TSTP Solution File: LCL449^7 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : LCL449^7 : TPTP v6.1.0. Released v5.5.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n106.star.cs.uiowa.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory : 32286.75MB
% OS : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:25:39 EDT 2014
% Result : Timeout 300.06s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : LCL449^7 : TPTP v6.1.0. Released v5.5.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n106.star.cs.uiowa.edu
% % Model : x86_64 x86_64
% % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory : 32286.75MB
% % OS : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 09:42:36 CDT 2014
% % CPUTime : 300.06
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% Failed to open /home/cristobal/cocATP/CASC/TPTP/Axioms/LCL015^0.ax, trying next directory
% FOF formula (<kernel.Constant object at 0x10e44d0>, <kernel.Type object at 0x10e4908>) of role type named mu_type
% Using role type
% Declaring mu:Type
% FOF formula (<kernel.Constant object at 0x10e46c8>, <kernel.DependentProduct object at 0x10e44d0>) of role type named qmltpeq_type
% Using role type
% Declaring qmltpeq:(mu->(mu->(fofType->Prop)))
% FOF formula (<kernel.Constant object at 0x10e3ea8>, <kernel.DependentProduct object at 0x10e3c20>) of role type named meq_prop_type
% Using role type
% Declaring meq_prop:((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) meq_prop) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (W:fofType)=> (((eq Prop) (X W)) (Y W)))) of role definition named meq_prop
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) meq_prop) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (W:fofType)=> (((eq Prop) (X W)) (Y W))))
% Defined: meq_prop:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (W:fofType)=> (((eq Prop) (X W)) (Y W)))
% FOF formula (<kernel.Constant object at 0x10e33b0>, <kernel.DependentProduct object at 0x10e3d88>) of role type named mnot_type
% Using role type
% Declaring mnot:((fofType->Prop)->(fofType->Prop))
% FOF formula (((eq ((fofType->Prop)->(fofType->Prop))) mnot) (fun (Phi:(fofType->Prop)) (W:fofType)=> ((Phi W)->False))) of role definition named mnot
% A new definition: (((eq ((fofType->Prop)->(fofType->Prop))) mnot) (fun (Phi:(fofType->Prop)) (W:fofType)=> ((Phi W)->False)))
% Defined: mnot:=(fun (Phi:(fofType->Prop)) (W:fofType)=> ((Phi W)->False))
% FOF formula (<kernel.Constant object at 0x10e3d88>, <kernel.DependentProduct object at 0x10e33f8>) of role type named mor_type
% Using role type
% Declaring mor:((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) mor) (fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop)) (W:fofType)=> ((or (Phi W)) (Psi W)))) of role definition named mor
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) mor) (fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop)) (W:fofType)=> ((or (Phi W)) (Psi W))))
% Defined: mor:=(fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop)) (W:fofType)=> ((or (Phi W)) (Psi W)))
% FOF formula (<kernel.Constant object at 0x10e33f8>, <kernel.DependentProduct object at 0x10e3200>) of role type named mbox_type
% Using role type
% Declaring mbox:((fofType->(fofType->Prop))->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->(fofType->Prop))->((fofType->Prop)->(fofType->Prop)))) mbox) (fun (R:(fofType->(fofType->Prop))) (Phi:(fofType->Prop)) (W:fofType)=> (forall (V:fofType), ((or (((R W) V)->False)) (Phi V))))) of role definition named mbox
% A new definition: (((eq ((fofType->(fofType->Prop))->((fofType->Prop)->(fofType->Prop)))) mbox) (fun (R:(fofType->(fofType->Prop))) (Phi:(fofType->Prop)) (W:fofType)=> (forall (V:fofType), ((or (((R W) V)->False)) (Phi V)))))
% Defined: mbox:=(fun (R:(fofType->(fofType->Prop))) (Phi:(fofType->Prop)) (W:fofType)=> (forall (V:fofType), ((or (((R W) V)->False)) (Phi V))))
% FOF formula (<kernel.Constant object at 0xbcc998>, <kernel.DependentProduct object at 0x10e3b00>) of role type named mforall_prop_type
% Using role type
% Declaring mforall_prop:(((fofType->Prop)->(fofType->Prop))->(fofType->Prop))
% FOF formula (((eq (((fofType->Prop)->(fofType->Prop))->(fofType->Prop))) mforall_prop) (fun (Phi:((fofType->Prop)->(fofType->Prop))) (W:fofType)=> (forall (P:(fofType->Prop)), ((Phi P) W)))) of role definition named mforall_prop
% A new definition: (((eq (((fofType->Prop)->(fofType->Prop))->(fofType->Prop))) mforall_prop) (fun (Phi:((fofType->Prop)->(fofType->Prop))) (W:fofType)=> (forall (P:(fofType->Prop)), ((Phi P) W))))
% Defined: mforall_prop:=(fun (Phi:((fofType->Prop)->(fofType->Prop))) (W:fofType)=> (forall (P:(fofType->Prop)), ((Phi P) W)))
% FOF formula (<kernel.Constant object at 0xbcc908>, <kernel.DependentProduct object at 0x10e3638>) of role type named mtrue_type
% Using role type
% Declaring mtrue:(fofType->Prop)
% FOF formula (((eq (fofType->Prop)) mtrue) (fun (W:fofType)=> True)) of role definition named mtrue
% A new definition: (((eq (fofType->Prop)) mtrue) (fun (W:fofType)=> True))
% Defined: mtrue:=(fun (W:fofType)=> True)
% FOF formula (<kernel.Constant object at 0xbcc908>, <kernel.DependentProduct object at 0x10e3638>) of role type named mfalse_type
% Using role type
% Declaring mfalse:(fofType->Prop)
% FOF formula (((eq (fofType->Prop)) mfalse) (mnot mtrue)) of role definition named mfalse
% A new definition: (((eq (fofType->Prop)) mfalse) (mnot mtrue))
% Defined: mfalse:=(mnot mtrue)
% FOF formula (<kernel.Constant object at 0xbccfc8>, <kernel.DependentProduct object at 0x10e3fc8>) of role type named mand_type
% Using role type
% Declaring mand:((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) mand) (fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> (mnot ((mor (mnot Phi)) (mnot Psi))))) of role definition named mand
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) mand) (fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> (mnot ((mor (mnot Phi)) (mnot Psi)))))
% Defined: mand:=(fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> (mnot ((mor (mnot Phi)) (mnot Psi))))
% FOF formula (<kernel.Constant object at 0x10e3fc8>, <kernel.DependentProduct object at 0x10e3200>) of role type named mimplies_type
% Using role type
% Declaring mimplies:((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) mimplies) (fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> ((mor (mnot Phi)) Psi))) of role definition named mimplies
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) mimplies) (fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> ((mor (mnot Phi)) Psi)))
% Defined: mimplies:=(fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> ((mor (mnot Phi)) Psi))
% FOF formula (<kernel.Constant object at 0x10e3200>, <kernel.DependentProduct object at 0x10e3f38>) of role type named mimplied_type
% Using role type
% Declaring mimplied:((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) mimplied) (fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> ((mor (mnot Psi)) Phi))) of role definition named mimplied
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) mimplied) (fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> ((mor (mnot Psi)) Phi)))
% Defined: mimplied:=(fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> ((mor (mnot Psi)) Phi))
% FOF formula (<kernel.Constant object at 0x10e3f38>, <kernel.DependentProduct object at 0x10e3b00>) of role type named mequiv_type
% Using role type
% Declaring mequiv:((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) mequiv) (fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> ((mand ((mimplies Phi) Psi)) ((mimplies Psi) Phi)))) of role definition named mequiv
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) mequiv) (fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> ((mand ((mimplies Phi) Psi)) ((mimplies Psi) Phi))))
% Defined: mequiv:=(fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> ((mand ((mimplies Phi) Psi)) ((mimplies Psi) Phi)))
% FOF formula (<kernel.Constant object at 0x10e3b00>, <kernel.DependentProduct object at 0x10e3d88>) of role type named mxor_type
% Using role type
% Declaring mxor:((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) mxor) (fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> (mnot ((mequiv Phi) Psi)))) of role definition named mxor
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) mxor) (fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> (mnot ((mequiv Phi) Psi))))
% Defined: mxor:=(fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> (mnot ((mequiv Phi) Psi)))
% FOF formula (<kernel.Constant object at 0x10e3d88>, <kernel.DependentProduct object at 0x10e3b00>) of role type named mdia_type
% Using role type
% Declaring mdia:((fofType->(fofType->Prop))->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->(fofType->Prop))->((fofType->Prop)->(fofType->Prop)))) mdia) (fun (R:(fofType->(fofType->Prop))) (Phi:(fofType->Prop))=> (mnot ((mbox R) (mnot Phi))))) of role definition named mdia
% A new definition: (((eq ((fofType->(fofType->Prop))->((fofType->Prop)->(fofType->Prop)))) mdia) (fun (R:(fofType->(fofType->Prop))) (Phi:(fofType->Prop))=> (mnot ((mbox R) (mnot Phi)))))
% Defined: mdia:=(fun (R:(fofType->(fofType->Prop))) (Phi:(fofType->Prop))=> (mnot ((mbox R) (mnot Phi))))
% FOF formula (<kernel.Constant object at 0x10e3d88>, <kernel.DependentProduct object at 0xced830>) of role type named exists_in_world_type
% Using role type
% Declaring exists_in_world:(mu->(fofType->Prop))
% FOF formula (forall (V:fofType), ((ex mu) (fun (X:mu)=> ((exists_in_world X) V)))) of role axiom named nonempty_ax
% A new axiom: (forall (V:fofType), ((ex mu) (fun (X:mu)=> ((exists_in_world X) V))))
% FOF formula (<kernel.Constant object at 0x10e3b00>, <kernel.DependentProduct object at 0xced560>) of role type named mforall_ind_type
% Using role type
% Declaring mforall_ind:((mu->(fofType->Prop))->(fofType->Prop))
% FOF formula (((eq ((mu->(fofType->Prop))->(fofType->Prop))) mforall_ind) (fun (Phi:(mu->(fofType->Prop))) (W:fofType)=> (forall (X:mu), (((exists_in_world X) W)->((Phi X) W))))) of role definition named mforall_ind
% A new definition: (((eq ((mu->(fofType->Prop))->(fofType->Prop))) mforall_ind) (fun (Phi:(mu->(fofType->Prop))) (W:fofType)=> (forall (X:mu), (((exists_in_world X) W)->((Phi X) W)))))
% Defined: mforall_ind:=(fun (Phi:(mu->(fofType->Prop))) (W:fofType)=> (forall (X:mu), (((exists_in_world X) W)->((Phi X) W))))
% FOF formula (<kernel.Constant object at 0xced758>, <kernel.DependentProduct object at 0xced3f8>) of role type named mexists_ind_type
% Using role type
% Declaring mexists_ind:((mu->(fofType->Prop))->(fofType->Prop))
% FOF formula (((eq ((mu->(fofType->Prop))->(fofType->Prop))) mexists_ind) (fun (Phi:(mu->(fofType->Prop)))=> (mnot (mforall_ind (fun (X:mu)=> (mnot (Phi X))))))) of role definition named mexists_ind
% A new definition: (((eq ((mu->(fofType->Prop))->(fofType->Prop))) mexists_ind) (fun (Phi:(mu->(fofType->Prop)))=> (mnot (mforall_ind (fun (X:mu)=> (mnot (Phi X)))))))
% Defined: mexists_ind:=(fun (Phi:(mu->(fofType->Prop)))=> (mnot (mforall_ind (fun (X:mu)=> (mnot (Phi X))))))
% FOF formula (<kernel.Constant object at 0xced3f8>, <kernel.DependentProduct object at 0xced680>) of role type named mexists_prop_type
% Using role type
% Declaring mexists_prop:(((fofType->Prop)->(fofType->Prop))->(fofType->Prop))
% FOF formula (((eq (((fofType->Prop)->(fofType->Prop))->(fofType->Prop))) mexists_prop) (fun (Phi:((fofType->Prop)->(fofType->Prop)))=> (mnot (mforall_prop (fun (P:(fofType->Prop))=> (mnot (Phi P))))))) of role definition named mexists_prop
% A new definition: (((eq (((fofType->Prop)->(fofType->Prop))->(fofType->Prop))) mexists_prop) (fun (Phi:((fofType->Prop)->(fofType->Prop)))=> (mnot (mforall_prop (fun (P:(fofType->Prop))=> (mnot (Phi P)))))))
% Defined: mexists_prop:=(fun (Phi:((fofType->Prop)->(fofType->Prop)))=> (mnot (mforall_prop (fun (P:(fofType->Prop))=> (mnot (Phi P))))))
% FOF formula (<kernel.Constant object at 0xced560>, <kernel.DependentProduct object at 0xcedcf8>) of role type named mreflexive_type
% Using role type
% Declaring mreflexive:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) mreflexive) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType), ((R S) S)))) of role definition named mreflexive
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) mreflexive) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType), ((R S) S))))
% Defined: mreflexive:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType), ((R S) S)))
% FOF formula (<kernel.Constant object at 0xcedcf8>, <kernel.DependentProduct object at 0xcedc68>) of role type named msymmetric_type
% Using role type
% Declaring msymmetric:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) msymmetric) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType), (((R S) T)->((R T) S))))) of role definition named msymmetric
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) msymmetric) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType), (((R S) T)->((R T) S)))))
% Defined: msymmetric:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType), (((R S) T)->((R T) S))))
% FOF formula (<kernel.Constant object at 0xcedc68>, <kernel.DependentProduct object at 0xcedd40>) of role type named mserial_type
% Using role type
% Declaring mserial:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) mserial) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType), ((ex fofType) (fun (T:fofType)=> ((R S) T)))))) of role definition named mserial
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) mserial) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType), ((ex fofType) (fun (T:fofType)=> ((R S) T))))))
% Defined: mserial:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType), ((ex fofType) (fun (T:fofType)=> ((R S) T)))))
% FOF formula (<kernel.Constant object at 0xcedd40>, <kernel.DependentProduct object at 0xcedf80>) of role type named mtransitive_type
% Using role type
% Declaring mtransitive:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) mtransitive) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R T) U))->((R S) U))))) of role definition named mtransitive
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) mtransitive) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R T) U))->((R S) U)))))
% Defined: mtransitive:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R T) U))->((R S) U))))
% FOF formula (<kernel.Constant object at 0xcedf80>, <kernel.DependentProduct object at 0xced8c0>) of role type named meuclidean_type
% Using role type
% Declaring meuclidean:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) meuclidean) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->((R T) U))))) of role definition named meuclidean
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) meuclidean) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->((R T) U)))))
% Defined: meuclidean:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->((R T) U))))
% FOF formula (<kernel.Constant object at 0xced8c0>, <kernel.DependentProduct object at 0xceddd0>) of role type named mpartially_functional_type
% Using role type
% Declaring mpartially_functional:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) mpartially_functional) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->(((eq fofType) T) U))))) of role definition named mpartially_functional
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) mpartially_functional) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->(((eq fofType) T) U)))))
% Defined: mpartially_functional:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->(((eq fofType) T) U))))
% FOF formula (<kernel.Constant object at 0xceddd0>, <kernel.DependentProduct object at 0xcedfc8>) of role type named mfunctional_type
% Using role type
% Declaring mfunctional:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) mfunctional) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType), ((ex fofType) (fun (T:fofType)=> ((and ((R S) T)) (forall (U:fofType), (((R S) U)->(((eq fofType) T) U))))))))) of role definition named mfunctional
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) mfunctional) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType), ((ex fofType) (fun (T:fofType)=> ((and ((R S) T)) (forall (U:fofType), (((R S) U)->(((eq fofType) T) U)))))))))
% Defined: mfunctional:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType), ((ex fofType) (fun (T:fofType)=> ((and ((R S) T)) (forall (U:fofType), (((R S) U)->(((eq fofType) T) U))))))))
% FOF formula (<kernel.Constant object at 0xcedfc8>, <kernel.DependentProduct object at 0xcedea8>) of role type named mweakly_dense_type
% Using role type
% Declaring mweakly_dense:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) mweakly_dense) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType), (fofType->(((R S) T)->((ex fofType) (fun (U:fofType)=> ((and ((R S) U)) ((R U) T))))))))) of role definition named mweakly_dense
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) mweakly_dense) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType), (fofType->(((R S) T)->((ex fofType) (fun (U:fofType)=> ((and ((R S) U)) ((R U) T)))))))))
% Defined: mweakly_dense:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType), (fofType->(((R S) T)->((ex fofType) (fun (U:fofType)=> ((and ((R S) U)) ((R U) T))))))))
% FOF formula (<kernel.Constant object at 0xcedea8>, <kernel.DependentProduct object at 0xceda70>) of role type named mweakly_connected_type
% Using role type
% Declaring mweakly_connected:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) mweakly_connected) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->((or ((or ((R T) U)) (((eq fofType) T) U))) ((R U) T)))))) of role definition named mweakly_connected
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) mweakly_connected) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->((or ((or ((R T) U)) (((eq fofType) T) U))) ((R U) T))))))
% Defined: mweakly_connected:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->((or ((or ((R T) U)) (((eq fofType) T) U))) ((R U) T)))))
% FOF formula (<kernel.Constant object at 0xceda70>, <kernel.DependentProduct object at 0xcedfc8>) of role type named mweakly_directed_type
% Using role type
% Declaring mweakly_directed:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) mweakly_directed) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->((ex fofType) (fun (V:fofType)=> ((and ((R T) V)) ((R U) V)))))))) of role definition named mweakly_directed
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) mweakly_directed) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->((ex fofType) (fun (V:fofType)=> ((and ((R T) V)) ((R U) V))))))))
% Defined: mweakly_directed:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->((ex fofType) (fun (V:fofType)=> ((and ((R T) V)) ((R U) V)))))))
% FOF formula (<kernel.Constant object at 0xced3f8>, <kernel.DependentProduct object at 0xcedcf8>) of role type named mvalid_type
% Using role type
% Declaring mvalid:((fofType->Prop)->Prop)
% FOF formula (((eq ((fofType->Prop)->Prop)) mvalid) (fun (Phi:(fofType->Prop))=> (forall (W:fofType), (Phi W)))) of role definition named mvalid
% A new definition: (((eq ((fofType->Prop)->Prop)) mvalid) (fun (Phi:(fofType->Prop))=> (forall (W:fofType), (Phi W))))
% Defined: mvalid:=(fun (Phi:(fofType->Prop))=> (forall (W:fofType), (Phi W)))
% FOF formula (<kernel.Constant object at 0xceda70>, <kernel.DependentProduct object at 0x10d21b8>) of role type named msatisfiable_type
% Using role type
% Declaring msatisfiable:((fofType->Prop)->Prop)
% FOF formula (((eq ((fofType->Prop)->Prop)) msatisfiable) (fun (Phi:(fofType->Prop))=> ((ex fofType) (fun (W:fofType)=> (Phi W))))) of role definition named msatisfiable
% A new definition: (((eq ((fofType->Prop)->Prop)) msatisfiable) (fun (Phi:(fofType->Prop))=> ((ex fofType) (fun (W:fofType)=> (Phi W)))))
% Defined: msatisfiable:=(fun (Phi:(fofType->Prop))=> ((ex fofType) (fun (W:fofType)=> (Phi W))))
% FOF formula (<kernel.Constant object at 0xcedea8>, <kernel.DependentProduct object at 0x10d21b8>) of role type named mcountersatisfiable_type
% Using role type
% Declaring mcountersatisfiable:((fofType->Prop)->Prop)
% FOF formula (((eq ((fofType->Prop)->Prop)) mcountersatisfiable) (fun (Phi:(fofType->Prop))=> ((ex fofType) (fun (W:fofType)=> ((Phi W)->False))))) of role definition named mcountersatisfiable
% A new definition: (((eq ((fofType->Prop)->Prop)) mcountersatisfiable) (fun (Phi:(fofType->Prop))=> ((ex fofType) (fun (W:fofType)=> ((Phi W)->False)))))
% Defined: mcountersatisfiable:=(fun (Phi:(fofType->Prop))=> ((ex fofType) (fun (W:fofType)=> ((Phi W)->False))))
% FOF formula (<kernel.Constant object at 0x10d2098>, <kernel.DependentProduct object at 0x10d2128>) of role type named minvalid_type
% Using role type
% Declaring minvalid:((fofType->Prop)->Prop)
% FOF formula (((eq ((fofType->Prop)->Prop)) minvalid) (fun (Phi:(fofType->Prop))=> (forall (W:fofType), ((Phi W)->False)))) of role definition named minvalid
% A new definition: (((eq ((fofType->Prop)->Prop)) minvalid) (fun (Phi:(fofType->Prop))=> (forall (W:fofType), ((Phi W)->False))))
% Defined: minvalid:=(fun (Phi:(fofType->Prop))=> (forall (W:fofType), ((Phi W)->False)))
% Failed to open /home/cristobal/cocATP/CASC/TPTP/Axioms/LCL013^5.ax, trying next directory
% FOF formula (<kernel.Constant object at 0x10e4b90>, <kernel.DependentProduct object at 0x10e4f38>) of role type named rel_s4_type
% Using role type
% Declaring rel_s4:(fofType->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0x10e4b48>, <kernel.DependentProduct object at 0x10e4cb0>) of role type named mbox_s4_type
% Using role type
% Declaring mbox_s4:((fofType->Prop)->(fofType->Prop))
% FOF formula (((eq ((fofType->Prop)->(fofType->Prop))) mbox_s4) (fun (Phi:(fofType->Prop)) (W:fofType)=> (forall (V:fofType), ((or (((rel_s4 W) V)->False)) (Phi V))))) of role definition named mbox_s4
% A new definition: (((eq ((fofType->Prop)->(fofType->Prop))) mbox_s4) (fun (Phi:(fofType->Prop)) (W:fofType)=> (forall (V:fofType), ((or (((rel_s4 W) V)->False)) (Phi V)))))
% Defined: mbox_s4:=(fun (Phi:(fofType->Prop)) (W:fofType)=> (forall (V:fofType), ((or (((rel_s4 W) V)->False)) (Phi V))))
% FOF formula (<kernel.Constant object at 0x10e4cb0>, <kernel.DependentProduct object at 0x10e4908>) of role type named mdia_s4_type
% Using role type
% Declaring mdia_s4:((fofType->Prop)->(fofType->Prop))
% FOF formula (((eq ((fofType->Prop)->(fofType->Prop))) mdia_s4) (fun (Phi:(fofType->Prop))=> (mnot (mbox_s4 (mnot Phi))))) of role definition named mdia_s4
% A new definition: (((eq ((fofType->Prop)->(fofType->Prop))) mdia_s4) (fun (Phi:(fofType->Prop))=> (mnot (mbox_s4 (mnot Phi)))))
% Defined: mdia_s4:=(fun (Phi:(fofType->Prop))=> (mnot (mbox_s4 (mnot Phi))))
% FOF formula (mreflexive rel_s4) of role axiom named a1
% A new axiom: (mreflexive rel_s4)
% FOF formula (mtransitive rel_s4) of role axiom named a2
% A new axiom: (mtransitive rel_s4)
% Failed to open /home/cristobal/cocATP/CASC/TPTP/Axioms/LCL015^1.ax, trying next directory
% FOF formula (forall (X:mu) (V:fofType) (W:fofType), (((and ((exists_in_world X) V)) ((rel_s4 V) W))->((exists_in_world X) W))) of role axiom named cumulative_ax
% A new axiom: (forall (X:mu) (V:fofType) (W:fofType), (((and ((exists_in_world X) V)) ((rel_s4 V) W))->((exists_in_world X) W)))
% FOF formula (<kernel.Constant object at 0xd08950>, <kernel.DependentProduct object at 0xbd60e0>) of role type named substitution_of_equivalents_type
% Using role type
% Declaring substitution_of_equivalents:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x1144098>, <kernel.DependentProduct object at 0xbd60e0>) of role type named kn1_type
% Using role type
% Declaring kn1:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x10c5488>, <kernel.DependentProduct object at 0xbd6d40>) of role type named kn2_type
% Using role type
% Declaring kn2:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0xf9bfc8>, <kernel.DependentProduct object at 0xbd0cb0>) of role type named kn3_type
% Using role type
% Declaring kn3:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0xf9bfc8>, <kernel.DependentProduct object at 0xbd0cf8>) of role type named cn1_type
% Using role type
% Declaring cn1:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0xbd60e0>, <kernel.DependentProduct object at 0x10e49e0>) of role type named cn2_type
% Using role type
% Declaring cn2:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0xbd60e0>, <kernel.DependentProduct object at 0x10e4b48>) of role type named cn3_type
% Using role type
% Declaring cn3:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0xbd0cb0>, <kernel.DependentProduct object at 0x10e4d40>) of role type named r1_type
% Using role type
% Declaring r1:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0xbd0cb0>, <kernel.DependentProduct object at 0x10e4d88>) of role type named r2_type
% Using role type
% Declaring r2:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0xd086c8>, <kernel.DependentProduct object at 0x10e4cf8>) of role type named r3_type
% Using role type
% Declaring r3:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0xd08368>, <kernel.DependentProduct object at 0x10e4998>) of role type named r4_type
% Using role type
% Declaring r4:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0xd08950>, <kernel.DependentProduct object at 0x10e4dd0>) of role type named r5_type
% Using role type
% Declaring r5:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0xd08368>, <kernel.DependentProduct object at 0x10e4560>) of role type named op_and_type
% Using role type
% Declaring op_and:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0xd08368>, <kernel.DependentProduct object at 0x10e4518>) of role type named op_implies_or_type
% Using role type
% Declaring op_implies_or:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x10e4b48>, <kernel.DependentProduct object at 0x10e48c0>) of role type named op_or_type
% Using role type
% Declaring op_or:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x10e4d88>, <kernel.DependentProduct object at 0x10e4878>) of role type named op_implies_and_type
% Using role type
% Declaring op_implies_and:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x10e4cf8>, <kernel.DependentProduct object at 0x10e40e0>) of role type named op_equiv_type
% Using role type
% Declaring op_equiv:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x10e4518>, <kernel.DependentProduct object at 0x10e4098>) of role type named modus_ponens_type
% Using role type
% Declaring modus_ponens:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x10e48c0>, <kernel.DependentProduct object at 0x10e4680>) of role type named modus_tollens_type
% Using role type
% Declaring modus_tollens:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x10e4878>, <kernel.DependentProduct object at 0x10e4638>) of role type named implies_1_type
% Using role type
% Declaring implies_1:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x10e40e0>, <kernel.DependentProduct object at 0x10e4ab8>) of role type named implies_2_type
% Using role type
% Declaring implies_2:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x10e4098>, <kernel.DependentProduct object at 0x10e4a70>) of role type named implies_3_type
% Using role type
% Declaring implies_3:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x10e4680>, <kernel.DependentProduct object at 0x10e4a28>) of role type named and_1_type
% Using role type
% Declaring and_1:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x10e4638>, <kernel.DependentProduct object at 0x10e4b00>) of role type named and_2_type
% Using role type
% Declaring and_2:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x10e4ab8>, <kernel.DependentProduct object at 0x10e4248>) of role type named and_3_type
% Using role type
% Declaring and_3:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x10e4a70>, <kernel.DependentProduct object at 0x10e4fc8>) of role type named or_1_type
% Using role type
% Declaring or_1:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x10e4a28>, <kernel.DependentProduct object at 0x10e4050>) of role type named or_2_type
% Using role type
% Declaring or_2:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x10e4b00>, <kernel.DependentProduct object at 0x10e47a0>) of role type named or_3_type
% Using role type
% Declaring or_3:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x10e4248>, <kernel.DependentProduct object at 0x10e4f80>) of role type named equivalence_1_type
% Using role type
% Declaring equivalence_1:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x10e4fc8>, <kernel.DependentProduct object at 0x10e44d0>) of role type named equivalence_2_type
% Using role type
% Declaring equivalence_2:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x10e4050>, <kernel.DependentProduct object at 0x10e4908>) of role type named equivalence_3_type
% Using role type
% Declaring equivalence_3:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x10e47a0>, <kernel.DependentProduct object at 0x10e4fc8>) of role type named is_a_theorem_type
% Using role type
% Declaring is_a_theorem:(mu->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0x10e4f80>, <kernel.DependentProduct object at 0x10e49e0>) of role type named or_type
% Using role type
% Declaring _TPTP_or:(mu->(mu->mu))
% FOF formula (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((_TPTP_or V2) V1)) V)) of role axiom named existence_of_or_ax
% A new axiom: (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((_TPTP_or V2) V1)) V))
% FOF formula (<kernel.Constant object at 0x10e49e0>, <kernel.DependentProduct object at 0x10e4908>) of role type named implies_type
% Using role type
% Declaring implies:(mu->(mu->mu))
% FOF formula (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((implies V2) V1)) V)) of role axiom named existence_of_implies_ax
% A new axiom: (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((implies V2) V1)) V))
% FOF formula (<kernel.Constant object at 0x10e49e0>, <kernel.DependentProduct object at 0x10e46c8>) of role type named and_type
% Using role type
% Declaring _TPTP_and:(mu->(mu->mu))
% FOF formula (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((_TPTP_and V2) V1)) V)) of role axiom named existence_of_and_ax
% A new axiom: (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((_TPTP_and V2) V1)) V))
% FOF formula (<kernel.Constant object at 0x10e49e0>, <kernel.DependentProduct object at 0x10e4f38>) of role type named not_type
% Using role type
% Declaring _TPTP_not:(mu->mu)
% FOF formula (forall (V:fofType) (V1:mu), ((exists_in_world (_TPTP_not V1)) V)) of role axiom named existence_of_not_ax
% A new axiom: (forall (V:fofType) (V1:mu), ((exists_in_world (_TPTP_not V1)) V))
% FOF formula (<kernel.Constant object at 0x10e4f38>, <kernel.DependentProduct object at 0x10e4b90>) of role type named equiv_type
% Using role type
% Declaring equiv:(mu->(mu->mu))
% FOF formula (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((equiv V2) V1)) V)) of role axiom named existence_of_equiv_ax
% A new axiom: (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((equiv V2) V1)) V))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq X) X)))))) of role axiom named reflexivity
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq X) X))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq X) Y))) (mbox_s4 ((qmltpeq Y) X))))))))))) of role axiom named symmetry
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq X) Y))) (mbox_s4 ((qmltpeq Y) X)))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq X) Y))) (mbox_s4 ((qmltpeq Y) Z)))) (mbox_s4 ((qmltpeq X) Z)))))))))))))) of role axiom named transitivity
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq X) Y))) (mbox_s4 ((qmltpeq Y) Z)))) (mbox_s4 ((qmltpeq X) Z))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((_TPTP_and A) C)) ((_TPTP_and B) C))))))))))))))) of role axiom named and_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((_TPTP_and A) C)) ((_TPTP_and B) C)))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((_TPTP_and C) A)) ((_TPTP_and C) B))))))))))))))) of role axiom named and_substitution_2
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((_TPTP_and C) A)) ((_TPTP_and C) B)))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((equiv A) C)) ((equiv B) C))))))))))))))) of role axiom named equiv_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((equiv A) C)) ((equiv B) C)))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((equiv C) A)) ((equiv C) B))))))))))))))) of role axiom named equiv_substitution_2
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((equiv C) A)) ((equiv C) B)))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((implies A) C)) ((implies B) C))))))))))))))) of role axiom named implies_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((implies A) C)) ((implies B) C)))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((implies C) A)) ((implies C) B))))))))))))))) of role axiom named implies_substitution_2
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((implies C) A)) ((implies C) B)))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (_TPTP_not A)) (_TPTP_not B)))))))))))) of role axiom named not_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (_TPTP_not A)) (_TPTP_not B))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((_TPTP_or A) C)) ((_TPTP_or B) C))))))))))))))) of role axiom named or_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((_TPTP_or A) C)) ((_TPTP_or B) C)))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((_TPTP_or C) A)) ((_TPTP_or C) B))))))))))))))) of role axiom named or_substitution_2
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((_TPTP_or C) A)) ((_TPTP_or C) B)))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 (is_a_theorem A)))) (mbox_s4 (is_a_theorem B))))))))))) of role axiom named is_a_theorem_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 (is_a_theorem A)))) (mbox_s4 (is_a_theorem B)))))))))))
% FOF formula (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 modus_ponens)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (is_a_theorem X))) (mbox_s4 (is_a_theorem ((implies X) Y))))) (mbox_s4 (is_a_theorem Y))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (is_a_theorem X))) (mbox_s4 (is_a_theorem ((implies X) Y))))) (mbox_s4 (is_a_theorem Y))))))))))) (mbox_s4 modus_ponens))))) of role axiom named modus_ponens
% A new axiom: (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 modus_ponens)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (is_a_theorem X))) (mbox_s4 (is_a_theorem ((implies X) Y))))) (mbox_s4 (is_a_theorem Y))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (is_a_theorem X))) (mbox_s4 (is_a_theorem ((implies X) Y))))) (mbox_s4 (is_a_theorem Y))))))))))) (mbox_s4 modus_ponens)))))
% FOF formula (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 substitution_of_equivalents)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 (is_a_theorem ((equiv X) Y)))) (mbox_s4 ((qmltpeq X) Y))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 (is_a_theorem ((equiv X) Y)))) (mbox_s4 ((qmltpeq X) Y))))))))))) (mbox_s4 substitution_of_equivalents))))) of role axiom named substitution_of_equivalents
% A new axiom: (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 substitution_of_equivalents)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 (is_a_theorem ((equiv X) Y)))) (mbox_s4 ((qmltpeq X) Y))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 (is_a_theorem ((equiv X) Y)))) (mbox_s4 ((qmltpeq X) Y))))))))))) (mbox_s4 substitution_of_equivalents)))))
% FOF formula (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 modus_tollens)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies (_TPTP_not Y)) (_TPTP_not X))) ((implies X) Y))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies (_TPTP_not Y)) (_TPTP_not X))) ((implies X) Y))))))))))) (mbox_s4 modus_tollens))))) of role axiom named modus_tollens
% A new axiom: (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 modus_tollens)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies (_TPTP_not Y)) (_TPTP_not X))) ((implies X) Y))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies (_TPTP_not Y)) (_TPTP_not X))) ((implies X) Y))))))))))) (mbox_s4 modus_tollens)))))
% FOF formula (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 implies_1)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies X) ((implies Y) X))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies X) ((implies Y) X))))))))))) (mbox_s4 implies_1))))) of role axiom named implies_1
% A new axiom: (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 implies_1)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies X) ((implies Y) X))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies X) ((implies Y) X))))))))))) (mbox_s4 implies_1)))))
% FOF formula (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 implies_2)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies X) ((implies X) Y))) ((implies X) Y))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies X) ((implies X) Y))) ((implies X) Y))))))))))) (mbox_s4 implies_2))))) of role axiom named implies_2
% A new axiom: (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 implies_2)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies X) ((implies X) Y))) ((implies X) Y))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies X) ((implies X) Y))) ((implies X) Y))))))))))) (mbox_s4 implies_2)))))
% FOF formula (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 implies_3)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies X) Y)) ((implies ((implies Y) Z)) ((implies X) Z))))))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies X) Y)) ((implies ((implies Y) Z)) ((implies X) Z))))))))))))))) (mbox_s4 implies_3))))) of role axiom named implies_3
% A new axiom: (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 implies_3)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies X) Y)) ((implies ((implies Y) Z)) ((implies X) Z))))))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies X) Y)) ((implies ((implies Y) Z)) ((implies X) Z))))))))))))))) (mbox_s4 implies_3)))))
% FOF formula (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 and_1)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_and X) Y)) X)))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_and X) Y)) X)))))))))) (mbox_s4 and_1))))) of role axiom named and_1
% A new axiom: (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 and_1)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_and X) Y)) X)))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_and X) Y)) X)))))))))) (mbox_s4 and_1)))))
% FOF formula (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 and_2)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_and X) Y)) Y)))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_and X) Y)) Y)))))))))) (mbox_s4 and_2))))) of role axiom named and_2
% A new axiom: (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 and_2)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_and X) Y)) Y)))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_and X) Y)) Y)))))))))) (mbox_s4 and_2)))))
% FOF formula (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 and_3)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies X) ((implies Y) ((_TPTP_and X) Y)))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies X) ((implies Y) ((_TPTP_and X) Y)))))))))))) (mbox_s4 and_3))))) of role axiom named and_3
% A new axiom: (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 and_3)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies X) ((implies Y) ((_TPTP_and X) Y)))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies X) ((implies Y) ((_TPTP_and X) Y)))))))))))) (mbox_s4 and_3)))))
% FOF formula (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 or_1)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies X) ((_TPTP_or X) Y))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies X) ((_TPTP_or X) Y))))))))))) (mbox_s4 or_1))))) of role axiom named or_1
% A new axiom: (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 or_1)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies X) ((_TPTP_or X) Y))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies X) ((_TPTP_or X) Y))))))))))) (mbox_s4 or_1)))))
% FOF formula (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 or_2)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies Y) ((_TPTP_or X) Y))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies Y) ((_TPTP_or X) Y))))))))))) (mbox_s4 or_2))))) of role axiom named or_2
% A new axiom: (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 or_2)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies Y) ((_TPTP_or X) Y))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies Y) ((_TPTP_or X) Y))))))))))) (mbox_s4 or_2)))))
% FOF formula (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 or_3)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies X) Z)) ((implies ((implies Y) Z)) ((implies ((_TPTP_or X) Y)) Z))))))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies X) Z)) ((implies ((implies Y) Z)) ((implies ((_TPTP_or X) Y)) Z))))))))))))))) (mbox_s4 or_3))))) of role axiom named or_3
% A new axiom: (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 or_3)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies X) Z)) ((implies ((implies Y) Z)) ((implies ((_TPTP_or X) Y)) Z))))))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies X) Z)) ((implies ((implies Y) Z)) ((implies ((_TPTP_or X) Y)) Z))))))))))))))) (mbox_s4 or_3)))))
% FOF formula (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 equivalence_1)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((equiv X) Y)) ((implies X) Y))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((equiv X) Y)) ((implies X) Y))))))))))) (mbox_s4 equivalence_1))))) of role axiom named equivalence_1
% A new axiom: (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 equivalence_1)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((equiv X) Y)) ((implies X) Y))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((equiv X) Y)) ((implies X) Y))))))))))) (mbox_s4 equivalence_1)))))
% FOF formula (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 equivalence_2)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((equiv X) Y)) ((implies Y) X))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((equiv X) Y)) ((implies Y) X))))))))))) (mbox_s4 equivalence_2))))) of role axiom named equivalence_2
% A new axiom: (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 equivalence_2)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((equiv X) Y)) ((implies Y) X))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((equiv X) Y)) ((implies Y) X))))))))))) (mbox_s4 equivalence_2)))))
% FOF formula (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 equivalence_3)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies X) Y)) ((implies ((implies Y) X)) ((equiv X) Y)))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies X) Y)) ((implies ((implies Y) X)) ((equiv X) Y)))))))))))) (mbox_s4 equivalence_3))))) of role axiom named equivalence_3
% A new axiom: (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 equivalence_3)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies X) Y)) ((implies ((implies Y) X)) ((equiv X) Y)))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies X) Y)) ((implies ((implies Y) X)) ((equiv X) Y)))))))))))) (mbox_s4 equivalence_3)))))
% FOF formula (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 kn1)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (is_a_theorem ((implies P) ((_TPTP_and P) P)))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (is_a_theorem ((implies P) ((_TPTP_and P) P)))))))) (mbox_s4 kn1))))) of role axiom named kn1
% A new axiom: (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 kn1)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (is_a_theorem ((implies P) ((_TPTP_and P) P)))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (is_a_theorem ((implies P) ((_TPTP_and P) P)))))))) (mbox_s4 kn1)))))
% FOF formula (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 kn2)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_and P) Q)) P)))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_and P) Q)) P)))))))))) (mbox_s4 kn2))))) of role axiom named kn2
% A new axiom: (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 kn2)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_and P) Q)) P)))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_and P) Q)) P)))))))))) (mbox_s4 kn2)))))
% FOF formula (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 kn3)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (mforall_ind (fun (R:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies P) Q)) ((implies (_TPTP_not ((_TPTP_and Q) R))) (_TPTP_not ((_TPTP_and R) P)))))))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (mforall_ind (fun (R:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies P) Q)) ((implies (_TPTP_not ((_TPTP_and Q) R))) (_TPTP_not ((_TPTP_and R) P)))))))))))))))) (mbox_s4 kn3))))) of role axiom named kn3
% A new axiom: (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 kn3)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (mforall_ind (fun (R:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies P) Q)) ((implies (_TPTP_not ((_TPTP_and Q) R))) (_TPTP_not ((_TPTP_and R) P)))))))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (mforall_ind (fun (R:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies P) Q)) ((implies (_TPTP_not ((_TPTP_and Q) R))) (_TPTP_not ((_TPTP_and R) P)))))))))))))))) (mbox_s4 kn3)))))
% FOF formula (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 cn1)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (mforall_ind (fun (R:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies P) Q)) ((implies ((implies Q) R)) ((implies P) R))))))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (mforall_ind (fun (R:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies P) Q)) ((implies ((implies Q) R)) ((implies P) R))))))))))))))) (mbox_s4 cn1))))) of role axiom named cn1
% A new axiom: (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 cn1)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (mforall_ind (fun (R:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies P) Q)) ((implies ((implies Q) R)) ((implies P) R))))))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (mforall_ind (fun (R:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies P) Q)) ((implies ((implies Q) R)) ((implies P) R))))))))))))))) (mbox_s4 cn1)))))
% FOF formula (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 cn2)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (is_a_theorem ((implies P) ((implies (_TPTP_not P)) Q))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (is_a_theorem ((implies P) ((implies (_TPTP_not P)) Q))))))))))) (mbox_s4 cn2))))) of role axiom named cn2
% A new axiom: (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 cn2)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (is_a_theorem ((implies P) ((implies (_TPTP_not P)) Q))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (is_a_theorem ((implies P) ((implies (_TPTP_not P)) Q))))))))))) (mbox_s4 cn2)))))
% FOF formula (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 cn3)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies (_TPTP_not P)) P)) P))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies (_TPTP_not P)) P)) P))))))) (mbox_s4 cn3))))) of role axiom named cn3
% A new axiom: (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 cn3)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies (_TPTP_not P)) P)) P))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies (_TPTP_not P)) P)) P))))))) (mbox_s4 cn3)))))
% FOF formula (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 r1)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_or P) P)) P))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_or P) P)) P))))))) (mbox_s4 r1))))) of role axiom named r1
% A new axiom: (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 r1)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_or P) P)) P))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_or P) P)) P))))))) (mbox_s4 r1)))))
% FOF formula (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 r2)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (is_a_theorem ((implies Q) ((_TPTP_or P) Q))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (is_a_theorem ((implies Q) ((_TPTP_or P) Q))))))))))) (mbox_s4 r2))))) of role axiom named r2
% A new axiom: (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 r2)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (is_a_theorem ((implies Q) ((_TPTP_or P) Q))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (is_a_theorem ((implies Q) ((_TPTP_or P) Q))))))))))) (mbox_s4 r2)))))
% FOF formula (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 r3)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_or P) Q)) ((_TPTP_or Q) P))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_or P) Q)) ((_TPTP_or Q) P))))))))))) (mbox_s4 r3))))) of role axiom named r3
% A new axiom: (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 r3)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_or P) Q)) ((_TPTP_or Q) P))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_or P) Q)) ((_TPTP_or Q) P))))))))))) (mbox_s4 r3)))))
% FOF formula (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 r4)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (mforall_ind (fun (R:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_or P) ((_TPTP_or Q) R))) ((_TPTP_or Q) ((_TPTP_or P) R))))))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (mforall_ind (fun (R:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_or P) ((_TPTP_or Q) R))) ((_TPTP_or Q) ((_TPTP_or P) R))))))))))))))) (mbox_s4 r4))))) of role axiom named r4
% A new axiom: (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 r4)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (mforall_ind (fun (R:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_or P) ((_TPTP_or Q) R))) ((_TPTP_or Q) ((_TPTP_or P) R))))))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (mforall_ind (fun (R:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_or P) ((_TPTP_or Q) R))) ((_TPTP_or Q) ((_TPTP_or P) R))))))))))))))) (mbox_s4 r4)))))
% FOF formula (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 r5)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (mforall_ind (fun (R:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies Q) R)) ((implies ((_TPTP_or P) Q)) ((_TPTP_or P) R))))))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (mforall_ind (fun (R:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies Q) R)) ((implies ((_TPTP_or P) Q)) ((_TPTP_or P) R))))))))))))))) (mbox_s4 r5))))) of role axiom named r5
% A new axiom: (mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 r5)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (mforall_ind (fun (R:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies Q) R)) ((implies ((_TPTP_or P) Q)) ((_TPTP_or P) R))))))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (mforall_ind (fun (R:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies Q) R)) ((implies ((_TPTP_or P) Q)) ((_TPTP_or P) R))))))))))))))) (mbox_s4 r5)))))
% FOF formula (mvalid (mbox_s4 ((mimplies (mbox_s4 op_or)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((qmltpeq ((_TPTP_or X) Y)) (_TPTP_not ((_TPTP_and (_TPTP_not X)) (_TPTP_not Y)))))))))))))) of role axiom named op_or
% A new axiom: (mvalid (mbox_s4 ((mimplies (mbox_s4 op_or)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((qmltpeq ((_TPTP_or X) Y)) (_TPTP_not ((_TPTP_and (_TPTP_not X)) (_TPTP_not Y))))))))))))))
% FOF formula (mvalid (mbox_s4 ((mimplies (mbox_s4 op_and)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((qmltpeq ((_TPTP_and X) Y)) (_TPTP_not ((_TPTP_or (_TPTP_not X)) (_TPTP_not Y)))))))))))))) of role axiom named op_and
% A new axiom: (mvalid (mbox_s4 ((mimplies (mbox_s4 op_and)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((qmltpeq ((_TPTP_and X) Y)) (_TPTP_not ((_TPTP_or (_TPTP_not X)) (_TPTP_not Y))))))))))))))
% FOF formula (mvalid (mbox_s4 ((mimplies (mbox_s4 op_implies_and)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((qmltpeq ((implies X) Y)) (_TPTP_not ((_TPTP_and X) (_TPTP_not Y)))))))))))))) of role axiom named op_implies_and
% A new axiom: (mvalid (mbox_s4 ((mimplies (mbox_s4 op_implies_and)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((qmltpeq ((implies X) Y)) (_TPTP_not ((_TPTP_and X) (_TPTP_not Y))))))))))))))
% FOF formula (mvalid (mbox_s4 ((mimplies (mbox_s4 op_implies_or)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((qmltpeq ((implies X) Y)) ((_TPTP_or (_TPTP_not X)) Y)))))))))))) of role axiom named op_implies_or
% A new axiom: (mvalid (mbox_s4 ((mimplies (mbox_s4 op_implies_or)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((qmltpeq ((implies X) Y)) ((_TPTP_or (_TPTP_not X)) Y))))))))))))
% FOF formula (mvalid (mbox_s4 ((mimplies (mbox_s4 op_equiv)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((qmltpeq ((equiv X) Y)) ((_TPTP_and ((implies X) Y)) ((implies Y) X))))))))))))) of role axiom named op_equiv
% A new axiom: (mvalid (mbox_s4 ((mimplies (mbox_s4 op_equiv)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((qmltpeq ((equiv X) Y)) ((_TPTP_and ((implies X) Y)) ((implies Y) X)))))))))))))
% FOF formula (mvalid (mbox_s4 op_or)) of role axiom named hilbert_op_or
% A new axiom: (mvalid (mbox_s4 op_or))
% FOF formula (mvalid (mbox_s4 op_implies_and)) of role axiom named hilbert_op_implies_and
% A new axiom: (mvalid (mbox_s4 op_implies_and))
% FOF formula (mvalid (mbox_s4 op_equiv)) of role axiom named hilbert_op_equiv
% A new axiom: (mvalid (mbox_s4 op_equiv))
% FOF formula (mvalid (mbox_s4 modus_ponens)) of role axiom named hilbert_modus_ponens
% A new axiom: (mvalid (mbox_s4 modus_ponens))
% FOF formula (mvalid (mbox_s4 modus_tollens)) of role axiom named hilbert_modus_tollens
% A new axiom: (mvalid (mbox_s4 modus_tollens))
% FOF formula (mvalid (mbox_s4 implies_1)) of role axiom named hilbert_implies_1
% A new axiom: (mvalid (mbox_s4 implies_1))
% FOF formula (mvalid (mbox_s4 implies_2)) of role axiom named hilbert_implies_2
% A new axiom: (mvalid (mbox_s4 implies_2))
% FOF formula (mvalid (mbox_s4 implies_3)) of role axiom named hilbert_implies_3
% A new axiom: (mvalid (mbox_s4 implies_3))
% FOF formula (mvalid (mbox_s4 and_1)) of role axiom named hilbert_and_1
% A new axiom: (mvalid (mbox_s4 and_1))
% FOF formula (mvalid (mbox_s4 and_2)) of role axiom named hilbert_and_2
% A new axiom: (mvalid (mbox_s4 and_2))
% FOF formula (mvalid (mbox_s4 and_3)) of role axiom named hilbert_and_3
% A new axiom: (mvalid (mbox_s4 and_3))
% FOF formula (mvalid (mbox_s4 or_1)) of role axiom named hilbert_or_1
% A new axiom: (mvalid (mbox_s4 or_1))
% FOF formula (mvalid (mbox_s4 or_2)) of role axiom named hilbert_or_2
% A new axiom: (mvalid (mbox_s4 or_2))
% FOF formula (mvalid (mbox_s4 or_3)) of role axiom named hilbert_or_3
% A new axiom: (mvalid (mbox_s4 or_3))
% FOF formula (mvalid (mbox_s4 equivalence_1)) of role axiom named hilbert_equivalence_1
% A new axiom: (mvalid (mbox_s4 equivalence_1))
% FOF formula (mvalid (mbox_s4 equivalence_2)) of role axiom named hilbert_equivalence_2
% A new axiom: (mvalid (mbox_s4 equivalence_2))
% FOF formula (mvalid (mbox_s4 equivalence_3)) of role axiom named hilbert_equivalence_3
% A new axiom: (mvalid (mbox_s4 equivalence_3))
% FOF formula (mvalid (mbox_s4 ((mimplies ((mand (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (is_a_theorem ((equiv X) X))))))) ((mand (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 (is_a_theorem ((equiv X) Y)))) (mbox_s4 (is_a_theorem ((equiv (_TPTP_not X)) (_TPTP_not Y))))))))))))) ((mand (mbox_s4 (mforall_ind (fun (X1:mu)=> (mbox_s4 (mforall_ind (fun (X2:mu)=> (mbox_s4 (mforall_ind (fun (Y1:mu)=> (mbox_s4 (mforall_ind (fun (Y2:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (is_a_theorem ((equiv X1) X2)))) (mbox_s4 (is_a_theorem ((equiv Y1) Y2))))) (mbox_s4 (is_a_theorem ((equiv ((_TPTP_and X1) Y1)) ((_TPTP_and X2) Y2))))))))))))))))))) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (is_a_theorem X))) (mbox_s4 (is_a_theorem ((equiv X) Y))))) (mbox_s4 (is_a_theorem Y)))))))))))))) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 (is_a_theorem ((equiv X) Y)))) (mbox_s4 ((qmltpeq X) Y))))))))))))) of role axiom named make_subs_of_equiv
% A new axiom: (mvalid (mbox_s4 ((mimplies ((mand (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (is_a_theorem ((equiv X) X))))))) ((mand (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 (is_a_theorem ((equiv X) Y)))) (mbox_s4 (is_a_theorem ((equiv (_TPTP_not X)) (_TPTP_not Y))))))))))))) ((mand (mbox_s4 (mforall_ind (fun (X1:mu)=> (mbox_s4 (mforall_ind (fun (X2:mu)=> (mbox_s4 (mforall_ind (fun (Y1:mu)=> (mbox_s4 (mforall_ind (fun (Y2:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (is_a_theorem ((equiv X1) X2)))) (mbox_s4 (is_a_theorem ((equiv Y1) Y2))))) (mbox_s4 (is_a_theorem ((equiv ((_TPTP_and X1) Y1)) ((_TPTP_and X2) Y2))))))))))))))))))) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (is_a_theorem X))) (mbox_s4 (is_a_theorem ((equiv X) Y))))) (mbox_s4 (is_a_theorem Y)))))))))))))) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 (is_a_theorem ((equiv X) Y)))) (mbox_s4 ((qmltpeq X) Y)))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 (is_a_theorem ((equiv X) Y)))) (mbox_s4 ((qmltpeq X) Y))))))))))) of role conjecture named subs_of_equiv
% Conjecture to prove = (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 (is_a_theorem ((equiv X) Y)))) (mbox_s4 ((qmltpeq X) Y))))))))))):Prop
% Parameter mu_DUMMY:mu.
% Parameter fofType_DUMMY:fofType.
% We need to prove ['(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 (is_a_theorem ((equiv X) Y)))) (mbox_s4 ((qmltpeq X) Y)))))))))))']
% Parameter mu:Type.
% Parameter fofType:Type.
% Parameter qmltpeq:(mu->(mu->(fofType->Prop))).
% Definition meq_prop:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (W:fofType)=> (((eq Prop) (X W)) (Y W))):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))).
% Definition mnot:=(fun (Phi:(fofType->Prop)) (W:fofType)=> ((Phi W)->False)):((fofType->Prop)->(fofType->Prop)).
% Definition mor:=(fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop)) (W:fofType)=> ((or (Phi W)) (Psi W))):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))).
% Definition mbox:=(fun (R:(fofType->(fofType->Prop))) (Phi:(fofType->Prop)) (W:fofType)=> (forall (V:fofType), ((or (((R W) V)->False)) (Phi V)))):((fofType->(fofType->Prop))->((fofType->Prop)->(fofType->Prop))).
% Definition mforall_prop:=(fun (Phi:((fofType->Prop)->(fofType->Prop))) (W:fofType)=> (forall (P:(fofType->Prop)), ((Phi P) W))):(((fofType->Prop)->(fofType->Prop))->(fofType->Prop)).
% Definition mtrue:=(fun (W:fofType)=> True):(fofType->Prop).
% Definition mfalse:=(mnot mtrue):(fofType->Prop).
% Definition mand:=(fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> (mnot ((mor (mnot Phi)) (mnot Psi)))):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))).
% Definition mimplies:=(fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> ((mor (mnot Phi)) Psi)):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))).
% Definition mimplied:=(fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> ((mor (mnot Psi)) Phi)):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))).
% Definition mequiv:=(fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> ((mand ((mimplies Phi) Psi)) ((mimplies Psi) Phi))):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))).
% Definition mxor:=(fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> (mnot ((mequiv Phi) Psi))):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))).
% Definition mdia:=(fun (R:(fofType->(fofType->Prop))) (Phi:(fofType->Prop))=> (mnot ((mbox R) (mnot Phi)))):((fofType->(fofType->Prop))->((fofType->Prop)->(fofType->Prop))).
% Parameter exists_in_world:(mu->(fofType->Prop)).
% Axiom nonempty_ax:(forall (V:fofType), ((ex mu) (fun (X:mu)=> ((exists_in_world X) V)))).
% Definition mforall_ind:=(fun (Phi:(mu->(fofType->Prop))) (W:fofType)=> (forall (X:mu), (((exists_in_world X) W)->((Phi X) W)))):((mu->(fofType->Prop))->(fofType->Prop)).
% Definition mexists_ind:=(fun (Phi:(mu->(fofType->Prop)))=> (mnot (mforall_ind (fun (X:mu)=> (mnot (Phi X)))))):((mu->(fofType->Prop))->(fofType->Prop)).
% Definition mexists_prop:=(fun (Phi:((fofType->Prop)->(fofType->Prop)))=> (mnot (mforall_prop (fun (P:(fofType->Prop))=> (mnot (Phi P)))))):(((fofType->Prop)->(fofType->Prop))->(fofType->Prop)).
% Definition mreflexive:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType), ((R S) S))):((fofType->(fofType->Prop))->Prop).
% Definition msymmetric:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType), (((R S) T)->((R T) S)))):((fofType->(fofType->Prop))->Prop).
% Definition mserial:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType), ((ex fofType) (fun (T:fofType)=> ((R S) T))))):((fofType->(fofType->Prop))->Prop).
% Definition mtransitive:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R T) U))->((R S) U)))):((fofType->(fofType->Prop))->Prop).
% Definition meuclidean:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->((R T) U)))):((fofType->(fofType->Prop))->Prop).
% Definition mpartially_functional:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->(((eq fofType) T) U)))):((fofType->(fofType->Prop))->Prop).
% Definition mfunctional:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType), ((ex fofType) (fun (T:fofType)=> ((and ((R S) T)) (forall (U:fofType), (((R S) U)->(((eq fofType) T) U)))))))):((fofType->(fofType->Prop))->Prop).
% Definition mweakly_dense:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType), (fofType->(((R S) T)->((ex fofType) (fun (U:fofType)=> ((and ((R S) U)) ((R U) T)))))))):((fofType->(fofType->Prop))->Prop).
% Definition mweakly_connected:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->((or ((or ((R T) U)) (((eq fofType) T) U))) ((R U) T))))):((fofType->(fofType->Prop))->Prop).
% Definition mweakly_directed:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->((ex fofType) (fun (V:fofType)=> ((and ((R T) V)) ((R U) V))))))):((fofType->(fofType->Prop))->Prop).
% Definition mvalid:=(fun (Phi:(fofType->Prop))=> (forall (W:fofType), (Phi W))):((fofType->Prop)->Prop).
% Definition msatisfiable:=(fun (Phi:(fofType->Prop))=> ((ex fofType) (fun (W:fofType)=> (Phi W)))):((fofType->Prop)->Prop).
% Definition mcountersatisfiable:=(fun (Phi:(fofType->Prop))=> ((ex fofType) (fun (W:fofType)=> ((Phi W)->False)))):((fofType->Prop)->Prop).
% Definition minvalid:=(fun (Phi:(fofType->Prop))=> (forall (W:fofType), ((Phi W)->False))):((fofType->Prop)->Prop).
% Parameter rel_s4:(fofType->(fofType->Prop)).
% Definition mbox_s4:=(fun (Phi:(fofType->Prop)) (W:fofType)=> (forall (V:fofType), ((or (((rel_s4 W) V)->False)) (Phi V)))):((fofType->Prop)->(fofType->Prop)).
% Definition mdia_s4:=(fun (Phi:(fofType->Prop))=> (mnot (mbox_s4 (mnot Phi)))):((fofType->Prop)->(fofType->Prop)).
% Axiom a1:(mreflexive rel_s4).
% Axiom a2:(mtransitive rel_s4).
% Axiom cumulative_ax:(forall (X:mu) (V:fofType) (W:fofType), (((and ((exists_in_world X) V)) ((rel_s4 V) W))->((exists_in_world X) W))).
% Parameter substitution_of_equivalents:(fofType->Prop).
% Parameter kn1:(fofType->Prop).
% Parameter kn2:(fofType->Prop).
% Parameter kn3:(fofType->Prop).
% Parameter cn1:(fofType->Prop).
% Parameter cn2:(fofType->Prop).
% Parameter cn3:(fofType->Prop).
% Parameter r1:(fofType->Prop).
% Parameter r2:(fofType->Prop).
% Parameter r3:(fofType->Prop).
% Parameter r4:(fofType->Prop).
% Parameter r5:(fofType->Prop).
% Parameter op_and:(fofType->Prop).
% Parameter op_implies_or:(fofType->Prop).
% Parameter op_or:(fofType->Prop).
% Parameter op_implies_and:(fofType->Prop).
% Parameter op_equiv:(fofType->Prop).
% Parameter modus_ponens:(fofType->Prop).
% Parameter modus_tollens:(fofType->Prop).
% Parameter implies_1:(fofType->Prop).
% Parameter implies_2:(fofType->Prop).
% Parameter implies_3:(fofType->Prop).
% Parameter and_1:(fofType->Prop).
% Parameter and_2:(fofType->Prop).
% Parameter and_3:(fofType->Prop).
% Parameter or_1:(fofType->Prop).
% Parameter or_2:(fofType->Prop).
% Parameter or_3:(fofType->Prop).
% Parameter equivalence_1:(fofType->Prop).
% Parameter equivalence_2:(fofType->Prop).
% Parameter equivalence_3:(fofType->Prop).
% Parameter is_a_theorem:(mu->(fofType->Prop)).
% Parameter _TPTP_or:(mu->(mu->mu)).
% Axiom existence_of_or_ax:(forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((_TPTP_or V2) V1)) V)).
% Parameter implies:(mu->(mu->mu)).
% Axiom existence_of_implies_ax:(forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((implies V2) V1)) V)).
% Parameter _TPTP_and:(mu->(mu->mu)).
% Axiom existence_of_and_ax:(forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((_TPTP_and V2) V1)) V)).
% Parameter _TPTP_not:(mu->mu).
% Axiom existence_of_not_ax:(forall (V:fofType) (V1:mu), ((exists_in_world (_TPTP_not V1)) V)).
% Parameter equiv:(mu->(mu->mu)).
% Axiom existence_of_equiv_ax:(forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((equiv V2) V1)) V)).
% Axiom reflexivity:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq X) X)))))).
% Axiom symmetry:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq X) Y))) (mbox_s4 ((qmltpeq Y) X))))))))))).
% Axiom transitivity:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq X) Y))) (mbox_s4 ((qmltpeq Y) Z)))) (mbox_s4 ((qmltpeq X) Z)))))))))))))).
% Axiom and_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((_TPTP_and A) C)) ((_TPTP_and B) C))))))))))))))).
% Axiom and_substitution_2:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((_TPTP_and C) A)) ((_TPTP_and C) B))))))))))))))).
% Axiom equiv_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((equiv A) C)) ((equiv B) C))))))))))))))).
% Axiom equiv_substitution_2:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((equiv C) A)) ((equiv C) B))))))))))))))).
% Axiom implies_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((implies A) C)) ((implies B) C))))))))))))))).
% Axiom implies_substitution_2:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((implies C) A)) ((implies C) B))))))))))))))).
% Axiom not_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (_TPTP_not A)) (_TPTP_not B)))))))))))).
% Axiom or_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((_TPTP_or A) C)) ((_TPTP_or B) C))))))))))))))).
% Axiom or_substitution_2:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((_TPTP_or C) A)) ((_TPTP_or C) B))))))))))))))).
% Axiom is_a_theorem_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 (is_a_theorem A)))) (mbox_s4 (is_a_theorem B))))))))))).
% Axiom modus_ponens_TPTP_next:(mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 modus_ponens)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (is_a_theorem X))) (mbox_s4 (is_a_theorem ((implies X) Y))))) (mbox_s4 (is_a_theorem Y))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (is_a_theorem X))) (mbox_s4 (is_a_theorem ((implies X) Y))))) (mbox_s4 (is_a_theorem Y))))))))))) (mbox_s4 modus_ponens))))).
% Axiom substitution_of_equivalents_TPTP_next:(mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 substitution_of_equivalents)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 (is_a_theorem ((equiv X) Y)))) (mbox_s4 ((qmltpeq X) Y))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 (is_a_theorem ((equiv X) Y)))) (mbox_s4 ((qmltpeq X) Y))))))))))) (mbox_s4 substitution_of_equivalents))))).
% Axiom modus_tollens_TPTP_next:(mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 modus_tollens)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies (_TPTP_not Y)) (_TPTP_not X))) ((implies X) Y))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies (_TPTP_not Y)) (_TPTP_not X))) ((implies X) Y))))))))))) (mbox_s4 modus_tollens))))).
% Axiom implies_1_TPTP_next:(mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 implies_1)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies X) ((implies Y) X))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies X) ((implies Y) X))))))))))) (mbox_s4 implies_1))))).
% Axiom implies_2_TPTP_next:(mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 implies_2)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies X) ((implies X) Y))) ((implies X) Y))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies X) ((implies X) Y))) ((implies X) Y))))))))))) (mbox_s4 implies_2))))).
% Axiom implies_3_TPTP_next:(mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 implies_3)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies X) Y)) ((implies ((implies Y) Z)) ((implies X) Z))))))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies X) Y)) ((implies ((implies Y) Z)) ((implies X) Z))))))))))))))) (mbox_s4 implies_3))))).
% Axiom and_1_TPTP_next:(mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 and_1)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_and X) Y)) X)))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_and X) Y)) X)))))))))) (mbox_s4 and_1))))).
% Axiom and_2_TPTP_next:(mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 and_2)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_and X) Y)) Y)))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_and X) Y)) Y)))))))))) (mbox_s4 and_2))))).
% Axiom and_3_TPTP_next:(mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 and_3)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies X) ((implies Y) ((_TPTP_and X) Y)))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies X) ((implies Y) ((_TPTP_and X) Y)))))))))))) (mbox_s4 and_3))))).
% Axiom or_1_TPTP_next:(mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 or_1)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies X) ((_TPTP_or X) Y))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies X) ((_TPTP_or X) Y))))))))))) (mbox_s4 or_1))))).
% Axiom or_2_TPTP_next:(mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 or_2)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies Y) ((_TPTP_or X) Y))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies Y) ((_TPTP_or X) Y))))))))))) (mbox_s4 or_2))))).
% Axiom or_3_TPTP_next:(mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 or_3)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies X) Z)) ((implies ((implies Y) Z)) ((implies ((_TPTP_or X) Y)) Z))))))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies X) Z)) ((implies ((implies Y) Z)) ((implies ((_TPTP_or X) Y)) Z))))))))))))))) (mbox_s4 or_3))))).
% Axiom equivalence_1_TPTP_next:(mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 equivalence_1)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((equiv X) Y)) ((implies X) Y))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((equiv X) Y)) ((implies X) Y))))))))))) (mbox_s4 equivalence_1))))).
% Axiom equivalence_2_TPTP_next:(mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 equivalence_2)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((equiv X) Y)) ((implies Y) X))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((equiv X) Y)) ((implies Y) X))))))))))) (mbox_s4 equivalence_2))))).
% Axiom equivalence_3_TPTP_next:(mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 equivalence_3)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies X) Y)) ((implies ((implies Y) X)) ((equiv X) Y)))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies X) Y)) ((implies ((implies Y) X)) ((equiv X) Y)))))))))))) (mbox_s4 equivalence_3))))).
% Axiom kn1_TPTP_next:(mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 kn1)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (is_a_theorem ((implies P) ((_TPTP_and P) P)))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (is_a_theorem ((implies P) ((_TPTP_and P) P)))))))) (mbox_s4 kn1))))).
% Axiom kn2_TPTP_next:(mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 kn2)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_and P) Q)) P)))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_and P) Q)) P)))))))))) (mbox_s4 kn2))))).
% Axiom kn3_TPTP_next:(mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 kn3)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (mforall_ind (fun (R:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies P) Q)) ((implies (_TPTP_not ((_TPTP_and Q) R))) (_TPTP_not ((_TPTP_and R) P)))))))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (mforall_ind (fun (R:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies P) Q)) ((implies (_TPTP_not ((_TPTP_and Q) R))) (_TPTP_not ((_TPTP_and R) P)))))))))))))))) (mbox_s4 kn3))))).
% Axiom cn1_TPTP_next:(mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 cn1)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (mforall_ind (fun (R:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies P) Q)) ((implies ((implies Q) R)) ((implies P) R))))))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (mforall_ind (fun (R:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies P) Q)) ((implies ((implies Q) R)) ((implies P) R))))))))))))))) (mbox_s4 cn1))))).
% Axiom cn2_TPTP_next:(mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 cn2)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (is_a_theorem ((implies P) ((implies (_TPTP_not P)) Q))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (is_a_theorem ((implies P) ((implies (_TPTP_not P)) Q))))))))))) (mbox_s4 cn2))))).
% Axiom cn3_TPTP_next:(mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 cn3)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies (_TPTP_not P)) P)) P))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies (_TPTP_not P)) P)) P))))))) (mbox_s4 cn3))))).
% Axiom r1_TPTP_next:(mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 r1)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_or P) P)) P))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_or P) P)) P))))))) (mbox_s4 r1))))).
% Axiom r2_TPTP_next:(mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 r2)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (is_a_theorem ((implies Q) ((_TPTP_or P) Q))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (is_a_theorem ((implies Q) ((_TPTP_or P) Q))))))))))) (mbox_s4 r2))))).
% Axiom r3_TPTP_next:(mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 r3)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_or P) Q)) ((_TPTP_or Q) P))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_or P) Q)) ((_TPTP_or Q) P))))))))))) (mbox_s4 r3))))).
% Axiom r4_TPTP_next:(mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 r4)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (mforall_ind (fun (R:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_or P) ((_TPTP_or Q) R))) ((_TPTP_or Q) ((_TPTP_or P) R))))))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (mforall_ind (fun (R:mu)=> (mbox_s4 (is_a_theorem ((implies ((_TPTP_or P) ((_TPTP_or Q) R))) ((_TPTP_or Q) ((_TPTP_or P) R))))))))))))))) (mbox_s4 r4))))).
% Axiom r5_TPTP_next:(mvalid ((mand (mbox_s4 ((mimplies (mbox_s4 r5)) (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (mforall_ind (fun (R:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies Q) R)) ((implies ((_TPTP_or P) Q)) ((_TPTP_or P) R))))))))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (P:mu)=> (mbox_s4 (mforall_ind (fun (Q:mu)=> (mbox_s4 (mforall_ind (fun (R:mu)=> (mbox_s4 (is_a_theorem ((implies ((implies Q) R)) ((implies ((_TPTP_or P) Q)) ((_TPTP_or P) R))))))))))))))) (mbox_s4 r5))))).
% Axiom op_or_TPTP_next:(mvalid (mbox_s4 ((mimplies (mbox_s4 op_or)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((qmltpeq ((_TPTP_or X) Y)) (_TPTP_not ((_TPTP_and (_TPTP_not X)) (_TPTP_not Y)))))))))))))).
% Axiom op_and_TPTP_next:(mvalid (mbox_s4 ((mimplies (mbox_s4 op_and)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((qmltpeq ((_TPTP_and X) Y)) (_TPTP_not ((_TPTP_or (_TPTP_not X)) (_TPTP_not Y)))))))))))))).
% Axiom op_implies_and_TPTP_next:(mvalid (mbox_s4 ((mimplies (mbox_s4 op_implies_and)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((qmltpeq ((implies X) Y)) (_TPTP_not ((_TPTP_and X) (_TPTP_not Y)))))))))))))).
% Axiom op_implies_or_TPTP_next:(mvalid (mbox_s4 ((mimplies (mbox_s4 op_implies_or)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((qmltpeq ((implies X) Y)) ((_TPTP_or (_TPTP_not X)) Y)))))))))))).
% Axiom op_equiv_TPTP_next:(mvalid (mbox_s4 ((mimplies (mbox_s4 op_equiv)) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((qmltpeq ((equiv X) Y)) ((_TPTP_and ((implies X) Y)) ((implies Y) X))))))))))))).
% Axiom hilbert_op_or:(mvalid (mbox_s4 op_or)).
% Axiom hilbert_op_implies_and:(mvalid (mbox_s4 op_implies_and)).
% Axiom hilbert_op_equiv:(mvalid (mbox_s4 op_equiv)).
% Axiom hilbert_modus_ponens:(mvalid (mbox_s4 modus_ponens)).
% Axiom hilbert_modus_tollens:(mvalid (mbox_s4 modus_tollens)).
% Axiom hilbert_implies_1:(mvalid (mbox_s4 implies_1)).
% Axiom hilbert_implies_2:(mvalid (mbox_s4 implies_2)).
% Axiom hilbert_implies_3:(mvalid (mbox_s4 implies_3)).
% Axiom hilbert_and_1:(mvalid (mbox_s4 and_1)).
% Axiom hilbert_and_2:(mvalid (mbox_s4 and_2)).
% Axiom hilbert_and_3:(mvalid (mbox_s4 and_3)).
% Axiom hilbert_or_1:(mvalid (mbox_s4 or_1)).
% Axiom hilbert_or_2:(mvalid (mbox_s4 or_2)).
% Axiom hilbert_or_3:(mvalid (mbox_s4 or_3)).
% Axiom hilbert_equivalence_1:(mvalid (mbox_s4 equivalence_1)).
% Axiom hilbert_equivalence_2:(mvalid (mbox_s4 equivalence_2)).
% Axiom hilbert_equivalence_3:(mvalid (mbox_s4 equivalence_3)).
% Axiom make_subs_of_equiv:(mvalid (mbox_s4 ((mimplies ((mand (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (is_a_theorem ((equiv X) X))))))) ((mand (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 (is_a_theorem ((equiv X) Y)))) (mbox_s4 (is_a_theorem ((equiv (_TPTP_not X)) (_TPTP_not Y))))))))))))) ((mand (mbox_s4 (mforall_ind (fun (X1:mu)=> (mbox_s4 (mforall_ind (fun (X2:mu)=> (mbox_s4 (mforall_ind (fun (Y1:mu)=> (mbox_s4 (mforall_ind (fun (Y2:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (is_a_theorem ((equiv X1) X2)))) (mbox_s4 (is_a_theorem ((equiv Y1) Y2))))) (mbox_s4 (is_a_theorem ((equiv ((_TPTP_and X1) Y1)) ((_TPTP_and X2) Y2))))))))))))))))))) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (is_a_theorem X))) (mbox_s4 (is_a_theorem ((equiv X) Y))))) (mbox_s4 (is_a_theorem Y)))))))))))))) (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 (is_a_theorem ((equiv X) Y)))) (mbox_s4 ((qmltpeq X) Y))))))))))))).
% Trying to prove (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 (is_a_theorem ((equiv X) Y)))) (
% EOF
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