%----Output an unreadable proof
% SZS output start Proof for theBenchmark
thf(type_def_5, type, unsorted: $tType).
thf(type_def_6, type, mu: $tType).
thf(type_def_7, type, sTfun: ($tType * $tType) > $tType).
thf(type_def_8, type, d_unsorted: $tType).
thf(type_def_9, type, d_mu: $tType).
thf(func_def_0, type, meq_ind: (mu > mu > $i > $o)).
thf(func_def_1, type, meq_prop: (($i > $o) > ($i > $o) > $i > $o)).
thf(func_def_2, type, mnot: (($i > $o) > $i > $o)).
thf(func_def_3, type, mor: (($i > $o) > ($i > $o) > $i > $o)).
thf(func_def_4, type, mand: (($i > $o) > ($i > $o) > $i > $o)).
thf(func_def_5, type, mimplies: (($i > $o) > ($i > $o) > $i > $o)).
thf(func_def_6, type, mimplied: (($i > $o) > ($i > $o) > $i > $o)).
thf(func_def_7, type, mequiv: (($i > $o) > ($i > $o) > $i > $o)).
thf(func_def_8, type, mxor: (($i > $o) > ($i > $o) > $i > $o)).
thf(func_def_9, type, mforall_ind: ((mu > $i > $o) > $i > $o)).
thf(func_def_10, type, mforall_prop: ((($i > $o) > $i > $o) > $i > $o)).
thf(func_def_11, type, mexists_ind: ((mu > $i > $o) > $i > $o)).
thf(func_def_12, type, mexists_prop: ((($i > $o) > $i > $o) > $i > $o)).
thf(func_def_13, type, mtrue: ($i > $o)).
thf(func_def_14, type, mfalse: ($i > $o)).
thf(func_def_15, type, mbox: (($i > $i > $o) > ($i > $o) > $i > $o)).
thf(func_def_16, type, mdia: (($i > $i > $o) > ($i > $o) > $i > $o)).
thf(func_def_17, type, mreflexive: (($i > $i > $o) > $o)).
thf(func_def_18, type, msymmetric: (($i > $i > $o) > $o)).
thf(func_def_19, type, mserial: (($i > $i > $o) > $o)).
thf(func_def_20, type, mtransitive: (($i > $i > $o) > $o)).
thf(func_def_21, type, meuclidean: (($i > $i > $o) > $o)).
thf(func_def_22, type, mpartially_functional: (($i > $i > $o) > $o)).
thf(func_def_23, type, mfunctional: (($i > $i > $o) > $o)).
thf(func_def_24, type, mweakly_dense: (($i > $i > $o) > $o)).
thf(func_def_25, type, mweakly_connected: (($i > $i > $o) > $o)).
thf(func_def_26, type, mweakly_directed: (($i > $i > $o) > $o)).
thf(func_def_27, type, mvalid: (($i > $o) > $o)).
thf(func_def_28, type, minvalid: (($i > $o) > $o)).
thf(func_def_29, type, msatisfiable: (($i > $o) > $o)).
thf(func_def_30, type, mcountersatisfiable: (($i > $o) > $o)).
thf(func_def_31, type, d2unsorted: (d_unsorted > $i)).
thf(func_def_32, type, d2mu: (d_mu > mu)).
thf(func_def_33, type, d_unsorted_0: d_unsorted).
thf(func_def_34, type, d_mu_0: d_mu).
thf(func_def_36, type, db1: !>[X0: $tType]:(X0)).
thf(func_def_37, type, db0: !>[X0: $tType]:(X0)).
thf(func_def_38, type, vLAM: !>[X0: $tType, X1: $tType]:((X1) > (X0 > X1))).
thf(func_def_39, type, vPI: !>[X0: $tType]:(((X0 > $o) > $o))).
thf(func_def_40, type, vOR: ($o > $o > $o)).
thf(func_def_41, type, vNOT: ($o > $o)).
thf(func_def_42, type, db3: !>[X0: $tType]:(X0)).
thf(func_def_43, type, db2: !>[X0: $tType]:(X0)).
thf(func_def_44, type, db4: !>[X0: $tType]:(X0)).
thf(func_def_45, type, vEQ: !>[X0: $tType]:((X0 > X0 > $o))).
thf(func_def_48, type, sK0: (mu > d_mu)).
thf(func_def_49, type, sK1: ($i > d_unsorted)).
thf(func_def_50, type, sK2: ($i > $o)).
thf(func_def_51, type, sK3: ($i > $o)).
thf(f1,axiom,(
  (mforall_prop = (^[X7 : (($i > $o) > $i > $o), X8 : $i] : (! [X12 : ($i > $o)] : (X7 @ X12 @ X8)))) & ((^[X13 : ($i > $i > $o)] : (! [X15 : $i] : ~ ! [X16 : $i] : ~(X13 @ X15 @ X16))) = mserial) & ((^[X13 : ($i > $i > $o)] : (! [X15 : $i,X16 : $i] : (~(X13 @ X15 @ X16) | ~ ! [X18 : $i] : (~(X13 @ X15 @ X18) | ~(X13 @ X18 @ X16))))) = mweakly_dense) & ((^[X13 : ($i > $i > $o)] : (! [X15 : $i,X17 : $i,X16 : $i] : (~(X13 @ X16 @ X17) | ~(X13 @ X15 @ X16) | (X13 @ X15 @ X17)))) = mtransitive) & (mimplied = (^[X7 : ($i > $o), X9 : ($i > $o), X10 : $i] : (~(X9 @ X10) | (X7 @ X10)))) & ((^[X7 : ($i > $o)] : (! [X8 : $i] : ~(X7 @ X8))) = minvalid) & (mforall_ind = (^[X7 : (mu > $i > $o), X8 : $i] : (! [X11 : mu] : (X7 @ X11 @ X8)))) & ((^[X13 : ($i > $i > $o)] : (! [X15 : $i] : ~ ! [X16 : $i] : (~ ! [X17 : $i] : (~(X13 @ X15 @ X17) | (X16 = X17)) | ~(X13 @ X15 @ X16)))) = mfunctional) & ! [X1 : d_unsorted,X2 : d_unsorted] : ((((d2unsorted @ X1)) = ((d2unsorted @ X2))) => (X1 = X2)) & ! [X0 : d_unsorted] : (X0 = d_unsorted_0) & ((^[X7 : (($i > $o) > $i > $o), X10 : $i] : (~ ! [X12 : ($i > $o)] : ~(X7 @ X12 @ X10))) = mexists_prop) & ! [X17 : $i] : ? [X0 : d_unsorted] : (X17 = ((d2unsorted @ X0))) & (meq_ind @ (d2mu @ d_mu_0) @ (d2mu @ d_mu_0) @ (d2unsorted @ d_unsorted_0)) & ((^[X13 : ($i > $i > $o)] : (! [X17 : $i,X15 : $i,X16 : $i] : ((X13 @ X17 @ X16) | (X16 = X17) | ~(X13 @ X15 @ X16) | ~(X13 @ X15 @ X17) | (X13 @ X16 @ X17)))) = mweakly_connected) & ! [X5 : d_mu,X6 : d_mu] : ((((d2mu @ X5)) = ((d2mu @ X6))) => (X5 = X6)) & ((^[X7 : ($i > $o), X9 : ($i > $o), X10 : $i] : (~(~(X7 @ X10) | ~(X9 @ X10)))) = mand) & (mtrue @ (d2unsorted @ d_unsorted_0)) & ((^[X7 : (mu > $i > $o), X10 : $i] : (~ ! [X11 : mu] : ~(X7 @ X11 @ X10))) = mexists_ind) & ((^[X13 : ($i > $i > $o)] : (! [X15 : $i,X16 : $i] : ((X13 @ X16 @ X15) | ~(X13 @ X15 @ X16)))) = msymmetric) & ((^[X7 : ($i > $o), X8 : $i] : (~(X7 @ X8))) = mnot) & ((^[X7 : ($i > $o)] : (~ ! [X8 : $i] : (X7 @ X8))) = mcountersatisfiable) & (mreflexive = (^[X13 : ($i > $i > $o)] : (! [X15 : $i] : (X13 @ X15 @ X15)))) & (mbox = (^[X13 : ($i > $i > $o), X7 : ($i > $o), X8 : $i] : (! [X14 : $i] : ((X7 @ X14) | ~(X13 @ X8 @ X14))))) & (mpartially_functional = (^[X13 : ($i > $i > $o)] : (! [X17 : $i,X15 : $i,X16 : $i] : ((X16 = X17) | ~(X13 @ X15 @ X17) | ~(X13 @ X15 @ X16))))) & ~(mfalse @ (d2unsorted @ d_unsorted_0)) & ! [X3 : mu] : ? [X4 : d_mu] : (X3 = ((d2mu @ X4))) & ((^[X7 : ($i > $o), X9 : ($i > $o), X10 : $i] : ((X9 @ X10) | ~(X7 @ X10))) = mimplies) & ((^[X7 : ($i > $o), X9 : ($i > $o), X10 : $i] : (~((X9 @ X10) | ~(X7 @ X10)) | ~((X7 @ X10) | ~(X9 @ X10)))) = mxor) & (mdia = (^[X13 : ($i > $i > $o), X7 : ($i > $o), X10 : $i] : (~ ! [X14 : $i] : (~(X13 @ X10 @ X14) | ~(X7 @ X14))))) & (meq_prop = (^[X11 : ($i > $o), X19 : ($i > $o), X8 : $i] : ((((X11 @ X8)) = ((X19 @ X8)))))) & ((^[X7 : ($i > $o), X9 : ($i > $o), X10 : $i] : (~(~((X7 @ X10) | ~(X9 @ X10)) | ~(~(X7 @ X10) | (X9 @ X10))))) = mequiv) & (mor = (^[X7 : ($i > $o), X9 : ($i > $o), X8 : $i] : ((X9 @ X8) | (X7 @ X8)))) & ((^[X13 : ($i > $i > $o)] : (! [X16 : $i,X17 : $i,X15 : $i] : (~(X13 @ X15 @ X16) | ~(X13 @ X15 @ X17) | (X13 @ X16 @ X17)))) = meuclidean) & (msatisfiable = (^[X7 : ($i > $o)] : (~ ! [X8 : $i] : ~(X7 @ X8)))) & ((^[X7 : ($i > $o)] : (! [X8 : $i] : (X7 @ X8))) = mvalid) & ! [X4 : d_mu] : (X4 = d_mu_0) & ((^[X13 : ($i > $i > $o)] : (! [X16 : $i,X17 : $i,X15 : $i] : (~ ! [X14 : $i] : (~(X13 @ X16 @ X14) | ~(X13 @ X17 @ X14)) | ~(X13 @ X15 @ X16) | ~(X13 @ X15 @ X17)))) = mweakly_directed)),
  file('/export/starexec/sandbox/benchmark/theBenchmark.p',lcl698_1)).
thf(f2,conjecture,(
  ((^[X0 : ($i > $o), X1 : ($i > $o)] : ((mand @ (mimplies @ X0 @ X1) @ (mimplies @ X1 @ X0)))) = mequiv)),
  file('/export/starexec/sandbox/benchmark/theBenchmark.p',mequiv)).
thf(f3,negated_conjecture,(
  ~ ((^[X0 : ($i > $o), X1 : ($i > $o)] : ((mand @ (mimplies @ X0 @ X1) @ (mimplies @ X1 @ X0)))) = mequiv)),
  inference(negated_conjecture,[status(cth)],[f2])).
thf(f4,plain,(
  ~ (mequiv = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (mand @ (mimplies @ Y0 @ Y1) @ (mimplies @ Y1 @ Y0))))))),
  inference(fool_elimination,[],[f3])).
thf(f5,plain,(
  (mforall_prop = (^[X0 : (($i > $o) > $i > $o), X1 : $i] : (! [X2 : ($i > $o)] : (X0 @ X2 @ X1)))) & ((^[X3 : ($i > $i > $o)] : (! [X4 : $i] : ~ ! [X5 : $i] : ~(X3 @ X4 @ X5))) = mserial) & ((^[X6 : ($i > $i > $o)] : (! [X7 : $i,X8 : $i] : (~(X6 @ X7 @ X8) | ~ ! [X9 : $i] : (~(X6 @ X7 @ X9) | ~(X6 @ X9 @ X8))))) = mweakly_dense) & ((^[X10 : ($i > $i > $o)] : (! [X11 : $i,X12 : $i,X13 : $i] : (~(X10 @ X13 @ X12) | ~(X10 @ X11 @ X13) | (X10 @ X11 @ X12)))) = mtransitive) & (mimplied = (^[X14 : ($i > $o), X15 : ($i > $o), X16 : $i] : (~(X15 @ X16) | (X14 @ X16)))) & ((^[X17 : ($i > $o)] : (! [X18 : $i] : ~(X17 @ X18))) = minvalid) & (mforall_ind = (^[X19 : (mu > $i > $o), X20 : $i] : (! [X21 : mu] : (X19 @ X21 @ X20)))) & ((^[X22 : ($i > $i > $o)] : (! [X23 : $i] : ~ ! [X24 : $i] : (~ ! [X25 : $i] : (~(X22 @ X23 @ X25) | (X24 = X25)) | ~(X22 @ X23 @ X24)))) = mfunctional) & ! [X26 : d_unsorted,X27 : d_unsorted] : ((((d2unsorted @ X26)) = ((d2unsorted @ X27))) => (X26 = X27)) & ! [X28 : d_unsorted] : (d_unsorted_0 = X28) & ((^[X29 : (($i > $o) > $i > $o), X30 : $i] : (~ ! [X31 : ($i > $o)] : ~(X29 @ X31 @ X30))) = mexists_prop) & ! [X32 : $i] : ? [X33 : d_unsorted] : (((d2unsorted @ X33)) = X32) & (meq_ind @ (d2mu @ d_mu_0) @ (d2mu @ d_mu_0) @ (d2unsorted @ d_unsorted_0)) & ((^[X34 : ($i > $i > $o)] : (! [X35 : $i,X36 : $i,X37 : $i] : ((X34 @ X35 @ X37) | (X35 = X37) | ~(X34 @ X36 @ X37) | ~(X34 @ X36 @ X35) | (X34 @ X37 @ X35)))) = mweakly_connected) & ! [X38 : d_mu,X39 : d_mu] : ((((d2mu @ X38)) = ((d2mu @ X39))) => (X38 = X39)) & ((^[X40 : ($i > $o), X41 : ($i > $o), X42 : $i] : (~(~(X40 @ X42) | ~(X41 @ X42)))) = mand) & (mtrue @ (d2unsorted @ d_unsorted_0)) & ((^[X43 : (mu > $i > $o), X44 : $i] : (~ ! [X45 : mu] : ~(X43 @ X45 @ X44))) = mexists_ind) & ((^[X46 : ($i > $i > $o)] : (! [X47 : $i,X48 : $i] : ((X46 @ X48 @ X47) | ~(X46 @ X47 @ X48)))) = msymmetric) & ((^[X49 : ($i > $o), X50 : $i] : (~(X49 @ X50))) = mnot) & ((^[X51 : ($i > $o)] : (~ ! [X52 : $i] : (X51 @ X52))) = mcountersatisfiable) & (mreflexive = (^[X53 : ($i > $i > $o)] : (! [X54 : $i] : (X53 @ X54 @ X54)))) & (mbox = (^[X55 : ($i > $i > $o), X56 : ($i > $o), X57 : $i] : (! [X58 : $i] : ((X56 @ X58) | ~(X55 @ X57 @ X58))))) & (mpartially_functional = (^[X59 : ($i > $i > $o)] : (! [X60 : $i,X61 : $i,X62 : $i] : ((X60 = X62) | ~(X59 @ X61 @ X60) | ~(X59 @ X61 @ X62))))) & ~(mfalse @ (d2unsorted @ d_unsorted_0)) & ! [X63 : mu] : ? [X64 : d_mu] : (((d2mu @ X64)) = X63) & ((^[X65 : ($i > $o), X66 : ($i > $o), X67 : $i] : ((X66 @ X67) | ~(X65 @ X67))) = mimplies) & ((^[X68 : ($i > $o), X69 : ($i > $o), X70 : $i] : (~((X69 @ X70) | ~(X68 @ X70)) | ~((X68 @ X70) | ~(X69 @ X70)))) = mxor) & (mdia = (^[X71 : ($i > $i > $o), X72 : ($i > $o), X73 : $i] : (~ ! [X74 : $i] : (~(X71 @ X73 @ X74) | ~(X72 @ X74))))) & (meq_prop = (^[X75 : ($i > $o), X76 : ($i > $o), X77 : $i] : ((((X75 @ X77)) = ((X76 @ X77)))))) & ((^[X78 : ($i > $o), X79 : ($i > $o), X80 : $i] : (~(~((X78 @ X80) | ~(X79 @ X80)) | ~(~(X78 @ X80) | (X79 @ X80))))) = mequiv) & (mor = (^[X81 : ($i > $o), X82 : ($i > $o), X83 : $i] : ((X82 @ X83) | (X81 @ X83)))) & ((^[X84 : ($i > $i > $o)] : (! [X85 : $i,X86 : $i,X87 : $i] : (~(X84 @ X87 @ X85) | ~(X84 @ X87 @ X86) | (X84 @ X85 @ X86)))) = meuclidean) & (msatisfiable = (^[X88 : ($i > $o)] : (~ ! [X89 : $i] : ~(X88 @ X89)))) & ((^[X90 : ($i > $o)] : (! [X91 : $i] : (X90 @ X91))) = mvalid) & ! [X92 : d_mu] : (d_mu_0 = X92) & ((^[X93 : ($i > $i > $o)] : (! [X94 : $i,X95 : $i,X96 : $i] : (~ ! [X97 : $i] : (~(X93 @ X94 @ X97) | ~(X93 @ X95 @ X97)) | ~(X93 @ X96 @ X94) | ~(X93 @ X96 @ X95)))) = mweakly_directed)),
  inference(rectify,[],[f1])).
thf(f6,plain,(
  (mforall_prop = (^[Y0 : ($i > $o) > $i > $o]: ((^[Y1 : $i]: (!! @ ($i > $o) @ (^[Y2 : $i > $o]: (Y0 @ Y2 @ Y1))))))) & (mserial = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (~ (!! @ $i @ (^[Y2 : $i]: (~ (Y0 @ Y1 @ Y2))))))))) & (mweakly_dense = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((~ (!! @ $i @ (^[Y3 : $i]: ((~ (Y0 @ Y3 @ Y1)) | (~ (Y0 @ Y2 @ Y3)))))) | (~ (Y0 @ Y2 @ Y1))))))))) & (mtransitive = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((Y0 @ Y3 @ Y2) | (~ (Y0 @ Y3 @ Y1))) | (~ (Y0 @ Y1 @ Y2))))))))))) & (mimplied = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((Y0 @ Y2) | (~ (Y1 @ Y2))))))))) & (minvalid = (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (~ (Y0 @ Y1)))))) & (mforall_ind = (^[Y0 : mu > $i > $o]: ((^[Y1 : $i]: (!! @ mu @ (^[Y2 : mu]: (Y0 @ Y2 @ Y1))))))) & (mfunctional = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (~ (!! @ $i @ (^[Y2 : $i]: ((~ (Y0 @ Y1 @ Y2)) | (~ (!! @ $i @ (^[Y3 : $i]: ((Y2 = Y3) | (~ (Y0 @ Y1 @ Y3)))))))))))))) & ! [X26 : d_unsorted,X27 : d_unsorted] : ((((d2unsorted @ X26)) = ((d2unsorted @ X27))) => (X26 = X27)) & ! [X28 : d_unsorted] : (d_unsorted_0 = X28) & (mexists_prop = (^[Y0 : ($i > $o) > $i > $o]: ((^[Y1 : $i]: (~ (!! @ ($i > $o) @ (^[Y2 : $i > $o]: (~ (Y0 @ Y2 @ Y1))))))))) & ! [X32 : $i] : ? [X33 : d_unsorted] : (((d2unsorted @ X33)) = X32) & (((meq_ind @ (d2mu @ d_mu_0) @ (d2mu @ d_mu_0) @ (d2unsorted @ d_unsorted_0))) = $true) & (mweakly_connected = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((((Y0 @ Y1 @ Y3) | (~ (Y0 @ Y2 @ Y3))) | (~ (Y0 @ Y2 @ Y1))) | (Y3 = Y1)) | (Y0 @ Y3 @ Y1)))))))))) & ! [X38 : d_mu,X39 : d_mu] : ((((d2mu @ X38)) = ((d2mu @ X39))) => (X38 = X39)) & (mand = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ ((~ (Y1 @ Y2)) | (~ (Y0 @ Y2)))))))))) & (((mtrue @ (d2unsorted @ d_unsorted_0))) = $true) & (mexists_ind = (^[Y0 : mu > $i > $o]: ((^[Y1 : $i]: (~ (!! @ mu @ (^[Y2 : mu]: (~ (Y0 @ Y2 @ Y1))))))))) & (msymmetric = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((~ (Y0 @ Y2 @ Y1)) | (Y0 @ Y1 @ Y2)))))))) & (mnot = (^[Y0 : $i > $o]: ((^[Y1 : $i]: (~ (Y0 @ Y1)))))) & (mcountersatisfiable = (^[Y0 : $i > $o]: (~ (!! @ $i @ (^[Y1 : $i]: (Y0 @ Y1)))))) & (mreflexive = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (Y0 @ Y1 @ Y1))))) & (mbox = (^[Y0 : $i > $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: ((~ (Y0 @ Y2 @ Y3)) | (Y1 @ Y3)))))))))) & (mpartially_functional = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((~ (Y0 @ Y2 @ Y1)) | (~ (Y0 @ Y2 @ Y3))) | (Y3 = Y1)))))))))) & ~ (((mfalse @ (d2unsorted @ d_unsorted_0))) = $true) & ! [X63 : mu] : ? [X64 : d_mu] : (((d2mu @ X64)) = X63) & (mimplies = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((~ (Y0 @ Y2)) | (Y1 @ Y2)))))))) & (mxor = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((~ ((~ (Y1 @ Y2)) | (Y0 @ Y2))) | (~ ((~ (Y0 @ Y2)) | (Y1 @ Y2)))))))))) & (mdia = (^[Y0 : $i > $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ (!! @ $i @ (^[Y3 : $i]: ((~ (Y1 @ Y3)) | (~ (Y0 @ Y2 @ Y3)))))))))))) & (meq_prop = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((Y0 @ Y2) = (Y1 @ Y2)))))))) & (mequiv = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ ((~ ((Y1 @ Y2) | (~ (Y0 @ Y2)))) | (~ ((~ (Y1 @ Y2)) | (Y0 @ Y2))))))))))) & (mor = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((Y0 @ Y2) | (Y1 @ Y2)))))))) & (meuclidean = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((Y0 @ Y3 @ Y2) | (~ (Y0 @ Y1 @ Y2))) | (~ (Y0 @ Y1 @ Y3))))))))))) & (msatisfiable = (^[Y0 : $i > $o]: (~ (!! @ $i @ (^[Y1 : $i]: (~ (Y0 @ Y1))))))) & (mvalid = (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (Y0 @ Y1))))) & ! [X92 : d_mu] : (d_mu_0 = X92) & (mweakly_directed = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((~ (Y0 @ Y1 @ Y2)) | (~ (Y0 @ Y1 @ Y3))) | (~ (!! @ $i @ (^[Y4 : $i]: ((~ (Y0 @ Y2 @ Y4)) | (~ (Y0 @ Y3 @ Y4)))))))))))))))),
  inference(fool_elimination,[],[f5])).
thf(f7,plain,(
  (mequiv != (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (mand @ (mimplies @ Y0 @ Y1) @ (mimplies @ Y1 @ Y0))))))),
  inference(flattening,[],[f4])).
thf(f8,plain,(
  (mforall_prop = (^[Y0 : ($i > $o) > $i > $o]: ((^[Y1 : $i]: (!! @ ($i > $o) @ (^[Y2 : $i > $o]: (Y0 @ Y2 @ Y1))))))) & (mserial = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (~ (!! @ $i @ (^[Y2 : $i]: (~ (Y0 @ Y1 @ Y2))))))))) & (mweakly_dense = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((~ (!! @ $i @ (^[Y3 : $i]: ((~ (Y0 @ Y3 @ Y1)) | (~ (Y0 @ Y2 @ Y3)))))) | (~ (Y0 @ Y2 @ Y1))))))))) & (mtransitive = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((Y0 @ Y3 @ Y2) | (~ (Y0 @ Y3 @ Y1))) | (~ (Y0 @ Y1 @ Y2))))))))))) & (mimplied = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((Y0 @ Y2) | (~ (Y1 @ Y2))))))))) & (minvalid = (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (~ (Y0 @ Y1)))))) & (mforall_ind = (^[Y0 : mu > $i > $o]: ((^[Y1 : $i]: (!! @ mu @ (^[Y2 : mu]: (Y0 @ Y2 @ Y1))))))) & (mfunctional = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (~ (!! @ $i @ (^[Y2 : $i]: ((~ (Y0 @ Y1 @ Y2)) | (~ (!! @ $i @ (^[Y3 : $i]: ((Y2 = Y3) | (~ (Y0 @ Y1 @ Y3)))))))))))))) & ! [X0 : d_unsorted,X1 : d_unsorted] : ((((d2unsorted @ X1)) = ((d2unsorted @ X0))) => (X0 = X1)) & ! [X2 : d_unsorted] : (d_unsorted_0 = X2) & (mexists_prop = (^[Y0 : ($i > $o) > $i > $o]: ((^[Y1 : $i]: (~ (!! @ ($i > $o) @ (^[Y2 : $i > $o]: (~ (Y0 @ Y2 @ Y1))))))))) & ! [X3 : $i] : ? [X4 : d_unsorted] : (((d2unsorted @ X4)) = X3) & (((meq_ind @ (d2mu @ d_mu_0) @ (d2mu @ d_mu_0) @ (d2unsorted @ d_unsorted_0))) = $true) & (mweakly_connected = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((((Y0 @ Y1 @ Y3) | (~ (Y0 @ Y2 @ Y3))) | (~ (Y0 @ Y2 @ Y1))) | (Y3 = Y1)) | (Y0 @ Y3 @ Y1)))))))))) & ! [X5 : d_mu,X6 : d_mu] : ((((d2mu @ X5)) = ((d2mu @ X6))) => (X5 = X6)) & (mand = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ ((~ (Y1 @ Y2)) | (~ (Y0 @ Y2)))))))))) & (((mtrue @ (d2unsorted @ d_unsorted_0))) = $true) & (mexists_ind = (^[Y0 : mu > $i > $o]: ((^[Y1 : $i]: (~ (!! @ mu @ (^[Y2 : mu]: (~ (Y0 @ Y2 @ Y1))))))))) & (msymmetric = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((~ (Y0 @ Y2 @ Y1)) | (Y0 @ Y1 @ Y2)))))))) & (mnot = (^[Y0 : $i > $o]: ((^[Y1 : $i]: (~ (Y0 @ Y1)))))) & (mcountersatisfiable = (^[Y0 : $i > $o]: (~ (!! @ $i @ (^[Y1 : $i]: (Y0 @ Y1)))))) & (mreflexive = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (Y0 @ Y1 @ Y1))))) & (mbox = (^[Y0 : $i > $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: ((~ (Y0 @ Y2 @ Y3)) | (Y1 @ Y3)))))))))) & (mpartially_functional = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((~ (Y0 @ Y2 @ Y1)) | (~ (Y0 @ Y2 @ Y3))) | (Y3 = Y1)))))))))) & ~ (((mfalse @ (d2unsorted @ d_unsorted_0))) = $true) & ! [X7 : mu] : ? [X8 : d_mu] : (((d2mu @ X8)) = X7) & (mimplies = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((~ (Y0 @ Y2)) | (Y1 @ Y2)))))))) & (mxor = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((~ ((~ (Y1 @ Y2)) | (Y0 @ Y2))) | (~ ((~ (Y0 @ Y2)) | (Y1 @ Y2)))))))))) & (mdia = (^[Y0 : $i > $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ (!! @ $i @ (^[Y3 : $i]: ((~ (Y1 @ Y3)) | (~ (Y0 @ Y2 @ Y3)))))))))))) & (meq_prop = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((Y0 @ Y2) = (Y1 @ Y2)))))))) & (mequiv = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ ((~ ((Y1 @ Y2) | (~ (Y0 @ Y2)))) | (~ ((~ (Y1 @ Y2)) | (Y0 @ Y2))))))))))) & (mor = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((Y0 @ Y2) | (Y1 @ Y2)))))))) & (meuclidean = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((Y0 @ Y3 @ Y2) | (~ (Y0 @ Y1 @ Y2))) | (~ (Y0 @ Y1 @ Y3))))))))))) & (msatisfiable = (^[Y0 : $i > $o]: (~ (!! @ $i @ (^[Y1 : $i]: (~ (Y0 @ Y1))))))) & (mvalid = (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (Y0 @ Y1))))) & ! [X9 : d_mu] : (d_mu_0 = X9) & (mweakly_directed = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((~ (Y0 @ Y1 @ Y2)) | (~ (Y0 @ Y1 @ Y3))) | (~ (!! @ $i @ (^[Y4 : $i]: ((~ (Y0 @ Y2 @ Y4)) | (~ (Y0 @ Y3 @ Y4)))))))))))))))),
  inference(rectify,[],[f6])).
thf(f9,plain,(
  (mforall_ind = (^[Y0 : mu > $i > $o]: ((^[Y1 : $i]: (!! @ mu @ (^[Y2 : mu]: (Y0 @ Y2 @ Y1))))))) & (mor = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((Y0 @ Y2) | (Y1 @ Y2)))))))) & (mequiv = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ ((~ ((Y1 @ Y2) | (~ (Y0 @ Y2)))) | (~ ((~ (Y1 @ Y2)) | (Y0 @ Y2))))))))))) & (mcountersatisfiable = (^[Y0 : $i > $o]: (~ (!! @ $i @ (^[Y1 : $i]: (Y0 @ Y1)))))) & (mexists_prop = (^[Y0 : ($i > $o) > $i > $o]: ((^[Y1 : $i]: (~ (!! @ ($i > $o) @ (^[Y2 : $i > $o]: (~ (Y0 @ Y2 @ Y1))))))))) & ! [X0 : d_unsorted,X1 : d_unsorted] : ((((d2unsorted @ X1)) = ((d2unsorted @ X0))) => (X0 = X1)) & (((meq_ind @ (d2mu @ d_mu_0) @ (d2mu @ d_mu_0) @ (d2unsorted @ d_unsorted_0))) = $true) & (mserial = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (~ (!! @ $i @ (^[Y2 : $i]: (~ (Y0 @ Y1 @ Y2))))))))) & (mpartially_functional = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((~ (Y0 @ Y2 @ Y1)) | (~ (Y0 @ Y2 @ Y3))) | (Y3 = Y1)))))))))) & (mreflexive = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (Y0 @ Y1 @ Y1))))) & (mimplied = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((Y0 @ Y2) | (~ (Y1 @ Y2))))))))) & (mweakly_connected = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((((Y0 @ Y1 @ Y3) | (~ (Y0 @ Y2 @ Y3))) | (~ (Y0 @ Y2 @ Y1))) | (Y3 = Y1)) | (Y0 @ Y3 @ Y1)))))))))) & ! [X5 : d_mu,X6 : d_mu] : ((((d2mu @ X5)) = ((d2mu @ X6))) => (X5 = X6)) & (mexists_ind = (^[Y0 : mu > $i > $o]: ((^[Y1 : $i]: (~ (!! @ mu @ (^[Y2 : mu]: (~ (Y0 @ Y2 @ Y1))))))))) & (mfunctional = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (~ (!! @ $i @ (^[Y2 : $i]: ((~ (Y0 @ Y1 @ Y2)) | (~ (!! @ $i @ (^[Y3 : $i]: ((Y2 = Y3) | (~ (Y0 @ Y1 @ Y3)))))))))))))) & (mxor = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((~ ((~ (Y1 @ Y2)) | (Y0 @ Y2))) | (~ ((~ (Y0 @ Y2)) | (Y1 @ Y2)))))))))) & (mweakly_dense = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((~ (!! @ $i @ (^[Y3 : $i]: ((~ (Y0 @ Y3 @ Y1)) | (~ (Y0 @ Y2 @ Y3)))))) | (~ (Y0 @ Y2 @ Y1))))))))) & (minvalid = (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (~ (Y0 @ Y1)))))) & (mdia = (^[Y0 : $i > $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ (!! @ $i @ (^[Y3 : $i]: ((~ (Y1 @ Y3)) | (~ (Y0 @ Y2 @ Y3)))))))))))) & (mnot = (^[Y0 : $i > $o]: ((^[Y1 : $i]: (~ (Y0 @ Y1)))))) & (mforall_prop = (^[Y0 : ($i > $o) > $i > $o]: ((^[Y1 : $i]: (!! @ ($i > $o) @ (^[Y2 : $i > $o]: (Y0 @ Y2 @ Y1))))))) & (mtransitive = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((Y0 @ Y3 @ Y2) | (~ (Y0 @ Y3 @ Y1))) | (~ (Y0 @ Y1 @ Y2))))))))))) & (mweakly_directed = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((~ (Y0 @ Y1 @ Y2)) | (~ (Y0 @ Y1 @ Y3))) | (~ (!! @ $i @ (^[Y4 : $i]: ((~ (Y0 @ Y2 @ Y4)) | (~ (Y0 @ Y3 @ Y4))))))))))))))) & (mimplies = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((~ (Y0 @ Y2)) | (Y1 @ Y2)))))))) & (mbox = (^[Y0 : $i > $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: ((~ (Y0 @ Y2 @ Y3)) | (Y1 @ Y3)))))))))) & (mvalid = (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (Y0 @ Y1))))) & (msatisfiable = (^[Y0 : $i > $o]: (~ (!! @ $i @ (^[Y1 : $i]: (~ (Y0 @ Y1))))))) & (meuclidean = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((Y0 @ Y3 @ Y2) | (~ (Y0 @ Y1 @ Y2))) | (~ (Y0 @ Y1 @ Y3))))))))))) & (((mtrue @ (d2unsorted @ d_unsorted_0))) = $true) & ! [X7 : mu] : ? [X8 : d_mu] : (((d2mu @ X8)) = X7) & (((mfalse @ (d2unsorted @ d_unsorted_0))) != $true) & ! [X9 : d_mu] : (d_mu_0 = X9) & (mand = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ ((~ (Y1 @ Y2)) | (~ (Y0 @ Y2)))))))))) & (msymmetric = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((~ (Y0 @ Y2 @ Y1)) | (Y0 @ Y1 @ Y2)))))))) & ! [X2 : d_unsorted] : (d_unsorted_0 = X2) & ! [X3 : $i] : ? [X4 : d_unsorted] : (((d2unsorted @ X4)) = X3) & (meq_prop = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((Y0 @ Y2) = (Y1 @ Y2))))))))),
  inference(flattening,[],[f8])).
thf(f10,plain,(
  (mand = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ ((~ (Y1 @ Y2)) | (~ (Y0 @ Y2)))))))))) & (mtransitive = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((Y0 @ Y3 @ Y2) | (~ (Y0 @ Y3 @ Y1))) | (~ (Y0 @ Y1 @ Y2))))))))))) & (meuclidean = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((Y0 @ Y3 @ Y2) | (~ (Y0 @ Y1 @ Y2))) | (~ (Y0 @ Y1 @ Y3))))))))))) & (mbox = (^[Y0 : $i > $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: ((~ (Y0 @ Y2 @ Y3)) | (Y1 @ Y3)))))))))) & (mexists_prop = (^[Y0 : ($i > $o) > $i > $o]: ((^[Y1 : $i]: (~ (!! @ ($i > $o) @ (^[Y2 : $i > $o]: (~ (Y0 @ Y2 @ Y1))))))))) & (mequiv = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ ((~ ((Y1 @ Y2) | (~ (Y0 @ Y2)))) | (~ ((~ (Y1 @ Y2)) | (Y0 @ Y2))))))))))) & (mimplied = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((Y0 @ Y2) | (~ (Y1 @ Y2))))))))) & (mweakly_dense = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((~ (!! @ $i @ (^[Y3 : $i]: ((~ (Y0 @ Y3 @ Y1)) | (~ (Y0 @ Y2 @ Y3)))))) | (~ (Y0 @ Y2 @ Y1))))))))) & ! [X5 : d_mu,X6 : d_mu] : ((((d2mu @ X5)) != ((d2mu @ X6))) | (X5 = X6)) & (mweakly_connected = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((((Y0 @ Y1 @ Y3) | (~ (Y0 @ Y2 @ Y3))) | (~ (Y0 @ Y2 @ Y1))) | (Y3 = Y1)) | (Y0 @ Y3 @ Y1)))))))))) & ! [X9 : d_mu] : (d_mu_0 = X9) & (((mtrue @ (d2unsorted @ d_unsorted_0))) = $true) & (mor = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((Y0 @ Y2) | (Y1 @ Y2)))))))) & (mexists_ind = (^[Y0 : mu > $i > $o]: ((^[Y1 : $i]: (~ (!! @ mu @ (^[Y2 : mu]: (~ (Y0 @ Y2 @ Y1))))))))) & ! [X7 : mu] : ? [X8 : d_mu] : (((d2mu @ X8)) = X7) & (mserial = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (~ (!! @ $i @ (^[Y2 : $i]: (~ (Y0 @ Y1 @ Y2))))))))) & (mweakly_directed = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((~ (Y0 @ Y1 @ Y2)) | (~ (Y0 @ Y1 @ Y3))) | (~ (!! @ $i @ (^[Y4 : $i]: ((~ (Y0 @ Y2 @ Y4)) | (~ (Y0 @ Y3 @ Y4))))))))))))))) & (meq_prop = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((Y0 @ Y2) = (Y1 @ Y2)))))))) & ! [X3 : $i] : ? [X4 : d_unsorted] : (((d2unsorted @ X4)) = X3) & (msatisfiable = (^[Y0 : $i > $o]: (~ (!! @ $i @ (^[Y1 : $i]: (~ (Y0 @ Y1))))))) & (mdia = (^[Y0 : $i > $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ (!! @ $i @ (^[Y3 : $i]: ((~ (Y1 @ Y3)) | (~ (Y0 @ Y2 @ Y3)))))))))))) & (((meq_ind @ (d2mu @ d_mu_0) @ (d2mu @ d_mu_0) @ (d2unsorted @ d_unsorted_0))) = $true) & (mfunctional = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (~ (!! @ $i @ (^[Y2 : $i]: ((~ (Y0 @ Y1 @ Y2)) | (~ (!! @ $i @ (^[Y3 : $i]: ((Y2 = Y3) | (~ (Y0 @ Y1 @ Y3)))))))))))))) & (mimplies = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((~ (Y0 @ Y2)) | (Y1 @ Y2)))))))) & (minvalid = (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (~ (Y0 @ Y1)))))) & (((mfalse @ (d2unsorted @ d_unsorted_0))) != $true) & (mpartially_functional = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((~ (Y0 @ Y2 @ Y1)) | (~ (Y0 @ Y2 @ Y3))) | (Y3 = Y1)))))))))) & (msymmetric = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((~ (Y0 @ Y2 @ Y1)) | (Y0 @ Y1 @ Y2)))))))) & ! [X1 : d_unsorted,X0 : d_unsorted] : ((((d2unsorted @ X1)) != ((d2unsorted @ X0))) | (X0 = X1)) & (mxor = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((~ ((~ (Y1 @ Y2)) | (Y0 @ Y2))) | (~ ((~ (Y0 @ Y2)) | (Y1 @ Y2)))))))))) & (mreflexive = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (Y0 @ Y1 @ Y1))))) & ! [X2 : d_unsorted] : (d_unsorted_0 = X2) & (mvalid = (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (Y0 @ Y1))))) & (mforall_prop = (^[Y0 : ($i > $o) > $i > $o]: ((^[Y1 : $i]: (!! @ ($i > $o) @ (^[Y2 : $i > $o]: (Y0 @ Y2 @ Y1))))))) & (mcountersatisfiable = (^[Y0 : $i > $o]: (~ (!! @ $i @ (^[Y1 : $i]: (Y0 @ Y1)))))) & (mforall_ind = (^[Y0 : mu > $i > $o]: ((^[Y1 : $i]: (!! @ mu @ (^[Y2 : mu]: (Y0 @ Y2 @ Y1))))))) & (mnot = (^[Y0 : $i > $o]: ((^[Y1 : $i]: (~ (Y0 @ Y1))))))),
  inference(ennf_transformation,[],[f9])).
thf(f11,plain,(
  (mand = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ ((~ (Y1 @ Y2)) | (~ (Y0 @ Y2)))))))))) & (mtransitive = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((Y0 @ Y3 @ Y2) | (~ (Y0 @ Y3 @ Y1))) | (~ (Y0 @ Y1 @ Y2))))))))))) & (meuclidean = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((Y0 @ Y3 @ Y2) | (~ (Y0 @ Y1 @ Y2))) | (~ (Y0 @ Y1 @ Y3))))))))))) & (mbox = (^[Y0 : $i > $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: ((~ (Y0 @ Y2 @ Y3)) | (Y1 @ Y3)))))))))) & (mexists_prop = (^[Y0 : ($i > $o) > $i > $o]: ((^[Y1 : $i]: (~ (!! @ ($i > $o) @ (^[Y2 : $i > $o]: (~ (Y0 @ Y2 @ Y1))))))))) & (mequiv = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ ((~ ((Y1 @ Y2) | (~ (Y0 @ Y2)))) | (~ ((~ (Y1 @ Y2)) | (Y0 @ Y2))))))))))) & (mimplied = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((Y0 @ Y2) | (~ (Y1 @ Y2))))))))) & (mweakly_dense = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((~ (!! @ $i @ (^[Y3 : $i]: ((~ (Y0 @ Y3 @ Y1)) | (~ (Y0 @ Y2 @ Y3)))))) | (~ (Y0 @ Y2 @ Y1))))))))) & ! [X0 : d_mu,X1 : d_mu] : ((((d2mu @ X1)) != ((d2mu @ X0))) | (X0 = X1)) & (mweakly_connected = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((((Y0 @ Y1 @ Y3) | (~ (Y0 @ Y2 @ Y3))) | (~ (Y0 @ Y2 @ Y1))) | (Y3 = Y1)) | (Y0 @ Y3 @ Y1)))))))))) & ! [X2 : d_mu] : (d_mu_0 = X2) & (((mtrue @ (d2unsorted @ d_unsorted_0))) = $true) & (mor = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((Y0 @ Y2) | (Y1 @ Y2)))))))) & (mexists_ind = (^[Y0 : mu > $i > $o]: ((^[Y1 : $i]: (~ (!! @ mu @ (^[Y2 : mu]: (~ (Y0 @ Y2 @ Y1))))))))) & ! [X3 : mu] : ? [X4 : d_mu] : (((d2mu @ X4)) = X3) & (mserial = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (~ (!! @ $i @ (^[Y2 : $i]: (~ (Y0 @ Y1 @ Y2))))))))) & (mweakly_directed = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((~ (Y0 @ Y1 @ Y2)) | (~ (Y0 @ Y1 @ Y3))) | (~ (!! @ $i @ (^[Y4 : $i]: ((~ (Y0 @ Y2 @ Y4)) | (~ (Y0 @ Y3 @ Y4))))))))))))))) & (meq_prop = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((Y0 @ Y2) = (Y1 @ Y2)))))))) & ! [X5 : $i] : ? [X6 : d_unsorted] : (((d2unsorted @ X6)) = X5) & (msatisfiable = (^[Y0 : $i > $o]: (~ (!! @ $i @ (^[Y1 : $i]: (~ (Y0 @ Y1))))))) & (mdia = (^[Y0 : $i > $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ (!! @ $i @ (^[Y3 : $i]: ((~ (Y1 @ Y3)) | (~ (Y0 @ Y2 @ Y3)))))))))))) & (((meq_ind @ (d2mu @ d_mu_0) @ (d2mu @ d_mu_0) @ (d2unsorted @ d_unsorted_0))) = $true) & (mfunctional = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (~ (!! @ $i @ (^[Y2 : $i]: ((~ (Y0 @ Y1 @ Y2)) | (~ (!! @ $i @ (^[Y3 : $i]: ((Y2 = Y3) | (~ (Y0 @ Y1 @ Y3)))))))))))))) & (mimplies = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((~ (Y0 @ Y2)) | (Y1 @ Y2)))))))) & (minvalid = (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (~ (Y0 @ Y1)))))) & (((mfalse @ (d2unsorted @ d_unsorted_0))) != $true) & (mpartially_functional = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((~ (Y0 @ Y2 @ Y1)) | (~ (Y0 @ Y2 @ Y3))) | (Y3 = Y1)))))))))) & (msymmetric = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((~ (Y0 @ Y2 @ Y1)) | (Y0 @ Y1 @ Y2)))))))) & ! [X7 : d_unsorted,X8 : d_unsorted] : ((((d2unsorted @ X7)) != ((d2unsorted @ X8))) | (X7 = X8)) & (mxor = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((~ ((~ (Y1 @ Y2)) | (Y0 @ Y2))) | (~ ((~ (Y0 @ Y2)) | (Y1 @ Y2)))))))))) & (mreflexive = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (Y0 @ Y1 @ Y1))))) & ! [X9 : d_unsorted] : (d_unsorted_0 = X9) & (mvalid = (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (Y0 @ Y1))))) & (mforall_prop = (^[Y0 : ($i > $o) > $i > $o]: ((^[Y1 : $i]: (!! @ ($i > $o) @ (^[Y2 : $i > $o]: (Y0 @ Y2 @ Y1))))))) & (mcountersatisfiable = (^[Y0 : $i > $o]: (~ (!! @ $i @ (^[Y1 : $i]: (Y0 @ Y1)))))) & (mforall_ind = (^[Y0 : mu > $i > $o]: ((^[Y1 : $i]: (!! @ mu @ (^[Y2 : mu]: (Y0 @ Y2 @ Y1))))))) & (mnot = (^[Y0 : $i > $o]: ((^[Y1 : $i]: (~ (Y0 @ Y1))))))),
  inference(rectify,[],[f10])).
thf(f12,plain,(
  (mand = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ ((~ (Y1 @ Y2)) | (~ (Y0 @ Y2)))))))))) & (mtransitive = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((Y0 @ Y3 @ Y2) | (~ (Y0 @ Y3 @ Y1))) | (~ (Y0 @ Y1 @ Y2))))))))))) & (meuclidean = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((Y0 @ Y3 @ Y2) | (~ (Y0 @ Y1 @ Y2))) | (~ (Y0 @ Y1 @ Y3))))))))))) & (mbox = (^[Y0 : $i > $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: ((~ (Y0 @ Y2 @ Y3)) | (Y1 @ Y3)))))))))) & (mexists_prop = (^[Y0 : ($i > $o) > $i > $o]: ((^[Y1 : $i]: (~ (!! @ ($i > $o) @ (^[Y2 : $i > $o]: (~ (Y0 @ Y2 @ Y1))))))))) & (mequiv = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ ((~ ((Y1 @ Y2) | (~ (Y0 @ Y2)))) | (~ ((~ (Y1 @ Y2)) | (Y0 @ Y2))))))))))) & (mimplied = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((Y0 @ Y2) | (~ (Y1 @ Y2))))))))) & (mweakly_dense = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((~ (!! @ $i @ (^[Y3 : $i]: ((~ (Y0 @ Y3 @ Y1)) | (~ (Y0 @ Y2 @ Y3)))))) | (~ (Y0 @ Y2 @ Y1))))))))) & ! [X0 : d_mu,X1 : d_mu] : ((((d2mu @ X1)) != ((d2mu @ X0))) | (X0 = X1)) & (mweakly_connected = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((((Y0 @ Y1 @ Y3) | (~ (Y0 @ Y2 @ Y3))) | (~ (Y0 @ Y2 @ Y1))) | (Y3 = Y1)) | (Y0 @ Y3 @ Y1)))))))))) & ! [X2 : d_mu] : (d_mu_0 = X2) & (((mtrue @ (d2unsorted @ d_unsorted_0))) = $true) & (mor = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((Y0 @ Y2) | (Y1 @ Y2)))))))) & (mexists_ind = (^[Y0 : mu > $i > $o]: ((^[Y1 : $i]: (~ (!! @ mu @ (^[Y2 : mu]: (~ (Y0 @ Y2 @ Y1))))))))) & ! [X3 : mu] : (((d2mu @ (sK0 @ X3))) = X3) & (mserial = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (~ (!! @ $i @ (^[Y2 : $i]: (~ (Y0 @ Y1 @ Y2))))))))) & (mweakly_directed = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((~ (Y0 @ Y1 @ Y2)) | (~ (Y0 @ Y1 @ Y3))) | (~ (!! @ $i @ (^[Y4 : $i]: ((~ (Y0 @ Y2 @ Y4)) | (~ (Y0 @ Y3 @ Y4))))))))))))))) & (meq_prop = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((Y0 @ Y2) = (Y1 @ Y2)))))))) & ! [X5 : $i] : (((d2unsorted @ (sK1 @ X5))) = X5) & (msatisfiable = (^[Y0 : $i > $o]: (~ (!! @ $i @ (^[Y1 : $i]: (~ (Y0 @ Y1))))))) & (mdia = (^[Y0 : $i > $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ (!! @ $i @ (^[Y3 : $i]: ((~ (Y1 @ Y3)) | (~ (Y0 @ Y2 @ Y3)))))))))))) & (((meq_ind @ (d2mu @ d_mu_0) @ (d2mu @ d_mu_0) @ (d2unsorted @ d_unsorted_0))) = $true) & (mfunctional = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (~ (!! @ $i @ (^[Y2 : $i]: ((~ (Y0 @ Y1 @ Y2)) | (~ (!! @ $i @ (^[Y3 : $i]: ((Y2 = Y3) | (~ (Y0 @ Y1 @ Y3)))))))))))))) & (mimplies = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((~ (Y0 @ Y2)) | (Y1 @ Y2)))))))) & (minvalid = (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (~ (Y0 @ Y1)))))) & (((mfalse @ (d2unsorted @ d_unsorted_0))) != $true) & (mpartially_functional = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: (!! @ $i @ (^[Y3 : $i]: (((~ (Y0 @ Y2 @ Y1)) | (~ (Y0 @ Y2 @ Y3))) | (Y3 = Y1)))))))))) & (msymmetric = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (!! @ $i @ (^[Y2 : $i]: ((~ (Y0 @ Y2 @ Y1)) | (Y0 @ Y1 @ Y2)))))))) & ! [X7 : d_unsorted,X8 : d_unsorted] : ((((d2unsorted @ X7)) != ((d2unsorted @ X8))) | (X7 = X8)) & (mxor = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((~ ((~ (Y1 @ Y2)) | (Y0 @ Y2))) | (~ ((~ (Y0 @ Y2)) | (Y1 @ Y2)))))))))) & (mreflexive = (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (Y0 @ Y1 @ Y1))))) & ! [X9 : d_unsorted] : (d_unsorted_0 = X9) & (mvalid = (^[Y0 : $i > $o]: (!! @ $i @ (^[Y1 : $i]: (Y0 @ Y1))))) & (mforall_prop = (^[Y0 : ($i > $o) > $i > $o]: ((^[Y1 : $i]: (!! @ ($i > $o) @ (^[Y2 : $i > $o]: (Y0 @ Y2 @ Y1))))))) & (mcountersatisfiable = (^[Y0 : $i > $o]: (~ (!! @ $i @ (^[Y1 : $i]: (Y0 @ Y1)))))) & (mforall_ind = (^[Y0 : mu > $i > $o]: ((^[Y1 : $i]: (!! @ mu @ (^[Y2 : mu]: (Y0 @ Y2 @ Y1))))))) & (mnot = (^[Y0 : $i > $o]: ((^[Y1 : $i]: (~ (Y0 @ Y1))))))),
  inference(skolemize,[status(esa),new_symbols(skolem,[vAPP]),skolemize(X6,$thf(sK1 @ X5)),skolemize(X6,$thf(sK1 @ X5))],[f11])).
thf(f26,plain,(
  (mimplies = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: ((~ (Y0 @ Y2)) | (Y1 @ Y2))))))))),
  inference(cnf_transformation,[],[f12])).
thf(f44,plain,(
  (mequiv = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ ((~ ((Y1 @ Y2) | (~ (Y0 @ Y2)))) | (~ ((~ (Y1 @ Y2)) | (Y0 @ Y2)))))))))))),
  inference(cnf_transformation,[],[f12])).
thf(f49,plain,(
  (mand = (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ ((~ (Y1 @ Y2)) | (~ (Y0 @ Y2))))))))))),
  inference(cnf_transformation,[],[f12])).
thf(f50,plain,(
  (mequiv != (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: (mand @ (mimplies @ Y0 @ Y1) @ (mimplies @ Y1 @ Y0))))))),
  inference(cnf_transformation,[],[f7])).
thf(f53,plain,(
  ((^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i > $o]: ((^[Y3 : $i > $o]: ((^[Y4 : $i]: (~ ((~ (Y3 @ Y4)) | (~ (Y2 @ Y4))))))))) @ ((^[Y2 : $i > $o]: ((^[Y3 : $i > $o]: ((^[Y4 : $i]: ((~ (Y2 @ Y4)) | (Y3 @ Y4))))))) @ Y0 @ Y1) @ ((^[Y2 : $i > $o]: ((^[Y3 : $i > $o]: ((^[Y4 : $i]: ((~ (Y2 @ Y4)) | (Y3 @ Y4))))))) @ Y1 @ Y0))))) != (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ ((~ ((Y1 @ Y2) | (~ (Y0 @ Y2)))) | (~ ((~ (Y1 @ Y2)) | (Y0 @ Y2)))))))))))),
  inference(definition_unfolding,[],[f50,f44,f49,f26,f26])).
thf(f54,plain,(
  ((^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ ((~ ((~ (Y1 @ Y2)) | (Y0 @ Y2))) | (~ ((~ (Y0 @ Y2)) | (Y1 @ Y2)))))))))) != (^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ ((~ ((Y1 @ Y2) | (~ (Y0 @ Y2)))) | (~ ((~ (Y1 @ Y2)) | (Y0 @ Y2)))))))))))),
  inference(beta-eta_normalization,[],[f53])).
thf(f56,plain,(
  ((((^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ ((~ ((~ (Y1 @ Y2)) | (Y0 @ Y2))) | (~ ((~ (Y0 @ Y2)) | (Y1 @ Y2)))))))))) @ sK2)) != (((^[Y0 : $i > $o]: ((^[Y1 : $i > $o]: ((^[Y2 : $i]: (~ ((~ ((Y1 @ Y2) | (~ (Y0 @ Y2)))) | (~ ((~ (Y1 @ Y2)) | (Y0 @ Y2)))))))))) @ sK2)))),
  inference(negative_extensionality,[],[f54])).
thf(f57,plain,(
  ((^[Y0 : $i > $o]: ((^[Y1 : $i]: (~ ((~ ((~ (Y0 @ Y1)) | (sK2 @ Y1))) | (~ ((~ (sK2 @ Y1)) | (Y0 @ Y1)))))))) != (^[Y0 : $i > $o]: ((^[Y1 : $i]: (~ ((~ ((Y0 @ Y1) | (~ (sK2 @ Y1)))) | (~ ((~ (Y0 @ Y1)) | (sK2 @ Y1)))))))))),
  inference(beta-eta_normalization,[],[f56])).
thf(f148,plain,(
  ((((^[Y0 : $i > $o]: ((^[Y1 : $i]: (~ ((~ ((~ (Y0 @ Y1)) | (sK2 @ Y1))) | (~ ((~ (sK2 @ Y1)) | (Y0 @ Y1)))))))) @ sK3)) != (((^[Y0 : $i > $o]: ((^[Y1 : $i]: (~ ((~ ((Y0 @ Y1) | (~ (sK2 @ Y1)))) | (~ ((~ (Y0 @ Y1)) | (sK2 @ Y1)))))))) @ sK3)))),
  inference(negative_extensionality,[],[f57])).
thf(f149,plain,(
  ((^[Y0 : $i]: (~ ((~ ((sK3 @ Y0) | (~ (sK2 @ Y0)))) | (~ ((~ (sK3 @ Y0)) | (sK2 @ Y0)))))) != (^[Y0 : $i]: (~ ((~ ((~ (sK3 @ Y0)) | (sK2 @ Y0))) | (~ ((~ (sK2 @ Y0)) | (sK3 @ Y0)))))))),
  inference(beta-eta_normalization,[],[f148])).
thf(f150,plain,(
  ((((^[Y0 : $i]: (~ ((~ ((~ (sK3 @ Y0)) | (sK2 @ Y0))) | (~ ((~ (sK2 @ Y0)) | (sK3 @ Y0)))))) @ sK4)) != (((^[Y0 : $i]: (~ ((~ ((sK3 @ Y0) | (~ (sK2 @ Y0)))) | (~ ((~ (sK3 @ Y0)) | (sK2 @ Y0)))))) @ sK4)))),
  inference(negative_extensionality,[],[f149])).
thf(f151,plain,(
  ((((^[Y0 : $i]: (~ ((~ ((~ (sK3 @ Y0)) | (sK2 @ Y0))) | (~ ((~ (sK2 @ Y0)) | (sK3 @ Y0)))))) @ sK4)) = $true) | ($true = (((^[Y0 : $i]: (~ ((~ ((sK3 @ Y0) | (~ (sK2 @ Y0)))) | (~ ((~ (sK3 @ Y0)) | (sK2 @ Y0)))))) @ sK4)))),
  inference(xor_proxy_clausification,[],[f150])).
thf(f152,plain,(
  ((((^[Y0 : $i]: (~ ((~ ((~ (sK3 @ Y0)) | (sK2 @ Y0))) | (~ ((~ (sK2 @ Y0)) | (sK3 @ Y0)))))) @ sK4)) = $false) | ((((^[Y0 : $i]: (~ ((~ ((sK3 @ Y0) | (~ (sK2 @ Y0)))) | (~ ((~ (sK3 @ Y0)) | (sK2 @ Y0)))))) @ sK4)) = $false)),
  inference(xor_proxy_clausification,[],[f150])).
thf(f153,plain,(
  (((~ ((~ ((sK3 @ sK4) | (~ (sK2 @ sK4)))) | (~ ((~ (sK3 @ sK4)) | (sK2 @ sK4)))))) = $false) | (((~ ((~ ((~ (sK3 @ sK4)) | (sK2 @ sK4))) | (~ ((~ (sK2 @ sK4)) | (sK3 @ sK4)))))) = $false)),
  inference(beta-eta_normalization,[],[f152])).
thf(f154,plain,(
  (((~ ((~ ((~ (sK3 @ sK4)) | (sK2 @ sK4))) | (~ ((~ (sK2 @ sK4)) | (sK3 @ sK4)))))) = $false) | ($true = (((~ ((sK3 @ sK4) | (~ (sK2 @ sK4)))) | (~ ((~ (sK3 @ sK4)) | (sK2 @ sK4))))))),
  inference(not_proxy_clausification,[],[f153])).
thf(f155,plain,(
  ($true = (((~ ((sK3 @ sK4) | (~ (sK2 @ sK4)))) | (~ ((~ (sK3 @ sK4)) | (sK2 @ sK4)))))) | ($true = (((~ ((~ (sK3 @ sK4)) | (sK2 @ sK4))) | (~ ((~ (sK2 @ sK4)) | (sK3 @ sK4))))))),
  inference(not_proxy_clausification,[],[f154])).
thf(f156,plain,(
  ($true = ((~ ((~ (sK3 @ sK4)) | (sK2 @ sK4))))) | ($true = ((~ ((sK3 @ sK4) | (~ (sK2 @ sK4)))))) | ($true = (((~ ((~ (sK3 @ sK4)) | (sK2 @ sK4))) | (~ ((~ (sK2 @ sK4)) | (sK3 @ sK4))))))),
  inference(or_proxy_clausification,[],[f155])).
thf(f157,plain,(
  ($true = ((~ ((sK3 @ sK4) | (~ (sK2 @ sK4)))))) | ($true = (((~ ((~ (sK3 @ sK4)) | (sK2 @ sK4))) | (~ ((~ (sK2 @ sK4)) | (sK3 @ sK4)))))) | ((((~ (sK3 @ sK4)) | (sK2 @ sK4))) = $false)),
  inference(not_proxy_clausification,[],[f156])).
thf(f158,plain,(
  ($true = (((~ ((~ (sK3 @ sK4)) | (sK2 @ sK4))) | (~ ((~ (sK2 @ sK4)) | (sK3 @ sK4)))))) | ((((~ (sK3 @ sK4)) | (sK2 @ sK4))) = $false) | ((((sK3 @ sK4) | (~ (sK2 @ sK4)))) = $false)),
  inference(not_proxy_clausification,[],[f157])).
thf(f159,plain,(
  ($true = ((~ ((~ (sK2 @ sK4)) | (sK3 @ sK4))))) | ((((~ (sK3 @ sK4)) | (sK2 @ sK4))) = $false) | ((((sK3 @ sK4) | (~ (sK2 @ sK4)))) = $false) | ($true = ((~ ((~ (sK3 @ sK4)) | (sK2 @ sK4)))))),
  inference(or_proxy_clausification,[],[f158])).
thf(f160,plain,(
  ((((~ (sK3 @ sK4)) | (sK2 @ sK4))) = $false) | ($true = ((~ ((~ (sK3 @ sK4)) | (sK2 @ sK4))))) | ((((~ (sK2 @ sK4)) | (sK3 @ sK4))) = $false) | ((((sK3 @ sK4) | (~ (sK2 @ sK4)))) = $false)),
  inference(not_proxy_clausification,[],[f159])).
thf(f161,plain,(
  ((((sK3 @ sK4) | (~ (sK2 @ sK4)))) = $false) | ($true = ((~ ((~ (sK3 @ sK4)) | (sK2 @ sK4))))) | (((sK2 @ sK4)) = $false) | ((((~ (sK2 @ sK4)) | (sK3 @ sK4))) = $false)),
  inference(or_proxy_clausification,[],[f160])).
thf(f162,plain,(
  ($true = ((~ ((~ (sK3 @ sK4)) | (sK2 @ sK4))))) | ((((sK3 @ sK4) | (~ (sK2 @ sK4)))) = $false) | ((((~ (sK2 @ sK4)) | (sK3 @ sK4))) = $false) | (((~ (sK3 @ sK4))) = $false)),
  inference(or_proxy_clausification,[],[f160])).
thf(f163,plain,(
  ((((~ (sK3 @ sK4)) | (sK2 @ sK4))) = $false) | ((((sK3 @ sK4) | (~ (sK2 @ sK4)))) = $false) | (((~ (sK3 @ sK4))) = $false) | ((((~ (sK2 @ sK4)) | (sK3 @ sK4))) = $false)),
  inference(not_proxy_clausification,[],[f162])).
thf(f165,plain,(
  ((((~ (sK2 @ sK4)) | (sK3 @ sK4))) = $false) | (((~ (sK3 @ sK4))) = $false) | (((~ (sK3 @ sK4))) = $false) | ((((sK3 @ sK4) | (~ (sK2 @ sK4)))) = $false)),
  inference(or_proxy_clausification,[],[f163])).
thf(f167,plain,(
  (((~ (sK3 @ sK4))) = $false) | (((~ (sK3 @ sK4))) = $false) | ((((sK3 @ sK4) | (~ (sK2 @ sK4)))) = $false) | (((~ (sK2 @ sK4))) = $false)),
  inference(or_proxy_clausification,[],[f165])).
thf(f168,plain,(
  ((((sK3 @ sK4) | (~ (sK2 @ sK4)))) = $false) | (((~ (sK2 @ sK4))) = $false) | (((sK3 @ sK4)) = $true) | (((sK3 @ sK4)) = $true)),
  inference(not_proxy_clausification,[],[f167])).
thf(f169,plain,(
  (((~ (sK2 @ sK4))) = $false) | (((~ (sK2 @ sK4))) = $false) | (((sK3 @ sK4)) = $true) | (((sK3 @ sK4)) = $true)),
  inference(or_proxy_clausification,[],[f168])).
thf(f173,plain,(
  (((sK3 @ sK4)) = $true) | ($true = ((sK2 @ sK4))) | (((sK3 @ sK4)) = $true) | ($true = ((sK2 @ sK4)))),
  inference(not_proxy_clausification,[],[f169])).
thf(f174,plain,(
  (((sK3 @ sK4)) = $true) | ($true = ((sK2 @ sK4)))),
  inference(duplicate_literal_removal,[],[f173])).
thf(f197,plain,(
  (((sK2 @ sK4)) = $false) | ($true = ((~ ((~ (sK3 @ sK4)) | (sK2 @ sK4))))) | (((sK3 @ sK4)) = $false) | ((((~ (sK2 @ sK4)) | (sK3 @ sK4))) = $false)),
  inference(or_proxy_clausification,[],[f161])).
thf(f198,plain,(
  (((sK2 @ sK4)) = $false) | ((((~ (sK3 @ sK4)) | (sK2 @ sK4))) = $false) | ((((~ (sK2 @ sK4)) | (sK3 @ sK4))) = $false) | (((sK3 @ sK4)) = $false)),
  inference(not_proxy_clausification,[],[f197])).
thf(f199,plain,(
  (((sK2 @ sK4)) = $false) | ((((~ (sK2 @ sK4)) | (sK3 @ sK4))) = $false) | (((sK3 @ sK4)) = $false) | (((sK2 @ sK4)) = $false)),
  inference(or_proxy_clausification,[],[f198])).
thf(f206,plain,(
  (((sK3 @ sK4)) = $false) | (((sK2 @ sK4)) = $false) | (((sK2 @ sK4)) = $false) | (((sK3 @ sK4)) = $false)),
  inference(or_proxy_clausification,[],[f199])).
thf(f210,plain,(
  (((sK3 @ sK4)) = $false) | (((sK2 @ sK4)) = $false)),
  inference(duplicate_literal_removal,[],[f206])).
thf(f226,plain,(
  ($true = ((~ ((~ ((sK3 @ sK4) | (~ (sK2 @ sK4)))) | (~ ((~ (sK3 @ sK4)) | (sK2 @ sK4))))))) | ($true = ((~ ((~ ((~ (sK3 @ sK4)) | (sK2 @ sK4))) | (~ ((~ (sK2 @ sK4)) | (sK3 @ sK4)))))))),
  inference(beta-eta_normalization,[],[f151])).
thf(f227,plain,(
  ($true = ((~ ((~ ((~ (sK3 @ sK4)) | (sK2 @ sK4))) | (~ ((~ (sK2 @ sK4)) | (sK3 @ sK4))))))) | ((((~ ((sK3 @ sK4) | (~ (sK2 @ sK4)))) | (~ ((~ (sK3 @ sK4)) | (sK2 @ sK4))))) = $false)),
  inference(not_proxy_clausification,[],[f226])).
thf(f228,plain,(
  ((((~ ((~ (sK3 @ sK4)) | (sK2 @ sK4))) | (~ ((~ (sK2 @ sK4)) | (sK3 @ sK4))))) = $false) | ((((~ ((sK3 @ sK4) | (~ (sK2 @ sK4)))) | (~ ((~ (sK3 @ sK4)) | (sK2 @ sK4))))) = $false)),
  inference(not_proxy_clausification,[],[f227])).
thf(f229,plain,(
  (((~ ((~ (sK2 @ sK4)) | (sK3 @ sK4)))) = $false) | ((((~ ((sK3 @ sK4) | (~ (sK2 @ sK4)))) | (~ ((~ (sK3 @ sK4)) | (sK2 @ sK4))))) = $false)),
  inference(or_proxy_clausification,[],[f228])).
thf(f230,plain,(
  ((((~ ((sK3 @ sK4) | (~ (sK2 @ sK4)))) | (~ ((~ (sK3 @ sK4)) | (sK2 @ sK4))))) = $false) | (((~ ((~ (sK3 @ sK4)) | (sK2 @ sK4)))) = $false)),
  inference(or_proxy_clausification,[],[f228])).
thf(f231,plain,(
  (((~ ((~ (sK3 @ sK4)) | (sK2 @ sK4)))) = $false) | (((~ ((~ (sK3 @ sK4)) | (sK2 @ sK4)))) = $false)),
  inference(or_proxy_clausification,[],[f230])).
thf(f239,plain,(
  ($true = (((~ (sK3 @ sK4)) | (sK2 @ sK4)))) | ($true = (((~ (sK3 @ sK4)) | (sK2 @ sK4))))),
  inference(not_proxy_clausification,[],[f231])).
thf(f240,plain,(
  ($true = ((~ (sK3 @ sK4)))) | ($true = ((sK2 @ sK4))) | ($true = ((sK2 @ sK4)))),
  inference(or_proxy_clausification,[],[f239])).
thf(f241,plain,(
  (((sK3 @ sK4)) = $false) | ($true = ((sK2 @ sK4))) | ($true = ((sK2 @ sK4)))),
  inference(not_proxy_clausification,[],[f240])).
thf(f242,plain,(
  ($true = ((sK2 @ sK4))) | (((sK3 @ sK4)) = $false)),
  inference(duplicate_literal_removal,[],[f241])).
thf(f243,plain,(
  ((((~ ((sK3 @ sK4) | (~ (sK2 @ sK4)))) | (~ ((~ (sK3 @ sK4)) | (sK2 @ sK4))))) = $false) | ((((~ (sK2 @ sK4)) | (sK3 @ sK4))) = $true)),
  inference(not_proxy_clausification,[],[f229])).
thf(f245,plain,(
  (((~ ((sK3 @ sK4) | (~ (sK2 @ sK4))))) = $false) | ((((~ (sK2 @ sK4)) | (sK3 @ sK4))) = $true)),
  inference(or_proxy_clausification,[],[f243])).
thf(f246,plain,(
  ($true = (((sK3 @ sK4) | (~ (sK2 @ sK4))))) | ((((~ (sK2 @ sK4)) | (sK3 @ sK4))) = $true)),
  inference(not_proxy_clausification,[],[f245])).
thf(f247,plain,(
  ((((~ (sK2 @ sK4)) | (sK3 @ sK4))) = $true) | ($true = ((~ (sK2 @ sK4)))) | (((sK3 @ sK4)) = $true)),
  inference(or_proxy_clausification,[],[f246])).
thf(f248,plain,(
  (((sK3 @ sK4)) = $true) | ($true = ((~ (sK2 @ sK4)))) | ($true = ((~ (sK2 @ sK4)))) | (((sK3 @ sK4)) = $true)),
  inference(or_proxy_clausification,[],[f247])).
thf(f249,plain,(
  (((sK2 @ sK4)) = $false) | (((sK3 @ sK4)) = $true) | (((sK3 @ sK4)) = $true) | (((sK2 @ sK4)) = $false)),
  inference(not_proxy_clausification,[],[f248])).
thf(f250,plain,(
  (((sK2 @ sK4)) = $false) | (((sK3 @ sK4)) = $true)),
  inference(duplicate_literal_removal,[],[f249])).
thf(f257,definition,(
  spl5_1 <=> (((sK3 @ sK4)) = $true)),
  introduced(definition,[new_symbols(definition,[spl5_1])],[avatar_definition])).
thf(f259,plain,(
  (((sK3 @ sK4)) = $true) | ~spl5_1),
  inference(avatar_component_clause,[],[f257])).
thf(f261,definition,(
  spl5_2 <=> ($true = ((sK2 @ sK4)))),
  introduced(definition,[new_symbols(definition,[spl5_2])],[avatar_definition])).
thf(f263,plain,(
  ($true = ((sK2 @ sK4))) | ~spl5_2),
  inference(avatar_component_clause,[],[f261])).
thf(f264,plain,(
  spl5_1 | spl5_2),
  inference(avatar_split_clause,[],[f174,f261,f257])).
thf(f266,definition,(
  spl5_3 <=> (((sK3 @ sK4)) = $false)),
  introduced(definition,[new_symbols(definition,[spl5_3])],[avatar_definition])).
thf(f268,plain,(
  (((sK3 @ sK4)) = $false) | ~spl5_3),
  inference(avatar_component_clause,[],[f266])).
thf(f270,definition,(
  spl5_4 <=> (((sK2 @ sK4)) = $false)),
  introduced(definition,[new_symbols(definition,[spl5_4])],[avatar_definition])).
thf(f272,plain,(
  (((sK2 @ sK4)) = $false) | ~spl5_4),
  inference(avatar_component_clause,[],[f270])).
thf(f273,plain,(
  spl5_3 | spl5_4),
  inference(avatar_split_clause,[],[f210,f270,f266])).
thf(f274,plain,(
  spl5_2 | spl5_3),
  inference(avatar_split_clause,[],[f242,f266,f261])).
thf(f275,plain,(
  spl5_1 | spl5_4),
  inference(avatar_split_clause,[],[f250,f270,f257])).
thf(f276,plain,(
  ($true = $false) | (~spl5_2 | ~spl5_4)),
  inference(forward_demodulation,[],[f272,f263])).
thf(f277,plain,(
  $false | (~spl5_2 | ~spl5_4)),
  inference(trivial_inequality_removal,[],[f276])).
thf(f278,plain,(
  ~spl5_2 | ~spl5_4),
  inference(avatar_contradiction_clause,[],[f277])).
thf(f280,plain,(
  ($true = $false) | (~spl5_1 | ~spl5_3)),
  inference(forward_demodulation,[],[f268,f259])).
thf(f281,plain,(
  $false | (~spl5_1 | ~spl5_3)),
  inference(trivial_inequality_removal,[],[f280])).
thf(f282,plain,(
  ~spl5_1 | ~spl5_3),
  inference(avatar_contradiction_clause,[],[f281])).
cnf(s1, plain, spl5_1 | spl5_2, inference(sat_conversion,[],[f264])).
cnf(s2, plain, spl5_3 | spl5_4, inference(sat_conversion,[],[f273])).
cnf(s3, plain, spl5_2 | spl5_3, inference(sat_conversion,[],[f274])).
cnf(s4, plain, spl5_1 | spl5_4, inference(sat_conversion,[],[f275])).
cnf(s5, plain, ~spl5_2 | ~spl5_4, inference(sat_conversion,[],[f278])).
cnf(s6, plain, ~spl5_1 | ~spl5_3, inference(sat_conversion,[],[f282])).
cnf(s7, plain, spl5_1, inference(rat,[],[s5,s1,s4])).
cnf(s8, plain, ~spl5_3, inference(rat,[],[s6,s7])).
cnf(s9, plain, spl5_2, inference(rat,[],[s3,s8])).
cnf(s10, plain, spl5_4, inference(rat,[],[s2,s8])).
cnf(s11, plain, $false, inference(rat,[],[s5,s10,s9])).
thf(f283,plain,(
  $false),
  inference(avatar_sat_refutation,[],[s11])).
% SZS output end Proof for theBenchmark