%------------------------------------------------------------------------------ % File : cocATP---0.2.0 % Problem : SWV061^7 : TPTP v6.1.0. Released v5.5.0. % Transfm : none % Format : tptp:raw % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % Computer : n097.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 13:35:20 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 : SWV061^7 : TPTP v6.1.0. Released v5.5.0. % % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % % Computer : n097.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 07:54:56 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 0x1d594d0>, <kernel.Type object at 0x1d59fc8>) of role type named mu_type % Using role type % Declaring mu:Type % FOF formula (<kernel.Constant object at 0x1d59908>, <kernel.DependentProduct object at 0x1d594d0>) of role type named qmltpeq_type % Using role type % Declaring qmltpeq:(mu->(mu->(fofType->Prop))) % FOF formula (<kernel.Constant object at 0x1d593f8>, <kernel.DependentProduct object at 0x1d59cf8>) 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 0x1d59c68>, <kernel.DependentProduct object at 0x1d59e18>) 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 0x1d59e18>, <kernel.DependentProduct object at 0x1d59f80>) 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 0x1c21cb0>, <kernel.DependentProduct object at 0x1d59e18>) 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 0x1c21cb0>, <kernel.DependentProduct object at 0x1d597a0>) 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 0x1f3acb0>, <kernel.DependentProduct object at 0x1d59c68>) 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 0x1f3a0e0>, <kernel.DependentProduct object at 0x1d59c68>) 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 0x1f3a0e0>, <kernel.DependentProduct object at 0x1d59f38>) 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 0x1f3a8c0>, <kernel.DependentProduct object at 0x1d596c8>) 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 0x1d596c8>, <kernel.DependentProduct object at 0x1d599e0>) 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 0x1d599e0>, <kernel.DependentProduct object at 0x1d59f80>) 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 0x1d59f80>, <kernel.DependentProduct object at 0x1d59cb0>) 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 0x1d59cb0>, <kernel.DependentProduct object at 0x1d59d40>) 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 0x1d59cb0>, <kernel.DependentProduct object at 0x1d3e710>) 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 0x1d59e18>, <kernel.DependentProduct object at 0x1d3e5f0>) 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 0x1d3e950>, <kernel.DependentProduct object at 0x1d3e488>) 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 0x1d3e488>, <kernel.DependentProduct object at 0x1d3e6c8>) 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 0x1d3e5f0>, <kernel.DependentProduct object at 0x1d3ec68>) 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 0x1d3ec68>, <kernel.DependentProduct object at 0x1d3ebd8>) 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 0x1d3ebd8>, <kernel.DependentProduct object at 0x1d3ecb0>) 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 0x1d3ecb0>, <kernel.DependentProduct object at 0x1d3e878>) 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 0x1d3e878>, <kernel.DependentProduct object at 0x1d3e4d0>) 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 0x1d3e4d0>, <kernel.DependentProduct object at 0x1d3ee60>) 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 0x1d3ee60>, <kernel.DependentProduct object at 0x1d3e7a0>) 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 0x1d3e7a0>, <kernel.DependentProduct object at 0x1d3e5a8>) 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 0x1d3e5a8>, <kernel.DependentProduct object at 0x1d3ec68>) 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 0x1d3ec68>, <kernel.DependentProduct object at 0x1d3e950>) 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 0x1d3e488>, <kernel.DependentProduct object at 0x1d3eb90>) 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 0x1d3ec68>, <kernel.DependentProduct object at 0x1f29050>) 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 0x1d3e4d0>, <kernel.DependentProduct object at 0x1f29128>) 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 0x1d3eb90>, <kernel.DependentProduct object at 0x1f292d8>) 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 0x1c27cb0>, <kernel.DependentProduct object at 0x1c21bd8>) of role type named rel_s4_type % Using role type % Declaring rel_s4:(fofType->(fofType->Prop)) % FOF formula (<kernel.Constant object at 0x1c27cb0>, <kernel.DependentProduct object at 0x1f3ae60>) 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 0x2193950>, <kernel.DependentProduct object at 0x1f3a0e0>) 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 0x1f3c998>, <kernel.DependentProduct object at 0x1f1c128>) of role type named true_type % Using role type % Declaring true:(fofType->Prop) % FOF formula (<kernel.Constant object at 0x1f3c560>, <kernel.DependentProduct object at 0x1f1c248>) of role type named lt_type % Using role type % Declaring lt:(mu->(mu->(fofType->Prop))) % FOF formula (<kernel.Constant object at 0x1f3c998>, <kernel.DependentProduct object at 0x1f1c290>) of role type named geq_type % Using role type % Declaring geq:(mu->(mu->(fofType->Prop))) % FOF formula (<kernel.Constant object at 0x1f3c560>, <kernel.DependentProduct object at 0x1f1c170>) of role type named gt_type % Using role type % Declaring gt:(mu->(mu->(fofType->Prop))) % FOF formula (<kernel.Constant object at 0x1f3c050>, <kernel.DependentProduct object at 0x1f1c3f8>) of role type named leq_type % Using role type % Declaring leq:(mu->(mu->(fofType->Prop))) % FOF formula (<kernel.Constant object at 0x1f3c050>, <kernel.DependentProduct object at 0x1f1c4d0>) of role type named uniform_int_rnd_type % Using role type % Declaring uniform_int_rnd:(mu->(mu->mu)) % FOF formula (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((uniform_int_rnd V2) V1)) V)) of role axiom named existence_of_uniform_int_rnd_ax % A new axiom: (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((uniform_int_rnd V2) V1)) V)) % FOF formula (<kernel.Constant object at 0x1f1c050>, <kernel.DependentProduct object at 0x1f1c0e0>) of role type named tptp_const_array1_type % Using role type % Declaring tptp_const_array1:(mu->(mu->mu)) % FOF formula (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((tptp_const_array1 V2) V1)) V)) of role axiom named existence_of_tptp_const_array1_ax % A new axiom: (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((tptp_const_array1 V2) V1)) V)) % FOF formula (<kernel.Constant object at 0x1f1c320>, <kernel.DependentProduct object at 0x1f1c4d0>) of role type named dim_type % Using role type % Declaring dim:(mu->(mu->mu)) % FOF formula (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((dim V2) V1)) V)) of role axiom named existence_of_dim_ax % A new axiom: (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((dim V2) V1)) V)) % FOF formula (<kernel.Constant object at 0x1f1c200>, <kernel.DependentProduct object at 0x1f1c908>) of role type named tptp_const_array2_type % Using role type % Declaring tptp_const_array2:(mu->(mu->(mu->mu))) % FOF formula (forall (V:fofType) (V3:mu) (V2:mu) (V1:mu), ((exists_in_world (((tptp_const_array2 V3) V2) V1)) V)) of role axiom named existence_of_tptp_const_array2_ax % A new axiom: (forall (V:fofType) (V3:mu) (V2:mu) (V1:mu), ((exists_in_world (((tptp_const_array2 V3) V2) V1)) V)) % FOF formula (<kernel.Constant object at 0x1c21d88>, <kernel.DependentProduct object at 0x1f1c200>) of role type named inv_type % Using role type % Declaring inv:(mu->mu) % FOF formula (forall (V:fofType) (V1:mu), ((exists_in_world (inv V1)) V)) of role axiom named existence_of_inv_ax % A new axiom: (forall (V:fofType) (V1:mu), ((exists_in_world (inv V1)) V)) % FOF formula (<kernel.Constant object at 0x1c21440>, <kernel.DependentProduct object at 0x1f1c5f0>) of role type named tptp_msub_type % Using role type % Declaring tptp_msub:(mu->(mu->mu)) % FOF formula (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((tptp_msub V2) V1)) V)) of role axiom named existence_of_tptp_msub_ax % A new axiom: (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((tptp_msub V2) V1)) V)) % FOF formula (<kernel.Constant object at 0x1c21f38>, <kernel.DependentProduct object at 0x1f1c3b0>) of role type named trans_type % Using role type % Declaring trans:(mu->mu) % FOF formula (forall (V:fofType) (V1:mu), ((exists_in_world (trans V1)) V)) of role axiom named existence_of_trans_ax % A new axiom: (forall (V:fofType) (V1:mu), ((exists_in_world (trans V1)) V)) % FOF formula (<kernel.Constant object at 0x1f1c3b0>, <kernel.DependentProduct object at 0x1f1c248>) of role type named tptp_mmul_type % Using role type % Declaring tptp_mmul:(mu->(mu->mu)) % FOF formula (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((tptp_mmul V2) V1)) V)) of role axiom named existence_of_tptp_mmul_ax % A new axiom: (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((tptp_mmul V2) V1)) V)) % FOF formula (<kernel.Constant object at 0x1f1c3b0>, <kernel.DependentProduct object at 0x1f1c4d0>) of role type named tptp_madd_type % Using role type % Declaring tptp_madd:(mu->(mu->mu)) % FOF formula (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((tptp_madd V2) V1)) V)) of role axiom named existence_of_tptp_madd_ax % A new axiom: (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((tptp_madd V2) V1)) V)) % FOF formula (<kernel.Constant object at 0x1f1c3b0>, <kernel.DependentProduct object at 0x1f1c908>) of role type named sum_type % Using role type % Declaring sum:(mu->(mu->(mu->mu))) % FOF formula (forall (V:fofType) (V3:mu) (V2:mu) (V1:mu), ((exists_in_world (((sum V3) V2) V1)) V)) of role axiom named existence_of_sum_ax % A new axiom: (forall (V:fofType) (V3:mu) (V2:mu) (V1:mu), ((exists_in_world (((sum V3) V2) V1)) V)) % FOF formula (<kernel.Constant object at 0x1f1c4d0>, <kernel.Constant object at 0x1f1c3b0>) of role type named tptp_float_0_0_type % Using role type % Declaring tptp_float_0_0:mu % FOF formula (forall (V:fofType), ((exists_in_world tptp_float_0_0) V)) of role axiom named existence_of_tptp_float_0_0_ax % A new axiom: (forall (V:fofType), ((exists_in_world tptp_float_0_0) V)) % FOF formula (<kernel.Constant object at 0x1f1c488>, <kernel.DependentProduct object at 0x1f3acb0>) of role type named plus_type % Using role type % Declaring plus:(mu->(mu->mu)) % FOF formula (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((plus V2) V1)) V)) of role axiom named existence_of_plus_ax % A new axiom: (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((plus V2) V1)) V)) % FOF formula (<kernel.Constant object at 0x1f1c488>, <kernel.DependentProduct object at 0x1f3acb0>) of role type named pred_type % Using role type % Declaring pred:(mu->mu) % FOF formula (forall (V:fofType) (V1:mu), ((exists_in_world (pred V1)) V)) of role axiom named existence_of_pred_ax % A new axiom: (forall (V:fofType) (V1:mu), ((exists_in_world (pred V1)) V)) % FOF formula (<kernel.Constant object at 0x1f1c488>, <kernel.DependentProduct object at 0x1f3acb0>) of role type named minus_type % Using role type % Declaring minus:(mu->(mu->mu)) % FOF formula (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((minus V2) V1)) V)) of role axiom named existence_of_minus_ax % A new axiom: (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((minus V2) V1)) V)) % FOF formula (<kernel.Constant object at 0x1f3a1b8>, <kernel.DependentProduct object at 0x1d59878>) of role type named tptp_update3_type % Using role type % Declaring tptp_update3:(mu->(mu->(mu->(mu->mu)))) % FOF formula (forall (V:fofType) (V4:mu) (V3:mu) (V2:mu) (V1:mu), ((exists_in_world ((((tptp_update3 V4) V3) V2) V1)) V)) of role axiom named existence_of_tptp_update3_ax % A new axiom: (forall (V:fofType) (V4:mu) (V3:mu) (V2:mu) (V1:mu), ((exists_in_world ((((tptp_update3 V4) V3) V2) V1)) V)) % FOF formula (<kernel.Constant object at 0x1f3acb0>, <kernel.DependentProduct object at 0x1d59440>) of role type named a_select3_type % Using role type % Declaring a_select3:(mu->(mu->(mu->mu))) % FOF formula (forall (V:fofType) (V3:mu) (V2:mu) (V1:mu), ((exists_in_world (((a_select3 V3) V2) V1)) V)) of role axiom named existence_of_a_select3_ax % A new axiom: (forall (V:fofType) (V3:mu) (V2:mu) (V1:mu), ((exists_in_world (((a_select3 V3) V2) V1)) V)) % FOF formula (<kernel.Constant object at 0x1f3acb0>, <kernel.DependentProduct object at 0x1d595f0>) of role type named tptp_update2_type % Using role type % Declaring tptp_update2:(mu->(mu->(mu->mu))) % FOF formula (forall (V:fofType) (V3:mu) (V2:mu) (V1:mu), ((exists_in_world (((tptp_update2 V3) V2) V1)) V)) of role axiom named existence_of_tptp_update2_ax % A new axiom: (forall (V:fofType) (V3:mu) (V2:mu) (V1:mu), ((exists_in_world (((tptp_update2 V3) V2) V1)) V)) % FOF formula (<kernel.Constant object at 0x1d59320>, <kernel.DependentProduct object at 0x1d59d88>) of role type named a_select2_type % Using role type % Declaring a_select2:(mu->(mu->mu)) % FOF formula (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((a_select2 V2) V1)) V)) of role axiom named existence_of_a_select2_ax % A new axiom: (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((a_select2 V2) V1)) V)) % FOF formula (<kernel.Constant object at 0x1d59680>, <kernel.Constant object at 0x1d597a0>) of role type named use_type % Using role type % Declaring use:mu % FOF formula (forall (V:fofType), ((exists_in_world use) V)) of role axiom named existence_of_use_ax % A new axiom: (forall (V:fofType), ((exists_in_world use) V)) % FOF formula (<kernel.Constant object at 0x1d59fc8>, <kernel.Constant object at 0x1d59878>) of role type named def_type % Using role type % Declaring def:mu % FOF formula (forall (V:fofType), ((exists_in_world def) V)) of role axiom named existence_of_def_ax % A new axiom: (forall (V:fofType), ((exists_in_world def) V)) % FOF formula (<kernel.Constant object at 0x1d59e60>, <kernel.Constant object at 0x1d59518>) of role type named loopcounter_type % Using role type % Declaring loopcounter:mu % FOF formula (forall (V:fofType), ((exists_in_world loopcounter) V)) of role axiom named existence_of_loopcounter_ax % A new axiom: (forall (V:fofType), ((exists_in_world loopcounter) V)) % FOF formula (<kernel.Constant object at 0x1d593f8>, <kernel.Constant object at 0x1d59cf8>) of role type named tptp_minus_1_type % Using role type % Declaring tptp_minus_1:mu % FOF formula (forall (V:fofType), ((exists_in_world tptp_minus_1) V)) of role axiom named existence_of_tptp_minus_1_ax % A new axiom: (forall (V:fofType), ((exists_in_world tptp_minus_1) V)) % FOF formula (<kernel.Constant object at 0x1d59440>, <kernel.Constant object at 0x1d59710>) of role type named n4_type % Using role type % Declaring n4:mu % FOF formula (forall (V:fofType), ((exists_in_world n4) V)) of role axiom named existence_of_n4_ax % A new axiom: (forall (V:fofType), ((exists_in_world n4) V)) % FOF formula (<kernel.Constant object at 0x1d59d88>, <kernel.Constant object at 0x1d59488>) of role type named n5_type % Using role type % Declaring n5:mu % FOF formula (forall (V:fofType), ((exists_in_world n5) V)) of role axiom named existence_of_n5_ax % A new axiom: (forall (V:fofType), ((exists_in_world n5) V)) % FOF formula (<kernel.Constant object at 0x1d59f38>, <kernel.Constant object at 0x1d59b90>) of role type named n1_type % Using role type % Declaring n1:mu % FOF formula (forall (V:fofType), ((exists_in_world n1) V)) of role axiom named existence_of_n1_ax % A new axiom: (forall (V:fofType), ((exists_in_world n1) V)) % FOF formula (<kernel.Constant object at 0x1d597e8>, <kernel.Constant object at 0x1d59c20>) of role type named n2_type % Using role type % Declaring n2:mu % FOF formula (forall (V:fofType), ((exists_in_world n2) V)) of role axiom named existence_of_n2_ax % A new axiom: (forall (V:fofType), ((exists_in_world n2) V)) % FOF formula (<kernel.Constant object at 0x1d599e0>, <kernel.Constant object at 0x1d598c0>) of role type named n3_type % Using role type % Declaring n3:mu % FOF formula (forall (V:fofType), ((exists_in_world n3) V)) of role axiom named existence_of_n3_ax % A new axiom: (forall (V:fofType), ((exists_in_world n3) V)) % FOF formula (<kernel.Constant object at 0x1d59d40>, <kernel.Constant object at 0x1d59368>) of role type named n0_type % Using role type % Declaring n0:mu % FOF formula (forall (V:fofType), ((exists_in_world n0) V)) of role axiom named existence_of_n0_ax % A new axiom: (forall (V:fofType), ((exists_in_world n0) V)) % FOF formula (<kernel.Constant object at 0x1d596c8>, <kernel.DependentProduct object at 0x1d59320>) of role type named succ_type % Using role type % Declaring succ:(mu->mu) % FOF formula (forall (V:fofType) (V1:mu), ((exists_in_world (succ V1)) V)) of role axiom named existence_of_succ_ax % A new axiom: (forall (V:fofType) (V1:mu), ((exists_in_world (succ 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 ((a_select2 A) C)) ((a_select2 B) C))))))))))))))) of role axiom named a_select2_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 ((a_select2 A) C)) ((a_select2 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 ((a_select2 C) A)) ((a_select2 C) B))))))))))))))) of role axiom named a_select2_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 ((a_select2 C) A)) ((a_select2 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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((a_select3 A) C) D)) (((a_select3 B) C) D)))))))))))))))))) of role axiom named a_select3_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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((a_select3 A) C) D)) (((a_select3 B) C) D)))))))))))))))))) % 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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((a_select3 C) A) D)) (((a_select3 C) B) D)))))))))))))))))) of role axiom named a_select3_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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((a_select3 C) A) D)) (((a_select3 C) B) D)))))))))))))))))) % 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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((a_select3 C) D) A)) (((a_select3 C) D) B)))))))))))))))))) of role axiom named a_select3_substitution_3 % 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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((a_select3 C) D) A)) (((a_select3 C) D) 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 ((dim A) C)) ((dim B) C))))))))))))))) of role axiom named dim_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 ((dim A) C)) ((dim 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 ((dim C) A)) ((dim C) B))))))))))))))) of role axiom named dim_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 ((dim C) A)) ((dim 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 (inv A)) (inv B)))))))))))) of role axiom named inv_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 (inv A)) (inv 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 ((minus A) C)) ((minus B) C))))))))))))))) of role axiom named minus_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 ((minus A) C)) ((minus 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 ((minus C) A)) ((minus C) B))))))))))))))) of role axiom named minus_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 ((minus C) A)) ((minus 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 ((plus A) C)) ((plus B) C))))))))))))))) of role axiom named plus_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 ((plus A) C)) ((plus 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 ((plus C) A)) ((plus C) B))))))))))))))) of role axiom named plus_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 ((plus C) A)) ((plus 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 (pred A)) (pred B)))))))))))) of role axiom named pred_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 (pred A)) (pred 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 (succ A)) (succ B)))))))))))) of role axiom named succ_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 (succ A)) (succ 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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((sum A) C) D)) (((sum B) C) D)))))))))))))))))) of role axiom named sum_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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((sum A) C) D)) (((sum B) C) D)))))))))))))))))) % 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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((sum C) A) D)) (((sum C) B) D)))))))))))))))))) of role axiom named sum_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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((sum C) A) D)) (((sum C) B) D)))))))))))))))))) % 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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((sum C) D) A)) (((sum C) D) B)))))))))))))))))) of role axiom named sum_substitution_3 % 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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((sum C) D) A)) (((sum C) D) 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_const_array1 A) C)) ((tptp_const_array1 B) C))))))))))))))) of role axiom named tptp_const_array1_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_const_array1 A) C)) ((tptp_const_array1 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_const_array1 C) A)) ((tptp_const_array1 C) B))))))))))))))) of role axiom named tptp_const_array1_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_const_array1 C) A)) ((tptp_const_array1 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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((tptp_const_array2 A) C) D)) (((tptp_const_array2 B) C) D)))))))))))))))))) of role axiom named tptp_const_array2_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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((tptp_const_array2 A) C) D)) (((tptp_const_array2 B) C) D)))))))))))))))))) % 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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((tptp_const_array2 C) A) D)) (((tptp_const_array2 C) B) D)))))))))))))))))) of role axiom named tptp_const_array2_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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((tptp_const_array2 C) A) D)) (((tptp_const_array2 C) B) D)))))))))))))))))) % 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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((tptp_const_array2 C) D) A)) (((tptp_const_array2 C) D) B)))))))))))))))))) of role axiom named tptp_const_array2_substitution_3 % 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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((tptp_const_array2 C) D) A)) (((tptp_const_array2 C) D) 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_madd A) C)) ((tptp_madd B) C))))))))))))))) of role axiom named tptp_madd_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_madd A) C)) ((tptp_madd 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_madd C) A)) ((tptp_madd C) B))))))))))))))) of role axiom named tptp_madd_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_madd C) A)) ((tptp_madd 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 ((tptp_mmul A) C)) ((tptp_mmul B) C))))))))))))))) of role axiom named tptp_mmul_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_mmul A) C)) ((tptp_mmul 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_mmul C) A)) ((tptp_mmul C) B))))))))))))))) of role axiom named tptp_mmul_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_mmul C) A)) ((tptp_mmul 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 ((tptp_msub A) C)) ((tptp_msub B) C))))))))))))))) of role axiom named tptp_msub_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_msub A) C)) ((tptp_msub 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_msub C) A)) ((tptp_msub C) B))))))))))))))) of role axiom named tptp_msub_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_msub C) A)) ((tptp_msub 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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((tptp_update2 A) C) D)) (((tptp_update2 B) C) D)))))))))))))))))) of role axiom named tptp_update2_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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((tptp_update2 A) C) D)) (((tptp_update2 B) C) D)))))))))))))))))) % 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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((tptp_update2 C) A) D)) (((tptp_update2 C) B) D)))))))))))))))))) of role axiom named tptp_update2_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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((tptp_update2 C) A) D)) (((tptp_update2 C) B) D)))))))))))))))))) % 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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((tptp_update2 C) D) A)) (((tptp_update2 C) D) B)))))))))))))))))) of role axiom named tptp_update2_substitution_3 % 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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((tptp_update2 C) D) A)) (((tptp_update2 C) D) 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 (mforall_ind (fun (D:mu)=> (mbox_s4 (mforall_ind (fun (E:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((((tptp_update3 A) C) D) E)) ((((tptp_update3 B) C) D) E))))))))))))))))))))) of role axiom named tptp_update3_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 (mforall_ind (fun (D:mu)=> (mbox_s4 (mforall_ind (fun (E:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((((tptp_update3 A) C) D) E)) ((((tptp_update3 B) C) D) E))))))))))))))))))))) % 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 (mforall_ind (fun (D:mu)=> (mbox_s4 (mforall_ind (fun (E:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((((tptp_update3 C) A) D) E)) ((((tptp_update3 C) B) D) E))))))))))))))))))))) of role axiom named tptp_update3_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 (mforall_ind (fun (D:mu)=> (mbox_s4 (mforall_ind (fun (E:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((((tptp_update3 C) A) D) E)) ((((tptp_update3 C) B) D) E))))))))))))))))))))) % 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 (mforall_ind (fun (D:mu)=> (mbox_s4 (mforall_ind (fun (E:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((((tptp_update3 C) D) A) E)) ((((tptp_update3 C) D) B) E))))))))))))))))))))) of role axiom named tptp_update3_substitution_3 % 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 (mforall_ind (fun (D:mu)=> (mbox_s4 (mforall_ind (fun (E:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((((tptp_update3 C) D) A) E)) ((((tptp_update3 C) D) B) E))))))))))))))))))))) % 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 (mforall_ind (fun (D:mu)=> (mbox_s4 (mforall_ind (fun (E:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((((tptp_update3 C) D) E) A)) ((((tptp_update3 C) D) E) B))))))))))))))))))))) of role axiom named tptp_update3_substitution_4 % 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 (mforall_ind (fun (D:mu)=> (mbox_s4 (mforall_ind (fun (E:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((((tptp_update3 C) D) E) A)) ((((tptp_update3 C) D) E) 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 (trans A)) (trans B)))))))))))) of role axiom named trans_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 (trans A)) (trans 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 ((uniform_int_rnd A) C)) ((uniform_int_rnd B) C))))))))))))))) of role axiom named uniform_int_rnd_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 ((uniform_int_rnd A) C)) ((uniform_int_rnd 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 ((uniform_int_rnd C) A)) ((uniform_int_rnd C) B))))))))))))))) of role axiom named uniform_int_rnd_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 ((uniform_int_rnd C) A)) ((uniform_int_rnd 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 ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((geq A) C)))) (mbox_s4 ((geq B) C)))))))))))))) of role axiom named geq_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 ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((geq A) C)))) (mbox_s4 ((geq 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 ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((geq C) A)))) (mbox_s4 ((geq C) B)))))))))))))) of role axiom named geq_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 ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((geq C) A)))) (mbox_s4 ((geq 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 ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((gt A) C)))) (mbox_s4 ((gt B) C)))))))))))))) of role axiom named gt_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 ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((gt A) C)))) (mbox_s4 ((gt 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 ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((gt C) A)))) (mbox_s4 ((gt C) B)))))))))))))) of role axiom named gt_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 ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((gt C) A)))) (mbox_s4 ((gt 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 ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((leq A) C)))) (mbox_s4 ((leq B) C)))))))))))))) of role axiom named leq_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 ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((leq A) C)))) (mbox_s4 ((leq 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 ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((leq C) A)))) (mbox_s4 ((leq C) B)))))))))))))) of role axiom named leq_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 ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((leq C) A)))) (mbox_s4 ((leq 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 ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((lt A) C)))) (mbox_s4 ((lt B) C)))))))))))))) of role axiom named lt_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 ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((lt A) C)))) (mbox_s4 ((lt 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 ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((lt C) A)))) (mbox_s4 ((lt C) B)))))))))))))) of role axiom named lt_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 ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((lt C) A)))) (mbox_s4 ((lt C) B)))))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mor (mbox_s4 ((gt X) Y))) ((mor (mbox_s4 ((gt Y) X))) (mbox_s4 ((qmltpeq X) Y))))))))))) of role axiom named totality % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mor (mbox_s4 ((gt X) Y))) ((mor (mbox_s4 ((gt Y) X))) (mbox_s4 ((qmltpeq X) Y))))))))))) % 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 ((gt X) Y))) (mbox_s4 ((gt Y) Z)))) (mbox_s4 ((gt X) Z)))))))))))))) of role axiom named transitivity_gt % 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 ((gt X) Y))) (mbox_s4 ((gt Y) Z)))) (mbox_s4 ((gt X) Z)))))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mnot (mbox_s4 ((gt X) X)))))))) of role axiom named irreflexivity_gt % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mnot (mbox_s4 ((gt X) X)))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((leq X) X)))))) of role axiom named reflexivity_leq % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((leq X) 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 ((leq X) Y))) (mbox_s4 ((leq Y) Z)))) (mbox_s4 ((leq X) Z)))))))))))))) of role axiom named transitivity_leq % 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 ((leq X) Y))) (mbox_s4 ((leq Y) Z)))) (mbox_s4 ((leq X) Z)))))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((lt X) Y))) (mbox_s4 ((gt Y) X))))) (mbox_s4 ((mimplies (mbox_s4 ((gt Y) X))) (mbox_s4 ((lt X) Y)))))))))))) of role axiom named lt_gt % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((lt X) Y))) (mbox_s4 ((gt Y) X))))) (mbox_s4 ((mimplies (mbox_s4 ((gt Y) X))) (mbox_s4 ((lt X) Y)))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((geq X) Y))) (mbox_s4 ((leq Y) X))))) (mbox_s4 ((mimplies (mbox_s4 ((leq Y) X))) (mbox_s4 ((geq X) Y)))))))))))) of role axiom named leq_geq % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((geq X) Y))) (mbox_s4 ((leq Y) X))))) (mbox_s4 ((mimplies (mbox_s4 ((leq Y) X))) (mbox_s4 ((geq X) Y)))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((gt Y) X))) (mbox_s4 ((leq X) Y))))))))))) of role axiom named leq_gt1 % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((gt Y) X))) (mbox_s4 ((leq X) Y))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq X) Y))) (mbox_s4 (mnot (mbox_s4 ((qmltpeq X) Y)))))) (mbox_s4 ((gt Y) X))))))))))) of role axiom named leq_gt2 % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq X) Y))) (mbox_s4 (mnot (mbox_s4 ((qmltpeq X) Y)))))) (mbox_s4 ((gt Y) X))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((leq X) (pred Y)))) (mbox_s4 ((gt Y) X))))) (mbox_s4 ((mimplies (mbox_s4 ((gt Y) X))) (mbox_s4 ((leq X) (pred Y))))))))))))) of role axiom named leq_gt_pred % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((leq X) (pred Y)))) (mbox_s4 ((gt Y) X))))) (mbox_s4 ((mimplies (mbox_s4 ((gt Y) X))) (mbox_s4 ((leq X) (pred Y))))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((gt (succ X)) X)))))) of role axiom named gt_succ % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((gt (succ X)) X)))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((leq X) Y))) (mbox_s4 ((leq X) (succ Y)))))))))))) of role axiom named leq_succ % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((leq X) Y))) (mbox_s4 ((leq X) (succ Y)))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((leq X) Y))) (mbox_s4 ((gt (succ Y)) X))))) (mbox_s4 ((mimplies (mbox_s4 ((gt (succ Y)) X))) (mbox_s4 ((leq X) Y)))))))))))) of role axiom named leq_succ_gt_equiv % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((leq X) Y))) (mbox_s4 ((gt (succ Y)) X))))) (mbox_s4 ((mimplies (mbox_s4 ((gt (succ Y)) X))) (mbox_s4 ((leq X) Y)))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((leq n0) X))) (mbox_s4 ((leq ((uniform_int_rnd C) X)) X))))))))))) of role axiom named uniform_int_rand_ranges_hi % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((leq n0) X))) (mbox_s4 ((leq ((uniform_int_rnd C) X)) X))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((leq n0) X))) (mbox_s4 ((leq n0) ((uniform_int_rnd C) X)))))))))))) of role axiom named uniform_int_rand_ranges_lo % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((leq n0) X))) (mbox_s4 ((leq n0) ((uniform_int_rnd C) X)))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (L:mu)=> (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (Val:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq L) _TPTP_I))) (mbox_s4 ((leq _TPTP_I) U)))) (mbox_s4 ((qmltpeq ((a_select2 ((tptp_const_array1 ((dim L) U)) Val)) _TPTP_I)) Val))))))))))))))))) of role axiom named const_array1_select % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (L:mu)=> (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (Val:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq L) _TPTP_I))) (mbox_s4 ((leq _TPTP_I) U)))) (mbox_s4 ((qmltpeq ((a_select2 ((tptp_const_array1 ((dim L) U)) Val)) _TPTP_I)) Val))))))))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (L1:mu)=> (mbox_s4 (mforall_ind (fun (U1:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 (mforall_ind (fun (L2:mu)=> (mbox_s4 (mforall_ind (fun (U2:mu)=> (mbox_s4 (mforall_ind (fun (Val:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq L1) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) U1))) ((mand (mbox_s4 ((leq L2) J))) (mbox_s4 ((leq J) U2)))))) (mbox_s4 ((qmltpeq (((a_select3 (((tptp_const_array2 ((dim L1) U1)) ((dim L2) U2)) Val)) _TPTP_I) J)) Val)))))))))))))))))))))))))) of role axiom named const_array2_select % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (L1:mu)=> (mbox_s4 (mforall_ind (fun (U1:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 (mforall_ind (fun (L2:mu)=> (mbox_s4 (mforall_ind (fun (U2:mu)=> (mbox_s4 (mforall_ind (fun (Val:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq L1) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) U1))) ((mand (mbox_s4 ((leq L2) J))) (mbox_s4 ((leq J) U2)))))) (mbox_s4 ((qmltpeq (((a_select3 (((tptp_const_array2 ((dim L1) U1)) ((dim L2) U2)) Val)) _TPTP_I) J)) Val)))))))))))))))))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (N:mu)=> (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 A) _TPTP_I) J)) (((a_select3 A) J) _TPTP_I)))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 (trans A)) _TPTP_I) J)) (((a_select3 (trans A)) J) _TPTP_I)))))))))))))))))))) of role axiom named matrix_symm_trans % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (N:mu)=> (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 A) _TPTP_I) J)) (((a_select3 A) J) _TPTP_I)))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 (trans A)) _TPTP_I) J)) (((a_select3 (trans A)) J) _TPTP_I)))))))))))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (N:mu)=> (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 A) _TPTP_I) J)) (((a_select3 A) J) _TPTP_I)))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 (inv A)) _TPTP_I) J)) (((a_select3 (inv A)) J) _TPTP_I)))))))))))))))))))) of role axiom named matrix_symm_inv % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (N:mu)=> (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 A) _TPTP_I) J)) (((a_select3 A) J) _TPTP_I)))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 (inv A)) _TPTP_I) J)) (((a_select3 (inv A)) J) _TPTP_I)))))))))))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (N:mu)=> (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 A) _TPTP_I) J)) (((a_select3 A) J) _TPTP_I)))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 (mforall_ind (fun (K:mu)=> (mbox_s4 (mforall_ind (fun (VAL:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) ((mand (mbox_s4 ((leq J) N))) ((mand (mbox_s4 ((leq n0) K))) (mbox_s4 ((leq K) N)))))))) (mbox_s4 ((qmltpeq (((a_select3 ((((tptp_update3 A) K) K) VAL)) _TPTP_I) J)) (((a_select3 ((((tptp_update3 A) K) K) VAL)) J) _TPTP_I)))))))))))))))))))))))))) of role axiom named matrix_symm_update_diagonal % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (N:mu)=> (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 A) _TPTP_I) J)) (((a_select3 A) J) _TPTP_I)))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 (mforall_ind (fun (K:mu)=> (mbox_s4 (mforall_ind (fun (VAL:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) ((mand (mbox_s4 ((leq J) N))) ((mand (mbox_s4 ((leq n0) K))) (mbox_s4 ((leq K) N)))))))) (mbox_s4 ((qmltpeq (((a_select3 ((((tptp_update3 A) K) K) VAL)) _TPTP_I) J)) (((a_select3 ((((tptp_update3 A) K) K) VAL)) J) _TPTP_I)))))))))))))))))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (N:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 A) _TPTP_I) J)) (((a_select3 A) J) _TPTP_I)))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 B) _TPTP_I) J)) (((a_select3 B) J) _TPTP_I))))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 ((tptp_madd A) B)) _TPTP_I) J)) (((a_select3 ((tptp_madd A) B)) J) _TPTP_I))))))))))))))))))))))) of role axiom named matrix_symm_add % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (N:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 A) _TPTP_I) J)) (((a_select3 A) J) _TPTP_I)))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 B) _TPTP_I) J)) (((a_select3 B) J) _TPTP_I))))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 ((tptp_madd A) B)) _TPTP_I) J)) (((a_select3 ((tptp_madd A) B)) J) _TPTP_I))))))))))))))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (N:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 A) _TPTP_I) J)) (((a_select3 A) J) _TPTP_I)))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 B) _TPTP_I) J)) (((a_select3 B) J) _TPTP_I))))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 ((tptp_msub A) B)) _TPTP_I) J)) (((a_select3 ((tptp_msub A) B)) J) _TPTP_I))))))))))))))))))))))) of role axiom named matrix_symm_sub % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (N:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 A) _TPTP_I) J)) (((a_select3 A) J) _TPTP_I)))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 B) _TPTP_I) J)) (((a_select3 B) J) _TPTP_I))))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 ((tptp_msub A) B)) _TPTP_I) J)) (((a_select3 ((tptp_msub A) B)) J) _TPTP_I))))))))))))))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (N:mu)=> (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 B) _TPTP_I) J)) (((a_select3 B) J) _TPTP_I)))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 ((tptp_mmul A) ((tptp_mmul B) (trans A)))) _TPTP_I) J)) (((a_select3 ((tptp_mmul A) ((tptp_mmul B) (trans A)))) J) _TPTP_I))))))))))))))))))))))) of role axiom named matrix_symm_aba1 % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (N:mu)=> (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 B) _TPTP_I) J)) (((a_select3 B) J) _TPTP_I)))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 ((tptp_mmul A) ((tptp_mmul B) (trans A)))) _TPTP_I) J)) (((a_select3 ((tptp_mmul A) ((tptp_mmul B) (trans A)))) J) _TPTP_I))))))))))))))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (N:mu)=> (mbox_s4 (mforall_ind (fun (M:mu)=> (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) M))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) M)))))) (mbox_s4 ((qmltpeq (((a_select3 B) _TPTP_I) J)) (((a_select3 B) J) _TPTP_I)))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 ((tptp_mmul A) ((tptp_mmul B) (trans A)))) _TPTP_I) J)) (((a_select3 ((tptp_mmul A) ((tptp_mmul B) (trans A)))) J) _TPTP_I)))))))))))))))))))))))))) of role axiom named matrix_symm_aba2 % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (N:mu)=> (mbox_s4 (mforall_ind (fun (M:mu)=> (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) M))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) M)))))) (mbox_s4 ((qmltpeq (((a_select3 B) _TPTP_I) J)) (((a_select3 B) J) _TPTP_I)))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 ((tptp_mmul A) ((tptp_mmul B) (trans A)))) _TPTP_I) J)) (((a_select3 ((tptp_mmul A) ((tptp_mmul B) (trans A)))) J) _TPTP_I)))))))))))))))))))))))))) % 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 (mforall_ind (fun (D:mu)=> (mbox_s4 (mforall_ind (fun (E:mu)=> (mbox_s4 (mforall_ind (fun (F:mu)=> (mbox_s4 (mforall_ind (fun (N:mu)=> (mbox_s4 (mforall_ind (fun (M:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) M))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) M)))))) (mbox_s4 ((qmltpeq (((a_select3 D) _TPTP_I) J)) (((a_select3 D) J) _TPTP_I)))))))))))) ((mand (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 A) _TPTP_I) J)) (((a_select3 A) J) _TPTP_I)))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 F) _TPTP_I) J)) (((a_select3 F) J) _TPTP_I)))))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 ((tptp_madd A) ((tptp_mmul B) ((tptp_mmul ((tptp_madd ((tptp_mmul C) ((tptp_mmul D) (trans C)))) ((tptp_mmul E) ((tptp_mmul F) (trans E))))) (trans B))))) _TPTP_I) J)) (((a_select3 ((tptp_madd A) ((tptp_mmul B) ((tptp_mmul ((tptp_madd ((tptp_mmul C) ((tptp_mmul D) (trans C)))) ((tptp_mmul E) ((tptp_mmul F) (trans E))))) (trans B))))) J) _TPTP_I)))))))))))))))))))))))))))))))))))))) of role axiom named matrix_symm_joseph_update % 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 (mforall_ind (fun (D:mu)=> (mbox_s4 (mforall_ind (fun (E:mu)=> (mbox_s4 (mforall_ind (fun (F:mu)=> (mbox_s4 (mforall_ind (fun (N:mu)=> (mbox_s4 (mforall_ind (fun (M:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) M))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) M)))))) (mbox_s4 ((qmltpeq (((a_select3 D) _TPTP_I) J)) (((a_select3 D) J) _TPTP_I)))))))))))) ((mand (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 A) _TPTP_I) J)) (((a_select3 A) J) _TPTP_I)))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 F) _TPTP_I) J)) (((a_select3 F) J) _TPTP_I)))))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 ((tptp_madd A) ((tptp_mmul B) ((tptp_mmul ((tptp_madd ((tptp_mmul C) ((tptp_mmul D) (trans C)))) ((tptp_mmul E) ((tptp_mmul F) (trans E))))) (trans B))))) _TPTP_I) J)) (((a_select3 ((tptp_madd A) ((tptp_mmul B) ((tptp_mmul ((tptp_madd ((tptp_mmul C) ((tptp_mmul D) (trans C)))) ((tptp_mmul E) ((tptp_mmul F) (trans E))))) (trans B))))) J) _TPTP_I)))))))))))))))))))))))))))))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (Body:mu)=> (mbox_s4 ((qmltpeq (((sum n0) tptp_minus_1) Body)) n0)))))) of role axiom named sum_plus_base % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (Body:mu)=> (mbox_s4 ((qmltpeq (((sum n0) tptp_minus_1) Body)) n0)))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (Body:mu)=> (mbox_s4 ((qmltpeq tptp_float_0_0) (((sum n0) tptp_minus_1) Body))))))) of role axiom named sum_plus_base_float % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (Body:mu)=> (mbox_s4 ((qmltpeq tptp_float_0_0) (((sum n0) tptp_minus_1) Body))))))) % FOF formula (mvalid (mbox_s4 ((qmltpeq (succ tptp_minus_1)) n0))) of role axiom named succ_tptp_minus_1 % A new axiom: (mvalid (mbox_s4 ((qmltpeq (succ tptp_minus_1)) n0))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus X) n1)) (succ X))))))) of role axiom named succ_plus_1_r % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus X) n1)) (succ X))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus n1) X)) (succ X))))))) of role axiom named succ_plus_1_l % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus n1) X)) (succ X))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus X) n2)) (succ (succ X)))))))) of role axiom named succ_plus_2_r % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus X) n2)) (succ (succ X)))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus n2) X)) (succ (succ X)))))))) of role axiom named succ_plus_2_l % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus n2) X)) (succ (succ X)))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus X) n3)) (succ (succ (succ X))))))))) of role axiom named succ_plus_3_r % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus X) n3)) (succ (succ (succ X))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus n3) X)) (succ (succ (succ X))))))))) of role axiom named succ_plus_3_l % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus n3) X)) (succ (succ (succ X))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus X) n4)) (succ (succ (succ (succ X)))))))))) of role axiom named succ_plus_4_r % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus X) n4)) (succ (succ (succ (succ X)))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus n4) X)) (succ (succ (succ (succ X)))))))))) of role axiom named succ_plus_4_l % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus n4) X)) (succ (succ (succ (succ X)))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus X) n5)) (succ (succ (succ (succ (succ X))))))))))) of role axiom named succ_plus_5_r % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus X) n5)) (succ (succ (succ (succ (succ X))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus n5) X)) (succ (succ (succ (succ (succ X))))))))))) of role axiom named succ_plus_5_l % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus n5) X)) (succ (succ (succ (succ (succ X))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((minus X) n1)) (pred X))))))) of role axiom named pred_minus_1 % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((minus X) n1)) (pred X))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq (pred (succ X))) X)))))) of role axiom named pred_succ % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq (pred (succ X))) X)))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq (succ (pred X))) X)))))) of role axiom named succ_pred % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq (succ (pred X))) X)))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((leq (succ X)) (succ Y)))) (mbox_s4 ((leq X) Y))))) (mbox_s4 ((mimplies (mbox_s4 ((leq X) Y))) (mbox_s4 ((leq (succ X)) (succ Y))))))))))))) of role axiom named leq_succ_succ % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((leq (succ X)) (succ Y)))) (mbox_s4 ((leq X) Y))))) (mbox_s4 ((mimplies (mbox_s4 ((leq X) Y))) (mbox_s4 ((leq (succ X)) (succ Y))))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((leq (succ X)) Y))) (mbox_s4 ((gt Y) X))))))))))) of role axiom named leq_succ_gt % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((leq (succ X)) Y))) (mbox_s4 ((gt Y) X))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((leq ((minus X) Y)) X))) (mbox_s4 ((leq n0) Y))))))))))) of role axiom named leq_minus % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((leq ((minus X) Y)) X))) (mbox_s4 ((leq n0) Y))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (V:mu)=> (mbox_s4 (mforall_ind (fun (VAL:mu)=> (mbox_s4 ((qmltpeq (((a_select3 ((((tptp_update3 X) U) V) VAL)) U) V)) VAL))))))))))))))) of role axiom named sel3_update_1 % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (V:mu)=> (mbox_s4 (mforall_ind (fun (VAL:mu)=> (mbox_s4 ((qmltpeq (((a_select3 ((((tptp_update3 X) U) V) VAL)) U) V)) VAL))))))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (V:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (VAL:mu)=> (mbox_s4 (mforall_ind (fun (VAL2:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (mnot (mbox_s4 ((qmltpeq _TPTP_I) U))))) ((mand (mbox_s4 ((qmltpeq J) V))) (mbox_s4 ((qmltpeq (((a_select3 X) U) V)) VAL))))) (mbox_s4 ((qmltpeq (((a_select3 ((((tptp_update3 X) _TPTP_I) J) VAL2)) U) V)) VAL)))))))))))))))))))))))))) of role axiom named sel3_update_2 % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (V:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (VAL:mu)=> (mbox_s4 (mforall_ind (fun (VAL2:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (mnot (mbox_s4 ((qmltpeq _TPTP_I) U))))) ((mand (mbox_s4 ((qmltpeq J) V))) (mbox_s4 ((qmltpeq (((a_select3 X) U) V)) VAL))))) (mbox_s4 ((qmltpeq (((a_select3 ((((tptp_update3 X) _TPTP_I) J) VAL2)) U) V)) VAL)))))))))))))))))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (V:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (VAL:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (mforall_ind (fun (I0:mu)=> (mbox_s4 (mforall_ind (fun (J0:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) I0))) ((mand (mbox_s4 ((leq n0) J0))) ((mand (mbox_s4 ((leq I0) U))) (mbox_s4 ((leq J0) V)))))) (mbox_s4 ((qmltpeq (((a_select3 X) I0) J0)) VAL))))))))))) ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) U))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) V))))))) (mbox_s4 ((qmltpeq (((a_select3 ((((tptp_update3 X) U) V) VAL)) _TPTP_I) J)) VAL))))))))))))))))))))))) of role axiom named sel3_update_3 % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (V:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (VAL:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (mforall_ind (fun (I0:mu)=> (mbox_s4 (mforall_ind (fun (J0:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) I0))) ((mand (mbox_s4 ((leq n0) J0))) ((mand (mbox_s4 ((leq I0) U))) (mbox_s4 ((leq J0) V)))))) (mbox_s4 ((qmltpeq (((a_select3 X) I0) J0)) VAL))))))))))) ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) U))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) V))))))) (mbox_s4 ((qmltpeq (((a_select3 ((((tptp_update3 X) U) V) VAL)) _TPTP_I) J)) VAL))))))))))))))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (VAL:mu)=> (mbox_s4 ((qmltpeq ((a_select2 (((tptp_update2 X) U) VAL)) U)) VAL)))))))))))) of role axiom named sel2_update_1 % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (VAL:mu)=> (mbox_s4 ((qmltpeq ((a_select2 (((tptp_update2 X) U) VAL)) U)) VAL)))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (VAL:mu)=> (mbox_s4 (mforall_ind (fun (VAL2:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (mnot (mbox_s4 ((qmltpeq _TPTP_I) U))))) (mbox_s4 ((qmltpeq ((a_select2 X) U)) VAL)))) (mbox_s4 ((qmltpeq ((a_select2 (((tptp_update2 X) _TPTP_I) VAL2)) U)) VAL)))))))))))))))))))) of role axiom named sel2_update_2 % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (VAL:mu)=> (mbox_s4 (mforall_ind (fun (VAL2:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (mnot (mbox_s4 ((qmltpeq _TPTP_I) U))))) (mbox_s4 ((qmltpeq ((a_select2 X) U)) VAL)))) (mbox_s4 ((qmltpeq ((a_select2 (((tptp_update2 X) _TPTP_I) VAL2)) U)) VAL)))))))))))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (VAL:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (mforall_ind (fun (I0:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) I0))) (mbox_s4 ((leq I0) U)))) (mbox_s4 ((qmltpeq ((a_select2 X) I0)) VAL)))))))) ((mand (mbox_s4 ((leq n0) _TPTP_I))) (mbox_s4 ((leq _TPTP_I) U))))) (mbox_s4 ((qmltpeq ((a_select2 (((tptp_update2 X) U) VAL)) _TPTP_I)) VAL))))))))))))))))) of role axiom named sel2_update_3 % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (VAL:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (mforall_ind (fun (I0:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) I0))) (mbox_s4 ((leq I0) U)))) (mbox_s4 ((qmltpeq ((a_select2 X) I0)) VAL)))))))) ((mand (mbox_s4 ((leq n0) _TPTP_I))) (mbox_s4 ((leq _TPTP_I) U))))) (mbox_s4 ((qmltpeq ((a_select2 (((tptp_update2 X) U) VAL)) _TPTP_I)) VAL))))))))))))))))) % FOF formula (mvalid (mbox_s4 true)) of role axiom named ttrue % A new axiom: (mvalid (mbox_s4 true)) % FOF formula (mvalid (mbox_s4 (mnot (mbox_s4 ((qmltpeq def) use))))) of role axiom named defuse % A new axiom: (mvalid (mbox_s4 (mnot (mbox_s4 ((qmltpeq def) use))))) % FOF formula (mvalid (mbox_s4 ((gt n5) n4))) of role axiom named gt_5_4 % A new axiom: (mvalid (mbox_s4 ((gt n5) n4))) % FOF formula (mvalid (mbox_s4 ((gt n4) tptp_minus_1))) of role axiom named gt_4_tptp_minus_1 % A new axiom: (mvalid (mbox_s4 ((gt n4) tptp_minus_1))) % FOF formula (mvalid (mbox_s4 ((gt n5) tptp_minus_1))) of role axiom named gt_5_tptp_minus_1 % A new axiom: (mvalid (mbox_s4 ((gt n5) tptp_minus_1))) % FOF formula (mvalid (mbox_s4 ((gt n0) tptp_minus_1))) of role axiom named gt_0_tptp_minus_1 % A new axiom: (mvalid (mbox_s4 ((gt n0) tptp_minus_1))) % FOF formula (mvalid (mbox_s4 ((gt n1) tptp_minus_1))) of role axiom named gt_1_tptp_minus_1 % A new axiom: (mvalid (mbox_s4 ((gt n1) tptp_minus_1))) % FOF formula (mvalid (mbox_s4 ((gt n2) tptp_minus_1))) of role axiom named gt_2_tptp_minus_1 % A new axiom: (mvalid (mbox_s4 ((gt n2) tptp_minus_1))) % FOF formula (mvalid (mbox_s4 ((gt n3) tptp_minus_1))) of role axiom named gt_3_tptp_minus_1 % A new axiom: (mvalid (mbox_s4 ((gt n3) tptp_minus_1))) % FOF formula (mvalid (mbox_s4 ((gt n4) n0))) of role axiom named gt_4_0 % A new axiom: (mvalid (mbox_s4 ((gt n4) n0))) % FOF formula (mvalid (mbox_s4 ((gt n5) n0))) of role axiom named gt_5_0 % A new axiom: (mvalid (mbox_s4 ((gt n5) n0))) % FOF formula (mvalid (mbox_s4 ((gt n1) n0))) of role axiom named gt_1_0 % A new axiom: (mvalid (mbox_s4 ((gt n1) n0))) % FOF formula (mvalid (mbox_s4 ((gt n2) n0))) of role axiom named gt_2_0 % A new axiom: (mvalid (mbox_s4 ((gt n2) n0))) % FOF formula (mvalid (mbox_s4 ((gt n3) n0))) of role axiom named gt_3_0 % A new axiom: (mvalid (mbox_s4 ((gt n3) n0))) % FOF formula (mvalid (mbox_s4 ((gt n4) n1))) of role axiom named gt_4_1 % A new axiom: (mvalid (mbox_s4 ((gt n4) n1))) % FOF formula (mvalid (mbox_s4 ((gt n5) n1))) of role axiom named gt_5_1 % A new axiom: (mvalid (mbox_s4 ((gt n5) n1))) % FOF formula (mvalid (mbox_s4 ((gt n2) n1))) of role axiom named gt_2_1 % A new axiom: (mvalid (mbox_s4 ((gt n2) n1))) % FOF formula (mvalid (mbox_s4 ((gt n3) n1))) of role axiom named gt_3_1 % A new axiom: (mvalid (mbox_s4 ((gt n3) n1))) % FOF formula (mvalid (mbox_s4 ((gt n4) n2))) of role axiom named gt_4_2 % A new axiom: (mvalid (mbox_s4 ((gt n4) n2))) % FOF formula (mvalid (mbox_s4 ((gt n5) n2))) of role axiom named gt_5_2 % A new axiom: (mvalid (mbox_s4 ((gt n5) n2))) % FOF formula (mvalid (mbox_s4 ((gt n3) n2))) of role axiom named gt_3_2 % A new axiom: (mvalid (mbox_s4 ((gt n3) n2))) % FOF formula (mvalid (mbox_s4 ((gt n4) n3))) of role axiom named gt_4_3 % A new axiom: (mvalid (mbox_s4 ((gt n4) n3))) % FOF formula (mvalid (mbox_s4 ((gt n5) n3))) of role axiom named gt_5_3 % A new axiom: (mvalid (mbox_s4 ((gt n5) n3))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) X))) (mbox_s4 ((leq X) n4)))) ((mor (mbox_s4 ((qmltpeq X) n0))) ((mor (mbox_s4 ((qmltpeq X) n1))) ((mor (mbox_s4 ((qmltpeq X) n2))) ((mor (mbox_s4 ((qmltpeq X) n3))) (mbox_s4 ((qmltpeq X) n4)))))))))))) of role axiom named finite_domain_4 % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) X))) (mbox_s4 ((leq X) n4)))) ((mor (mbox_s4 ((qmltpeq X) n0))) ((mor (mbox_s4 ((qmltpeq X) n1))) ((mor (mbox_s4 ((qmltpeq X) n2))) ((mor (mbox_s4 ((qmltpeq X) n3))) (mbox_s4 ((qmltpeq X) n4)))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) X))) (mbox_s4 ((leq X) n5)))) ((mor (mbox_s4 ((qmltpeq X) n0))) ((mor (mbox_s4 ((qmltpeq X) n1))) ((mor (mbox_s4 ((qmltpeq X) n2))) ((mor (mbox_s4 ((qmltpeq X) n3))) ((mor (mbox_s4 ((qmltpeq X) n4))) (mbox_s4 ((qmltpeq X) n5))))))))))))) of role axiom named finite_domain_5 % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) X))) (mbox_s4 ((leq X) n5)))) ((mor (mbox_s4 ((qmltpeq X) n0))) ((mor (mbox_s4 ((qmltpeq X) n1))) ((mor (mbox_s4 ((qmltpeq X) n2))) ((mor (mbox_s4 ((qmltpeq X) n3))) ((mor (mbox_s4 ((qmltpeq X) n4))) (mbox_s4 ((qmltpeq X) n5))))))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) X))) (mbox_s4 ((leq X) n0)))) (mbox_s4 ((qmltpeq X) n0)))))))) of role axiom named finite_domain_0 % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) X))) (mbox_s4 ((leq X) n0)))) (mbox_s4 ((qmltpeq X) n0)))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) X))) (mbox_s4 ((leq X) n1)))) ((mor (mbox_s4 ((qmltpeq X) n0))) (mbox_s4 ((qmltpeq X) n1))))))))) of role axiom named finite_domain_1 % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) X))) (mbox_s4 ((leq X) n1)))) ((mor (mbox_s4 ((qmltpeq X) n0))) (mbox_s4 ((qmltpeq X) n1))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) X))) (mbox_s4 ((leq X) n2)))) ((mor (mbox_s4 ((qmltpeq X) n0))) ((mor (mbox_s4 ((qmltpeq X) n1))) (mbox_s4 ((qmltpeq X) n2)))))))))) of role axiom named finite_domain_2 % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) X))) (mbox_s4 ((leq X) n2)))) ((mor (mbox_s4 ((qmltpeq X) n0))) ((mor (mbox_s4 ((qmltpeq X) n1))) (mbox_s4 ((qmltpeq X) n2)))))))))) % FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) X))) (mbox_s4 ((leq X) n3)))) ((mor (mbox_s4 ((qmltpeq X) n0))) ((mor (mbox_s4 ((qmltpeq X) n1))) ((mor (mbox_s4 ((qmltpeq X) n2))) (mbox_s4 ((qmltpeq X) n3))))))))))) of role axiom named finite_domain_3 % A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) X))) (mbox_s4 ((leq X) n3)))) ((mor (mbox_s4 ((qmltpeq X) n0))) ((mor (mbox_s4 ((qmltpeq X) n1))) ((mor (mbox_s4 ((qmltpeq X) n2))) (mbox_s4 ((qmltpeq X) n3))))))))))) % FOF formula (mvalid (mbox_s4 ((qmltpeq (succ (succ (succ (succ n0))))) n4))) of role axiom named successor_4 % A new axiom: (mvalid (mbox_s4 ((qmltpeq (succ (succ (succ (succ n0))))) n4))) % FOF formula (mvalid (mbox_s4 ((qmltpeq (succ (succ (succ (succ (succ n0)))))) n5))) of role axiom named successor_5 % A new axiom: (mvalid (mbox_s4 ((qmltpeq (succ (succ (succ (succ (succ n0)))))) n5))) % FOF formula (mvalid (mbox_s4 ((qmltpeq (succ n0)) n1))) of role axiom named successor_1 % A new axiom: (mvalid (mbox_s4 ((qmltpeq (succ n0)) n1))) % FOF formula (mvalid (mbox_s4 ((qmltpeq (succ (succ n0))) n2))) of role axiom named successor_2 % A new axiom: (mvalid (mbox_s4 ((qmltpeq (succ (succ n0))) n2))) % FOF formula (mvalid (mbox_s4 ((qmltpeq (succ (succ (succ n0)))) n3))) of role axiom named successor_3 % A new axiom: (mvalid (mbox_s4 ((qmltpeq (succ (succ (succ n0)))) n3))) % FOF formula (mvalid (mbox_s4 ((mimplies (mbox_s4 true)) ((mand (mbox_s4 ((mimplies (mbox_s4 (mnot (mbox_s4 ((gt loopcounter) n1))))) (mbox_s4 true)))) (mbox_s4 ((mimplies (mbox_s4 ((gt loopcounter) n1))) (mbox_s4 true))))))) of role conjecture named cl5_nebula_array_0002 % Conjecture to prove = (mvalid (mbox_s4 ((mimplies (mbox_s4 true)) ((mand (mbox_s4 ((mimplies (mbox_s4 (mnot (mbox_s4 ((gt loopcounter) n1))))) (mbox_s4 true)))) (mbox_s4 ((mimplies (mbox_s4 ((gt loopcounter) n1))) (mbox_s4 true))))))):Prop % Parameter fofType_DUMMY:fofType. % We need to prove ['(mvalid (mbox_s4 ((mimplies (mbox_s4 true)) ((mand (mbox_s4 ((mimplies (mbox_s4 (mnot (mbox_s4 ((gt loopcounter) n1))))) (mbox_s4 true)))) (mbox_s4 ((mimplies (mbox_s4 ((gt loopcounter) n1))) (mbox_s4 true)))))))'] % 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 true:(fofType->Prop). % Parameter lt:(mu->(mu->(fofType->Prop))). % Parameter geq:(mu->(mu->(fofType->Prop))). % Parameter gt:(mu->(mu->(fofType->Prop))). % Parameter leq:(mu->(mu->(fofType->Prop))). % Parameter uniform_int_rnd:(mu->(mu->mu)). % Axiom existence_of_uniform_int_rnd_ax:(forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((uniform_int_rnd V2) V1)) V)). % Parameter tptp_const_array1:(mu->(mu->mu)). % Axiom existence_of_tptp_const_array1_ax:(forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((tptp_const_array1 V2) V1)) V)). % Parameter dim:(mu->(mu->mu)). % Axiom existence_of_dim_ax:(forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((dim V2) V1)) V)). % Parameter tptp_const_array2:(mu->(mu->(mu->mu))). % Axiom existence_of_tptp_const_array2_ax:(forall (V:fofType) (V3:mu) (V2:mu) (V1:mu), ((exists_in_world (((tptp_const_array2 V3) V2) V1)) V)). % Parameter inv:(mu->mu). % Axiom existence_of_inv_ax:(forall (V:fofType) (V1:mu), ((exists_in_world (inv V1)) V)). % Parameter tptp_msub:(mu->(mu->mu)). % Axiom existence_of_tptp_msub_ax:(forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((tptp_msub V2) V1)) V)). % Parameter trans:(mu->mu). % Axiom existence_of_trans_ax:(forall (V:fofType) (V1:mu), ((exists_in_world (trans V1)) V)). % Parameter tptp_mmul:(mu->(mu->mu)). % Axiom existence_of_tptp_mmul_ax:(forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((tptp_mmul V2) V1)) V)). % Parameter tptp_madd:(mu->(mu->mu)). % Axiom existence_of_tptp_madd_ax:(forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((tptp_madd V2) V1)) V)). % Parameter sum:(mu->(mu->(mu->mu))). % Axiom existence_of_sum_ax:(forall (V:fofType) (V3:mu) (V2:mu) (V1:mu), ((exists_in_world (((sum V3) V2) V1)) V)). % Parameter tptp_float_0_0:mu. % Axiom existence_of_tptp_float_0_0_ax:(forall (V:fofType), ((exists_in_world tptp_float_0_0) V)). % Parameter plus:(mu->(mu->mu)). % Axiom existence_of_plus_ax:(forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((plus V2) V1)) V)). % Parameter pred:(mu->mu). % Axiom existence_of_pred_ax:(forall (V:fofType) (V1:mu), ((exists_in_world (pred V1)) V)). % Parameter minus:(mu->(mu->mu)). % Axiom existence_of_minus_ax:(forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((minus V2) V1)) V)). % Parameter tptp_update3:(mu->(mu->(mu->(mu->mu)))). % Axiom existence_of_tptp_update3_ax:(forall (V:fofType) (V4:mu) (V3:mu) (V2:mu) (V1:mu), ((exists_in_world ((((tptp_update3 V4) V3) V2) V1)) V)). % Parameter a_select3:(mu->(mu->(mu->mu))). % Axiom existence_of_a_select3_ax:(forall (V:fofType) (V3:mu) (V2:mu) (V1:mu), ((exists_in_world (((a_select3 V3) V2) V1)) V)). % Parameter tptp_update2:(mu->(mu->(mu->mu))). % Axiom existence_of_tptp_update2_ax:(forall (V:fofType) (V3:mu) (V2:mu) (V1:mu), ((exists_in_world (((tptp_update2 V3) V2) V1)) V)). % Parameter a_select2:(mu->(mu->mu)). % Axiom existence_of_a_select2_ax:(forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((a_select2 V2) V1)) V)). % Parameter use:mu. % Axiom existence_of_use_ax:(forall (V:fofType), ((exists_in_world use) V)). % Parameter def:mu. % Axiom existence_of_def_ax:(forall (V:fofType), ((exists_in_world def) V)). % Parameter loopcounter:mu. % Axiom existence_of_loopcounter_ax:(forall (V:fofType), ((exists_in_world loopcounter) V)). % Parameter tptp_minus_1:mu. % Axiom existence_of_tptp_minus_1_ax:(forall (V:fofType), ((exists_in_world tptp_minus_1) V)). % Parameter n4:mu. % Axiom existence_of_n4_ax:(forall (V:fofType), ((exists_in_world n4) V)). % Parameter n5:mu. % Axiom existence_of_n5_ax:(forall (V:fofType), ((exists_in_world n5) V)). % Parameter n1:mu. % Axiom existence_of_n1_ax:(forall (V:fofType), ((exists_in_world n1) V)). % Parameter n2:mu. % Axiom existence_of_n2_ax:(forall (V:fofType), ((exists_in_world n2) V)). % Parameter n3:mu. % Axiom existence_of_n3_ax:(forall (V:fofType), ((exists_in_world n3) V)). % Parameter n0:mu. % Axiom existence_of_n0_ax:(forall (V:fofType), ((exists_in_world n0) V)). % Parameter succ:(mu->mu). % Axiom existence_of_succ_ax:(forall (V:fofType) (V1:mu), ((exists_in_world (succ 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 a_select2_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 ((a_select2 A) C)) ((a_select2 B) C))))))))))))))). % Axiom a_select2_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 ((a_select2 C) A)) ((a_select2 C) B))))))))))))))). % Axiom a_select3_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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((a_select3 A) C) D)) (((a_select3 B) C) D)))))))))))))))))). % Axiom a_select3_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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((a_select3 C) A) D)) (((a_select3 C) B) D)))))))))))))))))). % Axiom a_select3_substitution_3:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((a_select3 C) D) A)) (((a_select3 C) D) B)))))))))))))))))). % Axiom dim_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 ((dim A) C)) ((dim B) C))))))))))))))). % Axiom dim_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 ((dim C) A)) ((dim C) B))))))))))))))). % Axiom inv_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 (inv A)) (inv B)))))))))))). % Axiom minus_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 ((minus A) C)) ((minus B) C))))))))))))))). % Axiom minus_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 ((minus C) A)) ((minus C) B))))))))))))))). % Axiom plus_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 ((plus A) C)) ((plus B) C))))))))))))))). % Axiom plus_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 ((plus C) A)) ((plus C) B))))))))))))))). % Axiom pred_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 (pred A)) (pred B)))))))))))). % Axiom succ_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 (succ A)) (succ B)))))))))))). % Axiom sum_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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((sum A) C) D)) (((sum B) C) D)))))))))))))))))). % Axiom sum_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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((sum C) A) D)) (((sum C) B) D)))))))))))))))))). % Axiom sum_substitution_3:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((sum C) D) A)) (((sum C) D) B)))))))))))))))))). % Axiom tptp_const_array1_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_const_array1 A) C)) ((tptp_const_array1 B) C))))))))))))))). % Axiom tptp_const_array1_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_const_array1 C) A)) ((tptp_const_array1 C) B))))))))))))))). % Axiom tptp_const_array2_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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((tptp_const_array2 A) C) D)) (((tptp_const_array2 B) C) D)))))))))))))))))). % Axiom tptp_const_array2_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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((tptp_const_array2 C) A) D)) (((tptp_const_array2 C) B) D)))))))))))))))))). % Axiom tptp_const_array2_substitution_3:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((tptp_const_array2 C) D) A)) (((tptp_const_array2 C) D) B)))))))))))))))))). % Axiom tptp_madd_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_madd A) C)) ((tptp_madd B) C))))))))))))))). % Axiom tptp_madd_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_madd C) A)) ((tptp_madd C) B))))))))))))))). % Axiom tptp_mmul_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_mmul A) C)) ((tptp_mmul B) C))))))))))))))). % Axiom tptp_mmul_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_mmul C) A)) ((tptp_mmul C) B))))))))))))))). % Axiom tptp_msub_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_msub A) C)) ((tptp_msub B) C))))))))))))))). % Axiom tptp_msub_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_msub C) A)) ((tptp_msub C) B))))))))))))))). % Axiom tptp_update2_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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((tptp_update2 A) C) D)) (((tptp_update2 B) C) D)))))))))))))))))). % Axiom tptp_update2_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 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((tptp_update2 C) A) D)) (((tptp_update2 C) B) D)))))))))))))))))). % Axiom tptp_update2_substitution_3:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((tptp_update2 C) D) A)) (((tptp_update2 C) D) B)))))))))))))))))). % Axiom tptp_update3_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 (mforall_ind (fun (D:mu)=> (mbox_s4 (mforall_ind (fun (E:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((((tptp_update3 A) C) D) E)) ((((tptp_update3 B) C) D) E))))))))))))))))))))). % Axiom tptp_update3_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 (mforall_ind (fun (D:mu)=> (mbox_s4 (mforall_ind (fun (E:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((((tptp_update3 C) A) D) E)) ((((tptp_update3 C) B) D) E))))))))))))))))))))). % Axiom tptp_update3_substitution_3:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 (mforall_ind (fun (D:mu)=> (mbox_s4 (mforall_ind (fun (E:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((((tptp_update3 C) D) A) E)) ((((tptp_update3 C) D) B) E))))))))))))))))))))). % Axiom tptp_update3_substitution_4:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 (mforall_ind (fun (D:mu)=> (mbox_s4 (mforall_ind (fun (E:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((((tptp_update3 C) D) E) A)) ((((tptp_update3 C) D) E) B))))))))))))))))))))). % Axiom trans_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 (trans A)) (trans B)))))))))))). % Axiom uniform_int_rnd_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 ((uniform_int_rnd A) C)) ((uniform_int_rnd B) C))))))))))))))). % Axiom uniform_int_rnd_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 ((uniform_int_rnd C) A)) ((uniform_int_rnd C) B))))))))))))))). % Axiom geq_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 ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((geq A) C)))) (mbox_s4 ((geq B) C)))))))))))))). % Axiom geq_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 ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((geq C) A)))) (mbox_s4 ((geq C) B)))))))))))))). % Axiom gt_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 ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((gt A) C)))) (mbox_s4 ((gt B) C)))))))))))))). % Axiom gt_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 ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((gt C) A)))) (mbox_s4 ((gt C) B)))))))))))))). % Axiom leq_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 ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((leq A) C)))) (mbox_s4 ((leq B) C)))))))))))))). % Axiom leq_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 ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((leq C) A)))) (mbox_s4 ((leq C) B)))))))))))))). % Axiom lt_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 ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((lt A) C)))) (mbox_s4 ((lt B) C)))))))))))))). % Axiom lt_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 ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((lt C) A)))) (mbox_s4 ((lt C) B)))))))))))))). % Axiom totality:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mor (mbox_s4 ((gt X) Y))) ((mor (mbox_s4 ((gt Y) X))) (mbox_s4 ((qmltpeq X) Y))))))))))). % Axiom transitivity_gt:(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 ((gt X) Y))) (mbox_s4 ((gt Y) Z)))) (mbox_s4 ((gt X) Z)))))))))))))). % Axiom irreflexivity_gt:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mnot (mbox_s4 ((gt X) X)))))))). % Axiom reflexivity_leq:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((leq X) X)))))). % Axiom transitivity_leq:(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 ((leq X) Y))) (mbox_s4 ((leq Y) Z)))) (mbox_s4 ((leq X) Z)))))))))))))). % Axiom lt_gt:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((lt X) Y))) (mbox_s4 ((gt Y) X))))) (mbox_s4 ((mimplies (mbox_s4 ((gt Y) X))) (mbox_s4 ((lt X) Y)))))))))))). % Axiom leq_geq:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((geq X) Y))) (mbox_s4 ((leq Y) X))))) (mbox_s4 ((mimplies (mbox_s4 ((leq Y) X))) (mbox_s4 ((geq X) Y)))))))))))). % Axiom leq_gt1:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((gt Y) X))) (mbox_s4 ((leq X) Y))))))))))). % Axiom leq_gt2:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq X) Y))) (mbox_s4 (mnot (mbox_s4 ((qmltpeq X) Y)))))) (mbox_s4 ((gt Y) X))))))))))). % Axiom leq_gt_pred:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((leq X) (pred Y)))) (mbox_s4 ((gt Y) X))))) (mbox_s4 ((mimplies (mbox_s4 ((gt Y) X))) (mbox_s4 ((leq X) (pred Y))))))))))))). % Axiom gt_succ:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((gt (succ X)) X)))))). % Axiom leq_succ:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((leq X) Y))) (mbox_s4 ((leq X) (succ Y)))))))))))). % Axiom leq_succ_gt_equiv:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((leq X) Y))) (mbox_s4 ((gt (succ Y)) X))))) (mbox_s4 ((mimplies (mbox_s4 ((gt (succ Y)) X))) (mbox_s4 ((leq X) Y)))))))))))). % Axiom uniform_int_rand_ranges_hi:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((leq n0) X))) (mbox_s4 ((leq ((uniform_int_rnd C) X)) X))))))))))). % Axiom uniform_int_rand_ranges_lo:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((leq n0) X))) (mbox_s4 ((leq n0) ((uniform_int_rnd C) X)))))))))))). % Axiom const_array1_select:(mvalid (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (L:mu)=> (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (Val:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq L) _TPTP_I))) (mbox_s4 ((leq _TPTP_I) U)))) (mbox_s4 ((qmltpeq ((a_select2 ((tptp_const_array1 ((dim L) U)) Val)) _TPTP_I)) Val))))))))))))))))). % Axiom const_array2_select:(mvalid (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (L1:mu)=> (mbox_s4 (mforall_ind (fun (U1:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 (mforall_ind (fun (L2:mu)=> (mbox_s4 (mforall_ind (fun (U2:mu)=> (mbox_s4 (mforall_ind (fun (Val:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq L1) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) U1))) ((mand (mbox_s4 ((leq L2) J))) (mbox_s4 ((leq J) U2)))))) (mbox_s4 ((qmltpeq (((a_select3 (((tptp_const_array2 ((dim L1) U1)) ((dim L2) U2)) Val)) _TPTP_I) J)) Val)))))))))))))))))))))))))). % Axiom matrix_symm_trans:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (N:mu)=> (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 A) _TPTP_I) J)) (((a_select3 A) J) _TPTP_I)))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 (trans A)) _TPTP_I) J)) (((a_select3 (trans A)) J) _TPTP_I)))))))))))))))))))). % Axiom matrix_symm_inv:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (N:mu)=> (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 A) _TPTP_I) J)) (((a_select3 A) J) _TPTP_I)))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 (inv A)) _TPTP_I) J)) (((a_select3 (inv A)) J) _TPTP_I)))))))))))))))))))). % Axiom matrix_symm_update_diagonal:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (N:mu)=> (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 A) _TPTP_I) J)) (((a_select3 A) J) _TPTP_I)))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 (mforall_ind (fun (K:mu)=> (mbox_s4 (mforall_ind (fun (VAL:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) ((mand (mbox_s4 ((leq J) N))) ((mand (mbox_s4 ((leq n0) K))) (mbox_s4 ((leq K) N)))))))) (mbox_s4 ((qmltpeq (((a_select3 ((((tptp_update3 A) K) K) VAL)) _TPTP_I) J)) (((a_select3 ((((tptp_update3 A) K) K) VAL)) J) _TPTP_I)))))))))))))))))))))))))). % Axiom matrix_symm_add:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (N:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 A) _TPTP_I) J)) (((a_select3 A) J) _TPTP_I)))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 B) _TPTP_I) J)) (((a_select3 B) J) _TPTP_I))))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 ((tptp_madd A) B)) _TPTP_I) J)) (((a_select3 ((tptp_madd A) B)) J) _TPTP_I))))))))))))))))))))))). % Axiom matrix_symm_sub:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (N:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 A) _TPTP_I) J)) (((a_select3 A) J) _TPTP_I)))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 B) _TPTP_I) J)) (((a_select3 B) J) _TPTP_I))))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 ((tptp_msub A) B)) _TPTP_I) J)) (((a_select3 ((tptp_msub A) B)) J) _TPTP_I))))))))))))))))))))))). % Axiom matrix_symm_aba1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (N:mu)=> (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 B) _TPTP_I) J)) (((a_select3 B) J) _TPTP_I)))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 ((tptp_mmul A) ((tptp_mmul B) (trans A)))) _TPTP_I) J)) (((a_select3 ((tptp_mmul A) ((tptp_mmul B) (trans A)))) J) _TPTP_I))))))))))))))))))))))). % Axiom matrix_symm_aba2:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (N:mu)=> (mbox_s4 (mforall_ind (fun (M:mu)=> (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) M))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) M)))))) (mbox_s4 ((qmltpeq (((a_select3 B) _TPTP_I) J)) (((a_select3 B) J) _TPTP_I)))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 ((tptp_mmul A) ((tptp_mmul B) (trans A)))) _TPTP_I) J)) (((a_select3 ((tptp_mmul A) ((tptp_mmul B) (trans A)))) J) _TPTP_I)))))))))))))))))))))))))). % Axiom matrix_symm_joseph_update:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 (mforall_ind (fun (D:mu)=> (mbox_s4 (mforall_ind (fun (E:mu)=> (mbox_s4 (mforall_ind (fun (F:mu)=> (mbox_s4 (mforall_ind (fun (N:mu)=> (mbox_s4 (mforall_ind (fun (M:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) M))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) M)))))) (mbox_s4 ((qmltpeq (((a_select3 D) _TPTP_I) J)) (((a_select3 D) J) _TPTP_I)))))))))))) ((mand (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 A) _TPTP_I) J)) (((a_select3 A) J) _TPTP_I)))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 F) _TPTP_I) J)) (((a_select3 F) J) _TPTP_I)))))))))))))) (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) N))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) N)))))) (mbox_s4 ((qmltpeq (((a_select3 ((tptp_madd A) ((tptp_mmul B) ((tptp_mmul ((tptp_madd ((tptp_mmul C) ((tptp_mmul D) (trans C)))) ((tptp_mmul E) ((tptp_mmul F) (trans E))))) (trans B))))) _TPTP_I) J)) (((a_select3 ((tptp_madd A) ((tptp_mmul B) ((tptp_mmul ((tptp_madd ((tptp_mmul C) ((tptp_mmul D) (trans C)))) ((tptp_mmul E) ((tptp_mmul F) (trans E))))) (trans B))))) J) _TPTP_I)))))))))))))))))))))))))))))))))))))). % Axiom sum_plus_base:(mvalid (mbox_s4 (mforall_ind (fun (Body:mu)=> (mbox_s4 ((qmltpeq (((sum n0) tptp_minus_1) Body)) n0)))))). % Axiom sum_plus_base_float:(mvalid (mbox_s4 (mforall_ind (fun (Body:mu)=> (mbox_s4 ((qmltpeq tptp_float_0_0) (((sum n0) tptp_minus_1) Body))))))). % Axiom succ_tptp_minus_1:(mvalid (mbox_s4 ((qmltpeq (succ tptp_minus_1)) n0))). % Axiom succ_plus_1_r:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus X) n1)) (succ X))))))). % Axiom succ_plus_1_l:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus n1) X)) (succ X))))))). % Axiom succ_plus_2_r:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus X) n2)) (succ (succ X)))))))). % Axiom succ_plus_2_l:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus n2) X)) (succ (succ X)))))))). % Axiom succ_plus_3_r:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus X) n3)) (succ (succ (succ X))))))))). % Axiom succ_plus_3_l:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus n3) X)) (succ (succ (succ X))))))))). % Axiom succ_plus_4_r:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus X) n4)) (succ (succ (succ (succ X)))))))))). % Axiom succ_plus_4_l:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus n4) X)) (succ (succ (succ (succ X)))))))))). % Axiom succ_plus_5_r:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus X) n5)) (succ (succ (succ (succ (succ X))))))))))). % Axiom succ_plus_5_l:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((plus n5) X)) (succ (succ (succ (succ (succ X))))))))))). % Axiom pred_minus_1:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq ((minus X) n1)) (pred X))))))). % Axiom pred_succ:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq (pred (succ X))) X)))))). % Axiom succ_pred:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq (succ (pred X))) X)))))). % Axiom leq_succ_succ:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((leq (succ X)) (succ Y)))) (mbox_s4 ((leq X) Y))))) (mbox_s4 ((mimplies (mbox_s4 ((leq X) Y))) (mbox_s4 ((leq (succ X)) (succ Y))))))))))))). % Axiom leq_succ_gt:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((leq (succ X)) Y))) (mbox_s4 ((gt Y) X))))))))))). % Axiom leq_minus:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((leq ((minus X) Y)) X))) (mbox_s4 ((leq n0) Y))))))))))). % Axiom sel3_update_1:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (V:mu)=> (mbox_s4 (mforall_ind (fun (VAL:mu)=> (mbox_s4 ((qmltpeq (((a_select3 ((((tptp_update3 X) U) V) VAL)) U) V)) VAL))))))))))))))). % Axiom sel3_update_2:(mvalid (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (V:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (VAL:mu)=> (mbox_s4 (mforall_ind (fun (VAL2:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (mnot (mbox_s4 ((qmltpeq _TPTP_I) U))))) ((mand (mbox_s4 ((qmltpeq J) V))) (mbox_s4 ((qmltpeq (((a_select3 X) U) V)) VAL))))) (mbox_s4 ((qmltpeq (((a_select3 ((((tptp_update3 X) _TPTP_I) J) VAL2)) U) V)) VAL)))))))))))))))))))))))))). % Axiom sel3_update_3:(mvalid (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (J:mu)=> (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (V:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (VAL:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (mforall_ind (fun (I0:mu)=> (mbox_s4 (mforall_ind (fun (J0:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) I0))) ((mand (mbox_s4 ((leq n0) J0))) ((mand (mbox_s4 ((leq I0) U))) (mbox_s4 ((leq J0) V)))))) (mbox_s4 ((qmltpeq (((a_select3 X) I0) J0)) VAL))))))))))) ((mand (mbox_s4 ((leq n0) _TPTP_I))) ((mand (mbox_s4 ((leq _TPTP_I) U))) ((mand (mbox_s4 ((leq n0) J))) (mbox_s4 ((leq J) V))))))) (mbox_s4 ((qmltpeq (((a_select3 ((((tptp_update3 X) U) V) VAL)) _TPTP_I) J)) VAL))))))))))))))))))))))). % Axiom sel2_update_1:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (VAL:mu)=> (mbox_s4 ((qmltpeq ((a_select2 (((tptp_update2 X) U) VAL)) U)) VAL)))))))))))). % Axiom sel2_update_2:(mvalid (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (VAL:mu)=> (mbox_s4 (mforall_ind (fun (VAL2:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (mnot (mbox_s4 ((qmltpeq _TPTP_I) U))))) (mbox_s4 ((qmltpeq ((a_select2 X) U)) VAL)))) (mbox_s4 ((qmltpeq ((a_select2 (((tptp_update2 X) _TPTP_I) VAL2)) U)) VAL)))))))))))))))))))). % Axiom sel2_update_3:(mvalid (mbox_s4 (mforall_ind (fun (_TPTP_I:mu)=> (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (VAL:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 (mforall_ind (fun (I0:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) I0))) (mbox_s4 ((leq I0) U)))) (mbox_s4 ((qmltpeq ((a_select2 X) I0)) VAL)))))))) ((mand (mbox_s4 ((leq n0) _TPTP_I))) (mbox_s4 ((leq _TPTP_I) U))))) (mbox_s4 ((qmltpeq ((a_select2 (((tptp_update2 X) U) VAL)) _TPTP_I)) VAL))))))))))))))))). % Axiom ttrue:(mvalid (mbox_s4 true)). % Axiom defuse:(mvalid (mbox_s4 (mnot (mbox_s4 ((qmltpeq def) use))))). % Axiom gt_5_4:(mvalid (mbox_s4 ((gt n5) n4))). % Axiom gt_4_tptp_minus_1:(mvalid (mbox_s4 ((gt n4) tptp_minus_1))). % Axiom gt_5_tptp_minus_1:(mvalid (mbox_s4 ((gt n5) tptp_minus_1))). % Axiom gt_0_tptp_minus_1:(mvalid (mbox_s4 ((gt n0) tptp_minus_1))). % Axiom gt_1_tptp_minus_1:(mvalid (mbox_s4 ((gt n1) tptp_minus_1))). % Axiom gt_2_tptp_minus_1:(mvalid (mbox_s4 ((gt n2) tptp_minus_1))). % Axiom gt_3_tptp_minus_1:(mvalid (mbox_s4 ((gt n3) tptp_minus_1))). % Axiom gt_4_0:(mvalid (mbox_s4 ((gt n4) n0))). % Axiom gt_5_0:(mvalid (mbox_s4 ((gt n5) n0))). % Axiom gt_1_0:(mvalid (mbox_s4 ((gt n1) n0))). % Axiom gt_2_0:(mvalid (mbox_s4 ((gt n2) n0))). % Axiom gt_3_0:(mvalid (mbox_s4 ((gt n3) n0))). % Axiom gt_4_1:(mvalid (mbox_s4 ((gt n4) n1))). % Axiom gt_5_1:(mvalid (mbox_s4 ((gt n5) n1))). % Axiom gt_2_1:(mvalid (mbox_s4 ((gt n2) n1))). % Axiom gt_3_1:(mvalid (mbox_s4 ((gt n3) n1))). % Axiom gt_4_2:(mvalid (mbox_s4 ((gt n4) n2))). % Axiom gt_5_2:(mvalid (mbox_s4 ((gt n5) n2))). % Axiom gt_3_2:(mvalid (mbox_s4 ((gt n3) n2))). % Axiom gt_4_3:(mvalid (mbox_s4 ((gt n4) n3))). % Axiom gt_5_3:(mvalid (mbox_s4 ((gt n5) n3))). % Axiom finite_domain_4:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) X))) (mbox_s4 ((leq X) n4)))) ((mor (mbox_s4 ((qmltpeq X) n0))) ((mor (mbox_s4 ((qmltpeq X) n1))) ((mor (mbox_s4 ((qmltpeq X) n2))) ((mor (mbox_s4 ((qmltpeq X) n3))) (mbox_s4 ((qmltpeq X) n4)))))))))))). % Axiom finite_domain_5:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) X))) (mbox_s4 ((leq X) n5)))) ((mor (mbox_s4 ((qmltpeq X) n0))) ((mor (mbox_s4 ((qmltpeq X) n1))) ((mor (mbox_s4 ((qmltpeq X) n2))) ((mor (mbox_s4 ((qmltpeq X) n3))) ((mor (mbox_s4 ((qmltpeq X) n4))) (mbox_s4 ((qmltpeq X) n5))))))))))))). % Axiom finite_domain_0:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) X))) (mbox_s4 ((leq X) n0)))) (mbox_s4 ((qmltpeq X) n0)))))))). % Axiom finite_domain_1:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) X))) (mbox_s4 ((leq X) n1)))) ((mor (mbox_s4 ((qmltpeq X) n0))) (mbox_s4 ((qmltpeq X) n1))))))))). % Axiom finite_domain_2:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) X))) (mbox_s4 ((leq X) n2)))) ((mor (mbox_s4 ((qmltpeq X) n0))) ((mor (mbox_s4 ((qmltpeq X) n1))) (mbox_s4 ((qmltpeq X) n2)))))))))). % Axiom finite_domain_3:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((leq n0) X))) (mbox_s4 ((leq X) n3)))) ((mor (mbox_s4 ((qmltpeq X) n0))) ((mor (mbox_s4 ((qmltpeq X) n1))) ((mor (mbox_s4 ((qmltpeq X) n2))) (mbox_s4 ((qmltpeq X) n3))))))))))). % Axiom successor_4:(mvalid (mbox_s4 ((qmltpeq (succ (succ (succ (succ n0))))) n4))). % Axiom successor_5:(mvalid (mbox_s4 ((qmltpeq (succ (succ (succ (succ (succ n0)))))) n5))). % Axiom successor_1:(mvalid (mbox_s4 ((qmltpeq (succ n0)) n1))). % Axiom successor_2:(mvalid (mbox_s4 ((qmltpeq (succ (succ n0))) n2))). % Axiom successor_3:(mvalid (mbox_s4 ((qmltpeq (succ (succ (succ n0)))) n3))). % Trying to prove (mvalid (mbox_s4 ((mimplies (mbox_s4 true)) ((mand (mbox_s4 ((mimplies (mbox_s4 (mnot (mbox_s4 ((gt loopcounter) n1))))) (mbox_s4 true)))) (mbox_s4 ((mimplies (mbox_s4 ((gt loopcounter) n1))) (mbox_s4 true))))))) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (x1 (a1 V)) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of ((((rel_s4 V) V0)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (x1 (a1 V)) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of ((((rel_s4 V) V0)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (x1 (a1 V)) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of ((((rel_s4 V) V0)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (x1 (a1 V)) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of ((((rel_s4 V) V0)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (x1 (a1 V)) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of ((((rel_s4 V) V0)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (x1 (a1 V)) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of ((((rel_s4 V) V0)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (x1 (a1 V)) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of ((((rel_s4 V) V0)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (x1 (a1 V)) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of ((((rel_s4 V) V0)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (x3 (a1 V)) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of ((((rel_s4 V) V00)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (x3 (a1 V)) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of ((((rel_s4 V) V00)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (x1 (a1 V)) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of ((((rel_s4 V) V0)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (x1 (a1 V)) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of ((((rel_s4 V) V0)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (x3 (a1 V)) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of ((((rel_s4 V) V00)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (x3 (a1 V)) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of ((((rel_s4 V) V00)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (x1 (a1 V)) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of ((((rel_s4 V) V0)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (x1 (a1 V)) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of ((((rel_s4 V) V0)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (x3 (a1 V)) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of ((((rel_s4 V) V00)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (x1 (a1 V)) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of ((((rel_s4 V) V0)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (x3 (a1 V)) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of ((((rel_s4 V) V00)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (x1 (a1 V)) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of ((((rel_s4 V) V0)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (x1 (a1 V)) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of ((((rel_s4 V) V0)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (x3 (a1 V)) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of ((((rel_s4 V) V00)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (x3 (a1 V)) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of ((((rel_s4 V) V00)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (x1 (a1 V)) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of ((((rel_s4 V) V0)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (x1 (a1 V)) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of ((((rel_s4 V) V0)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (x1 (a1 V)) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of ((((rel_s4 V) V0)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (x3 (a1 V)) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of ((((rel_s4 V) V00)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (x3 (a1 V)) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of ((((rel_s4 V) V00)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (x1 (a1 V)) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of ((((rel_s4 V) V0)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (x1 (a1 V)) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of ((((rel_s4 V) V0)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (x3 (a1 V)) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of ((((rel_s4 V) V00)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (x3 (a1 V)) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of ((((rel_s4 V) V00)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (x1 (a1 V)) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of ((((rel_s4 V) V0)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (x1 (a1 V)) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of ((((rel_s4 V) V0)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (x3 (a1 V)) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of ((((rel_s4 V) V00)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (x3 (a1 V)) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of ((((rel_s4 V) V00)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (x1 (a1 V)) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of ((((rel_s4 V) V0)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (a1 V) as proof of ((rel_s4 V) V0) % Found (x1 (a1 V)) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of False % Found (fun (x1:(((rel_s4 V) V0)->False))=> (x1 (a1 V))) as proof of ((((rel_s4 V) V0)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (x3 (a1 V)) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of ((((rel_s4 V) V00)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (x3 (a1 V)) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of ((((rel_s4 V) V00)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (x3 (a1 V)) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of ((((rel_s4 V) V00)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (x3 (a1 V)) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 (a1 V))) as proof of ((((rel_s4 V) V00)->False)->False) % Found a10:=(a1 V):((rel_s4 V) V) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (a1 V) as proof of ((rel_s4 V) V00) % Found (x3 (a1 V)) as proof of False % Found (fun (x3:(((rel_s4 V) V00)->False))=> (x3 % EOF %------------------------------------------------------------------------------