%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU492^1 : TPTP v6.1.0. Released v3.6.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n116.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 13:32:17 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  : SEU492^1 : TPTP v6.1.0. Released v3.6.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n116.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 10:22:31 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/SET009^0.ax, trying next directory
% FOF formula (<kernel.Constant object at 0x1b087a0>, <kernel.DependentProduct object at 0x1ae98c0>) of role type named subrel_type
% Using role type
% Declaring subrel:((fofType->(fofType->Prop))->((fofType->(fofType->Prop))->Prop))
% FOF formula (((eq ((fofType->(fofType->Prop))->((fofType->(fofType->Prop))->Prop))) subrel) (fun (R:(fofType->(fofType->Prop))) (S:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->((S X) Y))))) of role definition named subrel
% A new definition: (((eq ((fofType->(fofType->Prop))->((fofType->(fofType->Prop))->Prop))) subrel) (fun (R:(fofType->(fofType->Prop))) (S:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->((S X) Y)))))
% Defined: subrel:=(fun (R:(fofType->(fofType->Prop))) (S:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->((S X) Y))))
% FOF formula (<kernel.Constant object at 0x1b08488>, <kernel.DependentProduct object at 0x1b08ea8>) of role type named inv_type
% Using role type
% Declaring inv:((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))
% FOF formula (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) inv) (fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((R Y) X))) of role definition named inverse
% A new definition: (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) inv) (fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((R Y) X)))
% Defined: inv:=(fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((R Y) X))
% FOF formula (<kernel.Constant object at 0x1b08ea8>, <kernel.DependentProduct object at 0x1926cb0>) of role type named idem_type
% Using role type
% Declaring idem:(((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))->Prop)
% FOF formula (((eq (((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))->Prop)) idem) (fun (F:((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))=> (forall (R:(fofType->(fofType->Prop))), (((eq (fofType->(fofType->Prop))) (F (F R))) (F R))))) of role definition named idempotent
% A new definition: (((eq (((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))->Prop)) idem) (fun (F:((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))=> (forall (R:(fofType->(fofType->Prop))), (((eq (fofType->(fofType->Prop))) (F (F R))) (F R)))))
% Defined: idem:=(fun (F:((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))=> (forall (R:(fofType->(fofType->Prop))), (((eq (fofType->(fofType->Prop))) (F (F R))) (F R))))
% FOF formula (<kernel.Constant object at 0x1926cb0>, <kernel.DependentProduct object at 0x19265f0>) of role type named infl_type
% Using role type
% Declaring infl:(((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))->Prop)
% FOF formula (((eq (((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))->Prop)) infl) (fun (F:((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))=> (forall (R:(fofType->(fofType->Prop))), ((subrel R) (F R))))) of role definition named inflationary
% A new definition: (((eq (((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))->Prop)) infl) (fun (F:((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))=> (forall (R:(fofType->(fofType->Prop))), ((subrel R) (F R)))))
% Defined: infl:=(fun (F:((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))=> (forall (R:(fofType->(fofType->Prop))), ((subrel R) (F R))))
% FOF formula (<kernel.Constant object at 0x1926710>, <kernel.DependentProduct object at 0x1926680>) of role type named mono_type
% Using role type
% Declaring mono:(((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))->Prop)
% FOF formula (((eq (((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))->Prop)) mono) (fun (F:((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))=> (forall (R:(fofType->(fofType->Prop))) (S:(fofType->(fofType->Prop))), (((subrel R) S)->((subrel (F R)) (F S)))))) of role definition named monotonic
% A new definition: (((eq (((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))->Prop)) mono) (fun (F:((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))=> (forall (R:(fofType->(fofType->Prop))) (S:(fofType->(fofType->Prop))), (((subrel R) S)->((subrel (F R)) (F S))))))
% Defined: mono:=(fun (F:((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))=> (forall (R:(fofType->(fofType->Prop))) (S:(fofType->(fofType->Prop))), (((subrel R) S)->((subrel (F R)) (F S)))))
% FOF formula (<kernel.Constant object at 0x1d5e6c8>, <kernel.DependentProduct object at 0x1926cb0>) of role type named refl_type
% Using role type
% Declaring refl:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) refl) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), ((R X) X)))) of role definition named reflexive
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) refl) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), ((R X) X))))
% Defined: refl:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), ((R X) X)))
% FOF formula (<kernel.Constant object at 0x1926c20>, <kernel.DependentProduct object at 0x1926680>) of role type named irrefl_type
% Using role type
% Declaring irrefl:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) irrefl) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), (((R X) X)->False)))) of role definition named irreflexive
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) irrefl) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), (((R X) X)->False))))
% Defined: irrefl:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), (((R X) X)->False)))
% FOF formula (<kernel.Constant object at 0x1926cb0>, <kernel.DependentProduct object at 0x1926dd0>) of role type named rc_type
% Using role type
% Declaring rc:((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))
% FOF formula (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) rc) (fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((or (((eq fofType) X) Y)) ((R X) Y)))) of role definition named reflexive_closure
% A new definition: (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) rc) (fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((or (((eq fofType) X) Y)) ((R X) Y))))
% Defined: rc:=(fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((or (((eq fofType) X) Y)) ((R X) Y)))
% FOF formula (<kernel.Constant object at 0x1926cb0>, <kernel.DependentProduct object at 0x1926440>) of role type named symm_type
% Using role type
% Declaring symm:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) symm) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->((R Y) X))))) of role definition named symmetric
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) symm) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->((R Y) X)))))
% Defined: symm:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->((R Y) X))))
% FOF formula (<kernel.Constant object at 0x1926710>, <kernel.DependentProduct object at 0x1926488>) of role type named antisymm_type
% Using role type
% Declaring antisymm:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) antisymm) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((and ((R X) Y)) ((R Y) X))->(((eq fofType) X) Y))))) of role definition named antisymmetric
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) antisymm) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((and ((R X) Y)) ((R Y) X))->(((eq fofType) X) Y)))))
% Defined: antisymm:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((and ((R X) Y)) ((R Y) X))->(((eq fofType) X) Y))))
% FOF formula (<kernel.Constant object at 0x1926cb0>, <kernel.DependentProduct object at 0x191c908>) of role type named asymm_type
% Using role type
% Declaring asymm:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) asymm) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->(((R Y) X)->False))))) of role definition named asymmetric
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) asymm) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->(((R Y) X)->False)))))
% Defined: asymm:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->(((R Y) X)->False))))
% FOF formula (<kernel.Constant object at 0x1926710>, <kernel.DependentProduct object at 0x191c638>) of role type named sc_type
% Using role type
% Declaring sc:((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))
% FOF formula (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) sc) (fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((or ((R Y) X)) ((R X) Y)))) of role definition named symmetric_closure
% A new definition: (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) sc) (fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((or ((R Y) X)) ((R X) Y))))
% Defined: sc:=(fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((or ((R Y) X)) ((R X) Y)))
% FOF formula (<kernel.Constant object at 0x191c638>, <kernel.DependentProduct object at 0x191c4d0>) of role type named trans_type
% Using role type
% Declaring trans:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) trans) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Y)) ((R Y) Z))->((R X) Z))))) of role definition named transitive
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) trans) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Y)) ((R Y) Z))->((R X) Z)))))
% Defined: trans:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Y)) ((R Y) Z))->((R X) Z))))
% FOF formula (<kernel.Constant object at 0x191c908>, <kernel.DependentProduct object at 0x191c5f0>) of role type named tc_type
% Using role type
% Declaring tc:((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))
% FOF formula (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) tc) (fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> (forall (S:(fofType->(fofType->Prop))), (((and (trans S)) ((subrel R) S))->((S X) Y))))) of role definition named transitive_closure
% A new definition: (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) tc) (fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> (forall (S:(fofType->(fofType->Prop))), (((and (trans S)) ((subrel R) S))->((S X) Y)))))
% Defined: tc:=(fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> (forall (S:(fofType->(fofType->Prop))), (((and (trans S)) ((subrel R) S))->((S X) Y))))
% FOF formula (<kernel.Constant object at 0x191c488>, <kernel.DependentProduct object at 0x191c2d8>) of role type named trc_type
% Using role type
% Declaring trc:((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))
% FOF formula (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) trc) (fun (R:(fofType->(fofType->Prop)))=> (rc (tc R)))) of role definition named transitive_reflexive_closure
% A new definition: (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) trc) (fun (R:(fofType->(fofType->Prop)))=> (rc (tc R))))
% Defined: trc:=(fun (R:(fofType->(fofType->Prop)))=> (rc (tc R)))
% FOF formula (<kernel.Constant object at 0x191c050>, <kernel.DependentProduct object at 0x191ca28>) of role type named trsc_type
% Using role type
% Declaring trsc:((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))
% FOF formula (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) trsc) (fun (R:(fofType->(fofType->Prop)))=> (sc (rc (tc R))))) of role definition named transitive_reflexive_symmetric_closure
% A new definition: (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) trsc) (fun (R:(fofType->(fofType->Prop)))=> (sc (rc (tc R)))))
% Defined: trsc:=(fun (R:(fofType->(fofType->Prop)))=> (sc (rc (tc R))))
% FOF formula (<kernel.Constant object at 0x191c050>, <kernel.DependentProduct object at 0x191c8c0>) of role type named po_type
% Using role type
% Declaring po:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) po) (fun (R:(fofType->(fofType->Prop)))=> ((and ((and (refl R)) (antisymm R))) (trans R)))) of role definition named partial_order
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) po) (fun (R:(fofType->(fofType->Prop)))=> ((and ((and (refl R)) (antisymm R))) (trans R))))
% Defined: po:=(fun (R:(fofType->(fofType->Prop)))=> ((and ((and (refl R)) (antisymm R))) (trans R)))
% FOF formula (<kernel.Constant object at 0x191c128>, <kernel.DependentProduct object at 0x191c6c8>) of role type named so_type
% Using role type
% Declaring so:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) so) (fun (R:(fofType->(fofType->Prop)))=> ((and (asymm R)) (trans R)))) of role definition named strict_order
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) so) (fun (R:(fofType->(fofType->Prop)))=> ((and (asymm R)) (trans R))))
% Defined: so:=(fun (R:(fofType->(fofType->Prop)))=> ((and (asymm R)) (trans R)))
% FOF formula (<kernel.Constant object at 0x191c8c0>, <kernel.DependentProduct object at 0x191c3f8>) of role type named total_type
% Using role type
% Declaring total:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) total) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), ((or ((or (((eq fofType) X) Y)) ((R X) Y))) ((R Y) X))))) of role definition named total
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) total) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), ((or ((or (((eq fofType) X) Y)) ((R X) Y))) ((R Y) X)))))
% Defined: total:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), ((or ((or (((eq fofType) X) Y)) ((R X) Y))) ((R Y) X))))
% FOF formula (<kernel.Constant object at 0x191c3f8>, <kernel.DependentProduct object at 0x191cb90>) of role type named term_type
% Using role type
% Declaring term:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) term) (fun (R:(fofType->(fofType->Prop)))=> (forall (A:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> (A X)))->((ex fofType) (fun (X:fofType)=> ((and (A X)) (forall (Y:fofType), ((A Y)->(((R X) Y)->False)))))))))) of role definition named terminating
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) term) (fun (R:(fofType->(fofType->Prop)))=> (forall (A:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> (A X)))->((ex fofType) (fun (X:fofType)=> ((and (A X)) (forall (Y:fofType), ((A Y)->(((R X) Y)->False))))))))))
% Defined: term:=(fun (R:(fofType->(fofType->Prop)))=> (forall (A:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> (A X)))->((ex fofType) (fun (X:fofType)=> ((and (A X)) (forall (Y:fofType), ((A Y)->(((R X) Y)->False)))))))))
% FOF formula (<kernel.Constant object at 0x191cc20>, <kernel.DependentProduct object at 0x191cfc8>) of role type named ind_type
% Using role type
% Declaring ind:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) ind) (fun (R:(fofType->(fofType->Prop)))=> (forall (P:(fofType->Prop)), ((forall (X:fofType), ((forall (Y:fofType), ((((tc R) X) Y)->(P Y)))->(P X)))->(forall (X:fofType), (P X)))))) of role definition named satisfying_the_induction_principle
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) ind) (fun (R:(fofType->(fofType->Prop)))=> (forall (P:(fofType->Prop)), ((forall (X:fofType), ((forall (Y:fofType), ((((tc R) X) Y)->(P Y)))->(P X)))->(forall (X:fofType), (P X))))))
% Defined: ind:=(fun (R:(fofType->(fofType->Prop)))=> (forall (P:(fofType->Prop)), ((forall (X:fofType), ((forall (Y:fofType), ((((tc R) X) Y)->(P Y)))->(P X)))->(forall (X:fofType), (P X)))))
% FOF formula (<kernel.Constant object at 0x191c200>, <kernel.DependentProduct object at 0x191cc20>) of role type named innf_type
% Using role type
% Declaring innf:((fofType->(fofType->Prop))->(fofType->Prop))
% FOF formula (((eq ((fofType->(fofType->Prop))->(fofType->Prop))) innf) (fun (R:(fofType->(fofType->Prop))) (X:fofType)=> (((ex fofType) (fun (Y:fofType)=> ((R X) Y)))->False))) of role definition named in_normal_form
% A new definition: (((eq ((fofType->(fofType->Prop))->(fofType->Prop))) innf) (fun (R:(fofType->(fofType->Prop))) (X:fofType)=> (((ex fofType) (fun (Y:fofType)=> ((R X) Y)))->False)))
% Defined: innf:=(fun (R:(fofType->(fofType->Prop))) (X:fofType)=> (((ex fofType) (fun (Y:fofType)=> ((R X) Y)))->False))
% FOF formula (<kernel.Constant object at 0x191cfc8>, <kernel.DependentProduct object at 0x191c050>) of role type named nfof_type
% Using role type
% Declaring nfof:((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))
% FOF formula (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) nfof) (fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((and (((trc R) Y) X)) ((innf R) X)))) of role definition named normal_form_of
% A new definition: (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) nfof) (fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((and (((trc R) Y) X)) ((innf R) X))))
% Defined: nfof:=(fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((and (((trc R) Y) X)) ((innf R) X)))
% FOF formula (<kernel.Constant object at 0x191c200>, <kernel.DependentProduct object at 0x191cd40>) of role type named norm_type
% Using role type
% Declaring norm:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) norm) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), ((ex fofType) (fun (Y:fofType)=> (((nfof R) Y) X)))))) of role definition named normalizing
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) norm) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), ((ex fofType) (fun (Y:fofType)=> (((nfof R) Y) X))))))
% Defined: norm:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), ((ex fofType) (fun (Y:fofType)=> (((nfof R) Y) X)))))
% FOF formula (<kernel.Constant object at 0x191c200>, <kernel.DependentProduct object at 0x191ca70>) of role type named join_type
% Using role type
% Declaring join:((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))
% FOF formula (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) join) (fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((ex fofType) (fun (Z:fofType)=> ((and (((trc R) X) Z)) (((trc R) Y) Z)))))) of role definition named joinable
% A new definition: (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) join) (fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((ex fofType) (fun (Z:fofType)=> ((and (((trc R) X) Z)) (((trc R) Y) Z))))))
% Defined: join:=(fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((ex fofType) (fun (Z:fofType)=> ((and (((trc R) X) Z)) (((trc R) Y) Z)))))
% FOF formula (<kernel.Constant object at 0x191c290>, <kernel.DependentProduct object at 0x1af6290>) of role type named lconfl_type
% Using role type
% Declaring lconfl:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) lconfl) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Z)) ((R X) Y))->(((join R) Z) Y))))) of role definition named locally_confluent
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) lconfl) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Z)) ((R X) Y))->(((join R) Z) Y)))))
% Defined: lconfl:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Z)) ((R X) Y))->(((join R) Z) Y))))
% FOF formula (<kernel.Constant object at 0x191c878>, <kernel.DependentProduct object at 0x1af63b0>) of role type named sconfl_type
% Using role type
% Declaring sconfl:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) sconfl) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Z)) (((trc R) X) Y))->(((join R) Z) Y))))) of role definition named semi_confluent
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) sconfl) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Z)) (((trc R) X) Y))->(((join R) Z) Y)))))
% Defined: sconfl:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Z)) (((trc R) X) Y))->(((join R) Z) Y))))
% FOF formula (<kernel.Constant object at 0x1af63b0>, <kernel.DependentProduct object at 0x1af6488>) of role type named confl_type
% Using role type
% Declaring confl:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) confl) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and (((trc R) X) Z)) (((trc R) X) Y))->(((join R) Z) Y))))) of role definition named confluent
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) confl) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and (((trc R) X) Z)) (((trc R) X) Y))->(((join R) Z) Y)))))
% Defined: confl:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and (((trc R) X) Z)) (((trc R) X) Y))->(((join R) Z) Y))))
% FOF formula (<kernel.Constant object at 0x1af65a8>, <kernel.DependentProduct object at 0x1af62d8>) of role type named cr_type
% Using role type
% Declaring cr:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) cr) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), ((((trsc R) X) Y)->(((join R) X) Y))))) of role definition named church_rosser
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) cr) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), ((((trsc R) X) Y)->(((join R) X) Y)))))
% Defined: cr:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), ((((trsc R) X) Y)->(((join R) X) Y))))
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) so) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) of role conjecture named alternative_definition_of_strict_order
% Conjecture to prove = (((eq ((fofType->(fofType->Prop))->Prop)) so) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['(((eq ((fofType->(fofType->Prop))->Prop)) so) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))']
% Parameter fofType:Type.
% Definition subrel:=(fun (R:(fofType->(fofType->Prop))) (S:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->((S X) Y)))):((fofType->(fofType->Prop))->((fofType->(fofType->Prop))->Prop)).
% Definition inv:=(fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((R Y) X)):((fofType->(fofType->Prop))->(fofType->(fofType->Prop))).
% Definition idem:=(fun (F:((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))=> (forall (R:(fofType->(fofType->Prop))), (((eq (fofType->(fofType->Prop))) (F (F R))) (F R)))):(((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))->Prop).
% Definition infl:=(fun (F:((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))=> (forall (R:(fofType->(fofType->Prop))), ((subrel R) (F R)))):(((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))->Prop).
% Definition mono:=(fun (F:((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))=> (forall (R:(fofType->(fofType->Prop))) (S:(fofType->(fofType->Prop))), (((subrel R) S)->((subrel (F R)) (F S))))):(((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))->Prop).
% Definition refl:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), ((R X) X))):((fofType->(fofType->Prop))->Prop).
% Definition irrefl:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), (((R X) X)->False))):((fofType->(fofType->Prop))->Prop).
% Definition rc:=(fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((or (((eq fofType) X) Y)) ((R X) Y))):((fofType->(fofType->Prop))->(fofType->(fofType->Prop))).
% Definition symm:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->((R Y) X)))):((fofType->(fofType->Prop))->Prop).
% Definition antisymm:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((and ((R X) Y)) ((R Y) X))->(((eq fofType) X) Y)))):((fofType->(fofType->Prop))->Prop).
% Definition asymm:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->(((R Y) X)->False)))):((fofType->(fofType->Prop))->Prop).
% Definition sc:=(fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((or ((R Y) X)) ((R X) Y))):((fofType->(fofType->Prop))->(fofType->(fofType->Prop))).
% Definition trans:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Y)) ((R Y) Z))->((R X) Z)))):((fofType->(fofType->Prop))->Prop).
% Definition tc:=(fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> (forall (S:(fofType->(fofType->Prop))), (((and (trans S)) ((subrel R) S))->((S X) Y)))):((fofType->(fofType->Prop))->(fofType->(fofType->Prop))).
% Definition trc:=(fun (R:(fofType->(fofType->Prop)))=> (rc (tc R))):((fofType->(fofType->Prop))->(fofType->(fofType->Prop))).
% Definition trsc:=(fun (R:(fofType->(fofType->Prop)))=> (sc (rc (tc R)))):((fofType->(fofType->Prop))->(fofType->(fofType->Prop))).
% Definition po:=(fun (R:(fofType->(fofType->Prop)))=> ((and ((and (refl R)) (antisymm R))) (trans R))):((fofType->(fofType->Prop))->Prop).
% Definition so:=(fun (R:(fofType->(fofType->Prop)))=> ((and (asymm R)) (trans R))):((fofType->(fofType->Prop))->Prop).
% Definition total:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), ((or ((or (((eq fofType) X) Y)) ((R X) Y))) ((R Y) X)))):((fofType->(fofType->Prop))->Prop).
% Definition term:=(fun (R:(fofType->(fofType->Prop)))=> (forall (A:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> (A X)))->((ex fofType) (fun (X:fofType)=> ((and (A X)) (forall (Y:fofType), ((A Y)->(((R X) Y)->False))))))))):((fofType->(fofType->Prop))->Prop).
% Definition ind:=(fun (R:(fofType->(fofType->Prop)))=> (forall (P:(fofType->Prop)), ((forall (X:fofType), ((forall (Y:fofType), ((((tc R) X) Y)->(P Y)))->(P X)))->(forall (X:fofType), (P X))))):((fofType->(fofType->Prop))->Prop).
% Definition innf:=(fun (R:(fofType->(fofType->Prop))) (X:fofType)=> (((ex fofType) (fun (Y:fofType)=> ((R X) Y)))->False)):((fofType->(fofType->Prop))->(fofType->Prop)).
% Definition nfof:=(fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((and (((trc R) Y) X)) ((innf R) X))):((fofType->(fofType->Prop))->(fofType->(fofType->Prop))).
% Definition norm:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), ((ex fofType) (fun (Y:fofType)=> (((nfof R) Y) X))))):((fofType->(fofType->Prop))->Prop).
% Definition join:=(fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((ex fofType) (fun (Z:fofType)=> ((and (((trc R) X) Z)) (((trc R) Y) Z))))):((fofType->(fofType->Prop))->(fofType->(fofType->Prop))).
% Definition lconfl:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Z)) ((R X) Y))->(((join R) Z) Y)))):((fofType->(fofType->Prop))->Prop).
% Definition sconfl:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Z)) (((trc R) X) Y))->(((join R) Z) Y)))):((fofType->(fofType->Prop))->Prop).
% Definition confl:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and (((trc R) X) Z)) (((trc R) X) Y))->(((join R) Z) Y)))):((fofType->(fofType->Prop))->Prop).
% Definition cr:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), ((((trsc R) X) Y)->(((join R) X) Y)))):((fofType->(fofType->Prop))->Prop).
% Trying to prove (((eq ((fofType->(fofType->Prop))->Prop)) so) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 so):(((eq ((fofType->(fofType->Prop))->Prop)) so) (fun (x:(fofType->(fofType->Prop)))=> (so x)))
% Found (eta_expansion_dep00 so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found x2:(P so)
% Found (fun (x2:(P so))=> x2) as proof of (P so)
% Found (fun (x2:(P so))=> x2) as proof of (P0 so)
% Found x2:(P so)
% Found (fun (x2:(P so))=> x2) as proof of (P so)
% Found (fun (x2:(P so))=> x2) as proof of (P0 so)
% Found x2:(P so)
% Found (fun (x2:(P so))=> x2) as proof of (P so)
% Found (fun (x2:(P so))=> x2) as proof of (P0 so)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found eta_expansion000:=(eta_expansion00 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) (fun (x:(fofType->(fofType->Prop)))=> ((and (irrefl x)) (trans x))))
% Found (eta_expansion00 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found ((eta_expansion0 Prop) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P0 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P0 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found x01:(P (so x))
% Found (fun (x01:(P (so x)))=> x01) as proof of (P (so x))
% Found (fun (x01:(P (so x)))=> x01) as proof of (P0 (so x))
% Found x01:(P (so x))
% Found (fun (x01:(P (so x)))=> x01) as proof of (P (so x))
% Found (fun (x01:(P (so x)))=> x01) as proof of (P0 (so x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found x01:(P ((and (irrefl x)) (trans x)))
% Found (fun (x01:(P ((and (irrefl x)) (trans x))))=> x01) as proof of (P ((and (irrefl x)) (trans x)))
% Found (fun (x01:(P ((and (irrefl x)) (trans x))))=> x01) as proof of (P0 ((and (irrefl x)) (trans x)))
% Found x01:(P ((and (irrefl x)) (trans x)))
% Found (fun (x01:(P ((and (irrefl x)) (trans x))))=> x01) as proof of (P ((and (irrefl x)) (trans x)))
% Found (fun (x01:(P ((and (irrefl x)) (trans x))))=> x01) as proof of (P0 ((and (irrefl x)) (trans x)))
% Found x:(P so)
% Instantiate: b:=so:((fofType->(fofType->Prop))->Prop)
% Found x as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) (fun (x:(fofType->(fofType->Prop)))=> ((and (irrefl x)) (trans x))))
% Found (eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found x:(P so)
% Instantiate: f:=so:((fofType->(fofType->Prop))->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (irrefl x0)) (trans x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (irrefl x0)) (trans x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (irrefl x0)) (trans x0)))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (irrefl x0)) (trans x0)))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((and (irrefl x)) (trans x))))
% Found x:(P so)
% Instantiate: f:=so:((fofType->(fofType->Prop))->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (irrefl x0)) (trans x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (irrefl x0)) (trans x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (irrefl x0)) (trans x0)))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (irrefl x0)) (trans x0)))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((and (irrefl x)) (trans x))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eta_expansion000:=(eta_expansion00 so):(((eq ((fofType->(fofType->Prop))->Prop)) so) (fun (x:(fofType->(fofType->Prop)))=> (so x)))
% Found (eta_expansion00 so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found ((eta_expansion0 Prop) so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found x:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Instantiate: b:=(fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))):((fofType->(fofType->Prop))->Prop)
% Found x as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 so):(((eq ((fofType->(fofType->Prop))->Prop)) so) so)
% Found (eq_ref0 so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found eta_expansion000:=(eta_expansion00 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) (fun (x:(fofType->(fofType->Prop)))=> ((and (irrefl x)) (trans x))))
% Found (eta_expansion00 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found ((eta_expansion0 Prop) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found x:(P0 b)
% Instantiate: b:=(fun (R:(fofType->(fofType->Prop)))=> ((and (asymm R)) (trans R))):((fofType->(fofType->Prop))->Prop)
% Found (fun (x:(P0 b))=> x) as proof of (P0 so)
% Found (fun (P0:(((fofType->(fofType->Prop))->Prop)->Prop)) (x:(P0 b))=> x) as proof of ((P0 b)->(P0 so))
% Found (fun (P0:(((fofType->(fofType->Prop))->Prop)->Prop)) (x:(P0 b))=> x) as proof of (P b)
% Found x3:(P so)
% Found (fun (x3:(P so))=> x3) as proof of (P so)
% Found (fun (x3:(P so))=> x3) as proof of (P0 so)
% Found eq_ref00:=(eq_ref0 so):(((eq ((fofType->(fofType->Prop))->Prop)) so) so)
% Found (eq_ref0 so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found x:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Instantiate: f:=(fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))):((fofType->(fofType->Prop))->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (so x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (so x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (so x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (so x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (so x)))
% Found x:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Instantiate: f:=(fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))):((fofType->(fofType->Prop))->Prop)
% Found x as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (so x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (so x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (so x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (so x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (so x)))
% Found x0:(P (so x))
% Instantiate: b:=(so x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found x0:(P (so x))
% Instantiate: b:=(so x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found eta_expansion000:=(eta_expansion00 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) (fun (x:(fofType->(fofType->Prop)))=> ((and (irrefl x)) (trans x))))
% Found (eta_expansion00 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found ((eta_expansion0 Prop) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) (fun (x:(fofType->(fofType->Prop)))=> ((and (irrefl x)) (trans x))))
% Found (eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) (fun (x:(fofType->(fofType->Prop)))=> ((and (irrefl x)) (trans x))))
% Found (eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 so):(((eq ((fofType->(fofType->Prop))->Prop)) so) so)
% Found (eq_ref0 so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found x2:(P1 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P1 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P1 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P1 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P2 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found x2:(P1 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P1 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P1 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P1 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P2 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found x2:(P1 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P1 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P1 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P1 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P2 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found x2:(P1 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P1 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P1 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P1 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P2 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found x01:(P1 (so x))
% Found (fun (x01:(P1 (so x)))=> x01) as proof of (P1 (so x))
% Found (fun (x01:(P1 (so x)))=> x01) as proof of (P2 (so x))
% Found x01:(P1 (so x))
% Found (fun (x01:(P1 (so x)))=> x01) as proof of (P1 (so x))
% Found (fun (x01:(P1 (so x)))=> x01) as proof of (P2 (so x))
% Found x01:(P1 (so x))
% Found (fun (x01:(P1 (so x)))=> x01) as proof of (P1 (so x))
% Found (fun (x01:(P1 (so x)))=> x01) as proof of (P2 (so x))
% Found x01:(P1 (so x))
% Found (fun (x01:(P1 (so x)))=> x01) as proof of (P1 (so x))
% Found (fun (x01:(P1 (so x)))=> x01) as proof of (P2 (so x))
% Found x3:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x3:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x3) as proof of (P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x3:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x3) as proof of (P0 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) (fun (x:(fofType->(fofType->Prop)))=> ((and (irrefl x)) (trans x))))
% Found (eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found eta_expansion_dep000:=(eta_expansion_dep00 so):(((eq ((fofType->(fofType->Prop))->Prop)) so) (fun (x:(fofType->(fofType->Prop)))=> (so x)))
% Found (eta_expansion_dep00 so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R))))
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R))))
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R))))
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R))))
% Found x2:(P so)
% Found (fun (x2:(P so))=> x2) as proof of (P so)
% Found (fun (x2:(P so))=> x2) as proof of (P0 so)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) (fun (x:(fofType->(fofType->Prop)))=> ((and (irrefl x)) (trans x))))
% Found (eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found x0:(P0 b)
% Instantiate: b:=((and (asymm x)) (trans x)):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (so x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (so x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found x0:(P0 b)
% Instantiate: b:=((and (asymm x)) (trans x)):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (so x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (so x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found x02:(P (so x))
% Found (fun (x02:(P (so x)))=> x02) as proof of (P (so x))
% Found (fun (x02:(P (so x)))=> x02) as proof of (P0 (so x))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found x02:(P (so x))
% Found (fun (x02:(P (so x)))=> x02) as proof of (P (so x))
% Found (fun (x02:(P (so x)))=> x02) as proof of (P0 (so x))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found x2:(P so)
% Found (fun (x2:(P so))=> x2) as proof of (P so)
% Found (fun (x2:(P so))=> x2) as proof of (P0 so)
% Found x2:(P so)
% Found (fun (x2:(P so))=> x2) as proof of (P so)
% Found (fun (x2:(P so))=> x2) as proof of (P0 so)
% Found x0:(P ((and (irrefl x)) (trans x)))
% Instantiate: b:=((and (irrefl x)) (trans x)):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (trans x))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (trans x))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (trans x))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (trans x))
% Found x0:(P ((and (irrefl x)) (trans x)))
% Instantiate: b:=((and (irrefl x)) (trans x)):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (trans x))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (trans x))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (trans x))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (trans x))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (trans x))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (trans x))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (trans x))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (trans x))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (trans x))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (trans x))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (trans x))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (trans x))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (trans x))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (trans x))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (trans x))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (trans x))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (trans x))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (trans x))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (trans x))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (trans x))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) (fun (x:(fofType->(fofType->Prop)))=> ((and (irrefl x)) (trans x))))
% Found (eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found x0:(P ((and (irrefl x)) (trans x)))
% Instantiate: b:=((and (irrefl x)) (trans x)):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found x0:(P ((and (irrefl x)) (trans x)))
% Instantiate: b:=((and (irrefl x)) (trans x)):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P0 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found eta_expansion000:=(eta_expansion00 so):(((eq ((fofType->(fofType->Prop))->Prop)) so) (fun (x:(fofType->(fofType->Prop)))=> (so x)))
% Found (eta_expansion00 so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found ((eta_expansion0 Prop) so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) so) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) so) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found x2:(P so)
% Found (fun (x2:(P so))=> x2) as proof of (P so)
% Found (fun (x2:(P so))=> x2) as proof of (P0 so)
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (irrefl x)) (trans x)))
% Found x2:(P so)
% Found (fun (x2:(P so))=> x2) as proof of (P so)
% Found (fun (x2:(P so))=> x2) as proof of (P0 so)
% Found x2:(P so)
% Found (fun (x2:(P so))=> x2) as proof of (P so)
% Found (fun (x2:(P so))=> x2) as proof of (P0 so)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P0 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P0 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P0 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P0 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eta_expansion000:=(eta_expansion00 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R)))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R)))) (fun (x:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))))
% Found (eta_expansion00 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R)))) b)
% Found ((eta_expansion0 Prop) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R)))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R)))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R)))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b0)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b0)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b0)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b0)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b0)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b0)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (irrefl x)) (trans x)))
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x01:(P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P0 (b x))
% Found x01:(P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P0 (b x))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x)))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x)))
% Found x01:(P1 ((and (irrefl x)) (trans x)))
% Found (fun (x01:(P1 ((and (irrefl x)) (trans x))))=> x01) as proof of (P1 ((and (irrefl x)) (trans x)))
% Found (fun (x01:(P1 ((and (irrefl x)) (trans x))))=> x01) as proof of (P2 ((and (irrefl x)) (trans x)))
% Found x01:(P1 ((and (irrefl x)) (trans x)))
% Found (fun (x01:(P1 ((and (irrefl x)) (trans x))))=> x01) as proof of (P1 ((and (irrefl x)) (trans x)))
% Found (fun (x01:(P1 ((and (irrefl x)) (trans x))))=> x01) as proof of (P2 ((and (irrefl x)) (trans x)))
% Found x01:(P1 ((and (irrefl x)) (trans x)))
% Found (fun (x01:(P1 ((and (irrefl x)) (trans x))))=> x01) as proof of (P1 ((and (irrefl x)) (trans x)))
% Found (fun (x01:(P1 ((and (irrefl x)) (trans x))))=> x01) as proof of (P2 ((and (irrefl x)) (trans x)))
% Found x01:(P1 ((and (irrefl x)) (trans x)))
% Found (fun (x01:(P1 ((and (irrefl x)) (trans x))))=> x01) as proof of (P1 ((and (irrefl x)) (trans x)))
% Found (fun (x01:(P1 ((and (irrefl x)) (trans x))))=> x01) as proof of (P2 ((and (irrefl x)) (trans x)))
% Found x01:(P1 ((and (irrefl x)) (trans x)))
% Found (fun (x01:(P1 ((and (irrefl x)) (trans x))))=> x01) as proof of (P1 ((and (irrefl x)) (trans x)))
% Found (fun (x01:(P1 ((and (irrefl x)) (trans x))))=> x01) as proof of (P2 ((and (irrefl x)) (trans x)))
% Found x01:(P1 ((and (irrefl x)) (trans x)))
% Found (fun (x01:(P1 ((and (irrefl x)) (trans x))))=> x01) as proof of (P1 ((and (irrefl x)) (trans x)))
% Found (fun (x01:(P1 ((and (irrefl x)) (trans x))))=> x01) as proof of (P2 ((and (irrefl x)) (trans x)))
% Found x01:(P1 ((and (irrefl x)) (trans x)))
% Found (fun (x01:(P1 ((and (irrefl x)) (trans x))))=> x01) as proof of (P1 ((and (irrefl x)) (trans x)))
% Found (fun (x01:(P1 ((and (irrefl x)) (trans x))))=> x01) as proof of (P2 ((and (irrefl x)) (trans x)))
% Found x01:(P1 ((and (irrefl x)) (trans x)))
% Found (fun (x01:(P1 ((and (irrefl x)) (trans x))))=> x01) as proof of (P1 ((and (irrefl x)) (trans x)))
% Found (fun (x01:(P1 ((and (irrefl x)) (trans x))))=> x01) as proof of (P2 ((and (irrefl x)) (trans x)))
% Found x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R))))
% Found (fun (x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R)))))=> x2) as proof of (P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R))))
% Found (fun (x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R)))))=> x2) as proof of (P0 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R))))
% Found x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R))))
% Found (fun (x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R)))))=> x2) as proof of (P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R))))
% Found (fun (x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R)))))=> x2) as proof of (P0 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((R x3) x3))))))))) (trans R))))
% Found x01:(P (so x))
% Found (fun (x01:(P (so x)))=> x01) as proof of (P (so x))
% Found (fun (x01:(P (so x)))=> x01) as proof of (P0 (so x))
% Found x01:(P (so x))
% Found (fun (x01:(P (so x)))=> x01) as proof of (P (so x))
% Found (fun (x01:(P (so x)))=> x01) as proof of (P0 (so x))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found x02:(P ((and (irrefl x)) (trans x)))
% Found (fun (x02:(P ((and (irrefl x)) (trans x))))=> x02) as proof of (P ((and (irrefl x)) (trans x)))
% Found (fun (x02:(P ((and (irrefl x)) (trans x))))=> x02) as proof of (P0 ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found x02:(P ((and (irrefl x)) (trans x)))
% Found (fun (x02:(P ((and (irrefl x)) (trans x))))=> x02) as proof of (P ((and (irrefl x)) (trans x)))
% Found (fun (x02:(P ((and (irrefl x)) (trans x))))=> x02) as proof of (P0 ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found x02:(P ((and (irrefl x)) (trans x)))
% Found (fun (x02:(P ((and (irrefl x)) (trans x))))=> x02) as proof of (P ((and (irrefl x)) (trans x)))
% Found (fun (x02:(P ((and (irrefl x)) (trans x))))=> x02) as proof of (P0 ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found x02:(P ((and (irrefl x)) (trans x)))
% Found (fun (x02:(P ((and (irrefl x)) (trans x))))=> x02) as proof of (P ((and (irrefl x)) (trans x)))
% Found (fun (x02:(P ((and (irrefl x)) (trans x))))=> x02) as proof of (P0 ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found x0:(P0 b)
% Instantiate: b:=((and (irrefl x)) (trans x)):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 ((and (irrefl x)) (trans x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 ((and (irrefl x)) (trans x))))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found x0:(P0 b)
% Instantiate: b:=((and (irrefl x)) (trans x)):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 ((and (irrefl x)) (trans x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 ((and (irrefl x)) (trans x))))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found x0:(P (so x))
% Instantiate: b:=(so x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found x0:(P (so x))
% Instantiate: b:=(so x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) (fun (x:(fofType->(fofType->Prop)))=> ((and (irrefl x)) (trans x))))
% Found (eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) so)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) (fun (x:(fofType->(fofType->Prop)))=> ((and (irrefl x)) (trans x))))
% Found (eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P0 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P0 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P0 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found (fun (x2:(P (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R)))))=> x2) as proof of (P0 (fun (R:(fofType->(fofType->Prop)))=> ((and (irrefl R)) (trans R))))
% Found x01:(P (so x))
% Found (fun (x01:(P (so x)))=> x01) as proof of (P (so x))
% Found (fun (x01:(P (so x)))=> x01) as proof of (P0 (so x))
% Found x01:(P (so x))
% Found (fun (x01:(P (so x)))=> x01) as proof of (P (so x))
% Found (fun (x01:(P (so x)))=> x01) as proof of (P0 (so x))
% Found x01:(P (so x))
% Found (fun (x01:(P (so x)))=> x01) as proof of (P (so x))
% Found (fun (x01:(P (so x)))=> x01) as proof of (P0 (so x))
% Found x01:(P (so x))
% Found (fun (x01:(P (so x)))=> x01) as proof of (P (so x))
% Found (fun (x01:(P (so x)))=> x01) as proof of (P0 (so x))
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl x)) (trans x))):(((eq Prop) ((and (irrefl x)) (trans x))) ((and (irrefl x)) (trans x)))
% Found (eq_ref0 ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found ((eq_ref Prop) ((and (irrefl x)) (trans x))) as proof of (((eq Prop) ((and (irrefl x)) (trans x))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found x01:(P ((and (irrefl x)) (trans x)))
% Found (fun (x01:(P ((and (irrefl x)) (trans x))))=> x01) as proof of (P ((and (irrefl x)) (trans x)))
% Found (fun (x01:(P ((and (irrefl x)) (trans x))))=> x01) as proof of (P0 ((and (irrefl x)) (trans x)))
% Found x01:(P ((and (irrefl x)) (trans x)))
% Found (fun (x01:(P ((and (irrefl x)) (trans x))))=> x01) as proof of (P ((and (irrefl x)) (trans x)))
% Found (fun (x01:(P ((and (irrefl x)) (trans x))))=> x01) as proof of (P0 ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))):(((eq Prop) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x)))
% Found (eq_ref0 ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) as proof of (((eq Prop) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) as proof of (((eq Prop) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) as proof of (((eq Prop) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) as proof of (((eq Prop) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))):(((eq Prop) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x)))
% Found (eq_ref0 ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) as proof of (((eq Prop) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) as proof of (((eq Prop) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) as proof of (((eq Prop) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) as proof of (((eq Prop) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (irrefl x)) (trans x)))
% Found eq_ref00:=(eq_ref0 (so x)):(((eq Prop) (so x)) (so x))
% Found (eq_ref0 (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found ((eq_ref Prop) (so x)) as proof of (((eq Prop) (so x)) b)
% Found eq_ref00:=(eq_ref0 ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))):(((eq Prop) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x)))
% Found (eq_ref0 ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) as proof of (((eq Prop) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) as proof of (((eq Prop) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) as proof of (((eq Prop) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) b)
% Found ((eq_ref Prop) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) as proof of (((eq Prop) ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (so x))
% Found eq_ref00:=(eq_ref0 ((and (irrefl (inv (inv (inv (inv (inv (fun (x3:fofType) (x20:fofType)=> ((x x3) x3))))))))) (trans x))):(((e
% EOF
%------------------------------------------------------------------------------