TSTP Solution File: PUZ140^2 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : PUZ140^2 : TPTP v6.4.0. Released v6.4.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n031.star.cs.uiowa.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory : 16091.75MB
% OS : Linux 3.10.0-327.10.1.el7.x86_64
% CPULimit : 300s
% DateTime : Mon Mar 28 10:08:52 EDT 2016
% Result : Theorem 153.38s
% Output : Proof 154.60s
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : PUZ140^2 : TPTP v6.4.0. Released v6.4.0.
% 0.00/0.03 % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.02/0.23 % Computer : n031.star.cs.uiowa.edu
% 0.02/0.23 % Model : x86_64 x86_64
% 0.02/0.23 % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% 0.02/0.23 % Memory : 16091.75MB
% 0.02/0.23 % OS : Linux 3.10.0-327.10.1.el7.x86_64
% 0.02/0.23 % CPULimit : 300
% 0.02/0.23 % DateTime : Fri Mar 25 14:10:27 CDT 2016
% 0.02/0.23 % CPUTime :
% 0.07/0.39 Python 2.7.8
% 8.51/9.09 Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% 8.51/9.09 FOF formula (<kernel.Constant object at 0x2ae1b5a87098>, <kernel.Type object at 0x2ae1b5a87c20>) of role type named syrup_type
% 8.51/9.09 Using role type
% 8.51/9.09 Declaring syrup:Type
% 8.51/9.09 FOF formula (<kernel.Constant object at 0x2ae1b5d4e2d8>, <kernel.Type object at 0x2ae1b5a87248>) of role type named beverage_type
% 8.51/9.09 Using role type
% 8.51/9.09 Declaring beverage:Type
% 8.51/9.09 FOF formula (<kernel.Constant object at 0x2ae1b5a87518>, <kernel.Constant object at 0x2ae1b5a87098>) of role type named coffee_type
% 8.51/9.09 Using role type
% 8.51/9.09 Declaring coffee:beverage
% 8.51/9.09 FOF formula (<kernel.Constant object at 0x2ae1b5a875f0>, <kernel.DependentProduct object at 0x2ae1b5a87c20>) of role type named mix_type
% 8.51/9.09 Using role type
% 8.51/9.09 Declaring mix:(beverage->(syrup->beverage))
% 8.51/9.09 FOF formula (<kernel.Constant object at 0x2ae1b5a87cf8>, <kernel.DependentProduct object at 0x2ae1b5aefd40>) of role type named coffee_mixture_type
% 8.51/9.09 Using role type
% 8.51/9.09 Declaring coffee_mixture:(syrup->beverage)
% 8.51/9.09 FOF formula (<kernel.Constant object at 0x2ae1b5a87d40>, <kernel.DependentProduct object at 0x2ae1b5aefea8>) of role type named hot_type
% 8.51/9.09 Using role type
% 8.51/9.09 Declaring hot:(beverage->Prop)
% 8.51/9.09 FOF formula (((eq (syrup->beverage)) coffee_mixture) (mix coffee)) of role definition named coffee_mixture_definition
% 8.51/9.09 A new definition: (((eq (syrup->beverage)) coffee_mixture) (mix coffee))
% 8.51/9.09 Defined: coffee_mixture:=(mix coffee)
% 8.51/9.09 FOF formula (forall (S:syrup), ((and (((eq beverage) (coffee_mixture S)) coffee)) (hot (coffee_mixture S)))) of role axiom named coffee_and_syrup_is_hot_coffee
% 8.51/9.09 A new axiom: (forall (S:syrup), ((and (((eq beverage) (coffee_mixture S)) coffee)) (hot (coffee_mixture S))))
% 8.51/9.09 FOF formula ((ex (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) of role conjecture named there_is_hot_coffee
% 8.51/9.09 Conjecture to prove = ((ex (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))):Prop
% 8.51/9.09 Parameter syrup_DUMMY:syrup.
% 8.51/9.09 We need to prove ['((ex (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B)))))))']
% 8.51/9.09 Parameter syrup:Type.
% 8.51/9.09 Parameter beverage:Type.
% 8.51/9.09 Parameter coffee:beverage.
% 8.51/9.09 Parameter mix:(beverage->(syrup->beverage)).
% 8.51/9.09 Definition coffee_mixture:=(mix coffee):(syrup->beverage).
% 8.51/9.09 Parameter hot:(beverage->Prop).
% 8.51/9.09 Axiom coffee_and_syrup_is_hot_coffee:(forall (S:syrup), ((and (((eq beverage) (coffee_mixture S)) coffee)) (hot (coffee_mixture S)))).
% 8.51/9.09 Trying to prove ((ex (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B)))))))
% 8.51/9.09 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))):(((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) (fun (x:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))))))
% 8.51/9.09 Found (eta_expansion_dep00 (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) b)
% 10.17/10.73 Found ((eta_expansion_dep0 (fun (x1:(syrup->beverage))=> Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) b)
% 10.17/10.73 Found (((eta_expansion_dep (syrup->beverage)) (fun (x1:(syrup->beverage))=> Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) b)
% 10.17/10.73 Found (((eta_expansion_dep (syrup->beverage)) (fun (x1:(syrup->beverage))=> Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) b)
% 10.17/10.73 Found (((eta_expansion_dep (syrup->beverage)) (fun (x1:(syrup->beverage))=> Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) b)
% 10.17/10.73 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 10.17/10.73 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 10.17/10.73 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 10.17/10.73 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 10.17/10.73 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 10.17/10.73 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 10.17/10.73 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 10.17/10.73 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 10.17/10.73 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 10.17/10.73 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 10.17/10.73 Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% 10.17/10.73 Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B))))))
% 10.17/10.73 Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B))))))
% 10.17/10.73 Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B))))))
% 10.17/10.73 Found (fun (x:(syrup->beverage))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B))))))
% 10.17/10.73 Found (fun (x:(syrup->beverage))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(syrup->beverage)), (((eq Prop) (f x)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))))))
% 10.17/10.73 Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% 10.17/10.73 Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B))))))
% 12.88/13.40 Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B))))))
% 12.88/13.40 Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B))))))
% 12.88/13.40 Found (fun (x:(syrup->beverage))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B))))))
% 12.88/13.40 Found (fun (x:(syrup->beverage))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(syrup->beverage)), (((eq Prop) (f x)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))))))
% 12.88/13.40 Found x1:(P x0)
% 12.88/13.40 Instantiate: x0:=x':beverage
% 12.88/13.40 Found (fun (x1:(P x0))=> x1) as proof of (P x')
% 12.88/13.40 Found (fun (P:(beverage->Prop)) (x1:(P x0))=> x1) as proof of ((P x0)->(P x'))
% 12.88/13.40 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop)) (x1:(P x0))=> x1) as proof of (((eq beverage) x0) x')
% 12.88/13.40 Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% 12.88/13.40 Found (eq_ref00 P) as proof of ((P x0)->(P x'))
% 12.88/13.40 Found ((eq_ref0 x0) P) as proof of ((P x0)->(P x'))
% 12.88/13.40 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 12.88/13.40 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 12.88/13.40 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 12.88/13.40 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of (((eq beverage) x0) x')
% 12.88/13.40 Found x1:(P x0)
% 12.88/13.40 Instantiate: x0:=x':beverage
% 12.88/13.40 Found (fun (x1:(P x0))=> x1) as proof of (P x')
% 12.88/13.40 Found (fun (P:(beverage->Prop)) (x1:(P x0))=> x1) as proof of ((P x0)->(P x'))
% 12.88/13.40 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop)) (x1:(P x0))=> x1) as proof of (((eq beverage) x0) x')
% 12.88/13.40 Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% 12.88/13.40 Found (eq_ref00 P) as proof of ((P x0)->(P x'))
% 12.88/13.40 Found ((eq_ref0 x0) P) as proof of ((P x0)->(P x'))
% 12.88/13.40 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 12.88/13.40 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 12.88/13.40 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 12.88/13.40 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of (((eq beverage) x0) x')
% 12.88/13.40 Found eq_ref00:=(eq_ref0 (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))):(((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))
% 12.88/13.40 Found (eq_ref0 (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) b)
% 12.88/13.40 Found ((eq_ref (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) b)
% 12.88/13.40 Found ((eq_ref (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) b)
% 20.39/20.95 Found ((eq_ref (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) b)
% 20.39/20.95 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 20.39/20.95 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) coffee))) (hot B)))))
% 20.39/20.95 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) coffee))) (hot B)))))
% 20.39/20.95 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) coffee))) (hot B)))))
% 20.39/20.95 Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) coffee))) (hot B)))))
% 20.39/20.95 Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:beverage), (((eq Prop) (f x)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) coffee))) (hot B))))))
% 20.39/20.95 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 20.39/20.95 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) coffee))) (hot B)))))
% 20.39/20.95 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) coffee))) (hot B)))))
% 20.39/20.95 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) coffee))) (hot B)))))
% 20.39/20.95 Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) coffee))) (hot B)))))
% 20.39/20.95 Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:beverage), (((eq Prop) (f x)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) coffee))) (hot B))))))
% 20.39/20.95 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 20.39/20.95 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 20.39/20.95 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 20.39/20.95 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 20.39/20.95 Found (fun (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) coffee))) (hot x1)))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 20.39/20.95 Found (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) coffee))) (hot x1)))=> ((eq_ref beverage) x0)) as proof of (((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) coffee))) (hot x1))->(((eq beverage) x0) x'))
% 20.39/20.95 Found (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) coffee))) (hot x1)))=> ((eq_ref beverage) x0)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) coffee))) (hot x))->(((eq beverage) x0) x')))
% 20.39/20.95 Found (ex_ind00 (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) coffee))) (hot x1)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 20.39/20.95 Found ((ex_ind0 (((eq beverage) x0) x')) (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) coffee))) (hot x1)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 20.39/20.95 Found (((fun (P:Prop) (x1:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) coffee))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))) P) x1) x00)) (((eq beverage) x0) x')) (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) coffee))) (hot x1)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 21.73/22.28 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> (((fun (P:Prop) (x1:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) coffee))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))) P) x1) x00)) (((eq beverage) x0) x')) (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) coffee))) (hot x1)))=> ((eq_ref beverage) x0)))) as proof of (((eq beverage) x0) x')
% 21.73/22.28 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 21.73/22.28 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 21.73/22.28 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 21.73/22.28 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 21.73/22.28 Found (fun (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) coffee))) (hot x1)))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 21.73/22.28 Found (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) coffee))) (hot x1)))=> ((eq_ref beverage) x0)) as proof of (((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) coffee))) (hot x1))->(((eq beverage) x0) x'))
% 21.73/22.28 Found (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) coffee))) (hot x1)))=> ((eq_ref beverage) x0)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) coffee))) (hot x))->(((eq beverage) x0) x')))
% 21.73/22.28 Found (ex_ind00 (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) coffee))) (hot x1)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 21.73/22.28 Found ((ex_ind0 (((eq beverage) x0) x')) (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) coffee))) (hot x1)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 21.73/22.28 Found (((fun (P:Prop) (x1:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) coffee))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))) P) x1) x00)) (((eq beverage) x0) x')) (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) coffee))) (hot x1)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 21.73/22.28 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> (((fun (P:Prop) (x1:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) coffee))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))) P) x1) x00)) (((eq beverage) x0) x')) (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) coffee))) (hot x1)))=> ((eq_ref beverage) x0)))) as proof of (((eq beverage) x0) x')
% 21.73/22.28 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 21.73/22.28 Found (eq_ref0 x') as proof of (((eq beverage) x') x0)
% 21.73/22.28 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 21.73/22.28 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 21.73/22.28 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 21.73/22.28 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 21.73/22.28 Found ((eq_sym00 x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 21.73/22.28 Found (((eq_sym0 x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 21.73/22.28 Found ((((eq_sym beverage) x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 21.73/22.28 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> ((((eq_sym beverage) x') x0) ((eq_ref beverage) x'))) as proof of (((eq beverage) x0) x')
% 21.73/22.28 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 21.73/22.28 Found (eq_ref0 x') as proof of (((eq beverage) x') x0)
% 21.73/22.28 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 21.73/22.28 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 22.74/23.27 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 22.74/23.27 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 22.74/23.27 Found ((eq_sym00 x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 22.74/23.27 Found (((eq_sym0 x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 22.74/23.27 Found ((((eq_sym beverage) x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 22.74/23.27 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> ((((eq_sym beverage) x') x0) ((eq_ref beverage) x'))) as proof of (((eq beverage) x0) x')
% 22.74/23.27 Found eta_expansion000:=(eta_expansion00 ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))):(((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) (fun (x:beverage)=> (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) x)))
% 22.74/23.27 Found (eta_expansion00 ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 22.74/23.27 Found ((eta_expansion0 Prop) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 22.74/23.27 Found (((eta_expansion beverage) Prop) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 22.74/23.27 Found (((eta_expansion beverage) Prop) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 22.74/23.27 Found (((eta_expansion beverage) Prop) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 22.74/23.27 Found eq_ref00:=(eq_ref0 ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))):(((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))))
% 22.74/23.27 Found (eq_ref0 ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 22.74/23.27 Found ((eq_ref (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 27.44/28.01 Found ((eq_ref (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 27.44/28.01 Found ((eq_ref (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 27.44/28.01 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 27.44/28.01 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) x0))
% 27.44/28.01 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) x0))
% 27.44/28.01 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) x0))
% 27.44/28.01 Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) x0))
% 27.44/28.01 Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:beverage), (((eq Prop) (f x)) (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) x)))
% 27.44/28.01 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 27.44/28.01 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) x0))
% 27.44/28.01 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) x0))
% 27.44/28.01 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) x0))
% 27.44/28.01 Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) x0))
% 27.44/28.01 Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:beverage), (((eq Prop) (f x)) (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) x)))
% 27.44/28.01 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 27.44/28.01 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) x0))
% 27.44/28.01 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) x0))
% 27.44/28.01 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) x0))
% 53.22/53.71 Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) x0))
% 53.22/53.71 Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:beverage), (((eq Prop) (f x)) (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) x)))
% 53.22/53.71 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 53.22/53.71 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) x0))
% 53.22/53.71 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) x0))
% 53.22/53.71 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) x0))
% 53.22/53.71 Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) x0))
% 53.22/53.71 Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:beverage), (((eq Prop) (f x)) (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) x)))
% 53.22/53.71 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 53.22/53.71 Found (eq_ref0 x0) as proof of (((eq beverage) x0) coffee)
% 53.22/53.71 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) coffee)
% 53.22/53.71 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) coffee)
% 53.22/53.71 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) coffee)
% 53.22/53.71 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 53.22/53.71 Found (eq_ref0 x0) as proof of (((eq beverage) x0) (x S))
% 53.22/53.71 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) (x S))
% 53.22/53.71 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) (x S))
% 53.22/53.71 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) (x S))
% 53.22/53.71 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 53.22/53.71 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 53.22/53.71 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 53.22/53.71 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 53.22/53.71 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 53.22/53.71 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 53.22/53.71 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 53.22/53.71 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 53.22/53.71 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 53.22/53.71 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 53.22/53.71 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 53.22/53.71 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 53.22/53.71 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 53.22/53.71 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 53.22/53.71 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 53.22/53.71 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 53.22/53.71 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 53.22/53.71 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 53.22/53.71 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 53.22/53.71 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 56.55/57.04 Found eq_ref00:=(eq_ref0 x1):(((eq beverage) x1) x1)
% 56.55/57.04 Found (eq_ref0 x1) as proof of (((eq beverage) x1) coffee)
% 56.55/57.04 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) coffee)
% 56.55/57.04 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) coffee)
% 56.55/57.04 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) coffee)
% 56.55/57.04 Found eq_ref00:=(eq_ref0 x1):(((eq beverage) x1) x1)
% 56.55/57.04 Found (eq_ref0 x1) as proof of (((eq beverage) x1) x0)
% 56.55/57.04 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) x0)
% 56.55/57.04 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) x0)
% 56.55/57.04 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) x0)
% 56.55/57.04 Found eq_ref00:=(eq_ref0 x1):(((eq beverage) x1) x1)
% 56.55/57.04 Found (eq_ref0 x1) as proof of (((eq beverage) x1) x0)
% 56.55/57.04 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) x0)
% 56.55/57.04 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) x0)
% 56.55/57.04 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) x0)
% 56.55/57.04 Found ((conj10 ((eq_ref beverage) x1)) ((eq_ref beverage) x1)) as proof of ((and (((eq beverage) x1) x0)) (((eq beverage) x1) coffee))
% 56.55/57.04 Found (((conj1 (((eq beverage) x1) coffee)) ((eq_ref beverage) x1)) ((eq_ref beverage) x1)) as proof of ((and (((eq beverage) x1) x0)) (((eq beverage) x1) coffee))
% 56.55/57.04 Found ((((conj (((eq beverage) x1) x0)) (((eq beverage) x1) coffee)) ((eq_ref beverage) x1)) ((eq_ref beverage) x1)) as proof of ((and (((eq beverage) x1) x0)) (((eq beverage) x1) coffee))
% 56.55/57.04 Found ((((conj (((eq beverage) x1) x0)) (((eq beverage) x1) coffee)) ((eq_ref beverage) x1)) ((eq_ref beverage) x1)) as proof of ((and (((eq beverage) x1) x0)) (((eq beverage) x1) coffee))
% 56.55/57.04 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 56.55/57.04 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 56.55/57.04 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 56.55/57.04 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 56.55/57.04 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 56.55/57.04 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 56.55/57.04 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 56.55/57.04 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 56.55/57.04 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 56.55/57.04 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 56.55/57.04 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 56.55/57.04 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 56.55/57.04 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 56.55/57.04 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 56.55/57.04 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 56.55/57.04 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 56.55/57.04 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 56.55/57.04 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 56.55/57.04 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 56.55/57.04 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 56.55/57.04 Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% 56.55/57.04 Found (eq_ref00 P) as proof of ((P x2)->(P x'))
% 56.55/57.04 Found ((eq_ref0 x2) P) as proof of ((P x2)->(P x'))
% 56.55/57.04 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 56.55/57.04 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 56.55/57.04 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of ((P x2)->(P x'))
% 56.55/57.04 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of (((eq beverage) x2) x')
% 56.55/57.04 Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% 56.55/57.04 Found (eq_ref00 P) as proof of ((P x2)->(P x'))
% 56.55/57.04 Found ((eq_ref0 x2) P) as proof of ((P x2)->(P x'))
% 56.55/57.04 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 56.87/57.35 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 56.87/57.35 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of ((P x2)->(P x'))
% 56.87/57.35 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of (((eq beverage) x2) x')
% 56.87/57.35 Found x3:=(x (fun (x3:beverage)=> (P x2))):((P x2)->(P x2))
% 56.87/57.35 Found (x (fun (x3:beverage)=> (P x2))) as proof of ((P x2)->(P x'))
% 56.87/57.35 Found (x (fun (x3:beverage)=> (P x2))) as proof of ((P x2)->(P x'))
% 56.87/57.35 Found (fun (P:(beverage->Prop))=> (x (fun (x3:beverage)=> (P x2)))) as proof of ((P x2)->(P x'))
% 56.87/57.35 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop))=> (x (fun (x3:beverage)=> (P x2)))) as proof of (((eq beverage) x2) x')
% 56.87/57.35 Found x3:(P x2)
% 56.87/57.35 Instantiate: x2:=x':beverage
% 56.87/57.35 Found (fun (x3:(P x2))=> x3) as proof of (P x')
% 56.87/57.35 Found (fun (P:(beverage->Prop)) (x3:(P x2))=> x3) as proof of ((P x2)->(P x'))
% 56.87/57.35 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop)) (x3:(P x2))=> x3) as proof of (((eq beverage) x2) x')
% 56.87/57.35 Found x3:=(x (fun (x3:beverage)=> (P x2))):((P x2)->(P x2))
% 56.87/57.35 Found (x (fun (x3:beverage)=> (P x2))) as proof of ((P x2)->(P x'))
% 56.87/57.35 Found (x (fun (x3:beverage)=> (P x2))) as proof of ((P x2)->(P x'))
% 56.87/57.35 Found (fun (P:(beverage->Prop))=> (x (fun (x3:beverage)=> (P x2)))) as proof of ((P x2)->(P x'))
% 56.87/57.35 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop))=> (x (fun (x3:beverage)=> (P x2)))) as proof of (((eq beverage) x2) x')
% 56.87/57.35 Found x3:(P x2)
% 56.87/57.35 Instantiate: x2:=x':beverage
% 56.87/57.35 Found (fun (x3:(P x2))=> x3) as proof of (P x')
% 56.87/57.35 Found (fun (P:(beverage->Prop)) (x3:(P x2))=> x3) as proof of ((P x2)->(P x'))
% 56.87/57.35 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop)) (x3:(P x2))=> x3) as proof of (((eq beverage) x2) x')
% 56.87/57.35 Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% 56.87/57.35 Found (eq_ref00 P) as proof of ((P x0)->(P x'))
% 56.87/57.35 Found ((eq_ref0 x0) P) as proof of ((P x0)->(P x'))
% 56.87/57.35 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 56.87/57.35 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 56.87/57.35 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 56.87/57.35 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of (((eq beverage) x0) x')
% 56.87/57.35 Found coffee_and_syrup_is_hot_coffee__proj101:=(coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0))):((P x0)->(P x0))
% 56.87/57.35 Found (coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0))) as proof of ((P x0)->(P x'))
% 56.87/57.35 Found (coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0))) as proof of ((P x0)->(P x'))
% 56.87/57.35 Found (fun (P:(beverage->Prop))=> (coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0)))) as proof of ((P x0)->(P x'))
% 56.87/57.35 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0)))) as proof of (((eq beverage) x0) x')
% 56.87/57.35 Found x1:(P x0)
% 56.87/57.35 Instantiate: x0:=x':beverage
% 56.87/57.35 Found (fun (x1:(P x0))=> x1) as proof of (P x')
% 56.87/57.35 Found (fun (P:(beverage->Prop)) (x1:(P x0))=> x1) as proof of ((P x0)->(P x'))
% 56.87/57.35 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop)) (x1:(P x0))=> x1) as proof of (((eq beverage) x0) x')
% 56.87/57.35 Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% 56.87/57.35 Found (eq_ref00 P) as proof of ((P x0)->(P x'))
% 56.87/57.35 Found ((eq_ref0 x0) P) as proof of ((P x0)->(P x'))
% 56.87/57.35 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 57.26/57.76 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 57.26/57.76 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 57.26/57.76 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of (((eq beverage) x0) x')
% 57.26/57.76 Found x1:(P x0)
% 57.26/57.76 Instantiate: x0:=x':beverage
% 57.26/57.76 Found (fun (x1:(P x0))=> x1) as proof of (P x')
% 57.26/57.76 Found (fun (P:(beverage->Prop)) (x1:(P x0))=> x1) as proof of ((P x0)->(P x'))
% 57.26/57.76 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop)) (x1:(P x0))=> x1) as proof of (((eq beverage) x0) x')
% 57.26/57.76 Found coffee_and_syrup_is_hot_coffee__proj101:=(coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0))):((P x0)->(P x0))
% 57.26/57.76 Found (coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0))) as proof of ((P x0)->(P x'))
% 57.26/57.76 Found (coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0))) as proof of ((P x0)->(P x'))
% 57.26/57.76 Found (fun (P:(beverage->Prop))=> (coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0)))) as proof of ((P x0)->(P x'))
% 57.26/57.76 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0)))) as proof of (((eq beverage) x0) x')
% 57.26/57.76 Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% 57.26/57.76 Found (eq_ref00 P) as proof of ((P x0)->(P x'))
% 57.26/57.76 Found ((eq_ref0 x0) P) as proof of ((P x0)->(P x'))
% 57.26/57.76 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 57.26/57.76 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 57.26/57.76 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 57.26/57.76 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of (((eq beverage) x0) x')
% 57.26/57.76 Found coffee_and_syrup_is_hot_coffee__proj101:=(coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0))):((P x0)->(P x0))
% 57.26/57.76 Found (coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0))) as proof of ((P x0)->(P x'))
% 57.26/57.76 Found (coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0))) as proof of ((P x0)->(P x'))
% 57.26/57.76 Found (fun (P:(beverage->Prop))=> (coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0)))) as proof of ((P x0)->(P x'))
% 57.26/57.76 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0)))) as proof of (((eq beverage) x0) x')
% 57.26/57.76 Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% 57.26/57.76 Found (eq_ref00 P) as proof of ((P x0)->(P x'))
% 57.26/57.76 Found ((eq_ref0 x0) P) as proof of ((P x0)->(P x'))
% 57.26/57.76 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 57.26/57.76 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 57.26/57.76 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 57.26/57.76 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of (((eq beverage) x0) x')
% 57.26/57.76 Found coffee_and_syrup_is_hot_coffee__proj101:=(coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0))):((P x0)->(P x0))
% 57.26/57.76 Found (coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0))) as proof of ((P x0)->(P x'))
% 57.26/57.76 Found (coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0))) as proof of ((P x0)->(P x'))
% 57.26/57.76 Found (fun (P:(beverage->Prop))=> (coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0)))) as proof of ((P x0)->(P x'))
% 57.26/57.76 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0)))) as proof of (((eq beverage) x0) x')
% 58.35/58.87 Found x1:(P x0)
% 58.35/58.87 Instantiate: x0:=x':beverage
% 58.35/58.87 Found (fun (x1:(P x0))=> x1) as proof of (P x')
% 58.35/58.87 Found (fun (P:(beverage->Prop)) (x1:(P x0))=> x1) as proof of ((P x0)->(P x'))
% 58.35/58.87 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop)) (x1:(P x0))=> x1) as proof of (((eq beverage) x0) x')
% 58.35/58.87 Found x1:(P x0)
% 58.35/58.87 Instantiate: x0:=x':beverage
% 58.35/58.87 Found (fun (x1:(P x0))=> x1) as proof of (P x')
% 58.35/58.87 Found (fun (P:(beverage->Prop)) (x1:(P x0))=> x1) as proof of ((P x0)->(P x'))
% 58.35/58.87 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop)) (x1:(P x0))=> x1) as proof of (((eq beverage) x0) x')
% 58.35/58.87 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 58.35/58.87 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 58.35/58.87 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 58.35/58.87 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 58.35/58.87 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 58.35/58.87 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 58.35/58.87 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 58.35/58.87 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 58.35/58.87 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 58.35/58.87 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 58.35/58.87 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 58.35/58.87 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 58.35/58.87 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 58.35/58.87 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 58.35/58.87 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 58.35/58.87 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 58.35/58.87 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 58.35/58.87 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 58.35/58.87 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 58.35/58.87 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 58.35/58.87 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 58.35/58.87 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 58.35/58.87 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 58.35/58.87 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 58.35/58.87 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 58.35/58.87 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 58.35/58.87 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 58.35/58.87 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 58.35/58.87 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 58.35/58.87 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 58.35/58.87 Found coffee_and_syrup_is_hot_coffee__proj100:=(coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0))):((P x0)->(P x0))
% 58.35/58.87 Found (coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0))) as proof of ((P x0)->(P x'))
% 58.35/58.87 Found coffee_and_syrup_is_hot_coffee__proj100:=(coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0))):((P x0)->(P x0))
% 58.35/58.87 Found (coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0))) as proof of ((P x0)->(P x'))
% 58.35/58.87 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 58.35/58.87 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 59.42/59.91 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 59.42/59.91 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 59.42/59.91 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 59.42/59.91 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 59.42/59.91 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 59.42/59.91 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 59.42/59.91 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 59.42/59.91 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 59.42/59.91 Found eq_ref00:=(eq_ref0 x3):(((eq beverage) x3) x3)
% 59.42/59.91 Found (eq_ref0 x3) as proof of (((eq beverage) x3) x')
% 59.42/59.91 Found ((eq_ref beverage) x3) as proof of (((eq beverage) x3) x')
% 59.42/59.91 Found ((eq_ref beverage) x3) as proof of (((eq beverage) x3) x')
% 59.42/59.91 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((eq_ref beverage) x3)) as proof of (((eq beverage) x3) x')
% 59.42/59.91 Found eq_ref00:=(eq_ref0 x3):(((eq beverage) x3) x3)
% 59.42/59.91 Found (eq_ref0 x3) as proof of (((eq beverage) x3) x')
% 59.42/59.91 Found ((eq_ref beverage) x3) as proof of (((eq beverage) x3) x')
% 59.42/59.91 Found ((eq_ref beverage) x3) as proof of (((eq beverage) x3) x')
% 59.42/59.91 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((eq_ref beverage) x3)) as proof of (((eq beverage) x3) x')
% 59.42/59.91 Found eq_ref00:=(eq_ref0 x3):(((eq beverage) x3) x3)
% 59.42/59.91 Found (eq_ref0 x3) as proof of (((eq beverage) x3) x')
% 59.42/59.91 Found ((eq_ref beverage) x3) as proof of (((eq beverage) x3) x')
% 59.42/59.91 Found ((eq_ref beverage) x3) as proof of (((eq beverage) x3) x')
% 59.42/59.91 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((eq_ref beverage) x3)) as proof of (((eq beverage) x3) x')
% 59.42/59.91 Found eq_ref00:=(eq_ref0 x3):(((eq beverage) x3) x3)
% 59.42/59.91 Found (eq_ref0 x3) as proof of (((eq beverage) x3) x')
% 59.42/59.91 Found ((eq_ref beverage) x3) as proof of (((eq beverage) x3) x')
% 59.42/59.91 Found ((eq_ref beverage) x3) as proof of (((eq beverage) x3) x')
% 59.42/59.91 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((eq_ref beverage) x3)) as proof of (((eq beverage) x3) x')
% 59.42/59.91 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 59.42/59.91 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 59.42/59.91 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 59.42/59.91 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 59.42/59.91 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 59.42/59.91 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 59.42/59.91 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 59.42/59.91 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 59.42/59.91 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 59.42/59.91 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 59.42/59.91 Found x01:=(x0 (fun (x3:beverage)=> (P x2))):((P x2)->(P x2))
% 59.42/59.91 Found (x0 (fun (x3:beverage)=> (P x2))) as proof of ((P x2)->(P x'))
% 59.42/59.91 Found (x0 (fun (x3:beverage)=> (P x2))) as proof of ((P x2)->(P x'))
% 59.42/59.91 Found (fun (P:(beverage->Prop))=> (x0 (fun (x3:beverage)=> (P x2)))) as proof of ((P x2)->(P x'))
% 59.42/59.91 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop))=> (x0 (fun (x3:beverage)=> (P x2)))) as proof of (((eq beverage) x2) x')
% 59.42/59.91 Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% 59.42/59.91 Found (eq_ref00 P) as proof of ((P x2)->(P x'))
% 59.42/59.91 Found ((eq_ref0 x2) P) as proof of ((P x2)->(P x'))
% 59.42/59.91 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 60.80/61.27 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 60.80/61.27 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of ((P x2)->(P x'))
% 60.80/61.27 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of (((eq beverage) x2) x')
% 60.80/61.27 Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% 60.80/61.27 Found (eq_ref00 P) as proof of ((P x2)->(P x'))
% 60.80/61.27 Found ((eq_ref0 x2) P) as proof of ((P x2)->(P x'))
% 60.80/61.27 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 60.80/61.27 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 60.80/61.27 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of ((P x2)->(P x'))
% 60.80/61.27 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of (((eq beverage) x2) x')
% 60.80/61.27 Found x3:(P x2)
% 60.80/61.27 Instantiate: x2:=x':beverage
% 60.80/61.27 Found (fun (x3:(P x2))=> x3) as proof of (P x')
% 60.80/61.27 Found (fun (P:(beverage->Prop)) (x3:(P x2))=> x3) as proof of ((P x2)->(P x'))
% 60.80/61.27 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop)) (x3:(P x2))=> x3) as proof of (((eq beverage) x2) x')
% 60.80/61.27 Found x3:(P x2)
% 60.80/61.27 Instantiate: x2:=x':beverage
% 60.80/61.27 Found (fun (x3:(P x2))=> x3) as proof of (P x')
% 60.80/61.27 Found (fun (P:(beverage->Prop)) (x3:(P x2))=> x3) as proof of ((P x2)->(P x'))
% 60.80/61.27 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop)) (x3:(P x2))=> x3) as proof of (((eq beverage) x2) x')
% 60.80/61.27 Found x01:=(x0 (fun (x3:beverage)=> (P x2))):((P x2)->(P x2))
% 60.80/61.27 Found (x0 (fun (x3:beverage)=> (P x2))) as proof of ((P x2)->(P x'))
% 60.80/61.27 Found (x0 (fun (x3:beverage)=> (P x2))) as proof of ((P x2)->(P x'))
% 60.80/61.27 Found (fun (P:(beverage->Prop))=> (x0 (fun (x3:beverage)=> (P x2)))) as proof of ((P x2)->(P x'))
% 60.80/61.27 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop))=> (x0 (fun (x3:beverage)=> (P x2)))) as proof of (((eq beverage) x2) x')
% 60.80/61.27 Found eq_ref00:=(eq_ref0 (x S)):(((eq beverage) (x S)) (x S))
% 60.80/61.27 Found (eq_ref0 (x S)) as proof of (((eq beverage) (x S)) x0)
% 60.80/61.27 Found ((eq_ref beverage) (x S)) as proof of (((eq beverage) (x S)) x0)
% 60.80/61.27 Found ((eq_ref beverage) (x S)) as proof of (((eq beverage) (x S)) x0)
% 60.80/61.27 Found ((eq_ref beverage) (x S)) as proof of (((eq beverage) (x S)) x0)
% 60.80/61.27 Found (eq_sym000 ((eq_ref beverage) (x S))) as proof of (((eq beverage) x0) (x S))
% 60.80/61.27 Found ((eq_sym00 x0) ((eq_ref beverage) (x S))) as proof of (((eq beverage) x0) (x S))
% 60.80/61.27 Found (((eq_sym0 (x S)) x0) ((eq_ref beverage) (x S))) as proof of (((eq beverage) x0) (x S))
% 60.80/61.27 Found ((((eq_sym beverage) (x S)) x0) ((eq_ref beverage) (x S))) as proof of (((eq beverage) x0) (x S))
% 60.80/61.27 Found ((((eq_sym beverage) (x S)) x0) ((eq_ref beverage) (x S))) as proof of (((eq beverage) x0) (x S))
% 60.80/61.27 Found ((conj10 ((((eq_sym beverage) (x S)) x0) ((eq_ref beverage) (x S)))) ((eq_ref beverage) x0)) as proof of ((and (((eq beverage) x0) (x S))) (((eq beverage) x0) coffee))
% 60.80/61.27 Found (((conj1 (((eq beverage) x0) coffee)) ((((eq_sym beverage) (x S)) x0) ((eq_ref beverage) (x S)))) ((eq_ref beverage) x0)) as proof of ((and (((eq beverage) x0) (x S))) (((eq beverage) x0) coffee))
% 60.80/61.27 Found ((((conj (((eq beverage) x0) (x S))) (((eq beverage) x0) coffee)) ((((eq_sym beverage) (x S)) x0) ((eq_ref beverage) (x S)))) ((eq_ref beverage) x0)) as proof of ((and (((eq beverage) x0) (x S))) (((eq beverage) x0) coffee))
% 60.80/61.27 Found ((((conj (((eq beverage) x0) (x S))) (((eq beverage) x0) coffee)) ((((eq_sym beverage) (x S)) x0) ((eq_ref beverage) (x S)))) ((eq_ref beverage) x0)) as proof of ((and (((eq beverage) x0) (x S))) (((eq beverage) x0) coffee))
% 60.80/61.27 Found eq_ref00:=(eq_ref0 x1):(((eq beverage) x1) x1)
% 60.80/61.27 Found (eq_ref0 x1) as proof of (((eq beverage) x1) x')
% 60.80/61.27 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) x')
% 60.80/61.27 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) x')
% 61.39/61.88 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((eq_ref beverage) x1)) as proof of (((eq beverage) x1) x')
% 61.39/61.88 Found eq_ref00:=(eq_ref0 x1):(((eq beverage) x1) x1)
% 61.39/61.88 Found (eq_ref0 x1) as proof of (((eq beverage) x1) x')
% 61.39/61.88 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) x')
% 61.39/61.88 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) x')
% 61.39/61.88 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((eq_ref beverage) x1)) as proof of (((eq beverage) x1) x')
% 61.39/61.88 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 61.39/61.88 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 61.39/61.88 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 61.39/61.88 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 61.39/61.88 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 61.39/61.88 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 61.39/61.88 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 61.39/61.88 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 61.39/61.88 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 61.39/61.88 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 61.39/61.88 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 61.39/61.88 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 61.39/61.88 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 61.39/61.88 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 61.39/61.88 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 61.39/61.88 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 61.39/61.88 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 61.39/61.88 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 61.39/61.88 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 61.39/61.88 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 61.39/61.88 Found x3:(P x2)
% 61.39/61.88 Instantiate: x2:=x':beverage
% 61.39/61.88 Found (fun (x3:(P x2))=> x3) as proof of (P x')
% 61.39/61.88 Found (fun (P:(beverage->Prop)) (x3:(P x2))=> x3) as proof of ((P x2)->(P x'))
% 61.39/61.88 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop)) (x3:(P x2))=> x3) as proof of (((eq beverage) x2) x')
% 61.39/61.88 Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% 61.39/61.88 Found (eq_ref00 P) as proof of ((P x2)->(P x'))
% 61.39/61.88 Found ((eq_ref0 x2) P) as proof of ((P x2)->(P x'))
% 61.39/61.88 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 61.39/61.88 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 61.39/61.88 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of ((P x2)->(P x'))
% 61.39/61.88 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of (((eq beverage) x2) x')
% 61.39/61.88 Found x3:(P x2)
% 61.39/61.88 Instantiate: x2:=x':beverage
% 61.39/61.88 Found (fun (x3:(P x2))=> x3) as proof of (P x')
% 61.39/61.88 Found (fun (P:(beverage->Prop)) (x3:(P x2))=> x3) as proof of ((P x2)->(P x'))
% 61.39/61.88 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop)) (x3:(P x2))=> x3) as proof of (((eq beverage) x2) x')
% 61.39/61.88 Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% 61.39/61.88 Found (eq_ref00 P) as proof of ((P x2)->(P x'))
% 61.39/61.88 Found ((eq_ref0 x2) P) as proof of ((P x2)->(P x'))
% 61.39/61.88 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 61.39/61.88 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 61.60/62.08 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of ((P x2)->(P x'))
% 61.60/62.08 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of (((eq beverage) x2) x')
% 61.60/62.08 Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% 61.60/62.08 Found (eq_ref00 P) as proof of ((P x2)->(P x'))
% 61.60/62.08 Found ((eq_ref0 x2) P) as proof of ((P x2)->(P x'))
% 61.60/62.08 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 61.60/62.08 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 61.60/62.08 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of ((P x2)->(P x'))
% 61.60/62.08 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of (((eq beverage) x2) x')
% 61.60/62.08 Found x3:(P x2)
% 61.60/62.08 Instantiate: x2:=x':beverage
% 61.60/62.08 Found (fun (x3:(P x2))=> x3) as proof of (P x')
% 61.60/62.08 Found (fun (P:(beverage->Prop)) (x3:(P x2))=> x3) as proof of ((P x2)->(P x'))
% 61.60/62.08 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop)) (x3:(P x2))=> x3) as proof of (((eq beverage) x2) x')
% 61.60/62.08 Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% 61.60/62.08 Found (eq_ref00 P) as proof of ((P x2)->(P x'))
% 61.60/62.08 Found ((eq_ref0 x2) P) as proof of ((P x2)->(P x'))
% 61.60/62.08 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 61.60/62.08 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 61.60/62.08 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of ((P x2)->(P x'))
% 61.60/62.08 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of (((eq beverage) x2) x')
% 61.60/62.08 Found x3:(P x2)
% 61.60/62.08 Instantiate: x2:=x':beverage
% 61.60/62.08 Found (fun (x3:(P x2))=> x3) as proof of (P x')
% 61.60/62.08 Found (fun (P:(beverage->Prop)) (x3:(P x2))=> x3) as proof of ((P x2)->(P x'))
% 61.60/62.08 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop)) (x3:(P x2))=> x3) as proof of (((eq beverage) x2) x')
% 61.60/62.08 Found x3:(P x0)
% 61.60/62.08 Instantiate: x0:=x':beverage
% 61.60/62.08 Found (fun (x3:(P x0))=> x3) as proof of (P x')
% 61.60/62.08 Found (fun (P:(beverage->Prop)) (x3:(P x0))=> x3) as proof of ((P x0)->(P x'))
% 61.60/62.08 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop)) (x3:(P x0))=> x3) as proof of (((eq beverage) x0) x')
% 61.60/62.08 Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% 61.60/62.08 Found (eq_ref00 P) as proof of ((P x0)->(P x'))
% 61.60/62.08 Found ((eq_ref0 x0) P) as proof of ((P x0)->(P x'))
% 61.60/62.08 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 61.60/62.08 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 61.60/62.08 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 61.60/62.08 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of (((eq beverage) x0) x')
% 61.60/62.08 Found x3:(P x0)
% 61.60/62.08 Instantiate: x0:=x':beverage
% 61.60/62.08 Found (fun (x3:(P x0))=> x3) as proof of (P x')
% 61.60/62.08 Found (fun (P:(beverage->Prop)) (x3:(P x0))=> x3) as proof of ((P x0)->(P x'))
% 61.60/62.08 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop)) (x3:(P x0))=> x3) as proof of (((eq beverage) x0) x')
% 61.60/62.08 Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% 61.60/62.08 Found (eq_ref00 P) as proof of ((P x0)->(P x'))
% 61.60/62.08 Found ((eq_ref0 x0) P) as proof of ((P x0)->(P x'))
% 61.60/62.08 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 61.60/62.08 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 61.60/62.08 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 61.60/62.08 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of (((eq beverage) x0) x')
% 61.72/62.21 Found x10:=(x1 (fun (x3:beverage)=> (P x0))):((P x0)->(P x0))
% 61.72/62.21 Found (x1 (fun (x3:beverage)=> (P x0))) as proof of ((P x0)->(P x'))
% 61.72/62.21 Found (x1 (fun (x3:beverage)=> (P x0))) as proof of ((P x0)->(P x'))
% 61.72/62.21 Found (fun (P:(beverage->Prop))=> (x1 (fun (x3:beverage)=> (P x0)))) as proof of ((P x0)->(P x'))
% 61.72/62.21 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop))=> (x1 (fun (x3:beverage)=> (P x0)))) as proof of (((eq beverage) x0) x')
% 61.72/62.21 Found x10:=(x1 (fun (x3:beverage)=> (P x0))):((P x0)->(P x0))
% 61.72/62.21 Found (x1 (fun (x3:beverage)=> (P x0))) as proof of ((P x0)->(P x'))
% 61.72/62.21 Found (x1 (fun (x3:beverage)=> (P x0))) as proof of ((P x0)->(P x'))
% 61.72/62.21 Found (fun (P:(beverage->Prop))=> (x1 (fun (x3:beverage)=> (P x0)))) as proof of ((P x0)->(P x'))
% 61.72/62.21 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop))=> (x1 (fun (x3:beverage)=> (P x0)))) as proof of (((eq beverage) x0) x')
% 61.72/62.21 Found x1:(P x0)
% 61.72/62.21 Instantiate: x0:=x':beverage
% 61.72/62.21 Found (fun (x1:(P x0))=> x1) as proof of (P x')
% 61.72/62.21 Found (fun (P:(beverage->Prop)) (x1:(P x0))=> x1) as proof of ((P x0)->(P x'))
% 61.72/62.21 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop)) (x1:(P x0))=> x1) as proof of (((eq beverage) x0) x')
% 61.72/62.21 Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% 61.72/62.21 Found (eq_ref00 P) as proof of ((P x0)->(P x'))
% 61.72/62.21 Found ((eq_ref0 x0) P) as proof of ((P x0)->(P x'))
% 61.72/62.21 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 61.72/62.21 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 61.72/62.21 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 61.72/62.21 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of (((eq beverage) x0) x')
% 61.72/62.21 Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% 61.72/62.21 Found (eq_ref00 P) as proof of ((P x0)->(P x'))
% 61.72/62.21 Found ((eq_ref0 x0) P) as proof of ((P x0)->(P x'))
% 61.72/62.21 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 61.72/62.21 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 61.72/62.21 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 61.72/62.21 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of (((eq beverage) x0) x')
% 61.72/62.21 Found coffee_and_syrup_is_hot_coffee__proj101:=(coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0))):((P x0)->(P x0))
% 61.72/62.21 Found (coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0))) as proof of ((P x0)->(P x'))
% 61.72/62.21 Found (coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0))) as proof of ((P x0)->(P x'))
% 61.72/62.21 Found (fun (P:(beverage->Prop))=> (coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0)))) as proof of ((P x0)->(P x'))
% 61.72/62.21 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0)))) as proof of (((eq beverage) x0) x')
% 61.72/62.21 Found coffee_and_syrup_is_hot_coffee__proj101:=(coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0))):((P x0)->(P x0))
% 61.72/62.21 Found (coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0))) as proof of ((P x0)->(P x'))
% 61.72/62.21 Found (coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0))) as proof of ((P x0)->(P x'))
% 61.72/62.21 Found (fun (P:(beverage->Prop))=> (coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0)))) as proof of ((P x0)->(P x'))
% 61.72/62.21 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (coffee_and_syrup_is_hot_coffee__proj10 (fun (x1:beverage)=> (P x0)))) as proof of (((eq beverage) x0) x')
% 61.85/62.37 Found x1:(P x0)
% 61.85/62.37 Instantiate: x0:=x':beverage
% 61.85/62.37 Found (fun (x1:(P x0))=> x1) as proof of (P x')
% 61.85/62.37 Found (fun (P:(beverage->Prop)) (x1:(P x0))=> x1) as proof of ((P x0)->(P x'))
% 61.85/62.37 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop)) (x1:(P x0))=> x1) as proof of (((eq beverage) x0) x')
% 61.85/62.37 Found x4:(P x2)
% 61.85/62.37 Instantiate: x2:=x':beverage
% 61.85/62.37 Found (fun (x4:(P x2))=> x4) as proof of (P x')
% 61.85/62.37 Found (fun (P:(beverage->Prop)) (x4:(P x2))=> x4) as proof of ((P x2)->(P x'))
% 61.85/62.37 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop)) (x4:(P x2))=> x4) as proof of (((eq beverage) x2) x')
% 61.85/62.37 Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% 61.85/62.37 Found (eq_ref00 P) as proof of ((P x2)->(P x'))
% 61.85/62.37 Found ((eq_ref0 x2) P) as proof of ((P x2)->(P x'))
% 61.85/62.37 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 61.85/62.37 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 61.85/62.37 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of ((P x2)->(P x'))
% 61.85/62.37 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of (((eq beverage) x2) x')
% 61.85/62.37 Found x4:(P x2)
% 61.85/62.37 Instantiate: x2:=x':beverage
% 61.85/62.37 Found (fun (x4:(P x2))=> x4) as proof of (P x')
% 61.85/62.37 Found (fun (P:(beverage->Prop)) (x4:(P x2))=> x4) as proof of ((P x2)->(P x'))
% 61.85/62.37 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop)) (x4:(P x2))=> x4) as proof of (((eq beverage) x2) x')
% 61.85/62.37 Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% 61.85/62.37 Found (eq_ref00 P) as proof of ((P x2)->(P x'))
% 61.85/62.37 Found ((eq_ref0 x2) P) as proof of ((P x2)->(P x'))
% 61.85/62.37 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 61.85/62.37 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 61.85/62.37 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of ((P x2)->(P x'))
% 61.85/62.37 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of (((eq beverage) x2) x')
% 61.85/62.37 Found x4:(P x3)
% 61.85/62.37 Instantiate: x3:=x':beverage
% 61.85/62.37 Found (fun (x4:(P x3))=> x4) as proof of (P x')
% 61.85/62.37 Found (fun (P:(beverage->Prop)) (x4:(P x3))=> x4) as proof of ((P x3)->(P x'))
% 61.85/62.37 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop)) (x4:(P x3))=> x4) as proof of (((eq beverage) x3) x')
% 61.85/62.37 Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% 61.85/62.37 Found (eq_ref00 P) as proof of ((P x3)->(P x'))
% 61.85/62.37 Found ((eq_ref0 x3) P) as proof of ((P x3)->(P x'))
% 61.85/62.37 Found (((eq_ref beverage) x3) P) as proof of ((P x3)->(P x'))
% 61.85/62.37 Found (((eq_ref beverage) x3) P) as proof of ((P x3)->(P x'))
% 61.85/62.37 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x3) P)) as proof of ((P x3)->(P x'))
% 61.85/62.37 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x3) P)) as proof of (((eq beverage) x3) x')
% 61.85/62.37 Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% 61.85/62.37 Found (eq_ref00 P) as proof of ((P x3)->(P x'))
% 61.85/62.37 Found ((eq_ref0 x3) P) as proof of ((P x3)->(P x'))
% 61.85/62.37 Found (((eq_ref beverage) x3) P) as proof of ((P x3)->(P x'))
% 61.85/62.37 Found (((eq_ref beverage) x3) P) as proof of ((P x3)->(P x'))
% 61.85/62.37 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x3) P)) as proof of ((P x3)->(P x'))
% 61.85/62.37 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x3) P)) as proof of (((eq beverage) x3) x')
% 61.85/62.37 Found x4:(P x3)
% 61.85/62.37 Instantiate: x3:=x':beverage
% 61.85/62.37 Found (fun (x4:(P x3))=> x4) as proof of (P x')
% 61.85/62.37 Found (fun (P:(beverage->Prop)) (x4:(P x3))=> x4) as proof of ((P x3)->(P x'))
% 63.78/64.26 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop)) (x4:(P x3))=> x4) as proof of (((eq beverage) x3) x')
% 63.78/64.26 Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% 63.78/64.26 Found (eq_ref00 P) as proof of ((P x3)->(P x'))
% 63.78/64.26 Found ((eq_ref0 x3) P) as proof of ((P x3)->(P x'))
% 63.78/64.26 Found (((eq_ref beverage) x3) P) as proof of ((P x3)->(P x'))
% 63.78/64.26 Found (((eq_ref beverage) x3) P) as proof of ((P x3)->(P x'))
% 63.78/64.26 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x3) P)) as proof of ((P x3)->(P x'))
% 63.78/64.26 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x3) P)) as proof of (((eq beverage) x3) x')
% 63.78/64.26 Found x4:(P x3)
% 63.78/64.26 Instantiate: x3:=x':beverage
% 63.78/64.26 Found (fun (x4:(P x3))=> x4) as proof of (P x')
% 63.78/64.26 Found (fun (P:(beverage->Prop)) (x4:(P x3))=> x4) as proof of ((P x3)->(P x'))
% 63.78/64.26 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop)) (x4:(P x3))=> x4) as proof of (((eq beverage) x3) x')
% 63.78/64.26 Found x4:(P x3)
% 63.78/64.26 Instantiate: x3:=x':beverage
% 63.78/64.26 Found (fun (x4:(P x3))=> x4) as proof of (P x')
% 63.78/64.26 Found (fun (P:(beverage->Prop)) (x4:(P x3))=> x4) as proof of ((P x3)->(P x'))
% 63.78/64.26 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop)) (x4:(P x3))=> x4) as proof of (((eq beverage) x3) x')
% 63.78/64.26 Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% 63.78/64.26 Found (eq_ref00 P) as proof of ((P x3)->(P x'))
% 63.78/64.26 Found ((eq_ref0 x3) P) as proof of ((P x3)->(P x'))
% 63.78/64.26 Found (((eq_ref beverage) x3) P) as proof of ((P x3)->(P x'))
% 63.78/64.26 Found (((eq_ref beverage) x3) P) as proof of ((P x3)->(P x'))
% 63.78/64.26 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x3) P)) as proof of ((P x3)->(P x'))
% 63.78/64.26 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x3) P)) as proof of (((eq beverage) x3) x')
% 63.78/64.26 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 63.78/64.26 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 63.78/64.26 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 63.78/64.26 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 63.78/64.26 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 63.78/64.26 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 63.78/64.26 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 63.78/64.26 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 63.78/64.26 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 63.78/64.26 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 63.78/64.26 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 63.78/64.26 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 63.78/64.26 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 63.78/64.26 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 63.78/64.26 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 63.78/64.26 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 63.78/64.26 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 63.78/64.26 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 63.78/64.26 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 63.78/64.26 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 63.78/64.26 Found x2:(P x1)
% 63.78/64.26 Instantiate: x1:=x':beverage
% 63.78/64.26 Found (fun (x2:(P x1))=> x2) as proof of (P x')
% 63.78/64.26 Found (fun (P:(beverage->Prop)) (x2:(P x1))=> x2) as proof of ((P x1)->(P x'))
% 63.96/64.44 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop)) (x2:(P x1))=> x2) as proof of (((eq beverage) x1) x')
% 63.96/64.44 Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% 63.96/64.44 Found (eq_ref00 P) as proof of ((P x1)->(P x'))
% 63.96/64.44 Found ((eq_ref0 x1) P) as proof of ((P x1)->(P x'))
% 63.96/64.44 Found (((eq_ref beverage) x1) P) as proof of ((P x1)->(P x'))
% 63.96/64.44 Found (((eq_ref beverage) x1) P) as proof of ((P x1)->(P x'))
% 63.96/64.44 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x1) P)) as proof of ((P x1)->(P x'))
% 63.96/64.44 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x1) P)) as proof of (((eq beverage) x1) x')
% 63.96/64.44 Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% 63.96/64.44 Found (eq_ref00 P) as proof of ((P x1)->(P x'))
% 63.96/64.44 Found ((eq_ref0 x1) P) as proof of ((P x1)->(P x'))
% 63.96/64.44 Found (((eq_ref beverage) x1) P) as proof of ((P x1)->(P x'))
% 63.96/64.44 Found (((eq_ref beverage) x1) P) as proof of ((P x1)->(P x'))
% 63.96/64.44 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x1) P)) as proof of ((P x1)->(P x'))
% 63.96/64.44 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x1) P)) as proof of (((eq beverage) x1) x')
% 63.96/64.44 Found x2:(P x1)
% 63.96/64.44 Instantiate: x1:=x':beverage
% 63.96/64.44 Found (fun (x2:(P x1))=> x2) as proof of (P x')
% 63.96/64.44 Found (fun (P:(beverage->Prop)) (x2:(P x1))=> x2) as proof of ((P x1)->(P x'))
% 63.96/64.44 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop)) (x2:(P x1))=> x2) as proof of (((eq beverage) x1) x')
% 63.96/64.44 Found x3:(P x2)
% 63.96/64.44 Instantiate: x2:=x':beverage
% 63.96/64.44 Found (fun (x3:(P x2))=> x3) as proof of (P x')
% 63.96/64.44 Found (fun (P:(beverage->Prop)) (x3:(P x2))=> x3) as proof of ((P x2)->(P x'))
% 63.96/64.44 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop)) (x3:(P x2))=> x3) as proof of (((eq beverage) x2) x')
% 63.96/64.44 Found x3:(P x2)
% 63.96/64.44 Instantiate: x2:=x':beverage
% 63.96/64.44 Found (fun (x3:(P x2))=> x3) as proof of (P x')
% 63.96/64.44 Found (fun (P:(beverage->Prop)) (x3:(P x2))=> x3) as proof of ((P x2)->(P x'))
% 63.96/64.44 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop)) (x3:(P x2))=> x3) as proof of (((eq beverage) x2) x')
% 63.96/64.44 Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% 63.96/64.44 Found (eq_ref00 P) as proof of ((P x2)->(P x'))
% 63.96/64.44 Found ((eq_ref0 x2) P) as proof of ((P x2)->(P x'))
% 63.96/64.44 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 63.96/64.44 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 63.96/64.44 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of ((P x2)->(P x'))
% 63.96/64.44 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of (((eq beverage) x2) x')
% 63.96/64.44 Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% 63.96/64.44 Found (eq_ref00 P) as proof of ((P x2)->(P x'))
% 63.96/64.44 Found ((eq_ref0 x2) P) as proof of ((P x2)->(P x'))
% 63.96/64.44 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 63.96/64.44 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 63.96/64.44 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of ((P x2)->(P x'))
% 63.96/64.44 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of (((eq beverage) x2) x')
% 63.96/64.44 Found x3:(P x2)
% 63.96/64.44 Instantiate: x2:=x':beverage
% 63.96/64.44 Found (fun (x3:(P x2))=> x3) as proof of (P x')
% 63.96/64.44 Found (fun (P:(beverage->Prop)) (x3:(P x2))=> x3) as proof of ((P x2)->(P x'))
% 63.96/64.44 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop)) (x3:(P x2))=> x3) as proof of (((eq beverage) x2) x')
% 66.05/66.53 Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% 66.05/66.53 Found (eq_ref00 P) as proof of ((P x2)->(P x'))
% 66.05/66.53 Found ((eq_ref0 x2) P) as proof of ((P x2)->(P x'))
% 66.05/66.53 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 66.05/66.53 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 66.05/66.53 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of ((P x2)->(P x'))
% 66.05/66.53 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of (((eq beverage) x2) x')
% 66.05/66.53 Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% 66.05/66.53 Found (eq_ref00 P) as proof of ((P x2)->(P x'))
% 66.05/66.53 Found ((eq_ref0 x2) P) as proof of ((P x2)->(P x'))
% 66.05/66.53 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 66.05/66.53 Found (((eq_ref beverage) x2) P) as proof of ((P x2)->(P x'))
% 66.05/66.53 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of ((P x2)->(P x'))
% 66.05/66.53 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x2) P)) as proof of (((eq beverage) x2) x')
% 66.05/66.53 Found x3:(P x2)
% 66.05/66.53 Instantiate: x2:=x':beverage
% 66.05/66.53 Found (fun (x3:(P x2))=> x3) as proof of (P x')
% 66.05/66.53 Found (fun (P:(beverage->Prop)) (x3:(P x2))=> x3) as proof of ((P x2)->(P x'))
% 66.05/66.53 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop)) (x3:(P x2))=> x3) as proof of (((eq beverage) x2) x')
% 66.05/66.53 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 66.05/66.53 Found (eq_ref0 x0) as proof of (forall (P:(beverage->Prop)), ((P x0)->(P x')))
% 66.05/66.53 Found ((eq_ref beverage) x0) as proof of (forall (P:(beverage->Prop)), ((P x0)->(P x')))
% 66.05/66.53 Found ((eq_ref beverage) x0) as proof of (forall (P:(beverage->Prop)), ((P x0)->(P x')))
% 66.05/66.53 Found ((eq_ref beverage) x0) as proof of (forall (P:(beverage->Prop)), ((P x0)->(P x')))
% 66.05/66.53 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 66.05/66.53 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 66.05/66.53 Found (eq_ref0 x0) as proof of (forall (P:(beverage->Prop)), ((P x0)->(P x')))
% 66.05/66.53 Found ((eq_ref beverage) x0) as proof of (forall (P:(beverage->Prop)), ((P x0)->(P x')))
% 66.05/66.53 Found ((eq_ref beverage) x0) as proof of (forall (P:(beverage->Prop)), ((P x0)->(P x')))
% 66.05/66.53 Found ((eq_ref beverage) x0) as proof of (forall (P:(beverage->Prop)), ((P x0)->(P x')))
% 66.05/66.53 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 66.05/66.53 Found x2:(P x0)
% 66.05/66.53 Instantiate: x0:=x':beverage
% 66.05/66.53 Found (fun (x2:(P x0))=> x2) as proof of (P x')
% 66.05/66.53 Found (fun (P:(beverage->Prop)) (x2:(P x0))=> x2) as proof of ((P x0)->(P x'))
% 66.05/66.53 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop)) (x2:(P x0))=> x2) as proof of (((eq beverage) x0) x')
% 66.05/66.53 Found x2:(P x0)
% 66.05/66.53 Instantiate: x0:=x':beverage
% 66.05/66.53 Found (fun (x2:(P x0))=> x2) as proof of (P x')
% 66.05/66.53 Found (fun (P:(beverage->Prop)) (x2:(P x0))=> x2) as proof of ((P x0)->(P x'))
% 66.05/66.53 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop)) (x2:(P x0))=> x2) as proof of (((eq beverage) x0) x')
% 66.05/66.53 Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% 66.05/66.53 Found (eq_ref00 P) as proof of ((P x0)->(P x'))
% 66.05/66.53 Found ((eq_ref0 x0) P) as proof of ((P x0)->(P x'))
% 66.05/66.53 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 66.05/66.53 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 66.05/66.53 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 66.05/66.53 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of (((eq beverage) x0) x')
% 68.65/69.19 Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% 68.65/69.19 Found (eq_ref00 P) as proof of ((P x0)->(P x'))
% 68.65/69.19 Found ((eq_ref0 x0) P) as proof of ((P x0)->(P x'))
% 68.65/69.19 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 68.65/69.19 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 68.65/69.19 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 68.65/69.19 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of (((eq beverage) x0) x')
% 68.65/69.19 Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% 68.65/69.19 Found (eq_ref00 P) as proof of ((P x0)->(P x'))
% 68.65/69.19 Found ((eq_ref0 x0) P) as proof of ((P x0)->(P x'))
% 68.65/69.19 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 68.65/69.19 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 68.65/69.19 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 68.65/69.19 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of (((eq beverage) x0) x')
% 68.65/69.19 Found x3:(P x0)
% 68.65/69.19 Instantiate: x0:=x':beverage
% 68.65/69.19 Found (fun (x3:(P x0))=> x3) as proof of (P x')
% 68.65/69.19 Found (fun (P:(beverage->Prop)) (x3:(P x0))=> x3) as proof of ((P x0)->(P x'))
% 68.65/69.19 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop)) (x3:(P x0))=> x3) as proof of (((eq beverage) x0) x')
% 68.65/69.19 Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% 68.65/69.19 Found (eq_ref00 P) as proof of ((P x0)->(P x'))
% 68.65/69.19 Found ((eq_ref0 x0) P) as proof of ((P x0)->(P x'))
% 68.65/69.19 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 68.65/69.19 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 68.65/69.19 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 68.65/69.19 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of (((eq beverage) x0) x')
% 68.65/69.19 Found x3:(P x0)
% 68.65/69.19 Instantiate: x0:=x':beverage
% 68.65/69.19 Found (fun (x3:(P x0))=> x3) as proof of (P x')
% 68.65/69.19 Found (fun (P:(beverage->Prop)) (x3:(P x0))=> x3) as proof of ((P x0)->(P x'))
% 68.65/69.19 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B))))) (P:(beverage->Prop)) (x3:(P x0))=> x3) as proof of (((eq beverage) x0) x')
% 68.65/69.19 Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% 68.65/69.19 Found (eq_ref00 P) as proof of ((P x0)->(P x'))
% 68.65/69.19 Found ((eq_ref0 x0) P) as proof of ((P x0)->(P x'))
% 68.65/69.19 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 68.65/69.19 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 68.65/69.19 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 68.65/69.19 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of (forall (P:(beverage->Prop)), ((P x0)->(P x')))
% 68.65/69.19 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of (((eq beverage) x0) x')
% 68.65/69.19 Found x1:(P x0)
% 68.65/69.19 Instantiate: x0:=x':beverage
% 68.65/69.19 Found (fun (x1:(P x0))=> x1) as proof of (P x')
% 68.65/69.19 Found (fun (P:(beverage->Prop)) (x1:(P x0))=> x1) as proof of ((P x0)->(P x'))
% 68.65/69.19 Found (fun (P:(beverage->Prop)) (x1:(P x0))=> x1) as proof of (forall (P:(beverage->Prop)), ((P x0)->(P x')))
% 68.65/69.19 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop)) (x1:(P x0))=> x1) as proof of (((eq beverage) x0) x')
% 68.65/69.19 Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% 68.65/69.19 Found (eq_ref00 P) as proof of ((P x0)->(P x'))
% 68.65/69.19 Found ((eq_ref0 x0) P) as proof of ((P x0)->(P x'))
% 68.65/69.19 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 68.65/69.19 Found (((eq_ref beverage) x0) P) as proof of ((P x0)->(P x'))
% 71.82/72.32 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 71.82/72.32 Found (fun (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of (forall (P:(beverage->Prop)), ((P x0)->(P x')))
% 71.82/72.32 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop))=> (((eq_ref beverage) x0) P)) as proof of (((eq beverage) x0) x')
% 71.82/72.32 Found x1:(P x0)
% 71.82/72.32 Instantiate: x0:=x':beverage
% 71.82/72.32 Found (fun (x1:(P x0))=> x1) as proof of (P x')
% 71.82/72.32 Found (fun (P:(beverage->Prop)) (x1:(P x0))=> x1) as proof of ((P x0)->(P x'))
% 71.82/72.32 Found (fun (P:(beverage->Prop)) (x1:(P x0))=> x1) as proof of (forall (P:(beverage->Prop)), ((P x0)->(P x')))
% 71.82/72.32 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop)) (x1:(P x0))=> x1) as proof of (((eq beverage) x0) x')
% 71.82/72.32 Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% 71.82/72.32 Found (eq_ref00 P) as proof of (P0 x0)
% 71.82/72.32 Found ((eq_ref0 x0) P) as proof of (P0 x0)
% 71.82/72.32 Found (((eq_ref beverage) x0) P) as proof of (P0 x0)
% 71.82/72.32 Found (((eq_ref beverage) x0) P) as proof of (P0 x0)
% 71.82/72.32 Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% 71.82/72.32 Found (eq_ref00 P) as proof of (P0 x0)
% 71.82/72.32 Found ((eq_ref0 x0) P) as proof of (P0 x0)
% 71.82/72.32 Found (((eq_ref beverage) x0) P) as proof of (P0 x0)
% 71.82/72.32 Found (((eq_ref beverage) x0) P) as proof of (P0 x0)
% 71.82/72.32 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 71.82/72.32 Found (eq_ref0 x') as proof of (((eq beverage) x') x0)
% 71.82/72.32 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 71.82/72.32 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 71.82/72.32 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 71.82/72.32 Found ((eq_sym0000 ((eq_ref beverage) x')) (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 71.82/72.32 Found ((eq_sym0000 ((eq_ref beverage) x')) (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 71.82/72.32 Found (((fun (x1:(((eq beverage) x') x0))=> ((eq_sym000 x1) (fun (x3:beverage)=> ((P x0)->(P x3))))) ((eq_ref beverage) x')) (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 71.82/72.32 Found (((fun (x1:(((eq beverage) x') x0))=> (((eq_sym00 x0) x1) (fun (x3:beverage)=> ((P x0)->(P x3))))) ((eq_ref beverage) x')) (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 71.82/72.32 Found (((fun (x1:(((eq beverage) x') x0))=> ((((eq_sym0 x') x0) x1) (fun (x3:beverage)=> ((P x0)->(P x3))))) ((eq_ref beverage) x')) (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 71.82/72.32 Found (((fun (x1:(((eq beverage) x') x0))=> (((((eq_sym beverage) x') x0) x1) (fun (x3:beverage)=> ((P x0)->(P x3))))) ((eq_ref beverage) x')) (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 71.82/72.32 Found (fun (P:(beverage->Prop))=> (((fun (x1:(((eq beverage) x') x0))=> (((((eq_sym beverage) x') x0) x1) (fun (x3:beverage)=> ((P x0)->(P x3))))) ((eq_ref beverage) x')) (((eq_ref beverage) x0) P))) as proof of ((P x0)->(P x'))
% 71.82/72.32 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop))=> (((fun (x1:(((eq beverage) x') x0))=> (((((eq_sym beverage) x') x0) x1) (fun (x3:beverage)=> ((P x0)->(P x3))))) ((eq_ref beverage) x')) (((eq_ref beverage) x0) P))) as proof of (((eq beverage) x0) x')
% 71.82/72.32 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 71.82/72.32 Found (eq_ref0 x') as proof of (((eq beverage) x') x0)
% 71.82/72.32 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 71.82/72.32 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 71.82/72.32 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 71.82/72.32 Found (eq_sym0000 ((eq_ref beverage) x')) as proof of ((P x0)->(P x'))
% 71.82/72.32 Found (eq_sym0000 ((eq_ref beverage) x')) as proof of ((P x0)->(P x'))
% 71.82/72.32 Found ((fun (x1:(((eq beverage) x') x0))=> ((eq_sym000 x1) P)) ((eq_ref beverage) x')) as proof of ((P x0)->(P x'))
% 71.82/72.32 Found ((fun (x1:(((eq beverage) x') x0))=> (((eq_sym00 x0) x1) P)) ((eq_ref beverage) x')) as proof of ((P x0)->(P x'))
% 71.82/72.32 Found ((fun (x1:(((eq beverage) x') x0))=> ((((eq_sym0 x') x0) x1) P)) ((eq_ref beverage) x')) as proof of ((P x0)->(P x'))
% 71.82/72.32 Found ((fun (x1:(((eq beverage) x') x0))=> (((((eq_sym beverage) x') x0) x1) P)) ((eq_ref beverage) x')) as proof of ((P x0)->(P x'))
% 73.47/74.02 Found (fun (P:(beverage->Prop))=> ((fun (x1:(((eq beverage) x') x0))=> (((((eq_sym beverage) x') x0) x1) P)) ((eq_ref beverage) x'))) as proof of ((P x0)->(P x'))
% 73.47/74.02 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop))=> ((fun (x1:(((eq beverage) x') x0))=> (((((eq_sym beverage) x') x0) x1) P)) ((eq_ref beverage) x'))) as proof of (((eq beverage) x0) x')
% 73.47/74.02 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 73.47/74.02 Found (eq_ref0 x') as proof of (((eq beverage) x') x0)
% 73.47/74.02 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 73.47/74.02 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 73.47/74.02 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 73.47/74.02 Found ((eq_sym0000 ((eq_ref beverage) x')) (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 73.47/74.02 Found ((eq_sym0000 ((eq_ref beverage) x')) (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 73.47/74.02 Found (((fun (x1:(((eq beverage) x') x0))=> ((eq_sym000 x1) (fun (x3:beverage)=> ((P x0)->(P x3))))) ((eq_ref beverage) x')) (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 73.47/74.02 Found (((fun (x1:(((eq beverage) x') x0))=> (((eq_sym00 x0) x1) (fun (x3:beverage)=> ((P x0)->(P x3))))) ((eq_ref beverage) x')) (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 73.47/74.02 Found (((fun (x1:(((eq beverage) x') x0))=> ((((eq_sym0 x') x0) x1) (fun (x3:beverage)=> ((P x0)->(P x3))))) ((eq_ref beverage) x')) (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 73.47/74.02 Found (((fun (x1:(((eq beverage) x') x0))=> (((((eq_sym beverage) x') x0) x1) (fun (x3:beverage)=> ((P x0)->(P x3))))) ((eq_ref beverage) x')) (((eq_ref beverage) x0) P)) as proof of ((P x0)->(P x'))
% 73.47/74.02 Found (fun (P:(beverage->Prop))=> (((fun (x1:(((eq beverage) x') x0))=> (((((eq_sym beverage) x') x0) x1) (fun (x3:beverage)=> ((P x0)->(P x3))))) ((eq_ref beverage) x')) (((eq_ref beverage) x0) P))) as proof of ((P x0)->(P x'))
% 73.47/74.02 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop))=> (((fun (x1:(((eq beverage) x') x0))=> (((((eq_sym beverage) x') x0) x1) (fun (x3:beverage)=> ((P x0)->(P x3))))) ((eq_ref beverage) x')) (((eq_ref beverage) x0) P))) as proof of (((eq beverage) x0) x')
% 73.47/74.02 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 73.47/74.02 Found (eq_ref0 x') as proof of (((eq beverage) x') x0)
% 73.47/74.02 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 73.47/74.02 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 73.47/74.02 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 73.47/74.02 Found (eq_sym0000 ((eq_ref beverage) x')) as proof of ((P x0)->(P x'))
% 73.47/74.02 Found (eq_sym0000 ((eq_ref beverage) x')) as proof of ((P x0)->(P x'))
% 73.47/74.02 Found ((fun (x1:(((eq beverage) x') x0))=> ((eq_sym000 x1) P)) ((eq_ref beverage) x')) as proof of ((P x0)->(P x'))
% 73.47/74.02 Found ((fun (x1:(((eq beverage) x') x0))=> (((eq_sym00 x0) x1) P)) ((eq_ref beverage) x')) as proof of ((P x0)->(P x'))
% 73.47/74.02 Found ((fun (x1:(((eq beverage) x') x0))=> ((((eq_sym0 x') x0) x1) P)) ((eq_ref beverage) x')) as proof of ((P x0)->(P x'))
% 73.47/74.02 Found ((fun (x1:(((eq beverage) x') x0))=> (((((eq_sym beverage) x') x0) x1) P)) ((eq_ref beverage) x')) as proof of ((P x0)->(P x'))
% 73.47/74.02 Found (fun (P:(beverage->Prop))=> ((fun (x1:(((eq beverage) x') x0))=> (((((eq_sym beverage) x') x0) x1) P)) ((eq_ref beverage) x'))) as proof of ((P x0)->(P x'))
% 73.47/74.02 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B))))) (P:(beverage->Prop))=> ((fun (x1:(((eq beverage) x') x0))=> (((((eq_sym beverage) x') x0) x1) P)) ((eq_ref beverage) x'))) as proof of (((eq beverage) x0) x')
% 73.47/74.02 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq ((syrup->beverage)->Prop)) a) (fun (x:(syrup->beverage))=> (a x)))
% 73.47/74.02 Found (eta_expansion_dep00 a) as proof of (((eq ((syrup->beverage)->Prop)) a) b)
% 73.47/74.02 Found ((eta_expansion_dep0 (fun (x1:(syrup->beverage))=> Prop)) a) as proof of (((eq ((syrup->beverage)->Prop)) a) b)
% 73.47/74.02 Found (((eta_expansion_dep (syrup->beverage)) (fun (x1:(syrup->beverage))=> Prop)) a) as proof of (((eq ((syrup->beverage)->Prop)) a) b)
% 74.38/74.85 Found (((eta_expansion_dep (syrup->beverage)) (fun (x1:(syrup->beverage))=> Prop)) a) as proof of (((eq ((syrup->beverage)->Prop)) a) b)
% 74.38/74.85 Found (((eta_expansion_dep (syrup->beverage)) (fun (x1:(syrup->beverage))=> Prop)) a) as proof of (((eq ((syrup->beverage)->Prop)) a) b)
% 74.38/74.85 Found eta_expansion000:=(eta_expansion00 b):(((eq ((syrup->beverage)->Prop)) b) (fun (x:(syrup->beverage))=> (b x)))
% 74.38/74.85 Found (eta_expansion00 b) as proof of (((eq ((syrup->beverage)->Prop)) b) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B)))))))
% 74.38/74.85 Found ((eta_expansion0 Prop) b) as proof of (((eq ((syrup->beverage)->Prop)) b) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B)))))))
% 74.38/74.85 Found (((eta_expansion (syrup->beverage)) Prop) b) as proof of (((eq ((syrup->beverage)->Prop)) b) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B)))))))
% 74.38/74.85 Found (((eta_expansion (syrup->beverage)) Prop) b) as proof of (((eq ((syrup->beverage)->Prop)) b) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B)))))))
% 74.38/74.85 Found (((eta_expansion (syrup->beverage)) Prop) b) as proof of (((eq ((syrup->beverage)->Prop)) b) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B)))))))
% 74.38/74.85 Found eq_ref00:=(eq_ref0 b):(((eq ((syrup->beverage)->Prop)) b) b)
% 74.38/74.85 Found (eq_ref0 b) as proof of (((eq ((syrup->beverage)->Prop)) b) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B)))))))
% 74.38/74.85 Found ((eq_ref ((syrup->beverage)->Prop)) b) as proof of (((eq ((syrup->beverage)->Prop)) b) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B)))))))
% 74.38/74.85 Found ((eq_ref ((syrup->beverage)->Prop)) b) as proof of (((eq ((syrup->beverage)->Prop)) b) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B)))))))
% 74.38/74.85 Found ((eq_ref ((syrup->beverage)->Prop)) b) as proof of (((eq ((syrup->beverage)->Prop)) b) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B)))))))
% 74.38/74.85 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 74.38/74.85 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 74.38/74.85 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 74.38/74.85 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 74.38/74.85 Found (fun (x2:(hot (coffee_mixture S)))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 74.38/74.85 Found (fun (x1:(((eq beverage) (coffee_mixture S)) coffee)) (x2:(hot (coffee_mixture S)))=> ((eq_ref beverage) x0)) as proof of ((hot (coffee_mixture S))->(((eq beverage) x0) x'))
% 74.38/74.85 Found (fun (x1:(((eq beverage) (coffee_mixture S)) coffee)) (x2:(hot (coffee_mixture S)))=> ((eq_ref beverage) x0)) as proof of ((((eq beverage) (coffee_mixture S)) coffee)->((hot (coffee_mixture S))->(((eq beverage) x0) x')))
% 74.38/74.85 Found (and_rect00 (fun (x1:(((eq beverage) (coffee_mixture S)) coffee)) (x2:(hot (coffee_mixture S)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 74.38/74.85 Found ((and_rect0 (((eq beverage) x0) x')) (fun (x1:(((eq beverage) (coffee_mixture S)) coffee)) (x2:(hot (coffee_mixture S)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 75.81/76.30 Found (((fun (P:Type) (x1:((((eq beverage) (coffee_mixture S)) coffee)->((hot (coffee_mixture S))->P)))=> (((((and_rect (((eq beverage) (coffee_mixture S)) coffee)) (hot (coffee_mixture S))) P) x1) coffee_and_syrup_is_hot_coffee0)) (((eq beverage) x0) x')) (fun (x1:(((eq beverage) (coffee_mixture S)) coffee)) (x2:(hot (coffee_mixture S)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 75.81/76.30 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 75.81/76.30 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 75.81/76.30 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 75.81/76.30 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 75.81/76.30 Found (fun (x2:(hot (coffee_mixture S)))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 75.81/76.30 Found (fun (x1:(((eq beverage) (coffee_mixture S)) coffee)) (x2:(hot (coffee_mixture S)))=> ((eq_ref beverage) x0)) as proof of ((hot (coffee_mixture S))->(((eq beverage) x0) x'))
% 75.81/76.30 Found (fun (x1:(((eq beverage) (coffee_mixture S)) coffee)) (x2:(hot (coffee_mixture S)))=> ((eq_ref beverage) x0)) as proof of ((((eq beverage) (coffee_mixture S)) coffee)->((hot (coffee_mixture S))->(((eq beverage) x0) x')))
% 75.81/76.30 Found (and_rect00 (fun (x1:(((eq beverage) (coffee_mixture S)) coffee)) (x2:(hot (coffee_mixture S)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 75.81/76.30 Found ((and_rect0 (((eq beverage) x0) x')) (fun (x1:(((eq beverage) (coffee_mixture S)) coffee)) (x2:(hot (coffee_mixture S)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 75.81/76.30 Found (((fun (P:Type) (x1:((((eq beverage) (coffee_mixture S)) coffee)->((hot (coffee_mixture S))->P)))=> (((((and_rect (((eq beverage) (coffee_mixture S)) coffee)) (hot (coffee_mixture S))) P) x1) coffee_and_syrup_is_hot_coffee0)) (((eq beverage) x0) x')) (fun (x1:(((eq beverage) (coffee_mixture S)) coffee)) (x2:(hot (coffee_mixture S)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 75.81/76.30 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 75.81/76.30 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 75.81/76.30 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 75.81/76.30 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 75.81/76.30 Found (fun (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 75.81/76.30 Found (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0)) as proof of (((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1))->(((eq beverage) x0) x'))
% 75.81/76.30 Found (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->(((eq beverage) x0) x')))
% 75.81/76.30 Found (ex_ind00 (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 75.81/76.30 Found ((ex_ind0 (((eq beverage) x0) x')) (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 75.81/76.30 Found (((fun (P:Prop) (x1:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x1) x00)) (((eq beverage) x0) x')) (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 75.81/76.30 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> (((fun (P:Prop) (x1:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x1) x00)) (((eq beverage) x0) x')) (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0)))) as proof of (((eq beverage) x0) x')
% 76.03/76.55 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 76.03/76.55 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 76.03/76.55 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 76.03/76.55 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 76.03/76.55 Found (fun (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 76.03/76.55 Found (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0)) as proof of (((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1))->(((eq beverage) x0) x'))
% 76.03/76.55 Found (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->(((eq beverage) x0) x')))
% 76.03/76.55 Found (ex_ind00 (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 76.03/76.55 Found ((ex_ind0 (((eq beverage) x0) x')) (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 76.03/76.55 Found (((fun (P:Prop) (x1:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x1) x00)) (((eq beverage) x0) x')) (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 76.03/76.55 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> (((fun (P:Prop) (x1:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x1) x00)) (((eq beverage) x0) x')) (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0)))) as proof of (((eq beverage) x0) x')
% 76.03/76.55 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 76.03/76.55 Found (eq_ref0 x0) as proof of (((eq beverage) x0) b)
% 76.03/76.55 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) b)
% 76.03/76.55 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) b)
% 76.03/76.55 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) b)
% 76.03/76.55 Found eq_ref00:=(eq_ref0 b):(((eq beverage) b) b)
% 76.03/76.55 Found (eq_ref0 b) as proof of (((eq beverage) b) x')
% 76.03/76.55 Found ((eq_ref beverage) b) as proof of (((eq beverage) b) x')
% 76.03/76.55 Found ((eq_ref beverage) b) as proof of (((eq beverage) b) x')
% 76.03/76.55 Found ((eq_ref beverage) b) as proof of (((eq beverage) b) x')
% 76.03/76.55 Found ((eq_trans0000 ((eq_ref beverage) x0)) ((eq_ref beverage) b)) as proof of (((eq beverage) x0) x')
% 76.03/76.55 Found (((eq_trans000 x') ((eq_ref beverage) x0)) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 76.03/76.55 Found ((((fun (b:beverage)=> ((eq_trans00 b) x')) x') ((eq_ref beverage) x0)) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 76.03/76.55 Found ((((fun (b:beverage)=> (((eq_trans0 x0) b) x')) x') ((eq_ref beverage) x0)) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 76.03/76.55 Found ((((fun (b:beverage)=> ((((eq_trans beverage) x0) b) x')) x') ((eq_ref beverage) x0)) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 76.03/76.55 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> ((((fun (b:beverage)=> ((((eq_trans beverage) x0) b) x')) x') ((eq_ref beverage) x0)) ((eq_ref beverage) x'))) as proof of (((eq beverage) x0) x')
% 76.32/76.82 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 76.32/76.82 Found (eq_ref0 x0) as proof of (((eq beverage) x0) b)
% 76.32/76.82 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) b)
% 76.32/76.82 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) b)
% 76.32/76.82 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) b)
% 76.32/76.82 Found eq_ref00:=(eq_ref0 b):(((eq beverage) b) b)
% 76.32/76.82 Found (eq_ref0 b) as proof of (((eq beverage) b) x')
% 76.32/76.82 Found ((eq_ref beverage) b) as proof of (((eq beverage) b) x')
% 76.32/76.82 Found ((eq_ref beverage) b) as proof of (((eq beverage) b) x')
% 76.32/76.82 Found ((eq_ref beverage) b) as proof of (((eq beverage) b) x')
% 76.32/76.82 Found ((eq_trans0000 ((eq_ref beverage) x0)) ((eq_ref beverage) b)) as proof of (((eq beverage) x0) x')
% 76.32/76.82 Found (((eq_trans000 x') ((eq_ref beverage) x0)) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 76.32/76.82 Found ((((fun (b:beverage)=> ((eq_trans00 b) x')) x') ((eq_ref beverage) x0)) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 76.32/76.82 Found ((((fun (b:beverage)=> (((eq_trans0 x0) b) x')) x') ((eq_ref beverage) x0)) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 76.32/76.82 Found ((((fun (b:beverage)=> ((((eq_trans beverage) x0) b) x')) x') ((eq_ref beverage) x0)) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 76.32/76.82 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> ((((fun (b:beverage)=> ((((eq_trans beverage) x0) b) x')) x') ((eq_ref beverage) x0)) ((eq_ref beverage) x'))) as proof of (((eq beverage) x0) x')
% 76.32/76.82 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 76.32/76.82 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 76.32/76.82 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 76.32/76.82 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 76.32/76.82 Found (fun (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 76.32/76.82 Found (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0)) as proof of (((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1))->(((eq beverage) x0) x'))
% 76.32/76.82 Found (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->(((eq beverage) x0) x')))
% 76.32/76.82 Found (ex_ind00 (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 76.32/76.82 Found ((ex_ind0 (((eq beverage) x0) x')) (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 76.32/76.82 Found (((fun (P:Prop) (x1:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x1) x00)) (((eq beverage) x0) x')) (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 76.32/76.82 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> (((fun (P:Prop) (x1:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x1) x00)) (((eq beverage) x0) x')) (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0)))) as proof of (((eq beverage) x0) x')
% 76.43/76.90 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 76.43/76.90 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 76.43/76.90 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 76.43/76.90 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 76.43/76.90 Found (fun (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 76.43/76.90 Found (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0)) as proof of (((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1))->(((eq beverage) x0) x'))
% 76.43/76.90 Found (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->(((eq beverage) x0) x')))
% 76.43/76.90 Found (ex_ind00 (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 76.43/76.90 Found ((ex_ind0 (((eq beverage) x0) x')) (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 76.43/76.90 Found (((fun (P:Prop) (x1:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x1) x00)) (((eq beverage) x0) x')) (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 76.43/76.90 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> (((fun (P:Prop) (x1:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x1) x00)) (((eq beverage) x0) x')) (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0)))) as proof of (((eq beverage) x0) x')
% 76.43/76.90 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 76.43/76.90 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 76.43/76.90 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 76.43/76.90 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 76.43/76.90 Found (fun (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 76.43/76.90 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3))->(((eq beverage) x2) x'))
% 76.43/76.90 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) coffee))) (hot x))->(((eq beverage) x2) x')))
% 76.43/76.90 Found (ex_ind00 (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 76.43/76.90 Found ((ex_ind0 (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 76.43/76.90 Found (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) coffee))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))) P) x3) x00)) (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 76.69/77.17 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) coffee))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))) P) x3) x00)) (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2)))) as proof of (((eq beverage) x2) x')
% 76.69/77.17 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 76.69/77.17 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 76.69/77.17 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 76.69/77.17 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 76.69/77.17 Found (fun (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 76.69/77.17 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3))->(((eq beverage) x2) x'))
% 76.69/77.17 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) coffee))) (hot x))->(((eq beverage) x2) x')))
% 76.69/77.17 Found (ex_ind00 (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 76.69/77.17 Found ((ex_ind0 (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 76.69/77.17 Found (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) coffee))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))) P) x3) x00)) (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 76.69/77.17 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) coffee))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))) P) x3) x00)) (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2)))) as proof of (((eq beverage) x2) x')
% 76.69/77.17 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))):(((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) (fun (x:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))))))
% 76.69/77.17 Found (eta_expansion_dep00 (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) b)
% 77.75/78.27 Found ((eta_expansion_dep0 (fun (x2:(syrup->beverage))=> Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) b)
% 77.75/78.27 Found (((eta_expansion_dep (syrup->beverage)) (fun (x2:(syrup->beverage))=> Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) b)
% 77.75/78.27 Found (((eta_expansion_dep (syrup->beverage)) (fun (x2:(syrup->beverage))=> Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) b)
% 77.75/78.27 Found (((eta_expansion_dep (syrup->beverage)) (fun (x2:(syrup->beverage))=> Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) b)
% 77.75/78.27 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 77.75/78.27 Found (eq_ref0 x') as proof of (((eq beverage) x') x0)
% 77.75/78.27 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 77.75/78.27 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 77.75/78.27 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 77.75/78.27 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 77.75/78.27 Found ((eq_sym00 x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 77.75/78.27 Found (((eq_sym0 x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 77.75/78.27 Found ((((eq_sym beverage) x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 77.75/78.27 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((((eq_sym beverage) x') x0) ((eq_ref beverage) x'))) as proof of (((eq beverage) x0) x')
% 77.75/78.27 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 77.75/78.27 Found (eq_ref0 x') as proof of (((eq beverage) x') x0)
% 77.75/78.27 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 77.75/78.27 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 77.75/78.27 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 77.75/78.27 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 77.75/78.27 Found ((eq_sym00 x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 77.75/78.27 Found (((eq_sym0 x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 77.75/78.27 Found ((((eq_sym beverage) x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 77.75/78.27 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((((eq_sym beverage) x') x0) ((eq_ref beverage) x'))) as proof of (((eq beverage) x0) x')
% 77.75/78.27 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 77.75/78.27 Found (eq_ref0 x') as proof of (((eq beverage) x') x0)
% 77.75/78.27 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 77.75/78.27 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 80.21/80.73 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 80.21/80.73 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 80.21/80.73 Found ((eq_sym00 x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 80.21/80.73 Found (((eq_sym0 x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 80.21/80.73 Found ((((eq_sym beverage) x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 80.21/80.73 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((((eq_sym beverage) x') x0) ((eq_ref beverage) x'))) as proof of (((eq beverage) x0) x')
% 80.21/80.73 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 80.21/80.73 Found (eq_ref0 x') as proof of (((eq beverage) x') x0)
% 80.21/80.73 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 80.21/80.73 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 80.21/80.73 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 80.21/80.73 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 80.21/80.73 Found ((eq_sym00 x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 80.21/80.73 Found (((eq_sym0 x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 80.21/80.73 Found ((((eq_sym beverage) x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 80.21/80.73 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((((eq_sym beverage) x') x0) ((eq_ref beverage) x'))) as proof of (((eq beverage) x0) x')
% 80.21/80.73 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 80.21/80.73 Found (eq_ref0 x') as proof of (((eq beverage) x') x2)
% 80.21/80.73 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 80.21/80.73 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 80.21/80.73 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 80.21/80.73 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 80.21/80.73 Found ((eq_sym00 x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 80.21/80.73 Found (((eq_sym0 x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 80.21/80.73 Found ((((eq_sym beverage) x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 80.21/80.73 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> ((((eq_sym beverage) x') x2) ((eq_ref beverage) x'))) as proof of (((eq beverage) x2) x')
% 80.21/80.73 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 80.21/80.73 Found (eq_ref0 x') as proof of (((eq beverage) x') x2)
% 80.21/80.73 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 80.21/80.73 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 80.21/80.73 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 80.21/80.73 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 80.21/80.73 Found ((eq_sym00 x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 80.21/80.73 Found (((eq_sym0 x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 80.21/80.73 Found ((((eq_sym beverage) x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 80.21/80.73 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> ((((eq_sym beverage) x') x2) ((eq_ref beverage) x'))) as proof of (((eq beverage) x2) x')
% 80.21/80.73 Found eta_expansion000:=(eta_expansion00 (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))):(((eq (beverage->Prop)) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))) (fun (x0:beverage)=> ((and ((and (((eq beverage) x0) (x S))) (((eq beverage) x0) coffee))) (hot x0))))
% 80.21/80.73 Found (eta_expansion00 (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))) b)
% 80.21/80.73 Found ((eta_expansion0 Prop) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))) b)
% 82.02/82.51 Found (((eta_expansion beverage) Prop) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))) b)
% 82.02/82.51 Found (((eta_expansion beverage) Prop) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))) b)
% 82.02/82.51 Found (((eta_expansion beverage) Prop) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))) b)
% 82.02/82.51 Found eq_ref00:=(eq_ref0 (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))):(((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))))
% 82.02/82.51 Found (eq_ref0 (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 82.02/82.51 Found ((eq_ref ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 82.02/82.51 Found ((eq_ref ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 82.02/82.51 Found ((eq_ref ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 82.02/82.51 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 82.02/82.51 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 82.02/82.51 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 82.02/82.51 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 82.02/82.51 Found (fun (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 82.02/82.51 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3))->(((eq beverage) x2) x'))
% 82.02/82.51 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) coffee))) (hot x))->(((eq beverage) x2) x')))
% 83.22/83.71 Found (ex_ind00 (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 83.22/83.71 Found ((ex_ind0 (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 83.22/83.71 Found (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) coffee))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))) P) x3) x00)) (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 83.22/83.71 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) coffee))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))) P) x3) x00)) (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2)))) as proof of (((eq beverage) x2) x')
% 83.22/83.71 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 83.22/83.71 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 83.22/83.71 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 83.22/83.71 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 83.22/83.71 Found (fun (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 83.22/83.71 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3))->(((eq beverage) x2) x'))
% 83.22/83.71 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) coffee))) (hot x))->(((eq beverage) x2) x')))
% 83.22/83.71 Found (ex_ind00 (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 83.22/83.71 Found ((ex_ind0 (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 83.22/83.71 Found (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) coffee))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))) P) x3) x00)) (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 83.22/83.71 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) coffee))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))) P) x3) x00)) (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x2)))) as proof of (((eq beverage) x2) x')
% 83.22/83.71 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))):(((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) (fun (x:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))))
% 83.46/83.93 Found (eta_expansion_dep00 (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 83.46/83.93 Found ((eta_expansion_dep0 (fun (x2:(syrup->beverage))=> Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 83.46/83.93 Found (((eta_expansion_dep (syrup->beverage)) (fun (x2:(syrup->beverage))=> Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 83.46/83.93 Found (((eta_expansion_dep (syrup->beverage)) (fun (x2:(syrup->beverage))=> Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 83.46/83.93 Found (((eta_expansion_dep (syrup->beverage)) (fun (x2:(syrup->beverage))=> Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 83.46/83.93 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 83.46/83.93 Found (eq_ref0 x') as proof of (((eq beverage) x') x2)
% 83.46/83.93 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 83.46/83.93 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 83.46/83.93 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 83.46/83.93 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 83.46/83.93 Found ((eq_sym00 x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 83.46/83.93 Found (((eq_sym0 x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 83.46/83.93 Found ((((eq_sym beverage) x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 83.46/83.93 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> ((((eq_sym beverage) x') x2) ((eq_ref beverage) x'))) as proof of (((eq beverage) x2) x')
% 83.46/83.93 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 83.46/83.93 Found (eq_ref0 x') as proof of (((eq beverage) x') x2)
% 83.46/83.93 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 84.35/84.84 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 84.35/84.84 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 84.35/84.84 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 84.35/84.84 Found ((eq_sym00 x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 84.35/84.84 Found (((eq_sym0 x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 84.35/84.84 Found ((((eq_sym beverage) x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 84.35/84.84 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> ((((eq_sym beverage) x') x2) ((eq_ref beverage) x'))) as proof of (((eq beverage) x2) x')
% 84.35/84.84 Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% 84.35/84.84 Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S))) (((eq beverage) B) coffee))) (hot B))))))
% 84.35/84.84 Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S))) (((eq beverage) B) coffee))) (hot B))))))
% 84.35/84.84 Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S))) (((eq beverage) B) coffee))) (hot B))))))
% 84.35/84.84 Found (fun (x1:(syrup->beverage))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S))) (((eq beverage) B) coffee))) (hot B))))))
% 84.35/84.84 Found (fun (x1:(syrup->beverage))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(syrup->beverage)), (((eq Prop) (f x)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))))))
% 84.35/84.84 Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% 84.35/84.84 Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S))) (((eq beverage) B) coffee))) (hot B))))))
% 84.35/84.84 Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S))) (((eq beverage) B) coffee))) (hot B))))))
% 84.35/84.84 Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S))) (((eq beverage) B) coffee))) (hot B))))))
% 84.35/84.84 Found (fun (x1:(syrup->beverage))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S))) (((eq beverage) B) coffee))) (hot B))))))
% 84.35/84.84 Found (fun (x1:(syrup->beverage))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(syrup->beverage)), (((eq Prop) (f x)) (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))))))
% 84.35/84.84 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))):(((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) (fun (x:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))))
% 84.35/84.84 Found (eta_expansion_dep00 (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 84.35/84.84 Found ((eta_expansion_dep0 (fun (x3:(syrup->beverage))=> Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 85.46/86.00 Found (((eta_expansion_dep (syrup->beverage)) (fun (x3:(syrup->beverage))=> Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 85.46/86.00 Found (((eta_expansion_dep (syrup->beverage)) (fun (x3:(syrup->beverage))=> Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 85.46/86.00 Found (((eta_expansion_dep (syrup->beverage)) (fun (x3:(syrup->beverage))=> Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq ((syrup->beverage)->Prop)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 85.46/86.00 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 85.46/86.00 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 85.46/86.00 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 85.46/86.00 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 85.46/86.00 Found (fun (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 85.46/86.00 Found (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0)) as proof of (((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1))->(((eq beverage) x0) x'))
% 85.46/86.00 Found (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->(((eq beverage) x0) x')))
% 85.46/86.00 Found (ex_ind00 (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 85.46/86.00 Found ((ex_ind0 (((eq beverage) x0) x')) (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 85.46/86.00 Found (((fun (P:Prop) (x1:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x1) x00)) (((eq beverage) x0) x')) (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 85.46/86.00 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> (((fun (P:Prop) (x1:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x1) x00)) (((eq beverage) x0) x')) (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0)))) as proof of (((eq beverage) x0) x')
% 86.17/86.67 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 86.17/86.67 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 86.17/86.67 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 86.17/86.67 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 86.17/86.67 Found (fun (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 86.17/86.67 Found (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0)) as proof of (((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1))->(((eq beverage) x0) x'))
% 86.17/86.67 Found (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->(((eq beverage) x0) x')))
% 86.17/86.67 Found (ex_ind00 (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 86.17/86.67 Found ((ex_ind0 (((eq beverage) x0) x')) (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 86.17/86.67 Found (((fun (P:Prop) (x1:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x1) x00)) (((eq beverage) x0) x')) (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 86.17/86.67 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> (((fun (P:Prop) (x1:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x1) x00)) (((eq beverage) x0) x')) (fun (x1:beverage) (x2:((and ((and (((eq beverage) x1) x')) (((eq beverage) x1) (coffee_mixture S)))) (hot x1)))=> ((eq_ref beverage) x0)))) as proof of (((eq beverage) x0) x')
% 86.17/86.67 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 86.17/86.67 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 86.17/86.67 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 86.17/86.67 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 86.17/86.67 Found (fun (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 86.17/86.67 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x0)) as proof of (((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3))->(((eq beverage) x0) x'))
% 86.17/86.67 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x0)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) coffee))) (hot x))->(((eq beverage) x0) x')))
% 86.17/86.67 Found (ex_ind00 (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 86.17/86.67 Found ((ex_ind0 (((eq beverage) x0) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 86.98/87.46 Found (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) coffee))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))) P) x3) x00)) (((eq beverage) x0) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 86.98/87.46 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) coffee))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))) P) x3) x00)) (((eq beverage) x0) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x0)))) as proof of (((eq beverage) x0) x')
% 86.98/87.46 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 86.98/87.46 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 86.98/87.46 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 86.98/87.46 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 86.98/87.46 Found (fun (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 86.98/87.46 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x0)) as proof of (((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3))->(((eq beverage) x0) x'))
% 86.98/87.46 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x0)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) coffee))) (hot x))->(((eq beverage) x0) x')))
% 86.98/87.46 Found (ex_ind00 (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 86.98/87.46 Found ((ex_ind0 (((eq beverage) x0) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 86.98/87.46 Found (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) coffee))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))) P) x3) x00)) (((eq beverage) x0) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 86.98/87.46 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) coffee))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))) P) x3) x00)) (((eq beverage) x0) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) coffee))) (hot x3)))=> ((eq_ref beverage) x0)))) as proof of (((eq beverage) x0) x')
% 86.98/87.46 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 86.98/87.46 Found (eq_ref0 x') as proof of (((eq beverage) x') x0)
% 86.98/87.46 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 86.98/87.46 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 86.98/87.46 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 86.98/87.46 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 86.98/87.46 Found ((eq_sym00 x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 86.98/87.46 Found (((eq_sym0 x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 86.98/87.46 Found ((((eq_sym beverage) x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 88.17/88.64 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((((eq_sym beverage) x') x0) ((eq_ref beverage) x'))) as proof of (((eq beverage) x0) x')
% 88.17/88.64 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 88.17/88.64 Found (eq_ref0 x') as proof of (((eq beverage) x') x0)
% 88.17/88.64 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 88.17/88.64 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 88.17/88.64 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 88.17/88.64 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 88.17/88.64 Found ((eq_sym00 x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 88.17/88.64 Found (((eq_sym0 x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 88.17/88.64 Found ((((eq_sym beverage) x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 88.17/88.64 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((((eq_sym beverage) x') x0) ((eq_ref beverage) x'))) as proof of (((eq beverage) x0) x')
% 88.17/88.64 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 88.17/88.64 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (((eq beverage) x0) (x S))) (((eq beverage) x0) coffee))) (hot x0)))
% 88.17/88.64 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (((eq beverage) x0) (x S))) (((eq beverage) x0) coffee))) (hot x0)))
% 88.17/88.64 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (((eq beverage) x0) (x S))) (((eq beverage) x0) coffee))) (hot x0)))
% 88.17/88.64 Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and ((and (((eq beverage) x0) (x S))) (((eq beverage) x0) coffee))) (hot x0)))
% 88.17/88.64 Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (forall (x0:beverage), (((eq Prop) (f x0)) ((and ((and (((eq beverage) x0) (x S))) (((eq beverage) x0) coffee))) (hot x0))))
% 88.17/88.64 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 88.17/88.64 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (((eq beverage) x0) (x S))) (((eq beverage) x0) coffee))) (hot x0)))
% 88.17/88.64 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (((eq beverage) x0) (x S))) (((eq beverage) x0) coffee))) (hot x0)))
% 88.17/88.64 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((and (((eq beverage) x0) (x S))) (((eq beverage) x0) coffee))) (hot x0)))
% 88.17/88.64 Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and ((and (((eq beverage) x0) (x S))) (((eq beverage) x0) coffee))) (hot x0)))
% 88.17/88.64 Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (forall (x0:beverage), (((eq Prop) (f x0)) ((and ((and (((eq beverage) x0) (x S))) (((eq beverage) x0) coffee))) (hot x0))))
% 88.17/88.64 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 88.17/88.64 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 88.17/88.64 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 88.17/88.64 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 88.17/88.64 Found (fun (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 88.17/88.64 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3))->(((eq beverage) x2) x'))
% 88.17/88.64 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->(((eq beverage) x2) x')))
% 88.17/88.64 Found (ex_ind00 (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 88.17/88.64 Found ((ex_ind0 (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 88.17/88.68 Found (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x3) x00)) (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 88.17/88.68 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x3) x00)) (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)))) as proof of (((eq beverage) x2) x')
% 88.17/88.68 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 88.17/88.68 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 88.17/88.68 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 88.17/88.68 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 88.17/88.68 Found (fun (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 88.17/88.68 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3))->(((eq beverage) x2) x'))
% 88.17/88.68 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->(((eq beverage) x2) x')))
% 88.17/88.68 Found (ex_ind00 (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 88.17/88.68 Found ((ex_ind0 (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 88.17/88.68 Found (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x3) x01)) (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 88.17/88.68 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x3) x01)) (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)))) as proof of (((eq beverage) x2) x')
% 88.17/88.68 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 88.17/88.68 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 88.17/88.68 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 88.17/88.68 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 88.17/88.68 Found (fun (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 88.25/88.72 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3))->(((eq beverage) x2) x'))
% 88.25/88.72 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->(((eq beverage) x2) x')))
% 88.25/88.72 Found (ex_ind00 (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 88.25/88.72 Found ((ex_ind0 (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 88.25/88.72 Found (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x3) x01)) (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 88.25/88.72 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x3) x01)) (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)))) as proof of (((eq beverage) x2) x')
% 88.25/88.72 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 88.25/88.72 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 88.25/88.72 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 88.25/88.72 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 88.25/88.72 Found (fun (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 88.25/88.72 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3))->(((eq beverage) x2) x'))
% 88.25/88.72 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->(((eq beverage) x2) x')))
% 88.25/88.72 Found (ex_ind00 (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 88.25/88.72 Found ((ex_ind0 (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 88.25/88.72 Found (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x3) x00)) (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 88.25/88.72 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x3) x00)) (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)))) as proof of (((eq beverage) x2) x')
% 88.93/89.40 Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% 88.93/89.40 Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 88.93/89.40 Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 88.93/89.40 Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 88.93/89.40 Found (fun (x:(syrup->beverage))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 88.93/89.40 Found (fun (x:(syrup->beverage))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(syrup->beverage)), (((eq Prop) (f x)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))))
% 88.93/89.40 Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% 88.93/89.40 Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 88.93/89.40 Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 88.93/89.40 Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 88.93/89.40 Found (fun (x:(syrup->beverage))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 88.93/89.40 Found (fun (x:(syrup->beverage))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(syrup->beverage)), (((eq Prop) (f x)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))))
% 88.93/89.40 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 88.93/89.40 Found (eq_ref0 x') as proof of (((eq beverage) x') x0)
% 88.93/89.40 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 88.93/89.40 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 88.93/89.40 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 88.93/89.40 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 88.93/89.40 Found ((eq_sym00 x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 88.93/89.40 Found (((eq_sym0 x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 88.93/89.40 Found ((((eq_sym beverage) x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 88.93/89.40 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> ((((eq_sym beverage) x') x0) ((eq_ref beverage) x'))) as proof of (((eq beverage) x0) x')
% 88.93/89.40 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 88.93/89.40 Found (eq_ref0 x') as proof of (((eq beverage) x') x0)
% 88.93/89.40 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 88.93/89.40 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 88.93/89.40 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 88.93/89.40 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 89.32/89.80 Found ((eq_sym00 x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 89.32/89.80 Found (((eq_sym0 x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 89.32/89.80 Found ((((eq_sym beverage) x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 89.32/89.80 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> ((((eq_sym beverage) x') x0) ((eq_ref beverage) x'))) as proof of (((eq beverage) x0) x')
% 89.32/89.80 Found eq_ref00:=(eq_ref0 x3):(((eq beverage) x3) x3)
% 89.32/89.80 Found (eq_ref0 x3) as proof of (((eq beverage) x3) x')
% 89.32/89.80 Found ((eq_ref beverage) x3) as proof of (((eq beverage) x3) x')
% 89.32/89.80 Found ((eq_ref beverage) x3) as proof of (((eq beverage) x3) x')
% 89.32/89.80 Found (fun (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3)) as proof of (((eq beverage) x3) x')
% 89.32/89.80 Found (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3)) as proof of (((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4))->(((eq beverage) x3) x'))
% 89.32/89.80 Found (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->(((eq beverage) x3) x')))
% 89.32/89.80 Found (ex_ind00 (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3))) as proof of (((eq beverage) x3) x')
% 89.32/89.80 Found ((ex_ind0 (((eq beverage) x3) x')) (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3))) as proof of (((eq beverage) x3) x')
% 89.32/89.80 Found (((fun (P:Prop) (x4:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x4) x00)) (((eq beverage) x3) x')) (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3))) as proof of (((eq beverage) x3) x')
% 89.32/89.80 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> (((fun (P:Prop) (x4:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x4) x00)) (((eq beverage) x3) x')) (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3)))) as proof of (((eq beverage) x3) x')
% 89.32/89.80 Found eq_ref00:=(eq_ref0 x3):(((eq beverage) x3) x3)
% 89.32/89.80 Found (eq_ref0 x3) as proof of (((eq beverage) x3) x')
% 89.32/89.80 Found ((eq_ref beverage) x3) as proof of (((eq beverage) x3) x')
% 89.32/89.80 Found ((eq_ref beverage) x3) as proof of (((eq beverage) x3) x')
% 89.32/89.80 Found (fun (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3)) as proof of (((eq beverage) x3) x')
% 89.32/89.80 Found (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3)) as proof of (((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4))->(((eq beverage) x3) x'))
% 89.32/89.80 Found (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->(((eq beverage) x3) x')))
% 89.32/89.80 Found (ex_ind00 (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3))) as proof of (((eq beverage) x3) x')
% 89.46/89.97 Found ((ex_ind0 (((eq beverage) x3) x')) (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3))) as proof of (((eq beverage) x3) x')
% 89.46/89.97 Found (((fun (P:Prop) (x4:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x4) x00)) (((eq beverage) x3) x')) (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3))) as proof of (((eq beverage) x3) x')
% 89.46/89.97 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> (((fun (P:Prop) (x4:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x4) x00)) (((eq beverage) x3) x')) (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3)))) as proof of (((eq beverage) x3) x')
% 89.46/89.97 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 89.46/89.97 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 89.46/89.97 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 89.46/89.97 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 89.46/89.97 Found (fun (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 89.46/89.97 Found (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x2)) as proof of (((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4))->(((eq beverage) x2) x'))
% 89.46/89.97 Found (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x2)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->(((eq beverage) x2) x')))
% 89.46/89.97 Found (ex_ind00 (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 89.46/89.97 Found ((ex_ind0 (((eq beverage) x2) x')) (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 89.46/89.97 Found (((fun (P:Prop) (x4:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x4) x00)) (((eq beverage) x2) x')) (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 89.46/89.97 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> (((fun (P:Prop) (x4:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x4) x00)) (((eq beverage) x2) x')) (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x2)))) as proof of (((eq beverage) x2) x')
% 89.46/89.97 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 89.46/89.97 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 89.46/89.97 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 89.46/89.97 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 89.60/90.09 Found (fun (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 89.60/90.09 Found (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x2)) as proof of (((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4))->(((eq beverage) x2) x'))
% 89.60/90.09 Found (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x2)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->(((eq beverage) x2) x')))
% 89.60/90.09 Found (ex_ind00 (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 89.60/90.09 Found ((ex_ind0 (((eq beverage) x2) x')) (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 89.60/90.09 Found (((fun (P:Prop) (x4:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x4) x00)) (((eq beverage) x2) x')) (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 89.60/90.09 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> (((fun (P:Prop) (x4:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x4) x00)) (((eq beverage) x2) x')) (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x2)))) as proof of (((eq beverage) x2) x')
% 89.60/90.09 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 89.60/90.09 Found (eq_ref0 x') as proof of (((eq beverage) x') x2)
% 89.60/90.09 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 89.60/90.09 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 89.60/90.09 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 89.60/90.09 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 89.60/90.09 Found ((eq_sym00 x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 89.60/90.09 Found (((eq_sym0 x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 89.60/90.09 Found ((((eq_sym beverage) x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 89.60/90.09 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((((eq_sym beverage) x') x2) ((eq_ref beverage) x'))) as proof of (((eq beverage) x2) x')
% 89.60/90.09 Found eq_ref00:=(eq_ref0 x3):(((eq beverage) x3) x3)
% 89.60/90.09 Found (eq_ref0 x3) as proof of (((eq beverage) x3) x')
% 89.60/90.09 Found ((eq_ref beverage) x3) as proof of (((eq beverage) x3) x')
% 89.60/90.09 Found ((eq_ref beverage) x3) as proof of (((eq beverage) x3) x')
% 89.60/90.09 Found (fun (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3)) as proof of (((eq beverage) x3) x')
% 89.60/90.09 Found (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3)) as proof of (((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4))->(((eq beverage) x3) x'))
% 89.60/90.09 Found (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->(((eq beverage) x3) x')))
% 89.76/90.25 Found (ex_ind00 (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3))) as proof of (((eq beverage) x3) x')
% 89.76/90.25 Found ((ex_ind0 (((eq beverage) x3) x')) (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3))) as proof of (((eq beverage) x3) x')
% 89.76/90.25 Found (((fun (P:Prop) (x4:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x4) x00)) (((eq beverage) x3) x')) (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3))) as proof of (((eq beverage) x3) x')
% 89.76/90.25 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> (((fun (P:Prop) (x4:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x4) x00)) (((eq beverage) x3) x')) (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3)))) as proof of (((eq beverage) x3) x')
% 89.76/90.25 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 89.76/90.25 Found (eq_ref0 x') as proof of (((eq beverage) x') x2)
% 89.76/90.25 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 89.76/90.25 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 89.76/90.25 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 89.76/90.25 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 89.76/90.25 Found ((eq_sym00 x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 89.76/90.25 Found (((eq_sym0 x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 89.76/90.25 Found ((((eq_sym beverage) x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 89.76/90.25 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((((eq_sym beverage) x') x2) ((eq_ref beverage) x'))) as proof of (((eq beverage) x2) x')
% 89.76/90.25 Found eq_ref00:=(eq_ref0 x3):(((eq beverage) x3) x3)
% 89.76/90.25 Found (eq_ref0 x3) as proof of (((eq beverage) x3) x')
% 89.76/90.25 Found ((eq_ref beverage) x3) as proof of (((eq beverage) x3) x')
% 89.76/90.25 Found ((eq_ref beverage) x3) as proof of (((eq beverage) x3) x')
% 89.76/90.25 Found (fun (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3)) as proof of (((eq beverage) x3) x')
% 89.76/90.25 Found (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3)) as proof of (((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4))->(((eq beverage) x3) x'))
% 89.76/90.25 Found (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->(((eq beverage) x3) x')))
% 89.76/90.25 Found (ex_ind00 (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3))) as proof of (((eq beverage) x3) x')
% 89.76/90.25 Found ((ex_ind0 (((eq beverage) x3) x')) (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3))) as proof of (((eq beverage) x3) x')
% 89.76/90.25 Found (((fun (P:Prop) (x4:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x4) x00)) (((eq beverage) x3) x')) (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3))) as proof of (((eq beverage) x3) x')
% 90.42/90.93 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> (((fun (P:Prop) (x4:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x4) x00)) (((eq beverage) x3) x')) (fun (x4:beverage) (x5:((and ((and (((eq beverage) x4) x')) (((eq beverage) x4) (coffee_mixture S)))) (hot x4)))=> ((eq_ref beverage) x3)))) as proof of (((eq beverage) x3) x')
% 90.42/90.93 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 90.42/90.93 Found (eq_ref0 x') as proof of (((eq beverage) x') x2)
% 90.42/90.93 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 90.42/90.93 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 90.42/90.93 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 90.42/90.93 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 90.42/90.93 Found ((eq_sym00 x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 90.42/90.93 Found (((eq_sym0 x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 90.42/90.93 Found ((((eq_sym beverage) x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 90.42/90.93 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((((eq_sym beverage) x') x2) ((eq_ref beverage) x'))) as proof of (((eq beverage) x2) x')
% 90.42/90.93 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 90.42/90.93 Found (eq_ref0 x') as proof of (((eq beverage) x') x2)
% 90.42/90.93 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 90.42/90.93 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 90.42/90.93 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 90.42/90.93 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 90.42/90.93 Found ((eq_sym00 x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 90.42/90.93 Found (((eq_sym0 x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 90.42/90.93 Found ((((eq_sym beverage) x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 90.42/90.93 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((((eq_sym beverage) x') x2) ((eq_ref beverage) x'))) as proof of (((eq beverage) x2) x')
% 90.42/90.93 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 90.42/90.93 Found (eq_ref0 x') as proof of (((eq beverage) x') x3)
% 90.42/90.93 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x3)
% 90.42/90.93 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x3)
% 90.42/90.93 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x3)
% 90.42/90.93 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x3) x')
% 90.42/90.93 Found ((eq_sym00 x3) ((eq_ref beverage) x')) as proof of (((eq beverage) x3) x')
% 90.42/90.93 Found (((eq_sym0 x') x3) ((eq_ref beverage) x')) as proof of (((eq beverage) x3) x')
% 90.42/90.93 Found ((((eq_sym beverage) x') x3) ((eq_ref beverage) x')) as proof of (((eq beverage) x3) x')
% 90.42/90.93 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((((eq_sym beverage) x') x3) ((eq_ref beverage) x'))) as proof of (((eq beverage) x3) x')
% 90.42/90.93 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 90.42/90.93 Found (eq_ref0 x') as proof of (((eq beverage) x') x3)
% 90.42/90.93 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x3)
% 90.42/90.93 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x3)
% 90.42/90.93 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x3)
% 90.42/90.93 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x3) x')
% 90.42/90.93 Found ((eq_sym00 x3) ((eq_ref beverage) x')) as proof of (((eq beverage) x3) x')
% 90.42/90.93 Found (((eq_sym0 x') x3) ((eq_ref beverage) x')) as proof of (((eq beverage) x3) x')
% 90.42/90.93 Found ((((eq_sym beverage) x') x3) ((eq_ref beverage) x')) as proof of (((eq beverage) x3) x')
% 90.42/90.93 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((((eq_sym beverage) x') x3) ((eq_ref beverage) x'))) as proof of (((eq beverage) x3) x')
% 91.73/92.20 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 91.73/92.20 Found (eq_ref0 x') as proof of (((eq beverage) x') x2)
% 91.73/92.20 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 91.73/92.20 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 91.73/92.20 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 91.73/92.20 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 91.73/92.20 Found ((eq_sym00 x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 91.73/92.20 Found (((eq_sym0 x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 91.73/92.20 Found ((((eq_sym beverage) x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 91.73/92.20 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((((eq_sym beverage) x') x2) ((eq_ref beverage) x'))) as proof of (((eq beverage) x2) x')
% 91.73/92.20 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 91.73/92.20 Found (eq_ref0 x') as proof of (((eq beverage) x') x2)
% 91.73/92.20 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 91.73/92.20 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 91.73/92.20 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 91.73/92.20 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 91.73/92.20 Found ((eq_sym00 x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 91.73/92.20 Found (((eq_sym0 x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 91.73/92.20 Found ((((eq_sym beverage) x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 91.73/92.20 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((((eq_sym beverage) x') x2) ((eq_ref beverage) x'))) as proof of (((eq beverage) x2) x')
% 91.73/92.20 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 91.73/92.20 Found (eq_ref0 x') as proof of (((eq beverage) x') x3)
% 91.73/92.20 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x3)
% 91.73/92.20 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x3)
% 91.73/92.20 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x3)
% 91.73/92.20 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x3) x')
% 91.73/92.20 Found ((eq_sym00 x3) ((eq_ref beverage) x')) as proof of (((eq beverage) x3) x')
% 91.73/92.20 Found (((eq_sym0 x') x3) ((eq_ref beverage) x')) as proof of (((eq beverage) x3) x')
% 91.73/92.20 Found ((((eq_sym beverage) x') x3) ((eq_ref beverage) x')) as proof of (((eq beverage) x3) x')
% 91.73/92.20 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((((eq_sym beverage) x') x3) ((eq_ref beverage) x'))) as proof of (((eq beverage) x3) x')
% 91.73/92.20 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 91.73/92.20 Found (eq_ref0 x') as proof of (((eq beverage) x') x3)
% 91.73/92.20 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x3)
% 91.73/92.20 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x3)
% 91.73/92.20 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x3)
% 91.73/92.20 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x3) x')
% 91.73/92.20 Found ((eq_sym00 x3) ((eq_ref beverage) x')) as proof of (((eq beverage) x3) x')
% 91.73/92.20 Found (((eq_sym0 x') x3) ((eq_ref beverage) x')) as proof of (((eq beverage) x3) x')
% 91.73/92.20 Found ((((eq_sym beverage) x') x3) ((eq_ref beverage) x')) as proof of (((eq beverage) x3) x')
% 91.73/92.20 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((((eq_sym beverage) x') x3) ((eq_ref beverage) x'))) as proof of (((eq beverage) x3) x')
% 91.73/92.20 Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% 91.73/92.20 Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 91.73/92.20 Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 92.97/93.52 Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 92.97/93.52 Found (fun (x1:(syrup->beverage))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 92.97/93.52 Found (fun (x1:(syrup->beverage))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(syrup->beverage)), (((eq Prop) (f x)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))))
% 92.97/93.52 Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% 92.97/93.52 Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 92.97/93.52 Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 92.97/93.52 Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 92.97/93.52 Found (fun (x1:(syrup->beverage))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 92.97/93.52 Found (fun (x1:(syrup->beverage))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(syrup->beverage)), (((eq Prop) (f x)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))))
% 92.97/93.52 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 92.97/93.52 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x2 S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 92.97/93.52 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x2 S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 92.97/93.52 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x2 S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 92.97/93.52 Found (fun (x2:(syrup->beverage))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x2 S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 92.97/93.52 Found (fun (x2:(syrup->beverage))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(syrup->beverage)), (((eq Prop) (f x)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))))
% 92.97/93.52 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 92.97/93.52 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x2 S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 92.97/93.52 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x2 S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 92.97/93.52 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x2 S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 92.97/93.52 Found (fun (x2:(syrup->beverage))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x2 S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 95.22/95.69 Found (fun (x2:(syrup->beverage))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(syrup->beverage)), (((eq Prop) (f x)) (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S0))) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))))
% 95.22/95.69 Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (beverage->Prop)) a) (fun (x:beverage)=> (a x)))
% 95.22/95.69 Found (eta_expansion_dep00 a) as proof of (((eq (beverage->Prop)) a) b)
% 95.22/95.69 Found ((eta_expansion_dep0 (fun (x1:beverage)=> Prop)) a) as proof of (((eq (beverage->Prop)) a) b)
% 95.22/95.69 Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) a) as proof of (((eq (beverage->Prop)) a) b)
% 95.22/95.69 Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) a) as proof of (((eq (beverage->Prop)) a) b)
% 95.22/95.69 Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) a) as proof of (((eq (beverage->Prop)) a) b)
% 95.22/95.69 Found eq_ref00:=(eq_ref0 b):(((eq (beverage->Prop)) b) b)
% 95.22/95.69 Found (eq_ref0 b) as proof of (((eq (beverage->Prop)) b) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))
% 95.22/95.69 Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))
% 95.22/95.69 Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))
% 95.22/95.69 Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))
% 95.22/95.69 Found eq_ref00:=(eq_ref0 b):(((eq (beverage->Prop)) b) b)
% 95.22/95.69 Found (eq_ref0 b) as proof of (((eq (beverage->Prop)) b) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))
% 95.22/95.69 Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))
% 95.22/95.69 Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))
% 95.22/95.69 Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))
% 95.22/95.69 Found eq_ref00:=(eq_ref0 x1):(((eq beverage) x1) x1)
% 95.22/95.69 Found (eq_ref0 x1) as proof of (((eq beverage) x1) x')
% 95.22/95.69 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) x')
% 95.22/95.69 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) x')
% 95.22/95.69 Found (fun (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x1)) as proof of (((eq beverage) x1) x')
% 95.22/95.69 Found (fun (x2:beverage) (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x1)) as proof of (((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2))->(((eq beverage) x1) x'))
% 95.22/95.69 Found (fun (x2:beverage) (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x1)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->(((eq beverage) x1) x')))
% 95.22/95.69 Found (ex_ind00 (fun (x2:beverage) (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x1))) as proof of (((eq beverage) x1) x')
% 95.22/95.69 Found ((ex_ind0 (((eq beverage) x1) x')) (fun (x2:beverage) (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x1))) as proof of (((eq beverage) x1) x')
% 95.22/95.69 Found (((fun (P:Prop) (x2:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x2) x01)) (((eq beverage) x1) x')) (fun (x2:beverage) (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x1))) as proof of (((eq beverage) x1) x')
% 95.66/96.19 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> (((fun (P:Prop) (x2:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x2) x01)) (((eq beverage) x1) x')) (fun (x2:beverage) (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x1)))) as proof of (((eq beverage) x1) x')
% 95.66/96.19 Found eq_ref00:=(eq_ref0 x1):(((eq beverage) x1) x1)
% 95.66/96.19 Found (eq_ref0 x1) as proof of (((eq beverage) x1) x')
% 95.66/96.19 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) x')
% 95.66/96.19 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) x')
% 95.66/96.19 Found (fun (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x1)) as proof of (((eq beverage) x1) x')
% 95.66/96.19 Found (fun (x2:beverage) (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x1)) as proof of (((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2))->(((eq beverage) x1) x'))
% 95.66/96.19 Found (fun (x2:beverage) (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x1)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->(((eq beverage) x1) x')))
% 95.66/96.19 Found (ex_ind00 (fun (x2:beverage) (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x1))) as proof of (((eq beverage) x1) x')
% 95.66/96.19 Found ((ex_ind0 (((eq beverage) x1) x')) (fun (x2:beverage) (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x1))) as proof of (((eq beverage) x1) x')
% 95.66/96.19 Found (((fun (P:Prop) (x2:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x2) x01)) (((eq beverage) x1) x')) (fun (x2:beverage) (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x1))) as proof of (((eq beverage) x1) x')
% 95.66/96.19 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> (((fun (P:Prop) (x2:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x2) x01)) (((eq beverage) x1) x')) (fun (x2:beverage) (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x1)))) as proof of (((eq beverage) x1) x')
% 95.66/96.19 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 95.66/96.19 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 95.66/96.19 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 95.66/96.19 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 95.66/96.19 Found (fun (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 95.66/96.19 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3))->(((eq beverage) x2) x'))
% 95.76/96.27 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->(((eq beverage) x2) x')))
% 95.76/96.27 Found (ex_ind00 (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 95.76/96.27 Found ((ex_ind0 (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 95.76/96.27 Found (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x3) x01)) (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 95.76/96.27 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x3) x01)) (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)))) as proof of (((eq beverage) x2) x')
% 95.76/96.27 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 95.76/96.27 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 95.76/96.27 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 95.76/96.27 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 95.76/96.27 Found (fun (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 95.76/96.27 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3))->(((eq beverage) x2) x'))
% 95.76/96.27 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->(((eq beverage) x2) x')))
% 95.76/96.27 Found (ex_ind00 (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 95.76/96.27 Found ((ex_ind0 (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 95.76/96.27 Found (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x3) x01)) (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 95.76/96.27 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x3) x01)) (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)))) as proof of (((eq beverage) x2) x')
% 95.97/96.47 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 95.97/96.47 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 95.97/96.47 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 95.97/96.47 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 95.97/96.47 Found (fun (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 95.97/96.47 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3))->(((eq beverage) x2) x'))
% 95.97/96.47 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->(((eq beverage) x2) x')))
% 95.97/96.47 Found (ex_ind00 (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 95.97/96.47 Found ((ex_ind0 (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 95.97/96.47 Found (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x3) x01)) (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 95.97/96.47 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x3) x01)) (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)))) as proof of (((eq beverage) x2) x')
% 95.97/96.47 Found eq_ref00:=(eq_ref0 x2):(((eq beverage) x2) x2)
% 95.97/96.47 Found (eq_ref0 x2) as proof of (((eq beverage) x2) x')
% 95.97/96.47 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 95.97/96.47 Found ((eq_ref beverage) x2) as proof of (((eq beverage) x2) x')
% 95.97/96.47 Found (fun (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (((eq beverage) x2) x')
% 95.97/96.47 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3))->(((eq beverage) x2) x'))
% 95.97/96.47 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->(((eq beverage) x2) x')))
% 95.97/96.47 Found (ex_ind00 (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 95.97/96.47 Found ((ex_ind0 (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 97.10/97.56 Found (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x3) x01)) (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2))) as proof of (((eq beverage) x2) x')
% 97.10/97.56 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x3) x01)) (((eq beverage) x2) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x2)))) as proof of (((eq beverage) x2) x')
% 97.10/97.56 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 97.10/97.56 Found (eq_ref0 x') as proof of (((eq beverage) x') x1)
% 97.10/97.56 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x1)
% 97.10/97.56 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x1)
% 97.10/97.56 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x1)
% 97.10/97.56 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x1) x')
% 97.10/97.56 Found ((eq_sym00 x1) ((eq_ref beverage) x')) as proof of (((eq beverage) x1) x')
% 97.10/97.56 Found (((eq_sym0 x') x1) ((eq_ref beverage) x')) as proof of (((eq beverage) x1) x')
% 97.10/97.56 Found ((((eq_sym beverage) x') x1) ((eq_ref beverage) x')) as proof of (((eq beverage) x1) x')
% 97.10/97.56 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((((eq_sym beverage) x') x1) ((eq_ref beverage) x'))) as proof of (((eq beverage) x1) x')
% 97.10/97.56 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 97.10/97.56 Found (eq_ref0 x') as proof of (((eq beverage) x') x1)
% 97.10/97.56 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x1)
% 97.10/97.56 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x1)
% 97.10/97.56 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x1)
% 97.10/97.56 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x1) x')
% 97.10/97.56 Found ((eq_sym00 x1) ((eq_ref beverage) x')) as proof of (((eq beverage) x1) x')
% 97.10/97.56 Found (((eq_sym0 x') x1) ((eq_ref beverage) x')) as proof of (((eq beverage) x1) x')
% 97.10/97.56 Found ((((eq_sym beverage) x') x1) ((eq_ref beverage) x')) as proof of (((eq beverage) x1) x')
% 97.10/97.56 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((((eq_sym beverage) x') x1) ((eq_ref beverage) x'))) as proof of (((eq beverage) x1) x')
% 97.10/97.56 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 97.10/97.56 Found (eq_ref0 x') as proof of (((eq beverage) x') x2)
% 97.10/97.56 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 97.10/97.56 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 97.10/97.56 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 97.10/97.56 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 97.10/97.56 Found ((eq_sym00 x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 97.10/97.56 Found (((eq_sym0 x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 97.10/97.56 Found ((((eq_sym beverage) x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 97.10/97.56 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((((eq_sym beverage) x') x2) ((eq_ref beverage) x'))) as proof of (((eq beverage) x2) x')
% 97.10/97.56 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 97.10/97.56 Found (eq_ref0 x') as proof of (((eq beverage) x') x2)
% 97.10/97.56 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 97.10/97.56 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 97.10/97.56 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 99.71/100.23 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 99.71/100.23 Found ((eq_sym00 x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 99.71/100.23 Found (((eq_sym0 x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 99.71/100.23 Found ((((eq_sym beverage) x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 99.71/100.23 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((((eq_sym beverage) x') x2) ((eq_ref beverage) x'))) as proof of (((eq beverage) x2) x')
% 99.71/100.23 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 99.71/100.23 Found (eq_ref0 x') as proof of (((eq beverage) x') x2)
% 99.71/100.23 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 99.71/100.23 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 99.71/100.23 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 99.71/100.23 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 99.71/100.23 Found ((eq_sym00 x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 99.71/100.23 Found (((eq_sym0 x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 99.71/100.23 Found ((((eq_sym beverage) x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 99.71/100.23 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((((eq_sym beverage) x') x2) ((eq_ref beverage) x'))) as proof of (((eq beverage) x2) x')
% 99.71/100.23 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 99.71/100.23 Found (eq_ref0 x') as proof of (((eq beverage) x') x2)
% 99.71/100.23 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 99.71/100.23 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 99.71/100.23 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x2)
% 99.71/100.23 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 99.71/100.23 Found ((eq_sym00 x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 99.71/100.23 Found (((eq_sym0 x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 99.71/100.23 Found ((((eq_sym beverage) x') x2) ((eq_ref beverage) x')) as proof of (((eq beverage) x2) x')
% 99.71/100.23 Found (fun (x01:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((((eq_sym beverage) x') x2) ((eq_ref beverage) x'))) as proof of (((eq beverage) x2) x')
% 99.71/100.23 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 99.71/100.23 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 99.71/100.23 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 99.71/100.23 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 99.71/100.23 Found (fun (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 99.71/100.23 Found (fun (x2:beverage) (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x0)) as proof of (((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2))->(((eq beverage) x0) x'))
% 99.71/100.23 Found (fun (x2:beverage) (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x0)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->(((eq beverage) x0) x')))
% 99.71/100.23 Found (ex_ind00 (fun (x2:beverage) (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 99.71/100.23 Found ((ex_ind0 (((eq beverage) x0) x')) (fun (x2:beverage) (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 99.71/100.23 Found (((fun (P:Prop) (x2:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x2) x00)) (((eq beverage) x0) x')) (fun (x2:beverage) (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 100.10/100.58 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> (((fun (P:Prop) (x2:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x2) x00)) (((eq beverage) x0) x')) (fun (x2:beverage) (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x0)))) as proof of (((eq beverage) x0) x')
% 100.10/100.58 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 100.10/100.58 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 100.10/100.58 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 100.10/100.58 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 100.10/100.58 Found (fun (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 100.10/100.58 Found (fun (x2:beverage) (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x0)) as proof of (((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2))->(((eq beverage) x0) x'))
% 100.10/100.58 Found (fun (x2:beverage) (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x0)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->(((eq beverage) x0) x')))
% 100.10/100.58 Found (ex_ind00 (fun (x2:beverage) (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 100.10/100.58 Found ((ex_ind0 (((eq beverage) x0) x')) (fun (x2:beverage) (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 100.10/100.58 Found (((fun (P:Prop) (x2:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x2) x00)) (((eq beverage) x0) x')) (fun (x2:beverage) (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 100.10/100.58 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> (((fun (P:Prop) (x2:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x2) x00)) (((eq beverage) x0) x')) (fun (x2:beverage) (x3:((and ((and (((eq beverage) x2) x')) (((eq beverage) x2) (coffee_mixture S)))) (hot x2)))=> ((eq_ref beverage) x0)))) as proof of (((eq beverage) x0) x')
% 100.10/100.58 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 100.10/100.58 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 100.10/100.58 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 100.10/100.58 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 100.10/100.58 Found (fun (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 100.10/100.58 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x0)) as proof of (((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3))->(((eq beverage) x0) x'))
% 100.10/100.58 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x0)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->(((eq beverage) x0) x')))
% 100.10/100.61 Found (ex_ind00 (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 100.10/100.61 Found ((ex_ind0 (((eq beverage) x0) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 100.10/100.61 Found (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x3) x00)) (((eq beverage) x0) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 100.10/100.61 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x3) x00)) (((eq beverage) x0) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x0)))) as proof of (((eq beverage) x0) x')
% 100.10/100.61 Found eq_ref00:=(eq_ref0 x0):(((eq beverage) x0) x0)
% 100.10/100.61 Found (eq_ref0 x0) as proof of (((eq beverage) x0) x')
% 100.10/100.61 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 100.10/100.61 Found ((eq_ref beverage) x0) as proof of (((eq beverage) x0) x')
% 100.10/100.61 Found (fun (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x0)) as proof of (((eq beverage) x0) x')
% 100.10/100.61 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x0)) as proof of (((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3))->(((eq beverage) x0) x'))
% 100.10/100.61 Found (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x0)) as proof of (forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->(((eq beverage) x0) x')))
% 100.10/100.61 Found (ex_ind00 (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 100.10/100.61 Found ((ex_ind0 (((eq beverage) x0) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 100.10/100.61 Found (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x3) x00)) (((eq beverage) x0) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x0))) as proof of (((eq beverage) x0) x')
% 100.10/100.61 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> (((fun (P:Prop) (x3:(forall (x:beverage), (((and ((and (((eq beverage) x) x')) (((eq beverage) x) (coffee_mixture S)))) (hot x))->P)))=> (((((ex_ind beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))) P) x3) x00)) (((eq beverage) x0) x')) (fun (x3:beverage) (x4:((and ((and (((eq beverage) x3) x')) (((eq beverage) x3) (coffee_mixture S)))) (hot x3)))=> ((eq_ref beverage) x0)))) as proof of (((eq beverage) x0) x')
% 101.33/101.84 Found eq_ref00:=(eq_ref0 (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))):(((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))
% 101.33/101.84 Found (eq_ref0 (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) b)
% 101.33/101.84 Found ((eq_ref (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) b)
% 101.33/101.84 Found ((eq_ref (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) b)
% 101.33/101.84 Found ((eq_ref (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) b)
% 101.33/101.84 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 101.33/101.84 Found (eq_ref0 x') as proof of (((eq beverage) x') x0)
% 101.33/101.84 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 101.33/101.84 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 101.33/101.84 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 101.33/101.84 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 101.33/101.84 Found ((eq_sym00 x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 101.33/101.84 Found (((eq_sym0 x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 101.33/101.84 Found ((((eq_sym beverage) x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 101.33/101.84 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((((eq_sym beverage) x') x0) ((eq_ref beverage) x'))) as proof of (((eq beverage) x0) x')
% 101.33/101.84 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 101.33/101.84 Found (eq_ref0 x') as proof of (((eq beverage) x') x0)
% 101.33/101.84 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 101.33/101.84 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 101.33/101.84 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 101.33/101.84 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 101.33/101.84 Found ((eq_sym00 x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 101.33/101.84 Found (((eq_sym0 x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 101.33/101.84 Found ((((eq_sym beverage) x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 101.33/101.84 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((((eq_sym beverage) x') x0) ((eq_ref beverage) x'))) as proof of (((eq beverage) x0) x')
% 101.33/101.84 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 101.33/101.84 Found (eq_ref0 x') as proof of (((eq beverage) x') x0)
% 101.33/101.84 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 101.33/101.84 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 101.33/101.84 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 101.33/101.84 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (forall (P:(beverage->Prop)), ((P x0)->(P x')))
% 101.33/101.84 Found ((eq_sym00 x0) ((eq_ref beverage) x')) as proof of (forall (P:(beverage->Prop)), ((P x0)->(P x')))
% 101.33/101.84 Found (((eq_sym0 x') x0) ((eq_ref beverage) x')) as proof of (forall (P:(beverage->Prop)), ((P x0)->(P x')))
% 102.06/102.52 Found ((((eq_sym beverage) x') x0) ((eq_ref beverage) x')) as proof of (forall (P:(beverage->Prop)), ((P x0)->(P x')))
% 102.06/102.52 Found ((((eq_sym beverage) x') x0) ((eq_ref beverage) x')) as proof of (forall (P:(beverage->Prop)), ((P x0)->(P x')))
% 102.06/102.52 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> ((((eq_sym beverage) x') x0) ((eq_ref beverage) x'))) as proof of (((eq beverage) x0) x')
% 102.06/102.52 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 102.06/102.52 Found (eq_ref0 x') as proof of (((eq beverage) x') x0)
% 102.06/102.52 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 102.06/102.52 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 102.06/102.52 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 102.06/102.52 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 102.06/102.52 Found ((eq_sym00 x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 102.06/102.52 Found (((eq_sym0 x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 102.06/102.52 Found ((((eq_sym beverage) x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 102.06/102.52 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((((eq_sym beverage) x') x0) ((eq_ref beverage) x'))) as proof of (((eq beverage) x0) x')
% 102.06/102.52 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 102.06/102.52 Found (eq_ref0 x') as proof of (((eq beverage) x') x0)
% 102.06/102.52 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 102.06/102.52 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 102.06/102.52 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 102.06/102.52 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 102.06/102.52 Found ((eq_sym00 x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 102.06/102.52 Found (((eq_sym0 x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 102.06/102.52 Found ((((eq_sym beverage) x') x0) ((eq_ref beverage) x')) as proof of (((eq beverage) x0) x')
% 102.06/102.52 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))=> ((((eq_sym beverage) x') x0) ((eq_ref beverage) x'))) as proof of (((eq beverage) x0) x')
% 102.06/102.52 Found eq_ref00:=(eq_ref0 x'):(((eq beverage) x') x')
% 102.06/102.52 Found (eq_ref0 x') as proof of (((eq beverage) x') x0)
% 102.06/102.52 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 102.06/102.52 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 102.06/102.52 Found ((eq_ref beverage) x') as proof of (((eq beverage) x') x0)
% 102.06/102.52 Found (eq_sym000 ((eq_ref beverage) x')) as proof of (forall (P:(beverage->Prop)), ((P x0)->(P x')))
% 102.06/102.52 Found ((eq_sym00 x0) ((eq_ref beverage) x')) as proof of (forall (P:(beverage->Prop)), ((P x0)->(P x')))
% 102.06/102.52 Found (((eq_sym0 x') x0) ((eq_ref beverage) x')) as proof of (forall (P:(beverage->Prop)), ((P x0)->(P x')))
% 102.06/102.52 Found ((((eq_sym beverage) x') x0) ((eq_ref beverage) x')) as proof of (forall (P:(beverage->Prop)), ((P x0)->(P x')))
% 102.06/102.52 Found ((((eq_sym beverage) x') x0) ((eq_ref beverage) x')) as proof of (forall (P:(beverage->Prop)), ((P x0)->(P x')))
% 102.06/102.52 Found (fun (x00:((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x')) (((eq beverage) B) coffee))) (hot B)))))=> ((((eq_sym beverage) x') x0) ((eq_ref beverage) x'))) as proof of (((eq beverage) x0) x')
% 102.06/102.52 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) coffee))) (hot B)))):(((eq (beverage->Prop)) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) coffee))) (hot B)))) (fun (x:beverage)=> ((and ((and (((eq beverage) x) x0)) (((eq beverage) x) coffee))) (hot x))))
% 102.06/102.52 Found (eta_expansion_dep00 (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) coffee))) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) coffee))) (hot B)))) b)
% 102.06/102.52 Found ((eta_expansion_dep0 (fun (x2:beverage)=> Prop)) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) coffee))) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) coffee))) (hot B)))) b)
% 104.08/104.54 Found (((eta_expansion_dep beverage) (fun (x2:beverage)=> Prop)) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) coffee))) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) coffee))) (hot B)))) b)
% 104.08/104.54 Found (((eta_expansion_dep beverage) (fun (x2:beverage)=> Prop)) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) coffee))) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) coffee))) (hot B)))) b)
% 104.08/104.54 Found (((eta_expansion_dep beverage) (fun (x2:beverage)=> Prop)) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) coffee))) (hot B)))) as proof of (((eq (beverage->Prop)) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) coffee))) (hot B)))) b)
% 104.08/104.54 Found eq_ref00:=(eq_ref0 (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))):(((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 104.08/104.54 Found (eq_ref0 (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 104.08/104.54 Found ((eq_ref (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 104.08/104.54 Found ((eq_ref (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 104.08/104.54 Found ((eq_ref (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 104.08/104.54 Found eta_expansion000:=(eta_expansion00 (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))):(((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) (fun (x:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 104.08/104.54 Found (eta_expansion00 (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 104.08/104.54 Found ((eta_expansion0 Prop) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 111.92/112.42 Found (((eta_expansion beverage) Prop) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 111.92/112.42 Found (((eta_expansion beverage) Prop) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 111.92/112.42 Found (((eta_expansion beverage) Prop) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 111.92/112.42 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))):(((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) (fun (x:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) coffee))) (hot B))))))
% 111.92/112.42 Found (eta_expansion_dep00 (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) b)
% 111.92/112.42 Found ((eta_expansion_dep0 (fun (x3:beverage)=> Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) b)
% 111.92/112.42 Found (((eta_expansion_dep beverage) (fun (x3:beverage)=> Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) b)
% 111.92/112.42 Found (((eta_expansion_dep beverage) (fun (x3:beverage)=> Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) b)
% 111.92/112.42 Found (((eta_expansion_dep beverage) (fun (x3:beverage)=> Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) b)
% 111.92/112.42 Found x0:(hot (coffee_mixture S))
% 111.92/112.42 Instantiate: x2:=(coffee_mixture S):beverage
% 111.92/112.42 Found x0 as proof of (hot x2)
% 111.92/112.42 Found eq_ref00:=(eq_ref0 (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))):(((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 111.92/112.42 Found (eq_ref0 (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 112.64/113.10 Found ((eq_ref (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 112.64/113.10 Found ((eq_ref (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 112.64/113.10 Found ((eq_ref (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 112.64/113.10 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 112.64/113.10 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) coffee))) (hot B)))))
% 112.64/113.10 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) coffee))) (hot B)))))
% 112.64/113.10 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) coffee))) (hot B)))))
% 112.64/113.10 Found (fun (x2:beverage)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) coffee))) (hot B)))))
% 112.64/113.10 Found (fun (x2:beverage)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:beverage), (((eq Prop) (f x)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) coffee))) (hot B))))))
% 112.64/113.10 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 112.64/113.10 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) coffee))) (hot B)))))
% 112.64/113.10 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) coffee))) (hot B)))))
% 112.64/113.10 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) coffee))) (hot B)))))
% 112.64/113.10 Found (fun (x2:beverage)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) coffee))) (hot B)))))
% 112.64/113.10 Found (fun (x2:beverage)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:beverage), (((eq Prop) (f x)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) coffee))) (hot B))))))
% 112.64/113.10 Found eq_ref00:=(eq_ref0 (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))):(((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 112.64/113.10 Found (eq_ref0 (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 114.23/114.70 Found ((eq_ref (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 114.23/114.70 Found ((eq_ref (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 114.23/114.70 Found ((eq_ref (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 114.23/114.70 Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% 114.23/114.70 Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and (((eq beverage) x1) x0)) (((eq beverage) x1) coffee))) (hot x1)))
% 114.23/114.70 Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and (((eq beverage) x1) x0)) (((eq beverage) x1) coffee))) (hot x1)))
% 114.23/114.70 Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and (((eq beverage) x1) x0)) (((eq beverage) x1) coffee))) (hot x1)))
% 114.23/114.70 Found (fun (x1:beverage)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((and (((eq beverage) x1) x0)) (((eq beverage) x1) coffee))) (hot x1)))
% 114.23/114.70 Found (fun (x1:beverage)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:beverage), (((eq Prop) (f x)) ((and ((and (((eq beverage) x) x0)) (((eq beverage) x) coffee))) (hot x))))
% 114.23/114.70 Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% 114.23/114.70 Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and (((eq beverage) x1) x0)) (((eq beverage) x1) coffee))) (hot x1)))
% 114.23/114.70 Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and (((eq beverage) x1) x0)) (((eq beverage) x1) coffee))) (hot x1)))
% 114.23/114.70 Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and (((eq beverage) x1) x0)) (((eq beverage) x1) coffee))) (hot x1)))
% 114.23/114.70 Found (fun (x1:beverage)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((and (((eq beverage) x1) x0)) (((eq beverage) x1) coffee))) (hot x1)))
% 114.23/114.70 Found (fun (x1:beverage)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:beverage), (((eq Prop) (f x)) ((and ((and (((eq beverage) x) x0)) (((eq beverage) x) coffee))) (hot x))))
% 114.23/114.70 Found eta_expansion000:=(eta_expansion00 (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))):(((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) (fun (x:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 114.23/114.70 Found (eta_expansion00 (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 114.23/114.70 Found ((eta_expansion0 Prop) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 114.23/114.70 Found (((eta_expansion beverage) Prop) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 116.07/116.52 Found (((eta_expansion beverage) Prop) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 116.07/116.52 Found (((eta_expansion beverage) Prop) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 116.07/116.52 Found eta_expansion000:=(eta_expansion00 (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))):(((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) (fun (x:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 116.07/116.52 Found (eta_expansion00 (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 116.07/116.52 Found ((eta_expansion0 Prop) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 116.07/116.52 Found (((eta_expansion beverage) Prop) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 116.07/116.52 Found (((eta_expansion beverage) Prop) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 116.07/116.52 Found (((eta_expansion beverage) Prop) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 116.07/116.52 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 116.07/116.52 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 116.07/116.52 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 116.07/116.52 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 116.07/116.52 Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 116.07/116.52 Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:beverage), (((eq Prop) (f x)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 120.86/121.34 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 120.86/121.34 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 120.86/121.34 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 120.86/121.34 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 120.86/121.34 Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 120.86/121.34 Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:beverage), (((eq Prop) (f x)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 120.86/121.34 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 120.86/121.34 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 120.86/121.34 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 120.86/121.34 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 120.86/121.34 Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 120.86/121.34 Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:beverage), (((eq Prop) (f x)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 120.86/121.34 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 120.86/121.34 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 120.86/121.34 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 120.86/121.34 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 120.86/121.34 Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x0)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 120.86/121.34 Found (fun (x0:beverage)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:beverage), (((eq Prop) (f x)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 120.86/121.34 Found x00:(hot (coffee_mixture S0))
% 120.86/121.34 Instantiate: x01:=(coffee_mixture S0):beverage
% 120.86/121.34 Found x00 as proof of (hot x01)
% 120.86/121.34 Found x10:=(x1 hot):((hot (coffee_mixture S0))->(hot coffee))
% 120.86/121.34 Found (x1 hot) as proof of ((hot (coffee_mixture S0))->(hot x0))
% 120.86/121.34 Found (fun (x1:(((eq beverage) (coffee_mixture S0)) coffee))=> (x1 hot)) as proof of ((hot (coffee_mixture S0))->(hot x0))
% 120.86/121.34 Found (fun (x1:(((eq beverage) (coffee_mixture S0)) coffee))=> (x1 hot)) as proof of ((((eq beverage) (coffee_mixture S0)) coffee)->((hot (coffee_mixture S0))->(hot x0)))
% 120.86/121.34 Found (and_rect00 (fun (x1:(((eq beverage) (coffee_mixture S0)) coffee))=> (x1 hot))) as proof of (hot x0)
% 120.86/121.34 Found ((and_rect0 (hot x0)) (fun (x1:(((eq beverage) (coffee_mixture S0)) coffee))=> (x1 hot))) as proof of (hot x0)
% 120.86/121.34 Found (((fun (P:Type) (x1:((((eq beverage) (coffee_mixture S0)) coffee)->((hot (coffee_mixture S0))->P)))=> (((((and_rect (((eq beverage) (coffee_mixture S0)) coffee)) (hot (coffee_mixture S0))) P) x1) coffee_and_syrup_is_hot_coffee0)) (hot x0)) (fun (x1:(((eq beverage) (coffee_mixture S0)) coffee))=> (x1 hot))) as proof of (hot x0)
% 121.60/122.09 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 121.60/122.09 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) coffee))) (hot B)))))
% 121.60/122.09 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) coffee))) (hot B)))))
% 121.60/122.09 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) coffee))) (hot B)))))
% 121.60/122.09 Found (fun (x2:beverage)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) coffee))) (hot B)))))
% 121.60/122.09 Found (fun (x2:beverage)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:beverage), (((eq Prop) (f x)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) coffee))) (hot B))))))
% 121.60/122.09 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 121.60/122.09 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) coffee))) (hot B)))))
% 121.60/122.09 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) coffee))) (hot B)))))
% 121.60/122.09 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) coffee))) (hot B)))))
% 121.60/122.09 Found (fun (x2:beverage)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) coffee))) (hot B)))))
% 121.60/122.09 Found (fun (x2:beverage)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:beverage), (((eq Prop) (f x)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) coffee))) (hot B))))))
% 121.60/122.09 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))):(((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) (fun (x:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 121.60/122.09 Found (eta_expansion_dep00 (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 121.60/122.09 Found ((eta_expansion_dep0 (fun (x2:beverage)=> Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 121.60/122.09 Found (((eta_expansion_dep beverage) (fun (x2:beverage)=> Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 121.60/122.09 Found (((eta_expansion_dep beverage) (fun (x2:beverage)=> Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 123.85/124.33 Found (((eta_expansion_dep beverage) (fun (x2:beverage)=> Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 123.85/124.33 Found x00:(hot (coffee_mixture S))
% 123.85/124.33 Instantiate: x2:=(coffee_mixture S):beverage
% 123.85/124.33 Found x00 as proof of (hot x2)
% 123.85/124.33 Found x2:(hot (coffee_mixture S0))
% 123.85/124.33 Instantiate: x0:=(coffee_mixture S0):beverage
% 123.85/124.33 Found (fun (x2:(hot (coffee_mixture S0)))=> x2) as proof of (hot x0)
% 123.85/124.33 Found (fun (x1:(((eq beverage) (coffee_mixture S0)) coffee)) (x2:(hot (coffee_mixture S0)))=> x2) as proof of ((hot (coffee_mixture S0))->(hot x0))
% 123.85/124.33 Found (fun (x1:(((eq beverage) (coffee_mixture S0)) coffee)) (x2:(hot (coffee_mixture S0)))=> x2) as proof of ((((eq beverage) (coffee_mixture S0)) coffee)->((hot (coffee_mixture S0))->(hot x0)))
% 123.85/124.33 Found (and_rect00 (fun (x1:(((eq beverage) (coffee_mixture S0)) coffee)) (x2:(hot (coffee_mixture S0)))=> x2)) as proof of (hot x0)
% 123.85/124.33 Found ((and_rect0 (hot x0)) (fun (x1:(((eq beverage) (coffee_mixture S0)) coffee)) (x2:(hot (coffee_mixture S0)))=> x2)) as proof of (hot x0)
% 123.85/124.33 Found (((fun (P:Type) (x1:((((eq beverage) (coffee_mixture S0)) coffee)->((hot (coffee_mixture S0))->P)))=> (((((and_rect (((eq beverage) (coffee_mixture S0)) coffee)) (hot (coffee_mixture S0))) P) x1) coffee_and_syrup_is_hot_coffee0)) (hot x0)) (fun (x1:(((eq beverage) (coffee_mixture S0)) coffee)) (x2:(hot (coffee_mixture S0)))=> x2)) as proof of (hot x0)
% 123.85/124.33 Found eq_ref00:=(eq_ref0 (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))):(((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 123.85/124.33 Found (eq_ref0 (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 123.85/124.33 Found ((eq_ref (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 123.85/124.33 Found ((eq_ref (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 123.85/124.33 Found ((eq_ref (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) as proof of (((eq (beverage->Prop)) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) b)
% 123.85/124.33 Found x0:(hot (coffee_mixture S))
% 123.85/124.33 Instantiate: x3:=(coffee_mixture S):beverage
% 123.85/124.33 Found x0 as proof of (hot x3)
% 123.85/124.33 Found x0:(hot (coffee_mixture S))
% 123.85/124.33 Instantiate: x3:=(coffee_mixture S):beverage
% 123.85/124.33 Found x0 as proof of (hot x3)
% 123.85/124.33 Found x0:(hot (coffee_mixture S))
% 123.85/124.33 Instantiate: x2:=(coffee_mixture S):beverage
% 123.85/124.33 Found x0 as proof of (hot x2)
% 123.85/124.33 Found x0:(hot (coffee_mixture S))
% 126.16/126.62 Instantiate: x3:=(coffee_mixture S):beverage
% 126.16/126.62 Found x0 as proof of (hot x3)
% 126.16/126.62 Found x10:(hot (coffee_mixture S0))
% 126.16/126.62 Instantiate: x0:=(coffee_mixture S0):beverage
% 126.16/126.62 Found x10 as proof of (hot x0)
% 126.16/126.62 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 126.16/126.62 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 126.16/126.62 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 126.16/126.62 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 126.16/126.62 Found (fun (x2:beverage)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 126.16/126.62 Found (fun (x2:beverage)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:beverage), (((eq Prop) (f x)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 126.16/126.62 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 126.16/126.62 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 126.16/126.62 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 126.16/126.62 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 126.16/126.62 Found (fun (x2:beverage)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 126.16/126.62 Found (fun (x2:beverage)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:beverage), (((eq Prop) (f x)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 126.16/126.62 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 126.16/126.62 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 126.16/126.62 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 126.16/126.62 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 126.16/126.62 Found (fun (x2:beverage)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 126.16/126.62 Found (fun (x2:beverage)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:beverage), (((eq Prop) (f x)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 126.16/126.62 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 126.16/126.62 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 126.16/126.62 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 126.16/126.62 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 126.16/126.62 Found (fun (x2:beverage)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 128.14/128.60 Found (fun (x2:beverage)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:beverage), (((eq Prop) (f x)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 128.14/128.60 Found eq_ref00:=(eq_ref0 a):(((eq (beverage->Prop)) a) a)
% 128.14/128.60 Found (eq_ref0 a) as proof of (((eq (beverage->Prop)) a) b)
% 128.14/128.60 Found ((eq_ref (beverage->Prop)) a) as proof of (((eq (beverage->Prop)) a) b)
% 128.14/128.60 Found ((eq_ref (beverage->Prop)) a) as proof of (((eq (beverage->Prop)) a) b)
% 128.14/128.60 Found ((eq_ref (beverage->Prop)) a) as proof of (((eq (beverage->Prop)) a) b)
% 128.14/128.60 Found eta_expansion000:=(eta_expansion00 b):(((eq (beverage->Prop)) b) (fun (x:beverage)=> (b x)))
% 128.14/128.60 Found (eta_expansion00 b) as proof of (((eq (beverage->Prop)) b) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))))
% 128.14/128.60 Found ((eta_expansion0 Prop) b) as proof of (((eq (beverage->Prop)) b) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))))
% 128.14/128.60 Found (((eta_expansion beverage) Prop) b) as proof of (((eq (beverage->Prop)) b) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))))
% 128.14/128.60 Found (((eta_expansion beverage) Prop) b) as proof of (((eq (beverage->Prop)) b) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))))
% 128.14/128.60 Found (((eta_expansion beverage) Prop) b) as proof of (((eq (beverage->Prop)) b) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))))
% 128.14/128.60 Found eq_ref00:=(eq_ref0 a):(((eq (beverage->Prop)) a) a)
% 128.14/128.60 Found (eq_ref0 a) as proof of (((eq (beverage->Prop)) a) b)
% 128.14/128.60 Found ((eq_ref (beverage->Prop)) a) as proof of (((eq (beverage->Prop)) a) b)
% 128.14/128.60 Found ((eq_ref (beverage->Prop)) a) as proof of (((eq (beverage->Prop)) a) b)
% 128.14/128.60 Found ((eq_ref (beverage->Prop)) a) as proof of (((eq (beverage->Prop)) a) b)
% 128.14/128.60 Found eq_ref00:=(eq_ref0 b):(((eq (beverage->Prop)) b) b)
% 128.14/128.60 Found (eq_ref0 b) as proof of (((eq (beverage->Prop)) b) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))))
% 128.14/128.60 Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))))
% 128.14/128.60 Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))))
% 128.14/128.60 Found ((eq_ref (beverage->Prop)) b) as proof of (((eq (beverage->Prop)) b) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))))
% 128.14/128.60 Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% 128.14/128.60 Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x3)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 128.14/128.60 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x3)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 128.14/128.60 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x3)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 128.14/128.60 Found (fun (x3:beverage)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x3)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 128.14/128.60 Found (fun (x3:beverage)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:beverage), (((eq Prop) (f x)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 136.56/137.08 Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% 136.56/137.08 Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x3)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 136.56/137.08 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x3)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 136.56/137.08 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x3)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 136.56/137.08 Found (fun (x3:beverage)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x3)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 136.56/137.08 Found (fun (x3:beverage)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:beverage), (((eq Prop) (f x)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 136.56/137.08 Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% 136.56/137.08 Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x3)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 136.56/137.08 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x3)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 136.56/137.08 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x3)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 136.56/137.08 Found (fun (x3:beverage)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x3)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 136.56/137.08 Found (fun (x3:beverage)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:beverage), (((eq Prop) (f x)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 136.56/137.08 Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% 136.56/137.08 Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x3)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 136.56/137.08 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x3)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 136.56/137.08 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x3)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 136.56/137.08 Found (fun (x3:beverage)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x3)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 136.56/137.08 Found (fun (x3:beverage)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:beverage), (((eq Prop) (f x)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 136.56/137.08 Found x000:(hot (coffee_mixture S0))
% 136.56/137.08 Instantiate: x01:=(coffee_mixture S0):beverage
% 136.56/137.08 Found x000 as proof of (hot x01)
% 136.56/137.08 Found x00:(hot (coffee_mixture S0))
% 136.56/137.08 Instantiate: x01:=(coffee_mixture S0):beverage
% 136.56/137.08 Found x00 as proof of (hot x01)
% 136.56/137.08 Found x00:(hot (coffee_mixture S0))
% 136.56/137.08 Instantiate: x02:=(coffee_mixture S0):beverage
% 136.56/137.08 Found x00 as proof of (hot x02)
% 136.56/137.08 Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% 136.56/137.08 Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x1)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 136.56/137.08 Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x1)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 136.56/137.08 Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x1)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 139.74/140.23 Found (fun (x1:beverage)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x1)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 139.74/140.23 Found (fun (x1:beverage)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:beverage), (((eq Prop) (f x)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 139.74/140.23 Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% 139.74/140.23 Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x1)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 139.74/140.23 Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x1)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 139.74/140.23 Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x1)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 139.74/140.23 Found (fun (x1:beverage)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x1)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 139.74/140.23 Found (fun (x1:beverage)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:beverage), (((eq Prop) (f x)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 139.74/140.23 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 139.74/140.23 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 139.74/140.23 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 139.74/140.23 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 139.74/140.23 Found (fun (x2:beverage)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 139.74/140.23 Found (fun (x2:beverage)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:beverage), (((eq Prop) (f x)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 139.74/140.23 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 139.74/140.23 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 139.74/140.23 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 139.74/140.23 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 139.74/140.23 Found (fun (x2:beverage)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x2)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))
% 139.74/140.23 Found (fun (x2:beverage)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:beverage), (((eq Prop) (f x)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) x)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))
% 139.74/140.23 Found eta_expansion000:=(eta_expansion00 ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))):(((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) (fun (x:beverage)=> (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) x)))
% 139.98/140.44 Found (eta_expansion00 ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 139.98/140.44 Found ((eta_expansion0 Prop) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 139.98/140.44 Found (((eta_expansion beverage) Prop) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 139.98/140.44 Found (((eta_expansion beverage) Prop) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 139.98/140.44 Found (((eta_expansion beverage) Prop) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 139.98/140.44 Found eta_expansion000:=(eta_expansion00 ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))):(((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) (fun (x:beverage)=> (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) x)))
% 139.98/140.44 Found (eta_expansion00 ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 139.98/140.44 Found ((eta_expansion0 Prop) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 139.98/140.44 Found (((eta_expansion beverage) Prop) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 139.98/140.44 Found (((eta_expansion beverage) Prop) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 139.98/140.44 Found (((eta_expansion beverage) Prop) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 142.21/142.66 Found x100:(hot (coffee_mixture S0))
% 142.21/142.66 Instantiate: x0:=(coffee_mixture S0):beverage
% 142.21/142.66 Found x100 as proof of (hot x0)
% 142.21/142.66 Found x:(((eq beverage) (coffee_mixture S)) coffee)
% 142.21/142.66 Instantiate: x2:=(coffee_mixture S):beverage
% 142.21/142.66 Found x as proof of (((eq beverage) x2) coffee)
% 142.21/142.66 Found x:(((eq beverage) (coffee_mixture S)) coffee)
% 142.21/142.66 Instantiate: x2:=(coffee_mixture S):beverage;x1:=(fun (x5:syrup)=> coffee):(syrup->beverage)
% 142.21/142.66 Found x as proof of (((eq beverage) x2) (x1 S0))
% 142.21/142.66 Found ((conj10 x) x) as proof of ((and (((eq beverage) x2) (x1 S0))) (((eq beverage) x2) coffee))
% 142.21/142.66 Found (((conj1 (((eq beverage) x2) coffee)) x) x) as proof of ((and (((eq beverage) x2) (x1 S0))) (((eq beverage) x2) coffee))
% 142.21/142.66 Found ((((conj (((eq beverage) x2) (x1 S0))) (((eq beverage) x2) coffee)) x) x) as proof of ((and (((eq beverage) x2) (x1 S0))) (((eq beverage) x2) coffee))
% 142.21/142.66 Found ((((conj (((eq beverage) x2) (x1 S0))) (((eq beverage) x2) coffee)) x) x) as proof of ((and (((eq beverage) x2) (x1 S0))) (((eq beverage) x2) coffee))
% 142.21/142.66 Found ((conj00 ((((conj (((eq beverage) x2) (x1 S0))) (((eq beverage) x2) coffee)) x) x)) x0) as proof of ((and ((and (((eq beverage) x2) (x1 S0))) (((eq beverage) x2) coffee))) (hot x2))
% 142.21/142.66 Found (((conj0 (hot x2)) ((((conj (((eq beverage) x2) (x1 S0))) (((eq beverage) x2) coffee)) x) x)) x0) as proof of ((and ((and (((eq beverage) x2) (x1 S0))) (((eq beverage) x2) coffee))) (hot x2))
% 142.21/142.66 Found ((((conj ((and (((eq beverage) x2) (x1 S0))) (((eq beverage) x2) coffee))) (hot x2)) ((((conj (((eq beverage) x2) (x1 S0))) (((eq beverage) x2) coffee)) x) x)) x0) as proof of ((and ((and (((eq beverage) x2) (x1 S0))) (((eq beverage) x2) coffee))) (hot x2))
% 142.21/142.66 Found ((((conj ((and (((eq beverage) x2) (x1 S0))) (((eq beverage) x2) coffee))) (hot x2)) ((((conj (((eq beverage) x2) (x1 S0))) (((eq beverage) x2) coffee)) x) x)) x0) as proof of ((and ((and (((eq beverage) x2) (x1 S0))) (((eq beverage) x2) coffee))) (hot x2))
% 142.21/142.66 Found (ex_intro100 ((((conj ((and (((eq beverage) x2) (x1 S0))) (((eq beverage) x2) coffee))) (hot x2)) ((((conj (((eq beverage) x2) (x1 S0))) (((eq beverage) x2) coffee)) x) x)) x0)) as proof of ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S0))) (((eq beverage) B) coffee))) (hot B))))
% 142.21/142.66 Found ((ex_intro10 (coffee_mixture S)) ((((conj ((and (((eq beverage) (coffee_mixture S)) (x1 S0))) (((eq beverage) (coffee_mixture S)) coffee))) (hot (coffee_mixture S))) ((((conj (((eq beverage) (coffee_mixture S)) (x1 S0))) (((eq beverage) (coffee_mixture S)) coffee)) x) x)) x0)) as proof of ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S0))) (((eq beverage) B) coffee))) (hot B))))
% 142.21/142.66 Found (((ex_intro1 (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S0))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S)) ((((conj ((and (((eq beverage) (coffee_mixture S)) (x1 S0))) (((eq beverage) (coffee_mixture S)) coffee))) (hot (coffee_mixture S))) ((((conj (((eq beverage) (coffee_mixture S)) (x1 S0))) (((eq beverage) (coffee_mixture S)) coffee)) x) x)) x0)) as proof of ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S0))) (((eq beverage) B) coffee))) (hot B))))
% 142.21/142.66 Found ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S0))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S)) ((((conj ((and (((eq beverage) (coffee_mixture S)) (x1 S0))) (((eq beverage) (coffee_mixture S)) coffee))) (hot (coffee_mixture S))) ((((conj (((eq beverage) (coffee_mixture S)) (x1 S0))) (((eq beverage) (coffee_mixture S)) coffee)) x) x)) x0)) as proof of ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S0))) (((eq beverage) B) coffee))) (hot B))))
% 142.21/142.66 Found (fun (S0:syrup)=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S0))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S)) ((((conj ((and (((eq beverage) (coffee_mixture S)) (x1 S0))) (((eq beverage) (coffee_mixture S)) coffee))) (hot (coffee_mixture S))) ((((conj (((eq beverage) (coffee_mixture S)) (x1 S0))) (((eq beverage) (coffee_mixture S)) coffee)) x) x)) x0))) as proof of ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S0))) (((eq beverage) B) coffee))) (hot B))))
% 142.45/142.88 Found (fun (S0:syrup)=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S0))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S)) ((((conj ((and (((eq beverage) (coffee_mixture S)) (x1 S0))) (((eq beverage) (coffee_mixture S)) coffee))) (hot (coffee_mixture S))) ((((conj (((eq beverage) (coffee_mixture S)) (x1 S0))) (((eq beverage) (coffee_mixture S)) coffee)) x) x)) x0))) as proof of (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S))) (((eq beverage) B) coffee))) (hot B)))))
% 142.45/142.88 Found (ex_intro000 (fun (S0:syrup)=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x1 S0))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S)) ((((conj ((and (((eq beverage) (coffee_mixture S)) (x1 S0))) (((eq beverage) (coffee_mixture S)) coffee))) (hot (coffee_mixture S))) ((((conj (((eq beverage) (coffee_mixture S)) (x1 S0))) (((eq beverage) (coffee_mixture S)) coffee)) x) x)) x0)))) as proof of ((ex (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B)))))))
% 142.45/142.88 Found ((ex_intro00 (fun (x5:syrup)=> coffee)) (fun (S0:syrup)=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S)) ((((conj ((and (((eq beverage) (coffee_mixture S)) ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) (coffee_mixture S)) coffee))) (hot (coffee_mixture S))) ((((conj (((eq beverage) (coffee_mixture S)) ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) (coffee_mixture S)) coffee)) x) x)) x0)))) as proof of ((ex (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B)))))))
% 142.45/142.88 Found (((ex_intro0 (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) (fun (x5:syrup)=> coffee)) (fun (S0:syrup)=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S)) ((((conj ((and (((eq beverage) (coffee_mixture S)) ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) (coffee_mixture S)) coffee))) (hot (coffee_mixture S))) ((((conj (((eq beverage) (coffee_mixture S)) ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) (coffee_mixture S)) coffee)) x) x)) x0)))) as proof of ((ex (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B)))))))
% 142.45/142.88 Found ((((ex_intro (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) (fun (x5:syrup)=> coffee)) (fun (S0:syrup)=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S)) ((((conj ((and (((eq beverage) (coffee_mixture S)) ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) (coffee_mixture S)) coffee))) (hot (coffee_mixture S))) ((((conj (((eq beverage) (coffee_mixture S)) ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) (coffee_mixture S)) coffee)) x) x)) x0)))) as proof of ((ex (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B)))))))
% 142.75/143.21 Found (fun (x0:(hot (coffee_mixture S)))=> ((((ex_intro (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) (fun (x5:syrup)=> coffee)) (fun (S0:syrup)=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S)) ((((conj ((and (((eq beverage) (coffee_mixture S)) ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) (coffee_mixture S)) coffee))) (hot (coffee_mixture S))) ((((conj (((eq beverage) (coffee_mixture S)) ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) (coffee_mixture S)) coffee)) x) x)) x0))))) as proof of ((ex (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B)))))))
% 142.75/143.21 Found (fun (x:(((eq beverage) (coffee_mixture S)) coffee)) (x0:(hot (coffee_mixture S)))=> ((((ex_intro (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) (fun (x5:syrup)=> coffee)) (fun (S0:syrup)=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S)) ((((conj ((and (((eq beverage) (coffee_mixture S)) ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) (coffee_mixture S)) coffee))) (hot (coffee_mixture S))) ((((conj (((eq beverage) (coffee_mixture S)) ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) (coffee_mixture S)) coffee)) x) x)) x0))))) as proof of ((hot (coffee_mixture S))->((ex (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B)
% (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))))
% 142.75/143.21 Found (fun (x:(((eq beverage) (coffee_mixture S)) coffee)) (x0:(hot (coffee_mixture S)))=> ((((ex_intro (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) (fun (x5:syrup)=> coffee)) (fun (S0:syrup)=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S)) ((((conj ((and (((eq beverage) (coffee_mixture S)) ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) (coffee_mixture S)) coffee))) (hot (coffee_mixture S))) ((((conj (((eq beverage) (coffee_mixture S)) ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) (coffee_mixture S)) coffee)) x) x)) x0))))) as proof of ((((eq beverage) (coffee_mixture S)) coffee)->((hot (coffee_mixture S))->((ex (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun
% (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B)))))))))
% 142.75/143.21 Found (and_rect00 (fun (x:(((eq beverage) (coffee_mixture S)) coffee)) (x0:(hot (coffee_mixture S)))=> ((((ex_intro (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) (fun (x5:syrup)=> coffee)) (fun (S0:syrup)=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S)) ((((conj ((and (((eq beverage) (coffee_mixture S)) ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) (coffee_mixture S)) coffee))) (hot (coffee_mixture S))) ((((conj (((eq beverage) (coffee_mixture S)) ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) (coffee_mixture S)) coffee)) x) x)) x0)))))) as proof of ((ex (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq
% beverage) B) coffee))) (hot B)))))))
% 144.46/144.91 Found ((and_rect0 ((ex (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B)))))))) (fun (x:(((eq beverage) (coffee_mixture S)) coffee)) (x0:(hot (coffee_mixture S)))=> ((((ex_intro (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) (fun (x5:syrup)=> coffee)) (fun (S0:syrup)=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S)) ((((conj ((and (((eq beverage) (coffee_mixture S)) ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) (coffee_mixture S)) coffee))) (hot (coffee_mixture S))) ((((conj (((eq beverage) (coffee_mixture S)) ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) (coffee_mixture S))
% coffee)) x) x)) x0)))))) as proof of ((ex (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B)))))))
% 144.46/144.91 Found (((fun (P:Type) (x:((((eq beverage) (coffee_mixture S)) coffee)->((hot (coffee_mixture S))->P)))=> (((((and_rect (((eq beverage) (coffee_mixture S)) coffee)) (hot (coffee_mixture S))) P) x) coffee_and_syrup_is_hot_coffee0)) ((ex (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) coffee))) (hot B)))))))) (fun (x:(((eq beverage) (coffee_mixture S)) coffee)) (x0:(hot (coffee_mixture S)))=> ((((ex_intro (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S0:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S0))) (((eq beverage) B) coffee))) (hot B))))))) (fun (x5:syrup)=> coffee)) (fun (S0:syrup)=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S)) ((((conj ((and (((eq beverage) (coffee_mixture S))
% ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) (coffee_mixture S)) coffee))) (hot (coffee_mixture S))) ((((conj (((eq beverage) (coffee_mixture S)) ((fun (x5:syrup)=> coffee) S0))) (((eq beverage) (coffee_mixture S)) coffee)) x) x)) x0)))))) as proof of ((ex (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B)))))))
% 144.46/144.91 Found x10:(hot (coffee_mixture S0))
% 144.46/144.91 Instantiate: x0:=(coffee_mixture S0):beverage
% 144.46/144.91 Found x10 as proof of (hot x0)
% 144.46/144.91 Found eq_ref00:=(eq_ref0 ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))):(((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))))
% 144.46/144.91 Found (eq_ref0 ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 144.46/144.91 Found ((eq_ref (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 144.84/145.27 Found ((eq_ref (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 144.84/145.27 Found ((eq_ref (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 144.84/145.27 Found eta_expansion_dep000:=(eta_expansion_dep00 ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))):(((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) (fun (x:beverage)=> (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) x)))
% 144.84/145.27 Found (eta_expansion_dep00 ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 144.84/145.27 Found ((eta_expansion_dep0 (fun (x1:beverage)=> Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 144.84/145.27 Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 144.84/145.27 Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 144.84/145.27 Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 144.84/145.27 Found eta_expansion_dep000:=(eta_expansion_dep00 ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))):(((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) (fun (x:beverage)=> (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))) x)))
% 144.90/145.40 Found (eta_expansion_dep00 ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 144.90/145.40 Found ((eta_expansion_dep0 (fun (x1:beverage)=> Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 144.90/145.40 Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 144.90/145.40 Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 144.90/145.40 Found (((eta_expansion_dep beverage) (fun (x1:beverage)=> Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 144.90/145.40 Found eq_ref00:=(eq_ref0 ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))):(((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B)))))))
% 144.90/145.40 Found (eq_ref0 ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 144.90/145.40 Found ((eq_ref (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 144.90/145.40 Found ((eq_ref (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 148.07/148.50 Found ((eq_ref (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) (coffee_mixture S)))) (hot B))))))) b)
% 148.07/148.50 Found eta_expansion_dep000:=(eta_expansion_dep00 ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))):(((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) (fun (x:beverage)=> (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) x)))
% 148.07/148.50 Found (eta_expansion_dep00 ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 148.07/148.50 Found ((eta_expansion_dep0 (fun (x3:beverage)=> Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 148.07/148.50 Found (((eta_expansion_dep beverage) (fun (x3:beverage)=> Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 148.07/148.50 Found (((eta_expansion_dep beverage) (fun (x3:beverage)=> Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 148.07/148.50 Found (((eta_expansion_dep beverage) (fun (x3:beverage)=> Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 148.07/148.50 Found eta_expansion000:=(eta_expansion00 ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))):(((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) (fun (x:beverage)=> (((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B)))))) x)))
% 148.07/148.50 Found (eta_expansion00 ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 148.07/148.50 Found ((eta_expansion0 Prop) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 152.72/153.16 Found (((eta_expansion beverage) Prop) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 152.72/153.16 Found (((eta_expansion beverage) Prop) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 152.72/153.16 Found (((eta_expansion beverage) Prop) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) as proof of (((eq (beverage->Prop)) ((unique beverage) (fun (y:beverage)=> ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) y)) (((eq beverage) B) coffee))) (hot B))))))) b)
% 152.72/153.16 Found x0:(((eq beverage) (coffee_mixture S0)) coffee)
% 152.72/153.16 Instantiate: x01:=(coffee_mixture S0):beverage
% 152.72/153.16 Found x0 as proof of (((eq beverage) x01) coffee)
% 152.72/153.16 Found eq_ref00:=(eq_ref0 x01):(((eq beverage) x01) x01)
% 152.72/153.16 Found (eq_ref0 x01) as proof of (((eq beverage) x01) (x S))
% 152.72/153.16 Found ((eq_ref beverage) x01) as proof of (((eq beverage) x01) (x S))
% 152.72/153.16 Found ((eq_ref beverage) x01) as proof of (((eq beverage) x01) (x S))
% 152.72/153.16 Found ((eq_ref beverage) x01) as proof of (((eq beverage) x01) (x S))
% 152.72/153.16 Found x1:(hot (coffee_mixture S))
% 152.72/153.16 Instantiate: x3:=(coffee_mixture S):beverage
% 152.72/153.16 Found x1 as proof of (hot x3)
% 152.72/153.16 Found x0:(((eq beverage) (coffee_mixture S0)) coffee)
% 152.72/153.16 Found x0 as proof of (((eq beverage) x01) coffee)
% 152.72/153.16 Found eq_ref00:=(eq_ref0 x1):(((eq beverage) x1) x1)
% 152.72/153.16 Found (eq_ref0 x1) as proof of (((eq beverage) x1) coffee)
% 152.72/153.16 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) coffee)
% 152.72/153.16 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) coffee)
% 152.72/153.16 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) coffee)
% 152.72/153.16 Found eq_ref00:=(eq_ref0 x1):(((eq beverage) x1) x1)
% 152.72/153.16 Found (eq_ref0 x1) as proof of (((eq beverage) x1) x0)
% 152.72/153.16 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) x0)
% 152.72/153.16 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) x0)
% 152.72/153.16 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) x0)
% 152.72/153.16 Found eq_ref00:=(eq_ref0 x1):(((eq beverage) x1) x1)
% 152.72/153.16 Found (eq_ref0 x1) as proof of (((eq beverage) x1) x0)
% 152.72/153.16 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) x0)
% 152.72/153.16 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) x0)
% 152.72/153.16 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) x0)
% 152.72/153.16 Found ((conj20 ((eq_ref beverage) x1)) ((eq_ref beverage) x1)) as proof of ((and (((eq beverage) x1) x0)) (((eq beverage) x1) coffee))
% 152.72/153.16 Found (((conj2 (((eq beverage) x1) coffee)) ((eq_ref beverage) x1)) ((eq_ref beverage) x1)) as proof of ((and (((eq beverage) x1) x0)) (((eq beverage) x1) coffee))
% 152.72/153.16 Found ((((conj (((eq beverage) x1) x0)) (((eq beverage) x1) coffee)) ((eq_ref beverage) x1)) ((eq_ref beverage) x1)) as proof of ((and (((eq beverage) x1) x0)) (((eq beverage) x1) coffee))
% 152.72/153.16 Found ((((conj (((eq beverage) x1) x0)) (((eq beverage) x1) coffee)) ((eq_ref beverage) x1)) ((eq_ref beverage) x1)) as proof of ((and (((eq beverage) x1) x0)) (((eq beverage) x1) coffee))
% 152.72/153.16 Found eq_ref00:=(eq_ref0 x1):(((eq beverage) x1) x1)
% 152.72/153.16 Found (eq_ref0 x1) as proof of (((eq beverage) x1) coffee)
% 152.72/153.16 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) coffee)
% 152.72/153.16 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) coffee)
% 152.72/153.16 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) coffee)
% 152.72/153.16 Found eq_ref00:=(eq_ref0 x1):(((eq beverage) x1) x1)
% 152.72/153.16 Found (eq_ref0 x1) as proof of (((eq beverage) x1) x0)
% 153.00/153.49 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) x0)
% 153.00/153.49 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) x0)
% 153.00/153.49 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) x0)
% 153.00/153.49 Found eq_ref00:=(eq_ref0 x1):(((eq beverage) x1) x1)
% 153.00/153.49 Found (eq_ref0 x1) as proof of (((eq beverage) x1) x0)
% 153.00/153.49 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) x0)
% 153.00/153.49 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) x0)
% 153.00/153.49 Found ((eq_ref beverage) x1) as proof of (((eq beverage) x1) x0)
% 153.00/153.49 Found ((conj20 ((eq_ref beverage) x1)) ((eq_ref beverage) x1)) as proof of ((and (((eq beverage) x1) x0)) (((eq beverage) x1) coffee))
% 153.00/153.49 Found (((conj2 (((eq beverage) x1) coffee)) ((eq_ref beverage) x1)) ((eq_ref beverage) x1)) as proof of ((and (((eq beverage) x1) x0)) (((eq beverage) x1) coffee))
% 153.00/153.49 Found ((((conj (((eq beverage) x1) x0)) (((eq beverage) x1) coffee)) ((eq_ref beverage) x1)) ((eq_ref beverage) x1)) as proof of ((and (((eq beverage) x1) x0)) (((eq beverage) x1) coffee))
% 153.00/153.49 Found ((((conj (((eq beverage) x1) x0)) (((eq beverage) x1) coffee)) ((eq_ref beverage) x1)) ((eq_ref beverage) x1)) as proof of ((and (((eq beverage) x1) x0)) (((eq beverage) x1) coffee))
% 153.00/153.49 Found eq_ref00:=(eq_ref0 x01):(((eq beverage) x01) x01)
% 153.00/153.49 Found (eq_ref0 x01) as proof of (((eq beverage) x01) (x S))
% 153.00/153.49 Found ((eq_ref beverage) x01) as proof of (((eq beverage) x01) (x S))
% 153.00/153.49 Found ((eq_ref beverage) x01) as proof of (((eq beverage) x01) (x S))
% 153.00/153.49 Found ((eq_ref beverage) x01) as proof of (((eq beverage) x01) (x S))
% 153.00/153.49 Found ((eq_ref beverage) x01) as proof of (((eq beverage) x01) (x S))
% 153.00/153.49 Found ((conj10 ((eq_ref beverage) x01)) x0) as proof of ((and (((eq beverage) x01) (x S))) (((eq beverage) x01) coffee))
% 153.00/153.49 Found (((conj1 (((eq beverage) x01) coffee)) ((eq_ref beverage) x01)) x0) as proof of ((and (((eq beverage) x01) (x S))) (((eq beverage) x01) coffee))
% 153.00/153.49 Found ((((conj (((eq beverage) x01) (x S))) (((eq beverage) x01) coffee)) ((eq_ref beverage) x01)) x0) as proof of ((and (((eq beverage) x01) (x S))) (((eq beverage) x01) coffee))
% 153.00/153.49 Found ((((conj (((eq beverage) x01) (x S))) (((eq beverage) x01) coffee)) ((eq_ref beverage) x01)) x0) as proof of ((and (((eq beverage) x01) (x S))) (((eq beverage) x01) coffee))
% 153.00/153.49 Found ((conj00 ((((conj (((eq beverage) x01) (x S))) (((eq beverage) x01) coffee)) ((eq_ref beverage) x01)) x0)) x00) as proof of ((and ((and (((eq beverage) x01) (x S))) (((eq beverage) x01) coffee))) (hot x01))
% 153.00/153.49 Found (((conj0 (hot x01)) ((((conj (((eq beverage) x01) (x S))) (((eq beverage) x01) coffee)) ((eq_ref beverage) x01)) x0)) x00) as proof of ((and ((and (((eq beverage) x01) (x S))) (((eq beverage) x01) coffee))) (hot x01))
% 153.00/153.49 Found ((((conj ((and (((eq beverage) x01) (x S))) (((eq beverage) x01) coffee))) (hot x01)) ((((conj (((eq beverage) x01) (x S))) (((eq beverage) x01) coffee)) ((eq_ref beverage) x01)) x0)) x00) as proof of ((and ((and (((eq beverage) x01) (x S))) (((eq beverage) x01) coffee))) (hot x01))
% 153.00/153.49 Found ((((conj ((and (((eq beverage) x01) (x S))) (((eq beverage) x01) coffee))) (hot x01)) ((((conj (((eq beverage) x01) (x S))) (((eq beverage) x01) coffee)) ((eq_ref beverage) x01)) x0)) x00) as proof of ((and ((and (((eq beverage) x01) (x S))) (((eq beverage) x01) coffee))) (hot x01))
% 153.00/153.49 Found (ex_intro100 ((((conj ((and (((eq beverage) x01) (x S))) (((eq beverage) x01) coffee))) (hot x01)) ((((conj (((eq beverage) x01) (x S))) (((eq beverage) x01) coffee)) ((eq_ref beverage) x01)) x0)) x00)) as proof of ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B))))
% 153.00/153.49 Found ((ex_intro10 (coffee_mixture S0)) ((((conj ((and (((eq beverage) (coffee_mixture S0)) (x S))) (((eq beverage) (coffee_mixture S0)) coffee))) (hot (coffee_mixture S0))) ((((conj (((eq beverage) (coffee_mixture S0)) (x S))) (((eq beverage) (coffee_mixture S0)) coffee)) ((eq_ref beverage) (coffee_mixture S0))) x0)) x00)) as proof of ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B))))
% 153.00/153.49 Found (((ex_intro1 (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S0)) ((((conj ((and (((eq beverage) (coffee_mixture S0)) (x S))) (((eq beverage) (coffee_mixture S0)) coffee))) (hot (coffee_mixture S0))) ((((conj (((eq beverage) (coffee_mixture S0)) (x S))) (((eq beverage) (coffee_mixture S0)) coffee)) ((eq_ref beverage) (coffee_mixture S0))) x0)) x00)) as proof of ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B))))
% 153.07/153.52 Found ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S0)) ((((conj ((and (((eq beverage) (coffee_mixture S0)) (x S))) (((eq beverage) (coffee_mixture S0)) coffee))) (hot (coffee_mixture S0))) ((((conj (((eq beverage) (coffee_mixture S0)) (x S))) (((eq beverage) (coffee_mixture S0)) coffee)) ((eq_ref beverage) (coffee_mixture S0))) x0)) x00)) as proof of ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B))))
% 153.07/153.52 Found (fun (x00:(hot (coffee_mixture S0)))=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S0)) ((((conj ((and (((eq beverage) (coffee_mixture S0)) (x S))) (((eq beverage) (coffee_mixture S0)) coffee))) (hot (coffee_mixture S0))) ((((conj (((eq beverage) (coffee_mixture S0)) (x S))) (((eq beverage) (coffee_mixture S0)) coffee)) ((eq_ref beverage) (coffee_mixture S0))) x0)) x00))) as proof of ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B))))
% 153.07/153.52 Found (fun (x0:(((eq beverage) (coffee_mixture S0)) coffee)) (x00:(hot (coffee_mixture S0)))=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S0)) ((((conj ((and (((eq beverage) (coffee_mixture S0)) (x S))) (((eq beverage) (coffee_mixture S0)) coffee))) (hot (coffee_mixture S0))) ((((conj (((eq beverage) (coffee_mixture S0)) (x S))) (((eq beverage) (coffee_mixture S0)) coffee)) ((eq_ref beverage) (coffee_mixture S0))) x0)) x00))) as proof of ((hot (coffee_mixture S0))->((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))))
% 153.07/153.52 Found (fun (x0:(((eq beverage) (coffee_mixture S0)) coffee)) (x00:(hot (coffee_mixture S0)))=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S0)) ((((conj ((and (((eq beverage) (coffee_mixture S0)) (x S))) (((eq beverage) (coffee_mixture S0)) coffee))) (hot (coffee_mixture S0))) ((((conj (((eq beverage) (coffee_mixture S0)) (x S))) (((eq beverage) (coffee_mixture S0)) coffee)) ((eq_ref beverage) (coffee_mixture S0))) x0)) x00))) as proof of ((((eq beverage) (coffee_mixture S0)) coffee)->((hot (coffee_mixture S0))->((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B))))))
% 153.07/153.52 Found (and_rect00 (fun (x0:(((eq beverage) (coffee_mixture S0)) coffee)) (x00:(hot (coffee_mixture S0)))=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S0)) ((((conj ((and (((eq beverage) (coffee_mixture S0)) (x S))) (((eq beverage) (coffee_mixture S0)) coffee))) (hot (coffee_mixture S0))) ((((conj (((eq beverage) (coffee_mixture S0)) (x S))) (((eq beverage) (coffee_mixture S0)) coffee)) ((eq_ref beverage) (coffee_mixture S0))) x0)) x00)))) as proof of ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B))))
% 153.07/153.52 Found ((and_rect0 ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B))))) (fun (x0:(((eq beverage) (coffee_mixture S0)) coffee)) (x00:(hot (coffee_mixture S0)))=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S0)) ((((conj ((and (((eq beverage) (coffee_mixture S0)) (x S))) (((eq beverage) (coffee_mixture S0)) coffee))) (hot (coffee_mixture S0))) ((((conj (((eq beverage) (coffee_mixture S0)) (x S))) (((eq beverage) (coffee_mixture S0)) coffee)) ((eq_ref beverage) (coffee_mixture S0))) x0)) x00)))) as proof of ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B))))
% 153.07/153.52 Found (((fun (P:Type) (x0:((((eq beverage) (coffee_mixture S0)) coffee)->((hot (coffee_mixture S0))->P)))=> (((((and_rect (((eq beverage) (coffee_mixture S0)) coffee)) (hot (coffee_mixture S0))) P) x0) coffee_and_syrup_is_hot_coffee0)) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B))))) (fun (x0:(((eq beverage) (coffee_mixture S0)) coffee)) (x00:(hot (coffee_mixture S0)))=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S0)) ((((conj ((and (((eq beverage) (coffee_mixture S0)) (x S))) (((eq beverage) (coffee_mixture S0)) coffee))) (hot (coffee_mixture S0))) ((((conj (((eq beverage) (coffee_mixture S0)) (x S))) (((eq beverage) (coffee_mixture S0)) coffee)) ((eq_ref beverage) (coffee_mixture S0))) x0)) x00)))) as proof of ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B))))
% 153.07/153.52 Found (((fun (P:Type) (x0:((((eq beverage) (coffee_mixture S)) coffee)->((hot (coffee_mixture S))->P)))=> (((((and_rect (((eq beverage) (coffee_mixture S)) coffee)) (hot (coffee_mixture S))) P) x0) (coffee_and_syrup_is_hot_coffee S))) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B))))) (fun (x0:(((eq beverage) (coffee_mixture S)) coffee)) (x00:(hot (coffee_mixture S)))=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S)) ((((conj ((and (((eq beverage) (coffee_mixture S)) (x S))) (((eq beverage) (coffee_mixture S)) coffee))) (hot (coffee_mixture S))) ((((conj (((eq beverage) (coffee_mixture S)) (x S))) (((eq beverage) (coffee_mixture S)) coffee)) ((eq_ref beverage) (coffee_mixture S))) x0)) x00)))) as proof of ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B))))
% 153.07/153.52 Found (fun (S:syrup)=> (((fun (P:Type) (x0:((((eq beverage) (coffee_mixture S)) coffee)->((hot (coffee_mixture S))->P)))=> (((((and_rect (((eq beverage) (coffee_mixture S)) coffee)) (hot (coffee_mixture S))) P) x0) (coffee_and_syrup_is_hot_coffee S))) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B))))) (fun (x0:(((eq beverage) (coffee_mixture S)) coffee)) (x00:(hot (coffee_mixture S)))=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S)) ((((conj ((and (((eq beverage) (coffee_mixture S)) (x S))) (((eq beverage) (coffee_mixture S)) coffee))) (hot (coffee_mixture S))) ((((conj (((eq beverage) (coffee_mixture S)) (x S))) (((eq beverage) (coffee_mixture S)) coffee)) ((eq_ref beverage) (coffee_mixture S))) x0)) x00))))) as proof of ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B))))
% 153.07/153.52 Found (fun (S:syrup)=> (((fun (P:Type) (x0:((((eq beverage) (coffee_mixture S)) coffee)->((hot (coffee_mixture S))->P)))=> (((((and_rect (((eq beverage) (coffee_mixture S)) coffee)) (hot (coffee_mixture S))) P) x0) (coffee_and_syrup_is_hot_coffee S))) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B))))) (fun (x0:(((eq beverage) (coffee_mixture S)) coffee)) (x00:(hot (coffee_mixture S)))=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S)) ((((conj ((and (((eq beverage) (coffee_mixture S)) (x S))) (((eq beverage) (coffee_mixture S)) coffee))) (hot (coffee_mixture S))) ((((conj (((eq beverage) (coffee_mixture S)) (x S))) (((eq beverage) (coffee_mixture S)) coffee)) ((eq_ref beverage) (coffee_mixture S))) x0)) x00))))) as proof of (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B)
% coffee))) (hot B)))))
% 153.38/153.85 Found (ex_intro000 (fun (S:syrup)=> (((fun (P:Type) (x0:((((eq beverage) (coffee_mixture S)) coffee)->((hot (coffee_mixture S))->P)))=> (((((and_rect (((eq beverage) (coffee_mixture S)) coffee)) (hot (coffee_mixture S))) P) x0) (coffee_and_syrup_is_hot_coffee S))) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B))))) (fun (x0:(((eq beverage) (coffee_mixture S)) coffee)) (x00:(hot (coffee_mixture S)))=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (x S))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S)) ((((conj ((and (((eq beverage) (coffee_mixture S)) (x S))) (((eq beverage) (coffee_mixture S)) coffee))) (hot (coffee_mixture S))) ((((conj (((eq beverage) (coffee_mixture S)) (x S))) (((eq beverage) (coffee_mixture S)) coffee)) ((eq_ref beverage) (coffee_mixture S))) x0)) x00)))))) as proof of ((ex (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun
% (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B)))))))
% 153.38/153.85 Found ((ex_intro00 (mix coffee)) (fun (S:syrup)=> (((fun (P:Type) (x0:((((eq beverage) (coffee_mixture S)) coffee)->((hot (coffee_mixture S))->P)))=> (((((and_rect (((eq beverage) (coffee_mixture S)) coffee)) (hot (coffee_mixture S))) P) x0) (coffee_and_syrup_is_hot_coffee S))) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) ((mix coffee) S))) (((eq beverage) B) coffee))) (hot B))))) (fun (x0:(((eq beverage) (coffee_mixture S)) coffee)) (x00:(hot (coffee_mixture S)))=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) ((mix coffee) S))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S)) ((((conj ((and (((eq beverage) (coffee_mixture S)) ((mix coffee) S))) (((eq beverage) (coffee_mixture S)) coffee))) (hot (coffee_mixture S))) ((((conj (((eq beverage) (coffee_mixture S)) ((mix coffee) S))) (((eq beverage) (coffee_mixture S)) coffee)) ((eq_ref beverage) (coffee_mixture S))) x0)) x00)))))) as proof of ((ex (syrup->beverage)) (fun
% (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B)))))))
% 153.38/153.85 Found (((ex_intro0 (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) (mix coffee)) (fun (S:syrup)=> (((fun (P:Type) (x0:((((eq beverage) (coffee_mixture S)) coffee)->((hot (coffee_mixture S))->P)))=> (((((and_rect (((eq beverage) (coffee_mixture S)) coffee)) (hot (coffee_mixture S))) P) x0) (coffee_and_syrup_is_hot_coffee S))) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) ((mix coffee) S))) (((eq beverage) B) coffee))) (hot B))))) (fun (x0:(((eq beverage) (coffee_mixture S)) coffee)) (x00:(hot (coffee_mixture S)))=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) ((mix coffee) S))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S)) ((((conj ((and (((eq beverage) (coffee_mixture S)) ((mix coffee) S))) (((eq beverage) (coffee_mixture S)) coffee))) (hot (coffee_mixture S))) ((((conj (((eq beverage)
% (coffee_mixture S)) ((mix coffee) S))) (((eq beverage) (coffee_mixture S)) coffee)) ((eq_ref beverage) (coffee_mixture S))) x0)) x00)))))) as proof of ((ex (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B)))))))
% 153.38/153.85 Found ((((ex_intro (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) (mix coffee)) (fun (S:syrup)=> (((fun (P:Type) (x0:((((eq beverage) (coffee_mixture S)) coffee)->((hot (coffee_mixture S))->P)))=> (((((and_rect (((eq beverage) (coffee_mixture S)) coffee)) (hot (coffee_mixture S))) P) x0) (coffee_and_syrup_is_hot_coffee S))) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) ((mix coffee) S))) (((eq beverage) B) coffee))) (hot B))))) (fun (x0:(((eq beverage) (coffee_mixture S)) coffee)) (x00:(hot (coffee_mixture S)))=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) ((mix coffee) S))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S)) ((((conj ((and (((eq beverage) (coffee_mixture S)) ((mix coffee) S))) (((eq beverage) (coffee_mixture S)) coffee))) (hot (coffee_mixture S))) ((((conj (((eq
% beverage) (coffee_mixture S)) ((mix coffee) S))) (((eq beverage) (coffee_mixture S)) coffee)) ((eq_ref beverage) (coffee_mixture S))) x0)) x00)))))) as proof of ((ex (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B)))))))
% 153.38/153.85 Found ((((ex_intro (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) (mix coffee)) (fun (S:syrup)=> (((fun (P:Type) (x0:((((eq beverage) (coffee_mixture S)) coffee)->((hot (coffee_mixture S))->P)))=> (((((and_rect (((eq beverage) (coffee_mixture S)) coffee)) (hot (coffee_mixture S))) P) x0) (coffee_and_syrup_is_hot_coffee S))) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) ((mix coffee) S))) (((eq beverage) B) coffee))) (hot B))))) (fun (x0:(((eq beverage) (coffee_mixture S)) coffee)) (x00:(hot (coffee_mixture S)))=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) ((mix coffee) S))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S)) ((((conj ((and (((eq beverage) (coffee_mixture S)) ((mix coffee) S))) (((eq beverage) (coffee_mixture S)) coffee))) (hot (coffee_mixture S))) ((((conj (((eq
% beverage) (coffee_mixture S)) ((mix coffee) S))) (((eq beverage) (coffee_mixture S)) coffee)) ((eq_ref beverage) (coffee_mixture S))) x0)) x00)))))) as proof of ((ex (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B)))))))
% 153.38/153.85 Got proof ((((ex_intro (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) (mix coffee)) (fun (S:syrup)=> (((fun (P:Type) (x0:((((eq beverage) (coffee_mixture S)) coffee)->((hot (coffee_mixture S))->P)))=> (((((and_rect (((eq beverage) (coffee_mixture S)) coffee)) (hot (coffee_mixture S))) P) x0) (coffee_and_syrup_is_hot_coffee S))) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) ((mix coffee) S))) (((eq beverage) B) coffee))) (hot B))))) (fun (x0:(((eq beverage) (coffee_mixture S)) coffee)) (x00:(hot (coffee_mixture S)))=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) ((mix coffee) S))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S)) ((((conj ((and (((eq beverage) (coffee_mixture S)) ((mix coffee) S))) (((eq beverage) (coffee_mixture S)) coffee))) (hot (coffee_mixture S))) ((((conj (((eq
% beverage) (coffee_mixture S)) ((mix coffee) S))) (((eq beverage) (coffee_mixture S)) coffee)) ((eq_ref beverage) (coffee_mixture S))) x0)) x00))))))
% 153.38/153.85 Time elapsed = 153.003233s
% 153.38/153.85 node=22743 cost=2590.000000 depth=30
% 153.38/153.85::::::::::::::::::::::
% 153.38/153.85 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% 153.38/153.85 % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% 153.38/153.85 ((((ex_intro (syrup->beverage)) (fun (SyrupMixer:(syrup->beverage))=> (forall (S:syrup), ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) (SyrupMixer S))) (((eq beverage) B) coffee))) (hot B))))))) (mix coffee)) (fun (S:syrup)=> (((fun (P:Type) (x0:((((eq beverage) (coffee_mixture S)) coffee)->((hot (coffee_mixture S))->P)))=> (((((and_rect (((eq beverage) (coffee_mixture S)) coffee)) (hot (coffee_mixture S))) P) x0) (coffee_and_syrup_is_hot_coffee S))) ((ex beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) ((mix coffee) S))) (((eq beverage) B) coffee))) (hot B))))) (fun (x0:(((eq beverage) (coffee_mixture S)) coffee)) (x00:(hot (coffee_mixture S)))=> ((((ex_intro beverage) (fun (B:beverage)=> ((and ((and (((eq beverage) B) ((mix coffee) S))) (((eq beverage) B) coffee))) (hot B)))) (coffee_mixture S)) ((((conj ((and (((eq beverage) (coffee_mixture S)) ((mix coffee) S))) (((eq beverage) (coffee_mixture S)) coffee))) (hot (coffee_mixture S))) ((((conj (((eq beverage)
% (coffee_mixture S)) ((mix coffee) S))) (((eq beverage) (coffee_mixture S)) coffee)) ((eq_ref beverage) (coffee_mixture S))) x0)) x00))))))
% 154.60/155.07 % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
%------------------------------------------------------------------------------