%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : PUZ140^1 : TPTP v6.1.0. Released v6.1.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n110.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 13:29:02 EDT 2014

% Result   : Timeout 300.01s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : PUZ140^1 : TPTP v6.1.0. Released v6.1.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n110.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 08:31:06 CDT 2014
% % CPUTime  : 300.01 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x1d1f098>, <kernel.Type object at 0x1d1f560>) of role type named syrup_type
% Using role type
% Declaring syrup:Type
% FOF formula (<kernel.Constant object at 0x182ad40>, <kernel.Type object at 0x1d1f518>) of role type named beverage_type
% Using role type
% Declaring beverage:Type
% FOF formula (<kernel.Constant object at 0x1d1f0e0>, <kernel.Constant object at 0x1d1f320>) of role type named coffee_type
% Using role type
% Declaring coffee:beverage
% FOF formula (<kernel.Constant object at 0x1d1f560>, <kernel.DependentProduct object at 0x1d9cd88>) of role type named heat_type
% Using role type
% Declaring heat:(beverage->beverage)
% FOF formula (<kernel.Constant object at 0x1d1f098>, <kernel.DependentProduct object at 0x1d9c7e8>) of role type named hot_type
% Using role type
% Declaring hot:(beverage->Prop)
% FOF formula (<kernel.Constant object at 0x1d1f0e0>, <kernel.DependentProduct object at 0x1d9ca28>) of role type named mix_type
% Using role type
% Declaring mix:(beverage->(syrup->beverage))
% FOF formula (<kernel.Constant object at 0x1d1f098>, <kernel.DependentProduct object at 0x1d9c830>) of role type named hot_mixture_type
% Using role type
% Declaring hot_mixture:(beverage->(syrup->beverage))
% FOF formula (<kernel.Constant object at 0x1d1f0e0>, <kernel.DependentProduct object at 0x1d9ccb0>) of role type named cold_mixture_type
% Using role type
% Declaring cold_mixture:(beverage->(syrup->beverage))
% FOF formula (((eq (beverage->(syrup->beverage))) hot_mixture) (fun (B:beverage) (S:syrup)=> (heat ((mix B) S)))) of role definition named hot_mixture_definition
% A new definition: (((eq (beverage->(syrup->beverage))) hot_mixture) (fun (B:beverage) (S:syrup)=> (heat ((mix B) S))))
% Defined: hot_mixture:=(fun (B:beverage) (S:syrup)=> (heat ((mix B) S)))
% FOF formula (((eq (beverage->(syrup->beverage))) cold_mixture) (fun (B:beverage) (S:syrup)=> ((mix B) S))) of role definition named cold_mixture_definition
% A new definition: (((eq (beverage->(syrup->beverage))) cold_mixture) (fun (B:beverage) (S:syrup)=> ((mix B) S)))
% Defined: cold_mixture:=(fun (B:beverage) (S:syrup)=> ((mix B) S))
% FOF formula (forall (B:beverage), (hot (heat B))) of role axiom named its_hot
% A new axiom: (forall (B:beverage), (hot (heat B)))
% FOF formula ((ex (beverage->(syrup->beverage))) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) of role conjecture named hot_coffee
% Conjecture to prove = ((ex (beverage->(syrup->beverage))) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))):Prop
% Parameter syrup_DUMMY:syrup.
% We need to prove ['((ex (beverage->(syrup->beverage))) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B)))))))']
% Parameter syrup:Type.
% Parameter beverage:Type.
% Parameter coffee:beverage.
% Parameter heat:(beverage->beverage).
% Parameter hot:(beverage->Prop).
% Parameter mix:(beverage->(syrup->beverage)).
% Definition hot_mixture:=(fun (B:beverage) (S:syrup)=> (heat ((mix B) S))):(beverage->(syrup->beverage)).
% Definition cold_mixture:=(fun (B:beverage) (S:syrup)=> ((mix B) S)):(beverage->(syrup->beverage)).
% Axiom its_hot:(forall (B:beverage), (hot (heat B))).
% Trying to prove ((ex (beverage->(syrup->beverage))) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B)))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))):(((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) (fun (x:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))))))
% Found (eta_expansion00 (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b)
% Found ((eta_expansion0 Prop) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f x)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f x)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))))))
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq ((beverage->(syrup->beverage))->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a) b)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) a) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a) b)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) a) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a) b)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) a) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((beverage->(syrup->beverage))->Prop)) b) (fun (x:(beverage->(syrup->beverage)))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B)))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B)))))))
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B)))))))
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B)))))))
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B)))))))
% Found eta_expansion000:=(eta_expansion00 (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))):(((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) (fun (x0:beverage)=> ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0))))
% Found (eta_expansion00 (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b)
% Found ((eta_expansion0 Prop) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b)
% Found (((eta_expansion beverage) Prop) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b)
% Found (((eta_expansion beverage) Prop) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b)
% Found (((eta_expansion beverage) Prop) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b)
% 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 (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (forall (x0:beverage), (((eq Prop) (f x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0))))
% 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 (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (forall (x0:beverage), (((eq Prop) (f x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (beverage->Prop)) a) (fun (x:beverage)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (beverage->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x1:beverage)=> Prop)) a) as proof of (((eq (beverage->Prop)) a) b)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) a) as proof of (((eq (beverage->Prop)) a) b)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) a) as proof of (((eq (beverage->Prop)) a) b)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) a) as proof of (((eq (beverage->Prop)) a) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (beverage->Prop)) b) (fun (x:beverage)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (beverage->Prop)) b) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (beverage->Prop)) b) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found (((eta_expansion beverage) Prop) b) as proof of (((eq (beverage->Prop)) b) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found (((eta_expansion beverage) Prop) b) as proof of (((eq (beverage->Prop)) b) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found (((eta_expansion beverage) Prop) b) as proof of (((eq (beverage->Prop)) b) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found eq_ref00:=(eq_ref0 b):(((eq (beverage->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (beverage->Prop)) b) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (hot x0)):(((eq Prop) (hot x0)) (hot x0))
% Found (eq_ref0 (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((beverage->(syrup->beverage))->Prop)) b) (fun (x:(beverage->(syrup->beverage)))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) b0)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) b0)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) b0)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((beverage->(syrup->beverage))->Prop)) b) (fun (x:(beverage->(syrup->beverage)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) b0)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) b0)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) b0)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of b
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((conj00 ((eq_ref beverage) ((x coffee) S))) or_comm_i) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found (((conj0 b) ((eq_ref beverage) ((x coffee) S))) or_comm_i) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b) ((eq_ref beverage) ((x coffee) S))) or_comm_i) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b) ((eq_ref beverage) ((x coffee) S))) or_comm_i) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b) ((eq_ref beverage) ((x coffee) S))) or_comm_i) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))):(((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B)))))))
% Found (eq_ref0 (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b0)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b0)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b0)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (hot x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot x0))
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 b):(((eq (beverage->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (beverage->Prop)) b) b0)
% Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) b0)
% Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) b0)
% Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:beverage), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:beverage), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:beverage), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:beverage), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found choice_operator:=(fun (A:Type) (a:A)=> ((((classical_choice (A->Prop)) A) (fun (x3:(A->Prop))=> x3)) a)):(forall (A:Type), (A->((ex ((A->Prop)->A)) (fun (co:((A->Prop)->A))=> (forall (P:(A->Prop)), (((ex A) (fun (x:A)=> (P x)))->(P (co P))))))))
% Instantiate: a:=(forall (A:Type), (A->((ex ((A->Prop)->A)) (fun (co:((A->Prop)->A))=> (forall (P:(A->Prop)), (((ex A) (fun (x:A)=> (P x)))->(P (co P)))))))):Prop
% Found choice_operator as proof of a
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((conj00 ((eq_ref beverage) ((x coffee) S))) choice_operator) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) a)
% Found (((conj0 a) ((eq_ref beverage) ((x coffee) S))) choice_operator) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) a)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) a) ((eq_ref beverage) ((x coffee) S))) choice_operator) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) a)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) a) ((eq_ref beverage) ((x coffee) S))) choice_operator) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) a)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) a) ((eq_ref beverage) ((x coffee) S))) choice_operator) as proof of (P a)
% Found eq_ref00:=(eq_ref0 b):(((eq (beverage->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (beverage->Prop)) b) b0)
% Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) b0)
% Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) b0)
% Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:beverage), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:beverage), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:beverage), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:beverage), (((eq Prop) (f0 x)) (f x)))
% Found x0:(P2 (f x))
% Instantiate: f0:=f:((beverage->(syrup->beverage))->Prop)
% Found (fun (x0:(P2 (f x)))=> x0) as proof of (P2 (f0 x))
% Found (fun (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of ((P2 (f x))->(P2 (f0 x)))
% Found (fun (x:(beverage->(syrup->beverage))) (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage))) (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f x)) (f0 x)))
% Found x0:(P2 (f x))
% Instantiate: f0:=f:((beverage->(syrup->beverage))->Prop)
% Found (fun (x0:(P2 (f x)))=> x0) as proof of (P2 (f0 x))
% Found (fun (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of ((P2 (f x))->(P2 (f0 x)))
% Found (fun (x:(beverage->(syrup->beverage))) (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage))) (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))):(((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) (fun (x:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))))))
% Found (eta_expansion_dep00 (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b0)
% Found ((eta_expansion_dep0 (fun (x1:(beverage->(syrup->beverage)))=> Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b0)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b0)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b0)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b0)
% Found x0:(P2 (f x))
% Instantiate: f0:=f:((beverage->(syrup->beverage))->Prop)
% Found (fun (x0:(P2 (f x)))=> x0) as proof of (P2 (f0 x))
% Found (fun (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of ((P2 (f x))->(P2 (f0 x)))
% Found (fun (x:(beverage->(syrup->beverage))) (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage))) (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f x)) (f0 x)))
% Found x0:(P2 (f x))
% Instantiate: f0:=f:((beverage->(syrup->beverage))->Prop)
% Found (fun (x0:(P2 (f x)))=> x0) as proof of (P2 (f0 x))
% Found (fun (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of ((P2 (f x))->(P2 (f0 x)))
% Found (fun (x:(beverage->(syrup->beverage))) (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage))) (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found choice_operator:=(fun (A:Type) (a:A)=> ((((classical_choice (A->Prop)) A) (fun (x3:(A->Prop))=> x3)) a)):(forall (A:Type), (A->((ex ((A->Prop)->A)) (fun (co:((A->Prop)->A))=> (forall (P:(A->Prop)), (((ex A) (fun (x:A)=> (P x)))->(P (co P))))))))
% Instantiate: b:=(forall (A:Type), (A->((ex ((A->Prop)->A)) (fun (co:((A->Prop)->A))=> (forall (P:(A->Prop)), (((ex A) (fun (x:A)=> (P x)))->(P (co P)))))))):Prop
% Found choice_operator as proof of b
% Found choice_operator as proof of a
% Found ((conj00 ((eq_ref beverage) ((x coffee) S))) choice_operator) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) a)
% Found (((conj0 a) ((eq_ref beverage) ((x coffee) S))) choice_operator) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) a)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) a) ((eq_ref beverage) ((x coffee) S))) choice_operator) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) a)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) a) ((eq_ref beverage) ((x coffee) S))) choice_operator) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) a)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) a) ((eq_ref beverage) ((x coffee) S))) choice_operator) as proof of (P a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))):(((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found (eq_ref0 (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b0)
% Found ((eq_ref (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b0)
% Found ((eq_ref (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b0)
% Found ((eq_ref (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))):(((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) (fun (x0:beverage)=> ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0))))
% Found (eta_expansion_dep00 (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b)
% Found ((eta_expansion_dep0 (fun (x1:beverage)=> Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))):(((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) (fun (x0:beverage)=> ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0))))
% Found (eta_expansion00 (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b)
% Found ((eta_expansion0 Prop) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b)
% Found (((eta_expansion beverage) Prop) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b)
% Found (((eta_expansion beverage) Prop) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b)
% Found (((eta_expansion beverage) Prop) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f x)) (f0 x)))
% 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 (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (forall (x0:beverage), (((eq Prop) (f x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0))))
% 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 (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (forall (x0:beverage), (((eq Prop) (f x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0))))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x0:beverage), (((eq Prop) (f0 x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0))))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x0:beverage), (((eq Prop) (f0 x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0))))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x0:beverage), (((eq Prop) (f0 x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0))))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x0:beverage), (((eq Prop) (f0 x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0))))
% Found x0:(P2 (f x))
% Instantiate: f0:=f:((beverage->(syrup->beverage))->Prop)
% Found (fun (x0:(P2 (f x)))=> x0) as proof of (P2 (f0 x))
% Found (fun (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of ((P2 (f x))->(P2 (f0 x)))
% Found (fun (x:(beverage->(syrup->beverage))) (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage))) (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f x)) (f0 x)))
% Found x0:(P2 (f x))
% Instantiate: f0:=f:((beverage->(syrup->beverage))->Prop)
% Found (fun (x0:(P2 (f x)))=> x0) as proof of (P2 (f0 x))
% Found (fun (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of ((P2 (f x))->(P2 (f0 x)))
% Found (fun (x:(beverage->(syrup->beverage))) (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage))) (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f x)) (f0 x)))
% Found x0:(P2 (f x))
% Instantiate: f0:=f:((beverage->(syrup->beverage))->Prop)
% Found (fun (x0:(P2 (f x)))=> x0) as proof of (P2 (f0 x))
% Found (fun (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of ((P2 (f x))->(P2 (f0 x)))
% Found (fun (x:(beverage->(syrup->beverage))) (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage))) (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f x)) (f0 x)))
% Found x0:(P2 (f x))
% Instantiate: f0:=f:((beverage->(syrup->beverage))->Prop)
% Found (fun (x0:(P2 (f x)))=> x0) as proof of (P2 (f0 x))
% Found (fun (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of ((P2 (f x))->(P2 (f0 x)))
% Found (fun (x:(beverage->(syrup->beverage))) (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage))) (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 (hot a)):(((eq Prop) (hot a)) (hot a))
% Found (eq_ref0 (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found eq_ref00:=(eq_ref0 (hot a)):(((eq Prop) (hot a)) (hot a))
% Found (eq_ref0 (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq ((beverage->(syrup->beverage))->Prop)) b0) (fun (x:(beverage->(syrup->beverage)))=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((beverage->(syrup->beverage))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq ((beverage->(syrup->beverage))->Prop)) a0) (fun (x:(beverage->(syrup->beverage)))=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x1:(beverage->(syrup->beverage)))=> Prop)) a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((beverage->(syrup->beverage))->Prop)) b0) (fun (x:(beverage->(syrup->beverage)))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found ((eta_expansion_dep0 (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((beverage->(syrup->beverage))->Prop)) b0) (fun (x:(beverage->(syrup->beverage)))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found ((eta_expansion_dep0 (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((beverage->(syrup->beverage))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq ((beverage->(syrup->beverage))->Prop)) b0) (fun (x:(beverage->(syrup->beverage)))=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq ((beverage->(syrup->beverage))->Prop)) b0) (fun (x:(beverage->(syrup->beverage)))=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 (hot x0)):(((eq Prop) (hot x0)) (hot x0))
% Found (eq_ref0 (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((beverage->(syrup->beverage))->Prop)) b0) (fun (x:(beverage->(syrup->beverage)))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found ((eta_expansion_dep0 (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((beverage->(syrup->beverage))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((beverage->(syrup->beverage))->Prop)) b0) (fun (x:(beverage->(syrup->beverage)))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found ((eta_expansion_dep0 (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq ((beverage->(syrup->beverage))->Prop)) a0) (fun (x:(beverage->(syrup->beverage)))=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq ((beverage->(syrup->beverage))->Prop)) a0) (fun (x:(beverage->(syrup->beverage)))=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x1:(beverage->(syrup->beverage)))=> Prop)) a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq ((beverage->(syrup->beverage))->Prop)) a0) (fun (x:(beverage->(syrup->beverage)))=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x1:(beverage->(syrup->beverage)))=> Prop)) a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) a0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((beverage->(syrup->beverage))->Prop)) b0) (fun (x:(beverage->(syrup->beverage)))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found ((eta_expansion_dep0 (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) a)
% 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)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:beverage), (((eq Prop) (f x)) (f0 x)))
% 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)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:beverage), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))):(((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found (eq_ref0 (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b0)
% Found ((eq_ref (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b0)
% Found ((eq_ref (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b0)
% Found ((eq_ref (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b0)
% Found x1:(P2 (f x0))
% Instantiate: f0:=f:(beverage->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (f0 x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (f0 x0)))
% Found (fun (x0:beverage) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:beverage) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:beverage), (((eq Prop) (f x)) (f0 x)))
% Found x1:(P2 (f x0))
% Instantiate: f0:=f:(beverage->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (f0 x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (f0 x0)))
% Found (fun (x0:beverage) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:beverage) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:beverage), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq ((beverage->(syrup->beverage))->Prop)) b0) (fun (x:(beverage->(syrup->beverage)))=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 b0):(((eq ((beverage->(syrup->beverage))->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (hot x0)):(((eq Prop) (hot x0)) (hot x0))
% Found (eq_ref0 (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (hot a)):(((eq Prop) (hot a)) (hot a))
% Found (eq_ref0 (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 (hot a)):(((eq Prop) (hot a)) (hot a))
% Found (eq_ref0 (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found eq_ref00:=(eq_ref0 (hot x0)):(((eq Prop) (hot x0)) (hot x0))
% Found (eq_ref0 (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found eq_ref00:=(eq_ref0 (hot x0)):(((eq Prop) (hot x0)) (hot x0))
% Found (eq_ref0 (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 (hot a)):(((eq Prop) (hot a)) (hot a))
% Found (eq_ref0 (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 (hot a)):(((eq Prop) (hot a)) (hot a))
% Found (eq_ref0 (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((conj00 ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) a)) b)
% Found (((conj0 b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) a)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) a)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) a)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of (P0 b)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((conj00 ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) a)) b)
% Found (((conj0 b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) a)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) a)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) a)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq ((beverage->(syrup->beverage))->Prop)) b0) (fun (x:(beverage->(syrup->beverage)))=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 b0):(((eq ((beverage->(syrup->beverage))->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Instantiate: b0:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of b0
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((conj00 ((eq_ref beverage) ((x coffee) S))) eq_sym) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b0)
% Found (((conj0 b0) ((eq_ref beverage) ((x coffee) S))) eq_sym) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b0)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b0) ((eq_ref beverage) ((x coffee) S))) eq_sym) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b0)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b0) ((eq_ref beverage) ((x coffee) S))) eq_sym) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b0)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b0) ((eq_ref beverage) ((x coffee) S))) eq_sym) as proof of (P0 b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq ((beverage->(syrup->beverage))->Prop)) b0) (fun (x:(beverage->(syrup->beverage)))=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 (hot x0)):(((eq Prop) (hot x0)) (hot x0))
% Found (eq_ref0 (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 (hot x0)):(((eq Prop) (hot x0)) (hot x0))
% Found (eq_ref0 (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% 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)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:beverage), (((eq Prop) (f x)) (f0 x)))
% 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)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:beverage), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% 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)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:beverage), (((eq Prop) (f x)) (f0 x)))
% 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)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:beverage), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))):(((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found (eq_ref0 (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b0)
% Found ((eq_ref (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b0)
% Found ((eq_ref (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b0)
% Found ((eq_ref (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b0)
% Found x1:(P2 (f x0))
% Instantiate: f0:=f:(beverage->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (f0 x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (f0 x0)))
% Found (fun (x0:beverage) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:beverage) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:beverage), (((eq Prop) (f x)) (f0 x)))
% Found x1:(P2 (f x0))
% Instantiate: f0:=f:(beverage->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (f0 x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (f0 x0)))
% Found (fun (x0:beverage) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:beverage) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:beverage), (((eq Prop) (f x)) (f0 x)))
% Found x1:(P2 (f x0))
% Instantiate: f0:=f:(beverage->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (f0 x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (f0 x0)))
% Found (fun (x0:beverage) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:beverage) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:beverage), (((eq Prop) (f x)) (f0 x)))
% Found x1:(P2 (f x0))
% Instantiate: f0:=f:(beverage->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (f0 x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (f0 x0)))
% Found (fun (x0:beverage) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:beverage) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:beverage), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))):(((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B)))))))
% Found (eq_ref0 (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b0)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b0)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b0)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq ((beverage->(syrup->beverage))->Prop)) b0) (fun (x:(beverage->(syrup->beverage)))=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found eq_ref00:=(eq_ref0 (fun (B0:beverage)=> ((and (((eq beverage) ((x coffee) S)) B0)) (hot B0)))):(((eq (beverage->Prop)) (fun (B0:beverage)=> ((and (((eq beverage) ((x coffee) S)) B0)) (hot B0)))) (fun (B0:beverage)=> ((and (((eq beverage) ((x coffee) S)) B0)) (hot B0))))
% Found (eq_ref0 (fun (B0:beverage)=> ((and (((eq beverage) ((x coffee) S)) B0)) (hot B0)))) as proof of (((eq (beverage->Prop)) (fun (B0:beverage)=> ((and (((eq beverage) ((x coffee) S)) B0)) (hot B0)))) b0)
% Found ((eq_ref (beverage->Prop)) (fun (B0:beverage)=> ((and (((eq beverage) ((x coffee) S)) B0)) (hot B0)))) as proof of (((eq (beverage->Prop)) (fun (B0:beverage)=> ((and (((eq beverage) ((x coffee) S)) B0)) (hot B0)))) b0)
% Found ((eq_ref (beverage->Prop)) (fun (B0:beverage)=> ((and (((eq beverage) ((x coffee) S)) B0)) (hot B0)))) as proof of (((eq (beverage->Prop)) (fun (B0:beverage)=> ((and (((eq beverage) ((x coffee) S)) B0)) (hot B0)))) b0)
% Found ((eq_ref (beverage->Prop)) (fun (B0:beverage)=> ((and (((eq beverage) ((x coffee) S)) B0)) (hot B0)))) as proof of (((eq (beverage->Prop)) (fun (B0:beverage)=> ((and (((eq beverage) ((x coffee) S)) B0)) (hot B0)))) b0)
% Found eq_ref00:=(eq_ref0 (hot x0)):(((eq Prop) (hot x0)) (hot x0))
% Found (eq_ref0 (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq ((beverage->(syrup->beverage))->Prop)) b0) (fun (x:(beverage->(syrup->beverage)))=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 (hot x0)):(((eq Prop) (hot x0)) (hot x0))
% Found (eq_ref0 (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found eq_ref00:=(eq_ref0 (hot x0)):(((eq Prop) (hot x0)) (hot x0))
% Found (eq_ref0 (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% 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 (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (forall (x0:beverage), (((eq Prop) (f x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0))))
% 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 (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0)))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (forall (x0:beverage), (((eq Prop) (f x0)) ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (beverage->Prop)) a) (fun (x:beverage)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (beverage->Prop)) a) b0)
% Found ((eta_expansion_dep0 (fun (x1:beverage)=> Prop)) a) as proof of (((eq (beverage->Prop)) a) b0)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) a) as proof of (((eq (beverage->Prop)) a) b0)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) a) as proof of (((eq (beverage->Prop)) a) b0)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) a) as proof of (((eq (beverage->Prop)) a) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (beverage->Prop)) b0) (fun (x:beverage)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (beverage->Prop)) b0) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found ((eta_expansion_dep0 (fun (x1:beverage)=> Prop)) b0) as proof of (((eq (beverage->Prop)) b0) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) b0) as proof of (((eq (beverage->Prop)) b0) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) b0) as proof of (((eq (beverage->Prop)) b0) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) b0) as proof of (((eq (beverage->Prop)) b0) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found eq_ref00:=(eq_ref0 b0):(((eq (beverage->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (beverage->Prop)) b0) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found ((eq_ref (beverage->Prop)) b0) as proof of (((eq (beverage->Prop)) b0) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found ((eq_ref (beverage->Prop)) b0) as proof of (((eq (beverage->Prop)) b0) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found ((eq_ref (beverage->Prop)) b0) as proof of (((eq (beverage->Prop)) b0) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f0 x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (beverage->Prop)) a) (fun (x:beverage)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (beverage->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x1:beverage)=> Prop)) a) as proof of (((eq (beverage->Prop)) a) b)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) a) as proof of (((eq (beverage->Prop)) a) b)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) a) as proof of (((eq (beverage->Prop)) a) b)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) a) as proof of (((eq (beverage->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (beverage->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (beverage->Prop)) b) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found eq_ref00:=(eq_ref0 b):(((eq (beverage->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (beverage->Prop)) b) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (beverage->Prop)) a) (fun (x:beverage)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (beverage->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x1:beverage)=> Prop)) a) as proof of (((eq (beverage->Prop)) a) b)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) a) as proof of (((eq (beverage->Prop)) a) b)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) a) as proof of (((eq (beverage->Prop)) a) b)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) a) as proof of (((eq (beverage->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (beverage->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (beverage->Prop)) b) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found eq_ref00:=(eq_ref0 b):(((eq (beverage->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (beverage->Prop)) b) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B))))
% Found eq_ref00:=(eq_ref0 (hot x0)):(((eq Prop) (hot x0)) (hot x0))
% Found (eq_ref0 (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Instantiate: b0:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of b0
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((conj00 ((eq_ref beverage) ((x coffee) S))) eq_sym) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b0)
% Found (((conj0 b0) ((eq_ref beverage) ((x coffee) S))) eq_sym) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b0)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b0) ((eq_ref beverage) ((x coffee) S))) eq_sym) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b0)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b0) ((eq_ref beverage) ((x coffee) S))) eq_sym) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b0)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b0) ((eq_ref beverage) ((x coffee) S))) eq_sym) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (hot x0)):(((eq Prop) (hot x0)) (hot x0))
% Found (eq_ref0 (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found eq_ref00:=(eq_ref0 (hot x0)):(((eq Prop) (hot x0)) (hot x0))
% Found (eq_ref0 (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b0)
% Found eq_ref00:=(eq_ref0 (hot x0)):(((eq Prop) (hot x0)) (hot x0))
% Found (eq_ref0 (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((beverage->(syrup->beverage))->Prop)) b0) (fun (x:(beverage->(syrup->beverage)))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found ((eta_expansion_dep0 (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found eq_ref00:=(eq_ref0 (hot x0)):(((eq Prop) (hot x0)) (hot x0))
% Found (eq_ref0 (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found ((eq_ref Prop) (hot x0)) as proof of (((eq Prop) (hot x0)) b)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((conj00 ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) a)) b)
% Found (((conj0 b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) a)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) a)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) a)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of (P0 b)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((conj00 ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) a)) b)
% Found (((conj0 b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) a)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) a)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) a)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of (P0 b)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((conj00 ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) a)) b)
% Found (((conj0 b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) a)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) a)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) a)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((conj00 ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) a)) b)
% Found (((conj0 b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) a)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) a)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) a)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((conj00 ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found (((conj0 b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((conj00 ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found (((conj0 b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b) ((eq_ref beverage) ((x coffee) S))) iff_sym) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eta_expansion000:=(eta_expansion00 (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))):(((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) (fun (x:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))))))
% Found (eta_expansion00 (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b1)
% Found ((eta_expansion0 Prop) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b1)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Instantiate: b0:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of b0
% Found eq_sym as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Instantiate: b0:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of b0
% Found eq_sym as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% 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)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:beverage), (((eq Prop) (f x)) (f0 x)))
% 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)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:beverage), (((eq Prop) (f x)) (f0 x)))
% Found x1:(P2 (f x0))
% Instantiate: f0:=f:(beverage->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (f0 x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (f0 x0)))
% Found (fun (x0:beverage) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:beverage) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:beverage), (((eq Prop) (f x)) (f0 x)))
% Found x1:(P2 (f x0))
% Instantiate: f0:=f:(beverage->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (f0 x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (f0 x0)))
% Found (fun (x0:beverage) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:beverage) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:beverage), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (beverage->Prop)) b0) (fun (x:beverage)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found ((eta_expansion_dep0 (fun (x1:beverage)=> Prop)) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq (beverage->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (beverage->Prop)) a0) b0)
% Found ((eq_ref (beverage->Prop)) a0) as proof of (((eq (beverage->Prop)) a0) b0)
% Found ((eq_ref (beverage->Prop)) a0) as proof of (((eq (beverage->Prop)) a0) b0)
% Found ((eq_ref (beverage->Prop)) a0) as proof of (((eq (beverage->Prop)) a0) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (beverage->Prop)) b0) (fun (x:beverage)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found (((eta_expansion beverage) Prop) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found (((eta_expansion beverage) Prop) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found (((eta_expansion beverage) Prop) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq (beverage->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (beverage->Prop)) a0) b0)
% Found ((eq_ref (beverage->Prop)) a0) as proof of (((eq (beverage->Prop)) a0) b0)
% Found ((eq_ref (beverage->Prop)) a0) as proof of (((eq (beverage->Prop)) a0) b0)
% Found ((eq_ref (beverage->Prop)) a0) as proof of (((eq (beverage->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (beverage->Prop)) a0) (fun (x:beverage)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (beverage->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x1:beverage)=> Prop)) a0) as proof of (((eq (beverage->Prop)) a0) b0)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) a0) as proof of (((eq (beverage->Prop)) a0) b0)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) a0) as proof of (((eq (beverage->Prop)) a0) b0)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) a0) as proof of (((eq (beverage->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (beverage->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found ((eq_ref (beverage->Prop)) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found ((eq_ref (beverage->Prop)) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found ((eq_ref (beverage->Prop)) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (beverage->Prop)) b0) (fun (x:beverage)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found ((eta_expansion_dep0 (fun (x1:beverage)=> Prop)) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Instantiate: b0:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of b0
% Found eq_sym as proof of (P0 b0)
% Found eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Instantiate: b0:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of b0
% Found eq_sym as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found relational_choice:(forall (A:Type) (B:Type) (R:(A->(B->Prop))), ((forall (x:A), ((ex B) (fun (y:B)=> ((R x) y))))->((ex (A->(B->Prop))) (fun (R':(A->(B->Prop)))=> ((and ((((subrelation A) B) R') R)) (forall (x:A), ((ex B) ((unique B) (fun (y:B)=> ((R' x) y))))))))))
% Instantiate: b0:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), ((forall (x:A), ((ex B) (fun (y:B)=> ((R x) y))))->((ex (A->(B->Prop))) (fun (R':(A->(B->Prop)))=> ((and ((((subrelation A) B) R') R)) (forall (x:A), ((ex B) ((unique B) (fun (y:B)=> ((R' x) y)))))))))):Prop
% Found relational_choice as proof of b0
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((conj00 ((eq_ref beverage) ((x coffee) S))) relational_choice) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b0)
% Found (((conj0 b0) ((eq_ref beverage) ((x coffee) S))) relational_choice) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b0)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b0) ((eq_ref beverage) ((x coffee) S))) relational_choice) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b0)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b0) ((eq_ref beverage) ((x coffee) S))) relational_choice) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b0)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b0) ((eq_ref beverage) ((x coffee) S))) relational_choice) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (hot a))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot a))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot a))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot a))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (hot a))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot a))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot a))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot a))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: b:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of b
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((conj00 ((eq_ref beverage) ((x coffee) S))) iff_refl) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found (((conj0 b) ((eq_ref beverage) ((x coffee) S))) iff_refl) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b) ((eq_ref beverage) ((x coffee) S))) iff_refl) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b) ((eq_ref beverage) ((x coffee) S))) iff_refl) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b) ((eq_ref beverage) ((x coffee) S))) iff_refl) as proof of (P0 b)
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: b:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of b
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((conj00 ((eq_ref beverage) ((x coffee) S))) iff_refl) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found (((conj0 b) ((eq_ref beverage) ((x coffee) S))) iff_refl) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b) ((eq_ref beverage) ((x coffee) S))) iff_refl) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b) ((eq_ref beverage) ((x coffee) S))) iff_refl) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b) ((eq_ref beverage) ((x coffee) S))) iff_refl) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (beverage->Prop)) b0) (fun (x:beverage)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (beverage->Prop)) b0) b1)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (beverage->Prop)) b0) b1)
% Found (((eta_expansion beverage) Prop) b0) as proof of (((eq (beverage->Prop)) b0) b1)
% Found (((eta_expansion beverage) Prop) b0) as proof of (((eq (beverage->Prop)) b0) b1)
% Found (((eta_expansion beverage) Prop) b0) as proof of (((eq (beverage->Prop)) b0) b1)
% Found relational_choice:(forall (A:Type) (B:Type) (R:(A->(B->Prop))), ((forall (x:A), ((ex B) (fun (y:B)=> ((R x) y))))->((ex (A->(B->Prop))) (fun (R':(A->(B->Prop)))=> ((and ((((subrelation A) B) R') R)) (forall (x:A), ((ex B) ((unique B) (fun (y:B)=> ((R' x) y))))))))))
% Instantiate: b0:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), ((forall (x:A), ((ex B) (fun (y:B)=> ((R x) y))))->((ex (A->(B->Prop))) (fun (R':(A->(B->Prop)))=> ((and ((((subrelation A) B) R') R)) (forall (x:A), ((ex B) ((unique B) (fun (y:B)=> ((R' x) y)))))))))):Prop
% Found relational_choice as proof of b0
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((conj00 ((eq_ref beverage) ((x coffee) S))) relational_choice) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b0)
% Found (((conj0 b0) ((eq_ref beverage) ((x coffee) S))) relational_choice) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b0)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b0) ((eq_ref beverage) ((x coffee) S))) relational_choice) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b0)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b0) ((eq_ref beverage) ((x coffee) S))) relational_choice) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b0)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b0) ((eq_ref beverage) ((x coffee) S))) relational_choice) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: b:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of b
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((conj00 ((eq_ref beverage) ((x coffee) S))) iff_refl) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found (((conj0 b) ((eq_ref beverage) ((x coffee) S))) iff_refl) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b) ((eq_ref beverage) ((x coffee) S))) iff_refl) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b) ((eq_ref beverage) ((x coffee) S))) iff_refl) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b) ((eq_ref beverage) ((x coffee) S))) iff_refl) as proof of (P0 b)
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: b:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of b
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((conj00 ((eq_ref beverage) ((x coffee) S))) iff_refl) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found (((conj0 b) ((eq_ref beverage) ((x coffee) S))) iff_refl) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b) ((eq_ref beverage) ((x coffee) S))) iff_refl) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b) ((eq_ref beverage) ((x coffee) S))) iff_refl) as proof of ((and (((eq beverage) ((x coffee) S)) x0)) b)
% Found ((((conj (((eq beverage) ((x coffee) S)) x0)) b) ((eq_ref beverage) ((x coffee) S))) iff_refl) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:beverage), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:beverage), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:beverage), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:beverage), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:beverage), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:beverage), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:beverage), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:beverage), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (beverage->Prop)) b) (fun (x:beverage)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (beverage->Prop)) b) b00)
% Found ((eta_expansion_dep0 (fun (x1:beverage)=> Prop)) b) as proof of (((eq (beverage->Prop)) b) b00)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) b) as proof of (((eq (beverage->Prop)) b) b00)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) b) as proof of (((eq (beverage->Prop)) b) b00)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) b) as proof of (((eq (beverage->Prop)) b) b00)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq ((beverage->(syrup->beverage))->Prop)) b0) (fun (x:(beverage->(syrup->beverage)))=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (beverage->Prop)) b0) (fun (x:beverage)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found (((eta_expansion beverage) Prop) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found (((eta_expansion beverage) Prop) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found (((eta_expansion beverage) Prop) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq (beverage->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (beverage->Prop)) a0) b0)
% Found ((eq_ref (beverage->Prop)) a0) as proof of (((eq (beverage->Prop)) a0) b0)
% Found ((eq_ref (beverage->Prop)) a0) as proof of (((eq (beverage->Prop)) a0) b0)
% Found ((eq_ref (beverage->Prop)) a0) as proof of (((eq (beverage->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (beverage->Prop)) b0) (fun (x:beverage)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found ((eta_expansion_dep0 (fun (x1:beverage)=> Prop)) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq (beverage->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (beverage->Prop)) a0) b0)
% Found ((eq_ref (beverage->Prop)) a0) as proof of (((eq (beverage->Prop)) a0) b0)
% Found ((eq_ref (beverage->Prop)) a0) as proof of (((eq (beverage->Prop)) a0) b0)
% Found ((eq_ref (beverage->Prop)) a0) as proof of (((eq (beverage->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (beverage->Prop)) a0) (fun (x:beverage)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (beverage->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x1:beverage)=> Prop)) a0) as proof of (((eq (beverage->Prop)) a0) b0)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) a0) as proof of (((eq (beverage->Prop)) a0) b0)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) a0) as proof of (((eq (beverage->Prop)) a0) b0)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) a0) as proof of (((eq (beverage->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (beverage->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found ((eq_ref (beverage->Prop)) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found ((eq_ref (beverage->Prop)) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found ((eq_ref (beverage->Prop)) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (beverage->Prop)) b0) (fun (x:beverage)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found (((eta_expansion beverage) Prop) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found (((eta_expansion beverage) Prop) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found (((eta_expansion beverage) Prop) b0) as proof of (((eq (beverage->Prop)) b0) a)
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% 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 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (hot a))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot a))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot a))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot a))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (hot a))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot a))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot a))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot a))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (hot x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (hot x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot x0))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (hot a))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot a))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot a))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot a))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (hot a))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot a))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot a))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (hot a))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found or_second:=(fun (A:Prop) (B:Prop) (x:((or A) B))=> (((or_first B) A) (((or_comm_i A) B) x))):(forall (A:Prop) (B:Prop), (((or A) B)->((A->B)->B)))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((or A) B)->((A->B)->B))):Prop
% Found or_second as proof of a0
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((conj00 ((eq_ref beverage) ((x coffee) S))) or_second) as proof of ((and (((eq beverage) ((x coffee) S)) a)) a0)
% Found (((conj0 a0) ((eq_ref beverage) ((x coffee) S))) or_second) as proof of ((and (((eq beverage) ((x coffee) S)) a)) a0)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) a0) ((eq_ref beverage) ((x coffee) S))) or_second) as proof of ((and (((eq beverage) ((x coffee) S)) a)) a0)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) a0) ((eq_ref beverage) ((x coffee) S))) or_second) as proof of ((and (((eq beverage) ((x coffee) S)) a)) a0)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) a0) ((eq_ref beverage) ((x coffee) S))) or_second) as proof of (P0 a0)
% Found or_second:=(fun (A:Prop) (B:Prop) (x:((or A) B))=> (((or_first B) A) (((or_comm_i A) B) x))):(forall (A:Prop) (B:Prop), (((or A) B)->((A->B)->B)))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((or A) B)->((A->B)->B))):Prop
% Found or_second as proof of a0
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((conj00 ((eq_ref beverage) ((x coffee) S))) or_second) as proof of ((and (((eq beverage) ((x coffee) S)) a)) a0)
% Found (((conj0 a0) ((eq_ref beverage) ((x coffee) S))) or_second) as proof of ((and (((eq beverage) ((x coffee) S)) a)) a0)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) a0) ((eq_ref beverage) ((x coffee) S))) or_second) as proof of ((and (((eq beverage) ((x coffee) S)) a)) a0)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) a0) ((eq_ref beverage) ((x coffee) S))) or_second) as proof of ((and (((eq beverage) ((x coffee) S)) a)) a0)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) a0) ((eq_ref beverage) ((x coffee) S))) or_second) as proof of (P0 a0)
% Found or_second:=(fun (A:Prop) (B:Prop) (x:((or A) B))=> (((or_first B) A) (((or_comm_i A) B) x))):(forall (A:Prop) (B:Prop), (((or A) B)->((A->B)->B)))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((or A) B)->((A->B)->B))):Prop
% Found or_second as proof of a0
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((conj00 ((eq_ref beverage) ((x coffee) S))) or_second) as proof of ((and (((eq beverage) ((x coffee) S)) a)) a0)
% Found (((conj0 a0) ((eq_ref beverage) ((x coffee) S))) or_second) as proof of ((and (((eq beverage) ((x coffee) S)) a)) a0)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) a0) ((eq_ref beverage) ((x coffee) S))) or_second) as proof of ((and (((eq beverage) ((x coffee) S)) a)) a0)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) a0) ((eq_ref beverage) ((x coffee) S))) or_second) as proof of ((and (((eq beverage) ((x coffee) S)) a)) a0)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) a0) ((eq_ref beverage) ((x coffee) S))) or_second) as proof of (P0 a0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (beverage->Prop)) b0) (fun (x:beverage)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (beverage->Prop)) b0) b1)
% Found ((eta_expansion_dep0 (fun (x1:beverage)=> Prop)) b0) as proof of (((eq (beverage->Prop)) b0) b1)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) b0) as proof of (((eq (beverage->Prop)) b0) b1)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) b0) as proof of (((eq (beverage->Prop)) b0) b1)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) b0) as proof of (((eq (beverage->Prop)) b0) b1)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (beverage->Prop)) b0) (fun (x:beverage)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (beverage->Prop)) b0) b1)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (beverage->Prop)) b0) b1)
% Found (((eta_expansion beverage) Prop) b0) as proof of (((eq (beverage->Prop)) b0) b1)
% Found (((eta_expansion beverage) Prop) b0) as proof of (((eq (beverage->Prop)) b0) b1)
% Found (((eta_expansion beverage) Prop) b0) as proof of (((eq (beverage->Prop)) b0) b1)
% Found or_second:=(fun (A:Prop) (B:Prop) (x:((or A) B))=> (((or_first B) A) (((or_comm_i A) B) x))):(forall (A:Prop) (B:Prop), (((or A) B)->((A->B)->B)))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((or A) B)->((A->B)->B))):Prop
% Found or_second as proof of a0
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((conj00 ((eq_ref beverage) ((x coffee) S))) or_second) as proof of ((and (((eq beverage) ((x coffee) S)) a)) a0)
% Found (((conj0 a0) ((eq_ref beverage) ((x coffee) S))) or_second) as proof of ((and (((eq beverage) ((x coffee) S)) a)) a0)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) a0) ((eq_ref beverage) ((x coffee) S))) or_second) as proof of ((and (((eq beverage) ((x coffee) S)) a)) a0)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) a0) ((eq_ref beverage) ((x coffee) S))) or_second) as proof of ((and (((eq beverage) ((x coffee) S)) a)) a0)
% Found ((((conj (((eq beverage) ((x coffee) S)) a)) a0) ((eq_ref beverage) ((x coffee) S))) or_second) as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 (hot a)):(((eq Prop) (hot a)) (hot a))
% Found (eq_ref0 (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found eq_ref00:=(eq_ref0 (hot a)):(((eq Prop) (hot a)) (hot a))
% Found (eq_ref0 (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 (hot a)):(((eq Prop) (hot a)) (hot a))
% Found (eq_ref0 (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:beverage), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:beverage), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:beverage), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:beverage), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (hot a)):(((eq Prop) (hot a)) (hot a))
% Found (eq_ref0 (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found ((eq_ref Prop) (hot a)) as proof of (((eq Prop) (hot a)) b)
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:beverage), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:beverage), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:beverage), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:beverage), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:beverage), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:beverage), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:beverage), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:beverage), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:beverage), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:beverage), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:beverage), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:beverage)=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:beverage), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((beverage->(syrup->beverage))->Prop)) b0) (fun (x:(beverage->(syrup->beverage)))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found ((eta_expansion_dep0 (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b0) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b0) b1)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))):(((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) (fun (x0:beverage)=> ((and (((eq beverage) ((x coffee) S)) x0)) (hot x0))))
% Found (eta_expansion_dep00 (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b0)
% Found ((eta_expansion_dep0 (fun (x2:beverage)=> Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b0)
% Found (((eta_expansion_dep beverage) (fun (x2:beverage)=> Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b0)
% Found (((eta_expansion_dep beverage) (fun (x2:beverage)=> Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b0)
% Found (((eta_expansion_dep beverage) (fun (x2:beverage)=> Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))) b0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) a)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (beverage->Prop)) b) (fun (x:beverage)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (beverage->Prop)) b) b00)
% Found ((eta_expansion_dep0 (fun (x1:beverage)=> Prop)) b) as proof of (((eq (beverage->Prop)) b) b00)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) b) as proof of (((eq (beverage->Prop)) b) b00)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) b) as proof of (((eq (beverage->Prop)) b) b00)
% Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) b) as proof of (((eq (beverage->Prop)) b) b00)
% Found eq_ref00:=(eq_ref0 b):(((eq (beverage->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (beverage->Prop)) b) b00)
% Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) b00)
% Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) b00)
% Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) b00)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))):(((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) (fun (x:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))))))
% Found (eta_expansion_dep00 (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b1)
% Found ((eta_expansion_dep0 (fun (x1:(beverage->(syrup->beverage)))=> Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b1)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b1)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b1)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b1)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eta_expansion000:=(eta_expansion00 (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))):(((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) (fun (x:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((x coffee) S)) B)) (hot B)))))))
% Found (eta_expansion00 (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b1)
% Found ((eta_expansion0 Prop) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b1)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) (fun (Mixture:(beverage->(syrup->beverage)))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and (((eq beverage) ((Mixture coffee) S)) B)) (hot B))))))) b1)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq ((beverage->(syrup->beverage))->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a) b)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) a) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a) b)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) a) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq ((beverage->(syrup->beverage))->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a) b)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) a) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a) b)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) a) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((beverage->(syrup->beverage))->Prop)) b) (fun (x:(beverage->(syrup->beverage)))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) a0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) a0)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) a0)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) a0)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) a0)
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((beverage->(syrup->beverage))->Prop)) b) (fun (x:(beverage->(syrup->beverage)))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) a0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) a0)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) a0)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) a0)
% Found (((eta_expansion (beverage->(syrup->beverage))) Prop) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq ((beverage->(syrup->beverage))->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a) b)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) a) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a) b)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) a) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq ((beverage->(syrup->beverage))->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a) b)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) a) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a) b)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) a) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq ((beverage->(syrup->beverage))->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a) b)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) a) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a) b)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) a) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f x)))
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f1 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f1 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f1 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f1 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f1 x)))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (f1 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f1 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f1 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (f1 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f x)) (f1 x)))
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (f1 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f1 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f1 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (f1 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f x)) (f1 x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f1 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f1 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f1 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f1 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f1 x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((beverage->(syrup->beverage))->Prop)) b) (fun (x:(beverage->(syrup->beverage)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) a0)
% Found ((eta_expansion_dep0 (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) a0)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) a0)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) a0)
% Found (((eta_expansion_dep (beverage->(syrup->beverage))) (fun (x1:(beverage->(syrup->beverage)))=> Prop)) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) a0)
% Found eq_ref00:=(eq_ref0 b):(((eq ((beverage->(syrup->beverage))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) a0)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) a0)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) a0)
% Found ((eq_ref ((beverage->(syrup->beverage))->Prop)) b) as proof of (((eq ((beverage->(syrup->beverage))->Prop)) b) a0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 ((x coffee) S)):(((eq beverage) ((x coffee) S)) ((x coffee) S))
% Found (eq_ref0 ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found ((eq_ref beverage) ((x coffee) S)) as proof of (((eq beverage) ((x coffee) S)) a0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a0):(((eq beverage) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found ((eq_ref beverage) a0) as proof of (((eq beverage) a0) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq beverage) a) a)
% Found (eq_ref0 a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found ((eq_ref beverage) a) as proof of (((eq beverage) a) x0)
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f1 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (f1 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f1 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f1 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (f1 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f x)) (f1 x)))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (f1 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f1 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f1 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (f1 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f x)) (f1 x)))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (f1 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f1 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f1 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (f1 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f x)) (f1 x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f1 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f1 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f1 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f1 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f1 x)))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (f1 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f1 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f1 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (f1 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f x)) (f1 x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f1 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f1 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f1 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f1 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f1 x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f1 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f1 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f1 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f1 x))
% Found (fun (x:(beverage->(syrup->beverage)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(beverage->(syrup->beverage))), (((eq Prop) (f0 x)) (f1 x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f1 x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f1 x))
% Found ((
% EOF
%------------------------------------------------------------------------------