take the following subset of the input axioms:
% 0.20/0.66 fof(c0_CAMerideth, axiom, ![X, Y, Z, U, V]: is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Z, falsehood)), U), V), implies(implies(V, X), implies(Z, X))))).
% 0.20/0.66 fof(condensed_detachment, axiom, ![X2, Y2]: (~is_a_theorem(implies(X2, Y2)) | (~is_a_theorem(X2) | is_a_theorem(Y2)))).
% 0.20/0.66 fof(prove_c0_5, negated_conjecture, ~is_a_theorem(implies(implies(implies(a, falsehood), falsehood), a))).
% 0.20/0.66
% 0.20/0.66 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.66 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.66 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.66 fresh(y, y, x1...xn) = u
% 0.20/0.66 C => fresh(s, t, x1...xn) = v
% 0.20/0.66 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.66 variables of u and v.
% 0.20/0.66 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.66 input problem has no model of domain size 1).
% 0.20/0.66
% 0.20/0.66 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.66
% 0.20/0.66 Axiom 1 (condensed_detachment): fresh2(X, X, Y) = true.
% 0.20/0.66 Axiom 2 (condensed_detachment): fresh(X, X, Y, Z) = is_a_theorem(Z).
% 0.20/0.66 Axiom 3 (condensed_detachment): fresh(is_a_theorem(implies(X, Y)), true, X, Y) = fresh2(is_a_theorem(X), true, Y).
% 0.20/0.66 Axiom 4 (c0_CAMerideth): is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Z, falsehood)), W), V), implies(implies(V, X), implies(Z, X)))) = true.
% 0.20/0.66
% 0.20/0.66 Lemma 5: fresh2(is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, falsehood)), W), V)), true, implies(implies(V, X), implies(Z, X))) = is_a_theorem(implies(implies(V, X), implies(Z, X))).
% 0.20/0.66 Proof:
% 0.20/0.66 fresh2(is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, falsehood)), W), V)), true, implies(implies(V, X), implies(Z, X)))
% 0.20/0.66 = { by axiom 3 (condensed_detachment) R->L }
% 0.20/0.66 fresh(is_a_theorem(implies(implies(implies(implies(implies(X, Y), implies(Z, falsehood)), W), V), implies(implies(V, X), implies(Z, X)))), true, implies(implies(implies(implies(X, Y), implies(Z, falsehood)), W), V), implies(implies(V, X), implies(Z, X)))
% 0.20/0.66 = { by axiom 4 (c0_CAMerideth) }
% 0.20/0.66 fresh(true, true, implies(implies(implies(implies(X, Y), implies(Z, falsehood)), W), V), implies(implies(V, X), implies(Z, X)))
% 0.20/0.66 = { by axiom 2 (condensed_detachment) }
% 0.20/0.66 is_a_theorem(implies(implies(V, X), implies(Z, X)))
% 0.20/0.66
% 0.20/0.66 Lemma 6: is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), implies(Y, W)), implies(V, implies(Y, W)))) = true.
% 0.20/0.66 Proof:
% 0.20/0.66 is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), implies(Y, W)), implies(V, implies(Y, W))))
% 0.20/0.66 = { by lemma 5 R->L }
% 0.20/0.66 fresh2(is_a_theorem(implies(implies(implies(implies(implies(Y, W), implies(Z, falsehood)), implies(V, falsehood)), X), implies(implies(X, Y), implies(Z, Y)))), true, implies(implies(implies(implies(X, Y), implies(Z, Y)), implies(Y, W)), implies(V, implies(Y, W))))
% 0.20/0.66 = { by axiom 4 (c0_CAMerideth) }
% 0.20/0.66 fresh2(true, true, implies(implies(implies(implies(X, Y), implies(Z, Y)), implies(Y, W)), implies(V, implies(Y, W))))
% 0.20/0.66 = { by axiom 1 (condensed_detachment) }
% 0.20/0.66 true
% 0.20/0.66
% 0.20/0.66 Lemma 7: is_a_theorem(implies(implies(implies(falsehood, X), Y), implies(Z, Y))) = true.
% 0.20/0.66 Proof:
% 0.20/0.66 is_a_theorem(implies(implies(implies(falsehood, X), Y), implies(Z, Y)))
% 0.20/0.66 = { by lemma 5 R->L }
% 0.20/0.66 fresh2(is_a_theorem(implies(implies(implies(implies(Y, W), implies(Z, falsehood)), X), implies(falsehood, X))), true, implies(implies(implies(falsehood, X), Y), implies(Z, Y)))
% 0.20/0.66 = { by lemma 5 R->L }
% 0.20/0.66 fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(X, V), implies(falsehood, falsehood)), implies(Z, falsehood)), implies(implies(Y, W), implies(Z, falsehood)))), true, implies(implies(implies(implies(Y, W), implies(Z, falsehood)), X), implies(falsehood, X))), true, implies(implies(implies(falsehood, X), Y), implies(Z, Y)))
% 0.20/0.66 = { by lemma 5 R->L }
% 0.20/0.66 fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(implies(Z, falsehood), falsehood), implies(implies(Y, W), falsehood)), implies(falsehood, falsehood)), implies(implies(X, V), implies(falsehood, falsehood)))), true, implies(implies(implies(implies(X, V), implies(falsehood, falsehood)), implies(Z, falsehood)), implies(implies(Y, W), implies(Z, falsehood)))), true, implies(implies(implies(implies(Y, W), implies(Z, falsehood)), X), implies(falsehood, X))), true, implies(implies(implies(falsehood, X), Y), implies(Z, Y)))
% 0.20/0.66 = { by lemma 6 }
% 0.20/0.66 fresh2(fresh2(fresh2(true, true, implies(implies(implies(implies(X, V), implies(falsehood, falsehood)), implies(Z, falsehood)), implies(implies(Y, W), implies(Z, falsehood)))), true, implies(implies(implies(implies(Y, W), implies(Z, falsehood)), X), implies(falsehood, X))), true, implies(implies(implies(falsehood, X), Y), implies(Z, Y)))
% 0.20/0.66 = { by axiom 1 (condensed_detachment) }
% 0.20/0.66 fresh2(fresh2(true, true, implies(implies(implies(implies(Y, W), implies(Z, falsehood)), X), implies(falsehood, X))), true, implies(implies(implies(falsehood, X), Y), implies(Z, Y)))
% 0.20/0.66 = { by axiom 1 (condensed_detachment) }
% 0.20/0.66 fresh2(true, true, implies(implies(implies(falsehood, X), Y), implies(Z, Y)))
% 0.20/0.66 = { by axiom 1 (condensed_detachment) }
% 0.20/0.66 true
% 0.20/0.66
% 0.20/0.66 Lemma 8: is_a_theorem(implies(implies(implies(X, Y), Z), implies(implies(implies(Y, W), implies(X, falsehood)), Z))) = true.
% 0.20/0.66 Proof:
% 0.20/0.66 is_a_theorem(implies(implies(implies(X, Y), Z), implies(implies(implies(Y, W), implies(X, falsehood)), Z)))
% 0.20/0.66 = { by lemma 5 R->L }
% 0.20/0.66 fresh2(is_a_theorem(implies(implies(implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)), Y), implies(X, Y))), true, implies(implies(implies(X, Y), Z), implies(implies(implies(Y, W), implies(X, falsehood)), Z)))
% 0.20/0.66 = { by lemma 5 R->L }
% 0.20/0.67 fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(Y, W), implies(X, falsehood)), falsehood), implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)))), true, implies(implies(implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)), Y), implies(X, Y))), true, implies(implies(implies(X, Y), Z), implies(implies(implies(Y, W), implies(X, falsehood)), Z)))
% 0.20/0.67 = { by axiom 2 (condensed_detachment) R->L }
% 0.20/0.67 fresh2(fresh2(fresh(true, true, implies(implies(implies(implies(implies(U, T), implies(S, falsehood)), X2), Y2), implies(implies(Y2, U), implies(S, U))), implies(implies(implies(implies(Y, W), implies(X, falsehood)), falsehood), implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)))), true, implies(implies(implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)), Y), implies(X, Y))), true, implies(implies(implies(X, Y), Z), implies(implies(implies(Y, W), implies(X, falsehood)), Z)))
% 0.20/0.67 = { by axiom 1 (condensed_detachment) R->L }
% 0.20/0.67 fresh2(fresh2(fresh(fresh2(true, true, implies(implies(implies(implies(implies(implies(U, T), implies(S, falsehood)), X2), Y2), implies(implies(Y2, U), implies(S, U))), implies(implies(implies(implies(Y, W), implies(X, falsehood)), falsehood), implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood))))), true, implies(implies(implies(implies(implies(U, T), implies(S, falsehood)), X2), Y2), implies(implies(Y2, U), implies(S, U))), implies(implies(implies(implies(Y, W), implies(X, falsehood)), falsehood), implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)))), true, implies(implies(implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)), Y), implies(X, Y))), true, implies(implies(implies(X, Y), Z), implies(implies(implies(Y, W), implies(X, falsehood)), Z)))
% 0.20/0.67 = { by lemma 7 R->L }
% 0.20/0.67 fresh2(fresh2(fresh(fresh2(is_a_theorem(implies(implies(implies(falsehood, implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)), implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood))), implies(implies(implies(implies(Y, W), implies(X, falsehood)), falsehood), implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood))))), true, implies(implies(implies(implies(implies(implies(U, T), implies(S, falsehood)), X2), Y2), implies(implies(Y2, U), implies(S, U))), implies(implies(implies(implies(Y, W), implies(X, falsehood)), falsehood), implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood))))), true, implies(implies(implies(implies(implies(U, T), implies(S, falsehood)), X2), Y2), implies(implies(Y2, U), implies(S, U))), implies(implies(implies(implies(Y, W), implies(X, falsehood)), falsehood), implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)))), true, implies(implies(implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)), Y), implies(X, Y))), true, implies(implies(implies(X, Y), Z), implies(implies(implies(Y, W), implies(X, falsehood)), Z)))
% 0.20/0.67 = { by axiom 3 (condensed_detachment) R->L }
% 0.20/0.67 fresh2(fresh2(fresh(fresh(is_a_theorem(implies(implies(implies(implies(falsehood, implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)), implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood))), implies(implies(implies(implies(Y, W), implies(X, falsehood)), falsehood), implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)))), implies(implies(implies(implies(implies(implies(U, T), implies(S, falsehood)), X2), Y2), implies(implies(Y2, U), implies(S, U))), implies(implies(implies(implies(Y, W), implies(X, falsehood)), falsehood), implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)))))), true, implies(implies(implies(falsehood, implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)), implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood))), implies(implies(implies(implies(Y, W), implies(X, falsehood)), falsehood), implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)))), implies(implies(implies(implies(implies(implies(U, T), implies(S, falsehood)), X2), Y2), implies(implies(Y2, U), implies(S, U))), implies(implies(implies(implies(Y, W), implies(X, falsehood)), falsehood), implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood))))), true, implies(implies(implies(implies(implies(U, T), implies(S, falsehood)), X2), Y2), implies(implies(Y2, U), implies(S, U))), implies(implies(implies(implies(Y, W), implies(X, falsehood)), falsehood), implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)))), true, implies(implies(implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)), Y), implies(X, Y))), true, implies(implies(implies(X, Y), Z), implies(implies(implies(Y, W), implies(X, falsehood)), Z)))
% 0.20/0.67 = { by lemma 6 }
% 0.20/0.67 fresh2(fresh2(fresh(fresh(true, true, implies(implies(implies(falsehood, implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)), implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood))), implies(implies(implies(implies(Y, W), implies(X, falsehood)), falsehood), implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)))), implies(implies(implies(implies(implies(implies(U, T), implies(S, falsehood)), X2), Y2), implies(implies(Y2, U), implies(S, U))), implies(implies(implies(implies(Y, W), implies(X, falsehood)), falsehood), implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood))))), true, implies(implies(implies(implies(implies(U, T), implies(S, falsehood)), X2), Y2), implies(implies(Y2, U), implies(S, U))), implies(implies(implies(implies(Y, W), implies(X, falsehood)), falsehood), implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)))), true, implies(implies(implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)), Y), implies(X, Y))), true, implies(implies(implies(X, Y), Z), implies(implies(implies(Y, W), implies(X, falsehood)), Z)))
% 0.20/0.67 = { by axiom 2 (condensed_detachment) }
% 0.20/0.67 fresh2(fresh2(fresh(is_a_theorem(implies(implies(implies(implies(implies(implies(U, T), implies(S, falsehood)), X2), Y2), implies(implies(Y2, U), implies(S, U))), implies(implies(implies(implies(Y, W), implies(X, falsehood)), falsehood), implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood))))), true, implies(implies(implies(implies(implies(U, T), implies(S, falsehood)), X2), Y2), implies(implies(Y2, U), implies(S, U))), implies(implies(implies(implies(Y, W), implies(X, falsehood)), falsehood), implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)))), true, implies(implies(implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)), Y), implies(X, Y))), true, implies(implies(implies(X, Y), Z), implies(implies(implies(Y, W), implies(X, falsehood)), Z)))
% 0.20/0.67 = { by axiom 3 (condensed_detachment) }
% 0.20/0.67 fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(implies(U, T), implies(S, falsehood)), X2), Y2), implies(implies(Y2, U), implies(S, U)))), true, implies(implies(implies(implies(Y, W), implies(X, falsehood)), falsehood), implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)))), true, implies(implies(implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)), Y), implies(X, Y))), true, implies(implies(implies(X, Y), Z), implies(implies(implies(Y, W), implies(X, falsehood)), Z)))
% 0.20/0.67 = { by axiom 4 (c0_CAMerideth) }
% 0.20/0.67 fresh2(fresh2(fresh2(true, true, implies(implies(implies(implies(Y, W), implies(X, falsehood)), falsehood), implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)))), true, implies(implies(implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)), Y), implies(X, Y))), true, implies(implies(implies(X, Y), Z), implies(implies(implies(Y, W), implies(X, falsehood)), Z)))
% 0.20/0.67 = { by axiom 1 (condensed_detachment) }
% 0.20/0.67 fresh2(fresh2(true, true, implies(implies(implies(implies(Z, V), implies(implies(implies(Y, W), implies(X, falsehood)), falsehood)), Y), implies(X, Y))), true, implies(implies(implies(X, Y), Z), implies(implies(implies(Y, W), implies(X, falsehood)), Z)))
% 0.20/0.67 = { by axiom 1 (condensed_detachment) }
% 0.20/0.67 fresh2(true, true, implies(implies(implies(X, Y), Z), implies(implies(implies(Y, W), implies(X, falsehood)), Z)))
% 0.20/0.67 = { by axiom 1 (condensed_detachment) }
% 0.20/0.67 true
% 0.20/0.67
% 0.20/0.67 Lemma 9: fresh2(is_a_theorem(implies(implies(X, Y), Z)), true, implies(implies(implies(Y, W), implies(X, falsehood)), Z)) = is_a_theorem(implies(implies(implies(Y, W), implies(X, falsehood)), Z)).
% 0.20/0.67 Proof:
% 0.20/0.67 fresh2(is_a_theorem(implies(implies(X, Y), Z)), true, implies(implies(implies(Y, W), implies(X, falsehood)), Z))
% 0.20/0.67 = { by axiom 3 (condensed_detachment) R->L }
% 0.20/0.67 fresh(is_a_theorem(implies(implies(implies(X, Y), Z), implies(implies(implies(Y, W), implies(X, falsehood)), Z))), true, implies(implies(X, Y), Z), implies(implies(implies(Y, W), implies(X, falsehood)), Z))
% 0.20/0.67 = { by lemma 8 }
% 0.20/0.67 fresh(true, true, implies(implies(X, Y), Z), implies(implies(implies(Y, W), implies(X, falsehood)), Z))
% 0.20/0.67 = { by axiom 2 (condensed_detachment) }
% 0.20/0.67 is_a_theorem(implies(implies(implies(Y, W), implies(X, falsehood)), Z))
% 0.20/0.67
% 0.20/0.67 Lemma 10: is_a_theorem(implies(implies(implies(X, Y), Z), implies(Y, Z))) = true.
% 0.20/0.67 Proof:
% 0.20/0.67 is_a_theorem(implies(implies(implies(X, Y), Z), implies(Y, Z)))
% 0.20/0.67 = { by lemma 5 R->L }
% 0.20/0.67 fresh2(is_a_theorem(implies(implies(implies(implies(Z, W), implies(Y, falsehood)), Y), implies(X, Y))), true, implies(implies(implies(X, Y), Z), implies(Y, Z)))
% 0.20/0.67 = { by lemma 5 R->L }
% 0.20/0.67 fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(Y, falsehood), implies(X, falsehood)), implies(implies(falsehood, V), falsehood)), implies(implies(Z, W), implies(Y, falsehood)))), true, implies(implies(implies(implies(Z, W), implies(Y, falsehood)), Y), implies(X, Y))), true, implies(implies(implies(X, Y), Z), implies(Y, Z)))
% 0.20/0.67 = { by lemma 9 R->L }
% 0.20/0.67 fresh2(fresh2(fresh2(is_a_theorem(implies(implies(implies(falsehood, V), implies(Y, falsehood)), implies(implies(Z, W), implies(Y, falsehood)))), true, implies(implies(implies(implies(Y, falsehood), implies(X, falsehood)), implies(implies(falsehood, V), falsehood)), implies(implies(Z, W), implies(Y, falsehood)))), true, implies(implies(implies(implies(Z, W), implies(Y, falsehood)), Y), implies(X, Y))), true, implies(implies(implies(X, Y), Z), implies(Y, Z)))
% 0.20/0.67 = { by lemma 7 }
% 0.20/0.67 fresh2(fresh2(fresh2(true, true, implies(implies(implies(implies(Y, falsehood), implies(X, falsehood)), implies(implies(falsehood, V), falsehood)), implies(implies(Z, W), implies(Y, falsehood)))), true, implies(implies(implies(implies(Z, W), implies(Y, falsehood)), Y), implies(X, Y))), true, implies(implies(implies(X, Y), Z), implies(Y, Z)))
% 0.20/0.67 = { by axiom 1 (condensed_detachment) }
% 0.20/0.67 fresh2(fresh2(true, true, implies(implies(implies(implies(Z, W), implies(Y, falsehood)), Y), implies(X, Y))), true, implies(implies(implies(X, Y), Z), implies(Y, Z)))
% 0.20/0.67 = { by axiom 1 (condensed_detachment) }
% 0.20/0.67 fresh2(true, true, implies(implies(implies(X, Y), Z), implies(Y, Z)))
% 0.20/0.67 = { by axiom 1 (condensed_detachment) }
% 0.20/0.67 true
% 0.20/0.67
% 0.20/0.67 Goal 1 (prove_c0_5): is_a_theorem(implies(implies(implies(a, falsehood), falsehood), a)) = true.
% 0.20/0.67 Proof:
% 0.20/0.67 is_a_theorem(implies(implies(implies(a, falsehood), falsehood), a))
% 0.20/0.67 = { by axiom 2 (condensed_detachment) R->L }
% 0.20/0.67 fresh(true, true, implies(implies(a, falsehood), implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), falsehood)), implies(implies(implies(a, falsehood), falsehood), a))
% 0.20/0.67 = { by axiom 1 (condensed_detachment) R->L }
% 0.20/0.67 fresh(fresh2(true, true, implies(implies(implies(a, falsehood), implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), falsehood)), implies(implies(implies(a, falsehood), falsehood), a))), true, implies(implies(a, falsehood), implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), falsehood)), implies(implies(implies(a, falsehood), falsehood), a))
% 0.20/0.67 = { by lemma 10 R->L }
% 0.20/0.67 fresh(fresh2(is_a_theorem(implies(implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), a), implies(implies(implies(a, falsehood), falsehood), a))), true, implies(implies(implies(a, falsehood), implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), falsehood)), implies(implies(implies(a, falsehood), falsehood), a))), true, implies(implies(a, falsehood), implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), falsehood)), implies(implies(implies(a, falsehood), falsehood), a))
% 0.20/0.67 = { by lemma 9 }
% 0.20/0.67 fresh(is_a_theorem(implies(implies(implies(a, falsehood), implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), falsehood)), implies(implies(implies(a, falsehood), falsehood), a))), true, implies(implies(a, falsehood), implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), falsehood)), implies(implies(implies(a, falsehood), falsehood), a))
% 0.20/0.67 = { by axiom 3 (condensed_detachment) }
% 0.20/0.67 fresh2(is_a_theorem(implies(implies(a, falsehood), implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), falsehood))), true, implies(implies(implies(a, falsehood), falsehood), a))
% 0.20/0.67 = { by axiom 2 (condensed_detachment) R->L }
% 0.20/0.67 fresh2(fresh(true, true, implies(implies(implies(implies(a, falsehood), falsehood), falsehood), implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), falsehood)), implies(implies(a, falsehood), implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), falsehood))), true, implies(implies(implies(a, falsehood), falsehood), a))
% 0.20/0.67 = { by axiom 1 (condensed_detachment) R->L }
% 0.20/0.67 fresh2(fresh(fresh2(true, true, implies(implies(implies(implies(implies(a, falsehood), falsehood), falsehood), implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), falsehood)), implies(implies(a, falsehood), implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), falsehood)))), true, implies(implies(implies(implies(a, falsehood), falsehood), falsehood), implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), falsehood)), implies(implies(a, falsehood), implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), falsehood))), true, implies(implies(implies(a, falsehood), falsehood), a))
% 0.20/0.67 = { by lemma 10 R->L }
% 0.20/0.67 fresh2(fresh(fresh2(is_a_theorem(implies(implies(implies(implies(implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), falsehood), Y), implies(implies(a, falsehood), falsehood)), falsehood), implies(implies(implies(a, falsehood), falsehood), falsehood))), true, implies(implies(implies(implies(implies(a, falsehood), falsehood), falsehood), implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), falsehood)), implies(implies(a, falsehood), implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), falsehood)))), true, implies(implies(implies(implies(a, falsehood), falsehood), falsehood), implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), falsehood)), implies(implies(a, falsehood), implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), falsehood))), true, implies(implies(implies(a, falsehood), falsehood), a))
% 0.20/0.67 = { by lemma 5 }
% 0.20/0.68 fresh2(fresh(is_a_theorem(implies(implies(implies(implies(implies(a, falsehood), falsehood), falsehood), implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), falsehood)), implies(implies(a, falsehood), implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), falsehood)))), true, implies(implies(implies(implies(a, falsehood), falsehood), falsehood), implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), falsehood)), implies(implies(a, falsehood), implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), falsehood))), true, implies(implies(implies(a, falsehood), falsehood), a))
% 0.20/0.68 = { by axiom 3 (condensed_detachment) }
% 0.20/0.68 fresh2(fresh2(is_a_theorem(implies(implies(implies(implies(a, falsehood), falsehood), falsehood), implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), falsehood))), true, implies(implies(a, falsehood), implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), falsehood))), true, implies(implies(implies(a, falsehood), falsehood), a))
% 0.20/0.68 = { by lemma 8 }
% 0.20/0.68 fresh2(fresh2(true, true, implies(implies(a, falsehood), implies(implies(implies(falsehood, X), implies(implies(a, falsehood), falsehood)), falsehood))), true, implies(implies(implies(a, falsehood), falsehood), a))
% 0.20/0.68 = { by axiom 1 (condensed_detachment) }
% 0.20/0.68 fresh2(true, true, implies(implies(implies(a, falsehood), falsehood), a))
% 0.20/0.68 = { by axiom 1 (condensed_detachment) }
% 0.20/0.68 true
% 0.20/0.68 % SZS output end Proof
% 0.20/0.68
% 0.20/0.68 RESULT: Unsatisfiable (the axioms are contradictory).
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