TSTP Solution File: SEV277^5 by cocATP---0.2.0
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%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV277^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n089.star.cs.uiowa.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory : 32286.75MB
% OS : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:59 EDT 2014
% Result : Timeout 300.01s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV277^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n089.star.cs.uiowa.edu
% % Model : x86_64 x86_64
% % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory : 32286.75MB
% % OS : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 08:42:26 CDT 2014
% % CPUTime : 300.01
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0xbf83f8>, <kernel.Type object at 0xbf8d40>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0xe4f3f8>, <kernel.DependentProduct object at 0xe500e0>) of role type named cR
% Using role type
% Declaring cR:(a->(a->Prop))
% FOF formula ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((cR Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((cR Xy) Xx))))->(((eq a) Xy) Xz))))))))->(forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))) of role conjecture named cTHM545_pme
% Conjecture to prove = ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((cR Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((cR Xy) Xx))))->(((eq a) Xy) Xz))))))))->(forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz)))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((cR Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((cR Xy) Xx))))->(((eq a) Xy) Xz))))))))->(forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))']
% Parameter a:Type.
% Parameter cR:(a->(a->Prop)).
% Trying to prove ((forall (X:(a->Prop)), (((ex a) (fun (Xz:a)=> (X Xz)))->((ex a) (fun (Xz:a)=> ((and ((and (X Xz)) (forall (Xx:a), ((X Xx)->((cR Xz) Xx))))) (forall (Xy:a), (((and (X Xy)) (forall (Xx:a), ((X Xx)->((cR Xy) Xx))))->(((eq a) Xy) Xz))))))))->(forall (Xx:a) (Xy:a) (Xz:a), (((and ((cR Xx) Xy)) ((cR Xy) Xz))->((cR Xx) Xz))))
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found x1:((cR Xx) Xy)
% Instantiate: b:=Xy:a
% Found x1 as proof of (P b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% Found x1:((cR Xx) Xy)
% Instantiate: b:=Xy:a
% Found (fun (x2:((cR Xy) Xz))=> x1) as proof of (P b)
% Found (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1) as proof of (((cR Xy) Xz)->(P b))
% Found (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1) as proof of (((cR Xx) Xy)->(((cR Xy) Xz)->(P b)))
% Found (and_rect00 (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P b)
% Found ((and_rect0 (P b)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P b)
% Found (((fun (P0:Type) (x1:(((cR Xx) Xy)->(((cR Xy) Xz)->P0)))=> (((((and_rect ((cR Xx) Xy)) ((cR Xy) Xz)) P0) x1) x0)) (P b)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P b)
% Found (((fun (P0:Type) (x1:(((cR Xx) Xy)->(((cR Xy) Xz)->P0)))=> (((((and_rect ((cR Xx) Xy)) ((cR Xy) Xz)) P0) x1) x0)) (P b)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xz)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found x1:((cR Xx) Xy)
% Instantiate: a0:=Xy:a
% Found (fun (x2:((cR Xy) Xz))=> x1) as proof of (P a0)
% Found (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1) as proof of (((cR Xy) Xz)->(P a0))
% Found (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1) as proof of (((cR Xx) Xy)->(((cR Xy) Xz)->(P a0)))
% Found (and_rect00 (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P a0)
% Found ((and_rect0 (P a0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P a0)
% Found (((fun (P0:Type) (x1:(((cR Xx) Xy)->(((cR Xy) Xz)->P0)))=> (((((and_rect ((cR Xx) Xy)) ((cR Xy) Xz)) P0) x1) x0)) (P a0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P a0)
% Found (((fun (P0:Type) (x1:(((cR Xx) Xy)->(((cR Xy) Xz)->P0)))=> (((((and_rect ((cR Xx) Xy)) ((cR Xy) Xz)) P0) x1) x0)) (P a0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P a0)
% Found x1:((cR Xx) Xy)
% Instantiate: a0:=Xy:a
% Found (fun (x2:((cR Xy) Xz))=> x1) as proof of (P a0)
% Found (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1) as proof of (((cR Xy) Xz)->(P a0))
% Found (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1) as proof of (((cR Xx) Xy)->(((cR Xy) Xz)->(P a0)))
% Found (and_rect00 (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P a0)
% Found ((and_rect0 (P a0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P a0)
% Found (((fun (P0:Type) (x1:(((cR Xx) Xy)->(((cR Xy) Xz)->P0)))=> (((((and_rect ((cR Xx) Xy)) ((cR Xy) Xz)) P0) x1) x0)) (P a0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P a0)
% Found (((fun (P0:Type) (x1:(((cR Xx) Xy)->(((cR Xy) Xz)->P0)))=> (((((and_rect ((cR Xx) Xy)) ((cR Xy) Xz)) P0) x1) x0)) (P a0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P a0)
% Found x1:((cR Xx) Xy)
% Instantiate: a0:=Xy:a
% Found x1 as proof of (P a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xz)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xz)->(P0 Xz))
% Found (eq_ref00 P0) as proof of (P1 Xz)
% Found ((eq_ref0 Xz) P0) as proof of (P1 Xz)
% Found (((eq_ref a) Xz) P0) as proof of (P1 Xz)
% Found (((eq_ref a) Xz) P0) as proof of (P1 Xz)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xz)->(P0 Xz))
% Found (eq_ref00 P0) as proof of (P1 Xz)
% Found ((eq_ref0 Xz) P0) as proof of (P1 Xz)
% Found (((eq_ref a) Xz) P0) as proof of (P1 Xz)
% Found (((eq_ref a) Xz) P0) as proof of (P1 Xz)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P0 b0)
% Found ((eq_ref a) b0) as proof of (P0 b0)
% Found ((eq_ref a) b0) as proof of (P0 b0)
% Found ((eq_ref a) b0) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found x3:(P0 Xz)
% Instantiate: b0:=Xz:a
% Found x3 as proof of (P1 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref000:=(eq_ref00 P0):((P0 a0)->(P0 a0))
% Found (eq_ref00 P0) as proof of (P1 a0)
% Found ((eq_ref0 a0) P0) as proof of (P1 a0)
% Found (((eq_ref a) a0) P0) as proof of (P1 a0)
% Found (((eq_ref a) a0) P0) as proof of (P1 a0)
% Found x3:(P0 b)
% Instantiate: b0:=b:a
% Found x3 as proof of (P1 b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P0 b0)
% Found ((eq_ref a) b0) as proof of (P0 b0)
% Found ((eq_ref a) b0) as proof of (P0 b0)
% Found ((eq_ref a) b0) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P0 b0)
% Found ((eq_ref a) b) as proof of (P0 b0)
% Found ((eq_ref a) b) as proof of (P0 b0)
% Found ((eq_ref a) b) as proof of (P0 b0)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xz)->(P0 Xz))
% Found (eq_ref00 P0) as proof of (P1 Xz)
% Found ((eq_ref0 Xz) P0) as proof of (P1 Xz)
% Found (((eq_ref a) Xz) P0) as proof of (P1 Xz)
% Found (((eq_ref a) Xz) P0) as proof of (P1 Xz)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found eq_ref000:=(eq_ref00 P0):((P0 b0)->(P0 b0))
% Found (eq_ref00 P0) as proof of (P1 b0)
% Found ((eq_ref0 b0) P0) as proof of (P1 b0)
% Found (((eq_ref a) b0) P0) as proof of (P1 b0)
% Found (((eq_ref a) b0) P0) as proof of (P1 b0)
% Found eq_ref000:=(eq_ref00 P0):((P0 b)->(P0 b))
% Found (eq_ref00 P0) as proof of (P1 b)
% Found ((eq_ref0 b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found x1:((cR Xx) Xy)
% Instantiate: b0:=Xy:a
% Found (fun (x2:((cR Xy) Xz))=> x1) as proof of (P0 b0)
% Found (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1) as proof of (((cR Xy) Xz)->(P0 b0))
% Found (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1) as proof of (((cR Xx) Xy)->(((cR Xy) Xz)->(P0 b0)))
% Found (and_rect00 (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 b0)
% Found ((and_rect0 (P0 b0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 b0)
% Found (((fun (P1:Type) (x1:(((cR Xx) Xy)->(((cR Xy) Xz)->P1)))=> (((((and_rect ((cR Xx) Xy)) ((cR Xy) Xz)) P1) x1) x0)) (P0 b0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 b0)
% Found (((fun (P1:Type) (x1:(((cR Xx) Xy)->(((cR Xy) Xz)->P1)))=> (((((and_rect ((cR Xx) Xy)) ((cR Xy) Xz)) P1) x1) x0)) (P0 b0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 b0)
% Found x1:((cR Xx) Xy)
% Instantiate: b0:=Xy:a
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref000:=(eq_ref00 P0):((P0 a0)->(P0 a0))
% Found (eq_ref00 P0) as proof of (P1 a0)
% Found ((eq_ref0 a0) P0) as proof of (P1 a0)
% Found (((eq_ref a) a0) P0) as proof of (P1 a0)
% Found (((eq_ref a) a0) P0) as proof of (P1 a0)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xz)->(P0 Xz))
% Found (eq_ref00 P0) as proof of (P1 Xz)
% Found ((eq_ref0 Xz) P0) as proof of (P1 Xz)
% Found (((eq_ref a) Xz) P0) as proof of (P1 Xz)
% Found (((eq_ref a) Xz) P0) as proof of (P1 Xz)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found x3:(P0 Xz)
% Instantiate: b0:=Xz:a
% Found x3 as proof of (P1 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found x3:(P0 b)
% Instantiate: b0:=b:a
% Found x3 as proof of (P1 b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xz)->(P0 Xz))
% Found (eq_ref00 P0) as proof of (P1 Xz)
% Found ((eq_ref0 Xz) P0) as proof of (P1 Xz)
% Found (((eq_ref a) Xz) P0) as proof of (P1 Xz)
% Found (((eq_ref a) Xz) P0) as proof of (P1 Xz)
% Found x3:(P0 b)
% Instantiate: b0:=b:a
% Found x3 as proof of (P1 b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found eq_ref000:=(eq_ref00 P0):((P0 b)->(P0 b))
% Found (eq_ref00 P0) as proof of (P1 b)
% Found ((eq_ref0 b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P0 b0)
% Found ((eq_ref a) b0) as proof of (P0 b0)
% Found ((eq_ref a) b0) as proof of (P0 b0)
% Found ((eq_ref a) b0) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P0 b0)
% Found ((eq_ref a) b) as proof of (P0 b0)
% Found ((eq_ref a) b) as proof of (P0 b0)
% Found ((eq_ref a) b) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref000:=(eq_ref00 P0):((P0 b0)->(P0 b0))
% Found (eq_ref00 P0) as proof of (P1 b0)
% Found ((eq_ref0 b0) P0) as proof of (P1 b0)
% Found (((eq_ref a) b0) P0) as proof of (P1 b0)
% Found (((eq_ref a) b0) P0) as proof of (P1 b0)
% Found x1:((cR Xx) Xy)
% Instantiate: b0:=Xy:a
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found x1:((cR Xx) Xy)
% Instantiate: b0:=Xy:a
% Found (fun (x2:((cR Xy) Xz))=> x1) as proof of (P0 b0)
% Found (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1) as proof of (((cR Xy) Xz)->(P0 b0))
% Found (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1) as proof of (((cR Xx) Xy)->(((cR Xy) Xz)->(P0 b0)))
% Found (and_rect00 (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 b0)
% Found ((and_rect0 (P0 b0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 b0)
% Found (((fun (P1:Type) (x1:(((cR Xx) Xy)->(((cR Xy) Xz)->P1)))=> (((((and_rect ((cR Xx) Xy)) ((cR Xy) Xz)) P1) x1) x0)) (P0 b0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 b0)
% Found (((fun (P1:Type) (x1:(((cR Xx) Xy)->(((cR Xy) Xz)->P1)))=> (((((and_rect ((cR Xx) Xy)) ((cR Xy) Xz)) P1) x1) x0)) (P0 b0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 b0)
% Found x1:((cR Xx) Xy)
% Instantiate: b0:=Xy:a
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found eq_ref000:=(eq_ref00 P0):((P0 b)->(P0 b))
% Found (eq_ref00 P0) as proof of (P1 b)
% Found ((eq_ref0 b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x3:(P0 a0)
% Instantiate: b0:=a0:a
% Found x3 as proof of (P1 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found x1:((cR Xx) Xy)
% Instantiate: b0:=Xy:a
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found x1:((cR Xx) Xy)
% Instantiate: b0:=Xy:a
% Found (fun (x2:((cR Xy) Xz))=> x1) as proof of (P0 b0)
% Found (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1) as proof of (((cR Xy) Xz)->(P0 b0))
% Found (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1) as proof of (((cR Xx) Xy)->(((cR Xy) Xz)->(P0 b0)))
% Found (and_rect00 (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 b0)
% Found ((and_rect0 (P0 b0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 b0)
% Found (((fun (P1:Type) (x1:(((cR Xx) Xy)->(((cR Xy) Xz)->P1)))=> (((((and_rect ((cR Xx) Xy)) ((cR Xy) Xz)) P1) x1) x0)) (P0 b0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 b0)
% Found (((fun (P1:Type) (x1:(((cR Xx) Xy)->(((cR Xy) Xz)->P1)))=> (((((and_rect ((cR Xx) Xy)) ((cR Xy) Xz)) P1) x1) x0)) (P0 b0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 b0)
% Found x1:((cR Xx) Xy)
% Instantiate: a0:=Xy:a
% Found (fun (x2:((cR Xy) Xz))=> x1) as proof of (P0 a0)
% Found (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1) as proof of (((cR Xy) Xz)->(P0 a0))
% Found (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1) as proof of (((cR Xx) Xy)->(((cR Xy) Xz)->(P0 a0)))
% Found (and_rect00 (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 a0)
% Found ((and_rect0 (P0 a0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 a0)
% Found (((fun (P1:Type) (x1:(((cR Xx) Xy)->(((cR Xy) Xz)->P1)))=> (((((and_rect ((cR Xx) Xy)) ((cR Xy) Xz)) P1) x1) x0)) (P0 a0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 a0)
% Found (((fun (P1:Type) (x1:(((cR Xx) Xy)->(((cR Xy) Xz)->P1)))=> (((((and_rect ((cR Xx) Xy)) ((cR Xy) Xz)) P1) x1) x0)) (P0 a0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 a0)
% Found x3:(P0 a0)
% Found x3 as proof of (P1 a0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x1:((cR Xx) Xy)
% Instantiate: a0:=Xy:a
% Found (fun (x2:((cR Xy) Xz))=> x1) as proof of (P0 a0)
% Found (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1) as proof of (((cR Xy) Xz)->(P0 a0))
% Found (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1) as proof of (((cR Xx) Xy)->(((cR Xy) Xz)->(P0 a0)))
% Found (and_rect00 (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 a0)
% Found ((and_rect0 (P0 a0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 a0)
% Found (((fun (P1:Type) (x1:(((cR Xx) Xy)->(((cR Xy) Xz)->P1)))=> (((((and_rect ((cR Xx) Xy)) ((cR Xy) Xz)) P1) x1) x0)) (P0 a0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 a0)
% Found (((fun (P1:Type) (x1:(((cR Xx) Xy)->(((cR Xy) Xz)->P1)))=> (((((and_rect ((cR Xx) Xy)) ((cR Xy) Xz)) P1) x1) x0)) (P0 a0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 a0)
% Found x3:(P0 Xz)
% Instantiate: a0:=Xz:a
% Found x3 as proof of (P1 a0)
% Found x1:((cR Xx) Xy)
% Instantiate: a0:=Xy:a
% Found x1 as proof of (P0 a0)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xz)->(P1 Xz))
% Found (eq_ref00 P1) as proof of (P2 Xz)
% Found ((eq_ref0 Xz) P1) as proof of (P2 Xz)
% Found (((eq_ref a) Xz) P1) as proof of (P2 Xz)
% Found (((eq_ref a) Xz) P1) as proof of (P2 Xz)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x3:(P2 b)
% Instantiate: b0:=b:a
% Found x3 as proof of (P3 b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found x3:(P2 b)
% Instantiate: b0:=b:a
% Found x3 as proof of (P3 b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P2 b0)
% Found ((eq_ref a) b0) as proof of (P2 b0)
% Found ((eq_ref a) b0) as proof of (P2 b0)
% Found ((eq_ref a) b0) as proof of (P2 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P2 b0)
% Found ((eq_ref a) b0) as proof of (P2 b0)
% Found ((eq_ref a) b0) as proof of (P2 b0)
% Found ((eq_ref a) b0) as proof of (P2 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P2 b0)
% Found ((eq_ref a) b) as proof of (P2 b0)
% Found ((eq_ref a) b) as proof of (P2 b0)
% Found ((eq_ref a) b) as proof of (P2 b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P2 b0)
% Found ((eq_ref a) b) as proof of (P2 b0)
% Found ((eq_ref a) b) as proof of (P2 b0)
% Found ((eq_ref a) b) as proof of (P2 b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P0 b0)
% Found ((eq_ref a) b0) as proof of (P0 b0)
% Found ((eq_ref a) b0) as proof of (P0 b0)
% Found ((eq_ref a) b0) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found x3:(P0 b)
% Instantiate: b0:=b:a
% Found x3 as proof of (P1 b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref000:=(eq_ref00 P0):((P0 a0)->(P0 a0))
% Found (eq_ref00 P0) as proof of (P1 a0)
% Found ((eq_ref0 a0) P0) as proof of (P1 a0)
% Found (((eq_ref a) a0) P0) as proof of (P1 a0)
% Found (((eq_ref a) a0) P0) as proof of (P1 a0)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (P0 b1)
% Found ((eq_ref a) b1) as proof of (P0 b1)
% Found ((eq_ref a) b1) as proof of (P0 b1)
% Found ((eq_ref a) b1) as proof of (P0 b1)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (P0 a0)
% Found ((eq_ref a) a0) as proof of (P0 a0)
% Found ((eq_ref a) a0) as proof of (P0 a0)
% Found x3:(P0 b0)
% Instantiate: b00:=b0:a
% Found x3 as proof of (P1 b00)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x3:(P0 b)
% Instantiate: a0:=b:a
% Found x3 as proof of (P1 a0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (P0 b1)
% Found ((eq_ref a) b1) as proof of (P0 b1)
% Found ((eq_ref a) b1) as proof of (P0 b1)
% Found ((eq_ref a) b1) as proof of (P0 b1)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found x3:(P0 b0)
% Found x3 as proof of (P1 Xz)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found x3:(P0 Xz)
% Instantiate: b00:=Xz:a
% Found x3 as proof of (P1 b00)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref000:=(eq_ref00 P2):((P2 Xz)->(P2 Xz))
% Found (eq_ref00 P2) as proof of (P3 Xz)
% Found ((eq_ref0 Xz) P2) as proof of (P3 Xz)
% Found (((eq_ref a) Xz) P2) as proof of (P3 Xz)
% Found (((eq_ref a) Xz) P2) as proof of (P3 Xz)
% Found eq_ref000:=(eq_ref00 P2):((P2 Xz)->(P2 Xz))
% Found (eq_ref00 P2) as proof of (P3 Xz)
% Found ((eq_ref0 Xz) P2) as proof of (P3 Xz)
% Found (((eq_ref a) Xz) P2) as proof of (P3 Xz)
% Found (((eq_ref a) Xz) P2) as proof of (P3 Xz)
% Found x3:(P0 b)
% Instantiate: b0:=b:a
% Found x3 as proof of (P1 b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (P0 b00)
% Found ((eq_ref a) Xz) as proof of (P0 b00)
% Found ((eq_ref a) Xz) as proof of (P0 b00)
% Found ((eq_ref a) Xz) as proof of (P0 b00)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found eq_ref000:=(eq_ref00 P2):((P2 b)->(P2 b))
% Found (eq_ref00 P2) as proof of (P3 b)
% Found ((eq_ref0 b) P2) as proof of (P3 b)
% Found (((eq_ref a) b) P2) as proof of (P3 b)
% Found (((eq_ref a) b) P2) as proof of (P3 b)
% Found eq_ref000:=(eq_ref00 P2):((P2 b)->(P2 b))
% Found (eq_ref00 P2) as proof of (P3 b)
% Found ((eq_ref0 b) P2) as proof of (P3 b)
% Found (((eq_ref a) b) P2) as proof of (P3 b)
% Found (((eq_ref a) b) P2) as proof of (P3 b)
% Found eq_ref000:=(eq_ref00 P0):((P0 b)->(P0 b))
% Found (eq_ref00 P0) as proof of (P1 b)
% Found ((eq_ref0 b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xz)->(P0 Xz))
% Found (eq_ref00 P0) as proof of (P1 Xz)
% Found ((eq_ref0 Xz) P0) as proof of (P1 Xz)
% Found (((eq_ref a) Xz) P0) as proof of (P1 Xz)
% Found (((eq_ref a) Xz) P0) as proof of (P1 Xz)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (P0 b00)
% Found ((eq_ref a) b00) as proof of (P0 b00)
% Found ((eq_ref a) b00) as proof of (P0 b00)
% Found ((eq_ref a) b00) as proof of (P0 b00)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (P0 b00)
% Found ((eq_ref a) b00) as proof of (P0 b00)
% Found ((eq_ref a) b00) as proof of (P0 b00)
% Found ((eq_ref a) b00) as proof of (P0 b00)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b01)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b01)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b01)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b01)
% Found eq_ref00:=(eq_ref0 b01):(((eq a) b01) b01)
% Found (eq_ref0 b01) as proof of (((eq a) b01) b)
% Found ((eq_ref a) b01) as proof of (((eq a) b01) b)
% Found ((eq_ref a) b01) as proof of (((eq a) b01) b)
% Found ((eq_ref a) b01) as proof of (((eq a) b01) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xz)->(P0 Xz))
% Found (eq_ref00 P0) as proof of (P1 Xz)
% Found ((eq_ref0 Xz) P0) as proof of (P1 Xz)
% Found (((eq_ref a) Xz) P0) as proof of (P1 Xz)
% Found (((eq_ref a) Xz) P0) as proof of (P1 Xz)
% Found eq_ref000:=(eq_ref00 P0):((P0 b)->(P0 b))
% Found (eq_ref00 P0) as proof of (P1 b)
% Found ((eq_ref0 b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xz)->(P0 Xz))
% Found (eq_ref00 P0) as proof of (P1 Xz)
% Found ((eq_ref0 Xz) P0) as proof of (P1 Xz)
% Found (((eq_ref a) Xz) P0) as proof of (P1 Xz)
% Found (((eq_ref a) Xz) P0) as proof of (P1 Xz)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (P0 Xz)
% Found ((eq_ref a) Xz) as proof of (P0 Xz)
% Found ((eq_ref a) Xz) as proof of (P0 Xz)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found x3:(P0 b)
% Instantiate: b1:=b:a
% Found x3 as proof of (P1 b1)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found x3:(P0 b)
% Instantiate: b1:=b:a
% Found x3 as proof of (P1 b1)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P0 b1)
% Found ((eq_ref a) b) as proof of (P0 b1)
% Found ((eq_ref a) b) as proof of (P0 b1)
% Found ((eq_ref a) b) as proof of (P0 b1)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P0 b1)
% Found ((eq_ref a) b) as proof of (P0 b1)
% Found ((eq_ref a) b) as proof of (P0 b1)
% Found ((eq_ref a) b) as proof of (P0 b1)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) a0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) a0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) a0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) a0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) a0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) a0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) a0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found x3:(P0 b)
% Found x3 as proof of (P1 b)
% Found eq_ref000:=(eq_ref00 P0):((P0 b)->(P0 b))
% Found (eq_ref00 P0) as proof of (P1 b)
% Found ((eq_ref0 b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref000:=(eq_ref00 P0):((P0 b)->(P0 b))
% Found (eq_ref00 P0) as proof of (P1 b)
% Found ((eq_ref0 b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b10):(((eq a) b10) b10)
% Found (eq_ref0 b10) as proof of (((eq a) b10) b0)
% Found ((eq_ref a) b10) as proof of (((eq a) b10) b0)
% Found ((eq_ref a) b10) as proof of (((eq a) b10) b0)
% Found ((eq_ref a) b10) as proof of (((eq a) b10) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b10)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b10)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b10)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b10)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b01):(((eq a) b01) b01)
% Found (eq_ref0 b01) as proof of (((eq a) b01) Xz)
% Found ((eq_ref a) b01) as proof of (((eq a) b01) Xz)
% Found ((eq_ref a) b01) as proof of (((eq a) b01) Xz)
% Found ((eq_ref a) b01) as proof of (((eq a) b01) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b01)
% Found ((eq_ref a) b) as proof of (((eq a) b) b01)
% Found ((eq_ref a) b) as proof of (((eq a) b) b01)
% Found ((eq_ref a) b) as proof of (((eq a) b) b01)
% Found eq_ref00:=(eq_ref0 b01):(((eq a) b01) b01)
% Found (eq_ref0 b01) as proof of (((eq a) b01) b00)
% Found ((eq_ref a) b01) as proof of (((eq a) b01) b00)
% Found ((eq_ref a) b01) as proof of (((eq a) b01) b00)
% Found ((eq_ref a) b01) as proof of (((eq a) b01) b00)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b01)
% Found ((eq_ref a) b) as proof of (((eq a) b) b01)
% Found ((eq_ref a) b) as proof of (((eq a) b) b01)
% Found ((eq_ref a) b) as proof of (((eq a) b) b01)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xz)->(P1 Xz))
% Found (eq_ref00 P1) as proof of (P2 Xz)
% Found ((eq_ref0 Xz) P1) as proof of (P2 Xz)
% Found (((eq_ref a) Xz) P1) as proof of (P2 Xz)
% Found (((eq_ref a) Xz) P1) as proof of (P2 Xz)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b1)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b1)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b1)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b1)
% Found eq_ref00:=(eq_ref0 b000):(((eq a) b000) b000)
% Found (eq_ref0 b000) as proof of (((eq a) b000) b0)
% Found ((eq_ref a) b000) as proof of (((eq a) b000) b0)
% Found ((eq_ref a) b000) as proof of (((eq a) b000) b0)
% Found ((eq_ref a) b000) as proof of (((eq a) b000) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b000)
% Found ((eq_ref a) b) as proof of (((eq a) b) b000)
% Found ((eq_ref a) b) as proof of (((eq a) b) b000)
% Found ((eq_ref a) b) as proof of (((eq a) b) b000)
% Found eq_ref00:=(eq_ref0 b2):(((eq a) b2) b2)
% Found (eq_ref0 b2) as proof of (((eq a) b2) b1)
% Found ((eq_ref a) b2) as proof of (((eq a) b2) b1)
% Found ((eq_ref a) b2) as proof of (((eq a) b2) b1)
% Found ((eq_ref a) b2) as proof of (((eq a) b2) b1)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b2)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b2)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b2)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b2)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found x3:(P0 a0)
% Instantiate: b0:=a0:a
% Found x3 as proof of (P1 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found x1:((cR Xx) Xy)
% Instantiate: b0:=Xy:a
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x1:((cR Xx) Xy)
% Instantiate: b0:=Xy:a
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x1:((cR Xx) Xy)
% Instantiate: b0:=Xy:a
% Found (fun (x2:((cR Xy) Xz))=> x1) as proof of (P0 b0)
% Found (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1) as proof of (((cR Xy) Xz)->(P0 b0))
% Found (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1) as proof of (((cR Xx) Xy)->(((cR Xy) Xz)->(P0 b0)))
% Found (and_rect00 (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 b0)
% Found ((and_rect0 (P0 b0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 b0)
% Found (((fun (P1:Type) (x1:(((cR Xx) Xy)->(((cR Xy) Xz)->P1)))=> (((((and_rect ((cR Xx) Xy)) ((cR Xy) Xz)) P1) x1) x0)) (P0 b0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 b0)
% Found (((fun (P1:Type) (x1:(((cR Xx) Xy)->(((cR Xy) Xz)->P1)))=> (((((and_rect ((cR Xx) Xy)) ((cR Xy) Xz)) P1) x1) x0)) (P0 b0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 b0)
% Found x1:((cR Xx) Xy)
% Instantiate: a0:=Xy:a
% Found (fun (x2:((cR Xy) Xz))=> x1) as proof of (P0 a0)
% Found (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1) as proof of (((cR Xy) Xz)->(P0 a0))
% Found (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1) as proof of (((cR Xx) Xy)->(((cR Xy) Xz)->(P0 a0)))
% Found (and_rect00 (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 a0)
% Found ((and_rect0 (P0 a0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 a0)
% Found (((fun (P1:Type) (x1:(((cR Xx) Xy)->(((cR Xy) Xz)->P1)))=> (((((and_rect ((cR Xx) Xy)) ((cR Xy) Xz)) P1) x1) x0)) (P0 a0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 a0)
% Found (((fun (P1:Type) (x1:(((cR Xx) Xy)->(((cR Xy) Xz)->P1)))=> (((((and_rect ((cR Xx) Xy)) ((cR Xy) Xz)) P1) x1) x0)) (P0 a0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 a0)
% Found x3:(P0 Xz)
% Instantiate: a0:=Xz:a
% Found x3 as proof of (P1 a0)
% Found x1:((cR Xx) Xy)
% Instantiate: a0:=Xy:a
% Found (fun (x2:((cR Xy) Xz))=> x1) as proof of (P0 a0)
% Found (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1) as proof of (((cR Xy) Xz)->(P0 a0))
% Found (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1) as proof of (((cR Xx) Xy)->(((cR Xy) Xz)->(P0 a0)))
% Found (and_rect00 (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 a0)
% Found ((and_rect0 (P0 a0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 a0)
% Found (((fun (P1:Type) (x1:(((cR Xx) Xy)->(((cR Xy) Xz)->P1)))=> (((((and_rect ((cR Xx) Xy)) ((cR Xy) Xz)) P1) x1) x0)) (P0 a0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 a0)
% Found (((fun (P1:Type) (x1:(((cR Xx) Xy)->(((cR Xy) Xz)->P1)))=> (((((and_rect ((cR Xx) Xy)) ((cR Xy) Xz)) P1) x1) x0)) (P0 a0)) (fun (x1:((cR Xx) Xy)) (x2:((cR Xy) Xz))=> x1)) as proof of (P0 a0)
% Found x1:((cR Xx) Xy)
% Instantiate: a0:=Xy:a
% Found x1 as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x1:((cR Xx) Xy)
% Instantiate: a0:=Xy:a
% Found x1 as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P0 b0)
% Found ((eq_ref a) b0) as proof of (P0 b0)
% Found ((eq_ref a) b0) as proof of (P0 b0)
% Found ((eq_ref a) b0) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (P0 b1)
% Found ((eq_ref a) b1) as proof of (P0 b1)
% Found ((eq_ref a) b1) as proof of (P0 b1)
% Found ((eq_ref a) b1) as proof of (P0 b1)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found x3:(P0 a0)
% Found x3 as proof of (P1 a0)
% Found x3:(P2 b)
% Instantiate: b0:=b:a
% Found x3 as proof of (P3 b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found x3:(P2 b)
% Instantiate: b0:=b:a
% Found x3 as proof of (P3 b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found x3:(P0 b)
% Instantiate: b0:=b:a
% Found x3 as proof of (P1 b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref000:=(eq_ref00 P0):((P0 a0)->(P0 a0))
% Found (eq_ref00 P0) as proof of (P1 a0)
% Found ((eq_ref0 a0) P0) as proof of (P1 a0)
% Found (((eq_ref a) a0) P0) as proof of (P1 a0)
% Found (((eq_ref a) a0) P0) as proof of (P1 a0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (P0 a0)
% Found ((eq_ref a) a0) as proof of (P0 a0)
% Found ((eq_ref a) a0) as proof of (P0 a0)
% Found x3:(P0 Xz)
% Instantiate: b1:=Xz:a
% Found x3 as proof of (P1 b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x3:(P0 b)
% Instantiate: a0:=b:a
% Found x3 as proof of (P1 a0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xz)->(P1 Xz))
% Found (eq_ref00 P1) as proof of (P2 Xz)
% Found ((eq_ref0 Xz) P1) as proof of (P2 Xz)
% Found (((eq_ref a) Xz) P1) as proof of (P2 Xz)
% Found (((eq_ref a) Xz) P1) as proof of (P2 Xz)
% Found x3:(P0 Xz)
% Found x3 as proof of (P1 Xz)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref000:=(eq_ref00 P2):((P2 b0)->(P2 b0))
% Found (eq_ref00 P2) as proof of (P3 b0)
% Found ((eq_ref0 b0) P2) as proof of (P3 b0)
% Found (((eq_ref a) b0) P2) as proof of (P3 b0)
% Found (((eq_ref a) b0) P2) as proof of (P3 b0)
% Found eq_ref000:=(eq_ref00 P2):((P2 Xz)->(P2 Xz))
% Found (eq_ref00 P2) as proof of (P3 Xz)
% Found ((eq_ref0 Xz) P2) as proof of (P3 Xz)
% Found (((eq_ref a) Xz) P2) as proof of (P3 Xz)
% Found (((eq_ref a) Xz) P2) as proof of (P3 Xz)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found x3:(P2 b)
% Instantiate: b0:=b:a
% Found x3 as proof of (P3 b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found x3:(P2 b)
% Instantiate: b0:=b:a
% Found x3 as proof of (P3 b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P2 b0)
% Found ((eq_ref a) b0) as proof of (P2 b0)
% Found ((eq_ref a) b0) as proof of (P2 b0)
% Found ((eq_ref a) b0) as proof of (P2 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P2 b0)
% Found ((eq_ref a) b0) as proof of (P2 b0)
% Found ((eq_ref a) b0) as proof of (P2 b0)
% Found ((eq_ref a) b0) as proof of (P2 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P2 b0)
% Found ((eq_ref a) b) as proof of (P2 b0)
% Found ((eq_ref a) b) as proof of (P2 b0)
% Found ((eq_ref a) b) as proof of (P2 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P2 b0)
% Found ((eq_ref a) b) as proof of (P2 b0)
% Found ((eq_ref a) b) as proof of (P2 b0)
% Found ((eq_ref a) b) as proof of (P2 b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b0)
% Found eq_ref000:=(eq_ref00 P2):((P2 b)->(P2 b))
% Found (eq_ref00 P2) as proof of (P3 b)
% Found ((eq_ref0 b) P2) as proof of (P3 b)
% Found (((eq_ref a) b) P2) as proof of (P3 b)
% Found (((eq_ref a) b) P2) as proof of (P3 b)
% Found eq_ref000:=(eq_ref00 P2):((P2 b)->(P2 b))
% Found (eq_ref00 P2) as proof of (P3 b)
% Found ((eq_ref0 b) P2) as proof of (P3 b)
% Found (((eq_ref a) b) P2) as proof of (P3 b)
% Found (((eq_ref a) b) P2) as proof of (P3 b)
% Found eq_ref000:=(eq_ref00 P0):((P0 b)->(P0 b))
% Found (eq_ref00 P0) as proof of (P1 b)
% Found ((eq_ref0 b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found x3:(P0 b0)
% Instantiate: b1:=b0:a
% Found x3 as proof of (P1 b1)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (P0 b1)
% Found ((eq_ref a) b1) as proof of (P0 b1)
% Found ((eq_ref a) b1) as proof of (P0 b1)
% Found ((eq_ref a) b1) as proof of (P0 b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (P0 b00)
% Found ((eq_ref a) b00) as proof of (P0 b00)
% Found ((eq_ref a) b00) as proof of (P0 b00)
% Found ((eq_ref a) b00) as proof of (P0 b00)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P0 b1)
% Found ((eq_ref a) b0) as proof of (P0 b1)
% Found ((eq_ref a) b0) as proof of (P0 b1)
% Found ((eq_ref a) b0) as proof of (P0 b1)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b01):(((eq a) b01) b01)
% Found (eq_ref0 b01) as proof of (((eq a) b01) b)
% Found ((eq_ref a) b01) as proof of (((eq a) b01) b)
% Found ((eq_ref a) b01) as proof of (((eq a) b01) b)
% Found ((eq_ref a) b01) as proof of (((eq a) b01) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b01)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b01)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b01)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b01)
% Found x3:(P0 b)
% Instantiate: b0:=b:a
% Found x3 as proof of (P1 b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (P0 b1)
% Found ((eq_ref a) b1) as proof of (P0 b1)
% Found ((eq_ref a) b1) as proof of (P0 b1)
% Found ((eq_ref a) b1) as proof of (P0 b1)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xz)->(P0 Xz))
% Found (eq_ref00 P0) as proof of (P1 Xz)
% Found ((eq_ref0 Xz) P0) as proof of (P1 Xz)
% Found (((eq_ref a) Xz) P0) as proof of (P1 Xz)
% Found (((eq_ref a) Xz) P0) as proof of (P1 Xz)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xz)->(P0 Xz))
% Found (eq_ref00 P0) as proof of (P1 Xz)
% Found ((eq_ref0 Xz) P0) as proof of (P1 Xz)
% Found (((eq_ref a) Xz) P0) as proof of (P1 Xz)
% Found (((eq_ref a) Xz) P0) as proof of (P1 Xz)
% Found x3:(P0 b0)
% Instantiate: b00:=b0:a
% Found x3 as proof of (P1 b00)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (P0 Xz)
% Found ((eq_ref a) Xz) as proof of (P0 Xz)
% Found ((eq_ref a) Xz) as proof of (P0 Xz)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found x3:(P0 b0)
% Found x3 as proof of (P1 Xz)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found x3:(P0 b)
% Instantiate: a0:=b:a
% Found x3 as proof of (P1 a0)
% Found x3:(P0 b)
% Instantiate: b1:=b:a
% Found x3 as proof of (P1 b1)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P0 b1)
% Found ((eq_ref a) b) as proof of (P0 b1)
% Found ((eq_ref a) b) as proof of (P0 b1)
% Found ((eq_ref a) b) as proof of (P0 b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) a0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) a0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) a0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) a0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) a0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) a0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) a0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found x3:(P0 b)
% Found x3 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found x3:(P0 Xz)
% Instantiate: b00:=Xz:a
% Found x3 as proof of (P1 b00)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found eq_ref000:=(eq_ref00 P2):((P2 Xz)->(P2 Xz))
% Found (eq_ref00 P2) as proof of (P3 Xz)
% Found ((eq_ref0 Xz) P2) as proof of (P3 Xz)
% Found (((eq_ref a) Xz) P2) as proof of (P3 Xz)
% Found (((eq_ref a) Xz) P2) as proof of (P3 Xz)
% Found eq_ref000:=(eq_ref00 P2):((P2 Xz)->(P2 Xz))
% Found (eq_ref00 P2) as proof of (P3 Xz)
% Found ((eq_ref0 Xz) P2) as proof of (P3 Xz)
% Found (((eq_ref a) Xz) P2) as proof of (P3 Xz)
% Found (((eq_ref a) Xz) P2) as proof of (P3 Xz)
% Found eq_ref000:=(eq_ref00 P0):((P0 b)->(P0 b))
% Found (eq_ref00 P0) as proof of (P1 b)
% Found ((eq_ref0 b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (P0 b00)
% Found ((eq_ref a) Xz) as proof of (P0 b00)
% Found ((eq_ref a) Xz) as proof of (P0 b00)
% Found ((eq_ref a) Xz) as proof of (P0 b00)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref000:=(eq_ref00 P2):((P2 b)->(P2 b))
% Found (eq_ref00 P2) as proof of (P3 b)
% Found ((eq_ref0 b) P2) as proof of (P3 b)
% Found (((eq_ref a) b) P2) as proof of (P3 b)
% Found (((eq_ref a) b) P2) as proof of (P3 b)
% Found eq_ref000:=(eq_ref00 P2):((P2 b)->(P2 b))
% Found (eq_ref00 P2) as proof of (P3 b)
% Found ((eq_ref0 b) P2) as proof of (P3 b)
% Found (((eq_ref a) b) P2) as proof of (P3 b)
% Found (((eq_ref a) b) P2) as proof of (P3 b)
% Found eq_ref000:=(eq_ref00 P0):((P0 b)->(P0 b))
% Found (eq_ref00 P0) as proof of (P1 b)
% Found ((eq_ref0 b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b10):(((eq a) b10) b10)
% Found (eq_ref0 b10) as proof of (((eq a) b10) b0)
% Found ((eq_ref a) b10) as proof of (((eq a) b10) b0)
% Found ((eq_ref a) b10) as proof of (((eq a) b10) b0)
% Found ((eq_ref a) b10) as proof of (((eq a) b10) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b10)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b10)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b10)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b10)
% Found eq_ref000:=(eq_ref00 P0):((P0 b0)->(P0 b0))
% Found (eq_ref00 P0) as proof of (P1 b0)
% Found ((eq_ref0 b0) P0) as proof of (P1 b0)
% Found (((eq_ref a) b0) P0) as proof of (P1 b0)
% Found (((eq_ref a) b0) P0) as proof of (P1 b0)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (P0 b00)
% Found ((eq_ref a) b00) as proof of (P0 b00)
% Found ((eq_ref a) b00) as proof of (P0 b00)
% Found ((eq_ref a) b00) as proof of (P0 b00)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (P0 b00)
% Found ((eq_ref a) b00) as proof of (P0 b00)
% Found ((eq_ref a) b00) as proof of (P0 b00)
% Found ((eq_ref a) b00) as proof of (P0 b00)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b01):(((eq a) b01) b01)
% Found (eq_ref0 b01) as proof of (((eq a) b01) Xz)
% Found ((eq_ref a) b01) as proof of (((eq a) b01) Xz)
% Found ((eq_ref a) b01) as proof of (((eq a) b01) Xz)
% Found ((eq_ref a) b01) as proof of (((eq a) b01) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b01)
% Found ((eq_ref a) b) as proof of (((eq a) b) b01)
% Found ((eq_ref a) b) as proof of (((eq a) b) b01)
% Found ((eq_ref a) b) as proof of (((eq a) b) b01)
% Found eq_ref00:=(eq_ref0 b01):(((eq a) b01) b01)
% Found (eq_ref0 b01) as proof of (((eq a) b01) b00)
% Found ((eq_ref a) b01) as proof of (((eq a) b01) b00)
% Found ((eq_ref a) b01) as proof of (((eq a) b01) b00)
% Found ((eq_ref a) b01) as proof of (((eq a) b01) b00)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b01)
% Found ((eq_ref a) b) as proof of (((eq a) b) b01)
% Found ((eq_ref a) b) as proof of (((eq a) b) b01)
% Found ((eq_ref a) b) as proof of (((eq a) b) b01)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xz)->(P0 Xz))
% Found (eq_ref00 P0) as proof of (P1 Xz)
% Found ((eq_ref0 Xz) P0) as proof of (P1 Xz)
% Found (((eq_ref a) Xz) P0) as proof of (P1 Xz)
% Found (((eq_ref a) Xz) P0) as proof of (P1 Xz)
% Found eq_ref000:=(eq_ref00 P0):((P0 Xz)->(P0 Xz))
% Found (eq_ref00 P0) as proof of (P1 Xz)
% Found ((eq_ref0 Xz) P0) as proof of (P1 Xz)
% Found (((eq_ref a) Xz) P0) as proof of (P1 Xz)
% Found (((eq_ref a) Xz) P0) as proof of (P1 Xz)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (P0 Xz)
% Found ((eq_ref a) Xz) as proof of (P0 Xz)
% Found ((eq_ref a) Xz) as proof of (P0 Xz)
% Found x3:(P0 b)
% Found x3 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found x3:(P0 b)
% Instantiate: b1:=b:a
% Found x3 as proof of (P1 b1)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found x3:(P0 b)
% Instantiate: b1:=b:a
% Found x3 as proof of (P1 b1)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P0 b1)
% Found ((eq_ref a) b) as proof of (P0 b1)
% Found ((eq_ref a) b) as proof of (P0 b1)
% Found ((eq_ref a) b) as proof of (P0 b1)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P0 b1)
% Found ((eq_ref a) b) as proof of (P0 b1)
% Found ((eq_ref a) b) as proof of (P0 b1)
% Found ((eq_ref a) b) as proof of (P0 b1)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) a0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) a0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) a0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) a0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xz)->(P1 Xz))
% Found (eq_ref00 P1) as proof of (P2 Xz)
% Found ((eq_ref0 Xz) P1) as proof of (P2 Xz)
% Found (((eq_ref a) Xz) P1) as proof of (P2 Xz)
% Found (((eq_ref a) Xz) P1) as proof of (P2 Xz)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) a0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) a0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) a0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found x3:(P0 b)
% Found x3 as proof of (P1 b)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xz)->(P1 Xz))
% Found (eq_ref00 P1) as proof of (P2 Xz)
% Found ((eq_ref0 Xz) P1) as proof of (P2 Xz)
% Found (((eq_ref a) Xz) P1) as proof of (P2 Xz)
% Found (((eq_ref a) Xz) P1) as proof of (P2 Xz)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b1)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b1)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b1)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b1)
% Found eq_ref00:=(eq_ref0 b10):(((eq a) b10) b10)
% Found (eq_ref0 b10) as proof of (((eq a) b10) Xz)
% Found ((eq_ref a) b10) as proof of (((eq a) b10) Xz)
% Found ((eq_ref a) b10) as proof of (((eq a) b10) Xz)
% Found ((eq_ref a) b10) as proof of (((eq a) b10) Xz)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b10)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b10)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b10)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b10)
% Found eq_ref000:=(eq_ref00 P0):((P0 b)->(P0 b))
% Found (eq_ref00 P0) as proof of (P1 b)
% Found ((eq_ref0 b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found eq_ref000:=(eq_ref00 P0):((P0 b)->(P0 b))
% Found (eq_ref00 P0) as proof of (P1 b)
% Found ((eq_ref0 b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found x3:(P0 Xz)
% Instantiate: b00:=Xz:a
% Found x3 as proof of (P1 b00)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b2):(((eq a) b2) b2)
% Found (eq_ref0 b2) as proof of (((eq a) b2) b1)
% Found ((eq_ref a) b2) as proof of (((eq a) b2) b1)
% Found ((eq_ref a) b2) as proof of (((eq a) b2) b1)
% Found ((eq_ref a) b2) as proof of (((eq a) b2) b1)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b2)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b2)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b2)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b2)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) Xz)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref000:=(eq_ref00 P0):((P0 b)->(P0 b))
% Found (eq_ref00 P0) as proof of (P1 b)
% Found ((eq_ref0 b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found eq_ref000:=(eq_ref00 P0):((P0 b)->(P0 b))
% Found (eq_ref00 P0) as proof of (P1 b)
% Found ((eq_ref0 b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found (((eq_ref a) b) P0) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 b10):(((eq a) b10) b10)
% Found (eq_ref0 b10) as proof of (((eq a) b10) b0)
% Found ((eq_ref a) b10) as proof of (((eq a) b10) b0)
% Found ((eq_ref a) b10) as proof of (((eq a) b10) b0)
% Found ((eq_ref a) b10) as proof of (((eq a) b10) b0)
% Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% Found (eq_ref0 Xz) as proof of (((eq a) Xz) b10)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b10)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b10)
% Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b10)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xz
% EOF
%------------------------------------------------------------------------------