%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : MSC007^1.003.004 : TPTP v6.1.0. Released v5.4.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n109.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 13:26:41 EDT 2014

% Result   : Timeout 300.05s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : MSC007^1.003.004 : TPTP v6.1.0. Released v5.4.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n109.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 07:24:46 CDT 2014
% % CPUTime  : 300.05 
% 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 0x210cf38>, <kernel.Type object at 0x210cb00>) of role type named hole
% Using role type
% Declaring hole:Type
% FOF formula (<kernel.Constant object at 0x210ce60>, <kernel.Type object at 0x26031b8>) of role type named pigeon
% Using role type
% Declaring pigeon:Type
% FOF formula (<kernel.Constant object at 0x2603320>, <kernel.Constant object at 0x210cf38>) of role type named hole1
% Using role type
% Declaring hole1:hole
% FOF formula (<kernel.Constant object at 0x210c290>, <kernel.Constant object at 0x210cf38>) of role type named hole2
% Using role type
% Declaring hole2:hole
% FOF formula (<kernel.Constant object at 0x210ce60>, <kernel.Constant object at 0x210cb00>) of role type named hole3
% Using role type
% Declaring hole3:hole
% FOF formula (<kernel.Constant object at 0x210c290>, <kernel.Constant object at 0x2246b48>) of role type named pigeon1
% Using role type
% Declaring pigeon1:pigeon
% FOF formula (<kernel.Constant object at 0x210cb00>, <kernel.Constant object at 0x2246cb0>) of role type named pigeon2
% Using role type
% Declaring pigeon2:pigeon
% FOF formula (<kernel.Constant object at 0x210c290>, <kernel.Constant object at 0x2246d40>) of role type named pigeon3
% Using role type
% Declaring pigeon3:pigeon
% FOF formula (<kernel.Constant object at 0x210cf38>, <kernel.Constant object at 0x2246d40>) of role type named pigeon4
% Using role type
% Declaring pigeon4:pigeon
% FOF formula (<kernel.Constant object at 0x210cf38>, <kernel.DependentProduct object at 0x2246878>) of role type named pigeon_hole_t
% Using role type
% Declaring pigeon_hole:(pigeon->hole)
% FOF formula (forall (P:(hole->Prop)), (((and ((and (P hole1)) (P hole2))) (P hole3))->(forall (X:hole), (P X)))) of role axiom named holecover
% A new axiom: (forall (P:(hole->Prop)), (((and ((and (P hole1)) (P hole2))) (P hole3))->(forall (X:hole), (P X))))
% FOF formula (not (((eq pigeon) pigeon1) pigeon2)) of role axiom named pigeon1pigeon2
% A new axiom: (not (((eq pigeon) pigeon1) pigeon2))
% FOF formula (not (((eq pigeon) pigeon1) pigeon3)) of role axiom named pigeon1pigeon3
% A new axiom: (not (((eq pigeon) pigeon1) pigeon3))
% FOF formula (not (((eq pigeon) pigeon2) pigeon3)) of role axiom named pigeon2pigeon3
% A new axiom: (not (((eq pigeon) pigeon2) pigeon3))
% FOF formula (not (((eq pigeon) pigeon1) pigeon4)) of role axiom named pigeon1pigeon4
% A new axiom: (not (((eq pigeon) pigeon1) pigeon4))
% FOF formula (not (((eq pigeon) pigeon2) pigeon4)) of role axiom named pigeon2pigeon4
% A new axiom: (not (((eq pigeon) pigeon2) pigeon4))
% FOF formula (not (((eq pigeon) pigeon3) pigeon4)) of role axiom named pigeon3pigeon4
% A new axiom: (not (((eq pigeon) pigeon3) pigeon4))
% FOF formula ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))) of role conjecture named sharing_a_hole
% Conjecture to prove = ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))):Prop
% We need to prove ['((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))']
% Parameter hole:Type.
% Parameter pigeon:Type.
% Parameter hole1:hole.
% Parameter hole2:hole.
% Parameter hole3:hole.
% Parameter pigeon1:pigeon.
% Parameter pigeon2:pigeon.
% Parameter pigeon3:pigeon.
% Parameter pigeon4:pigeon.
% Parameter pigeon_hole:(pigeon->hole).
% Axiom holecover:(forall (P:(hole->Prop)), (((and ((and (P hole1)) (P hole2))) (P hole3))->(forall (X:hole), (P X)))).
% Axiom pigeon1pigeon2:(not (((eq pigeon) pigeon1) pigeon2)).
% Axiom pigeon1pigeon3:(not (((eq pigeon) pigeon1) pigeon3)).
% Axiom pigeon2pigeon3:(not (((eq pigeon) pigeon2) pigeon3)).
% Axiom pigeon1pigeon4:(not (((eq pigeon) pigeon1) pigeon4)).
% Axiom pigeon2pigeon4:(not (((eq pigeon) pigeon2) pigeon4)).
% Axiom pigeon3pigeon4:(not (((eq pigeon) pigeon3) pigeon4)).
% Trying to prove ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found eq_ref00:=(eq_ref0 (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))):(((eq (pigeon->Prop)) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found (eq_ref0 (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))) as proof of (((eq (pigeon->Prop)) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))) b)
% Found ((eq_ref (pigeon->Prop)) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))) as proof of (((eq (pigeon->Prop)) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))) b)
% Found ((eq_ref (pigeon->Prop)) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))) as proof of (((eq (pigeon->Prop)) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))) b)
% Found ((eq_ref (pigeon->Prop)) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))) as proof of (((eq (pigeon->Prop)) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))) b)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole x)) (pigeon_hole Y))) (not (((eq pigeon) x) Y))))))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole x)) (pigeon_hole Y))) (not (((eq pigeon) x) Y))))))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole x)) (pigeon_hole Y))) (not (((eq pigeon) x) Y))))))
% Found (fun (x:pigeon)=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole x)) (pigeon_hole Y))) (not (((eq pigeon) x) Y))))))
% Found (fun (x:pigeon)=> ((eq_ref Prop) (f x))) as proof of (forall (x:pigeon), (((eq Prop) (f x)) ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole x)) (pigeon_hole Y))) (not (((eq pigeon) x) Y)))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole x)) (pigeon_hole Y))) (not (((eq pigeon) x) Y))))))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole x)) (pigeon_hole Y))) (not (((eq pigeon) x) Y))))))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole x)) (pigeon_hole Y))) (not (((eq pigeon) x) Y))))))
% Found (fun (x:pigeon)=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole x)) (pigeon_hole Y))) (not (((eq pigeon) x) Y))))))
% Found (fun (x:pigeon)=> ((eq_ref Prop) (f x))) as proof of (forall (x:pigeon), (((eq Prop) (f x)) ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole x)) (pigeon_hole Y))) (not (((eq pigeon) x) Y)))))))
% Found eq_ref00:=(eq_ref0 ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))):(((eq Prop) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))
% Found (eq_ref0 ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) as proof of (((eq Prop) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) b)
% Found ((eq_ref Prop) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) as proof of (((eq Prop) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) b)
% Found ((eq_ref Prop) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) as proof of (((eq Prop) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) b)
% Found ((eq_ref Prop) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) as proof of (((eq Prop) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) b)
% Found eq_ref00:=(eq_ref0 ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))):(((eq Prop) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))
% Found (eq_ref0 ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) as proof of (((eq Prop) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) b)
% Found ((eq_ref Prop) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) as proof of (((eq Prop) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) b)
% Found ((eq_ref Prop) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) as proof of (((eq Prop) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) b)
% Found ((eq_ref Prop) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) as proof of (((eq Prop) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) b)
% Found eq_ref00:=(eq_ref0 ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))):(((eq Prop) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))
% Found (eq_ref0 ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) as proof of (((eq Prop) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) b)
% Found ((eq_ref Prop) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) as proof of (((eq Prop) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) b)
% Found ((eq_ref Prop) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) as proof of (((eq Prop) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) b)
% Found ((eq_ref Prop) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) as proof of (((eq Prop) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) b)
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of b
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of b
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of b
% Found eq_ref00:=(eq_ref0 ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))):(((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))))
% Found (eq_ref0 ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) b)
% Found ((eq_ref Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) b)
% Found ((eq_ref Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) b)
% Found ((eq_ref Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) b)
% Found eq_ref00:=(eq_ref0 ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))):(((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))))
% Found (eq_ref0 ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) b)
% Found ((eq_ref Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) b)
% Found ((eq_ref Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) b)
% Found ((eq_ref Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) b)
% Found eq_ref00:=(eq_ref0 ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))):(((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))))
% Found (eq_ref0 ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) b)
% Found ((eq_ref Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) b)
% Found ((eq_ref Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) b)
% Found ((eq_ref Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) b)
% Found eq_ref00:=(eq_ref0 ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))):(((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))))
% Found (eq_ref0 ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) b)
% Found ((eq_ref Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) b)
% Found ((eq_ref Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) b)
% Found ((eq_ref Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) b)
% Found eq_ref00:=(eq_ref0 ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))):(((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))))
% Found (eq_ref0 ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) b)
% Found ((eq_ref Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) b)
% Found ((eq_ref Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) b)
% Found ((eq_ref Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) b)
% Found eq_ref00:=(eq_ref0 ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))):(((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))))
% Found (eq_ref0 ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) b)
% Found ((eq_ref Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) b)
% Found ((eq_ref Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) b)
% Found ((eq_ref Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) as proof of (((eq Prop) ((and ((and ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) ((ex pigeon) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y))))))))) b)
% Found eta_expansion000:=(eta_expansion00 a):(((eq (pigeon->Prop)) a) (fun (x:pigeon)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found ((eta_expansion0 Prop) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found (((eta_expansion pigeon) Prop) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found (((eta_expansion pigeon) Prop) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found (((eta_expansion pigeon) Prop) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found eta_expansion000:=(eta_expansion00 a):(((eq (pigeon->Prop)) a) (fun (x:pigeon)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found ((eta_expansion0 Prop) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found (((eta_expansion pigeon) Prop) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found (((eta_expansion pigeon) Prop) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found (((eta_expansion pigeon) Prop) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found eta_expansion000:=(eta_expansion00 a):(((eq (pigeon->Prop)) a) (fun (x:pigeon)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found ((eta_expansion0 Prop) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found (((eta_expansion pigeon) Prop) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found (((eta_expansion pigeon) Prop) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found (((eta_expansion pigeon) Prop) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (pigeon->Prop)) a) (fun (x:pigeon)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found ((eta_expansion_dep0 (fun (x1:pigeon)=> Prop)) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found (((eta_expansion_dep pigeon) (fun (x1:pigeon)=> Prop)) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found (((eta_expansion_dep pigeon) (fun (x1:pigeon)=> Prop)) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found (((eta_expansion_dep pigeon) (fun (x1:pigeon)=> Prop)) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found eq_ref00:=(eq_ref0 a):(((eq (pigeon->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found ((eq_ref (pigeon->Prop)) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found ((eq_ref (pigeon->Prop)) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found ((eq_ref (pigeon->Prop)) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found eq_ref00:=(eq_ref0 a):(((eq (pigeon->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found ((eq_ref (pigeon->Prop)) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found ((eq_ref (pigeon->Prop)) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found ((eq_ref (pigeon->Prop)) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found eq_ref00:=(eq_ref0 a):(((eq (pigeon->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found ((eq_ref (pigeon->Prop)) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found ((eq_ref (pigeon->Prop)) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found ((eq_ref (pigeon->Prop)) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (pigeon->Prop)) a) (fun (x:pigeon)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found ((eta_expansion_dep0 (fun (x1:pigeon)=> Prop)) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found (((eta_expansion_dep pigeon) (fun (x1:pigeon)=> Prop)) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found (((eta_expansion_dep pigeon) (fun (x1:pigeon)=> Prop)) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found (((eta_expansion_dep pigeon) (fun (x1:pigeon)=> Prop)) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found eta_expansion000:=(eta_expansion00 a):(((eq (pigeon->Prop)) a) (fun (x:pigeon)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found ((eta_expansion0 Prop) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found (((eta_expansion pigeon) Prop) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found (((eta_expansion pigeon) Prop) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found (((eta_expansion pigeon) Prop) a) as proof of (((eq (pigeon->Prop)) a) (fun (X:pigeon)=> ((ex pigeon) (fun (Y:pigeon)=> ((and (((eq hole) (pigeon_hole X)) (pigeon_hole Y))) (not (((eq pigeon) X) Y)))))))
% Found pigeon2pigeon4:(not (((eq pigeon) pigeon2) pigeon4))
% Instantiate: x:=pigeon2:pigeon;x0:=pigeon4:pigeon
% Found pigeon2pigeon4 as proof of (not (((eq pigeon) x) x0))
% Found eq_ref00:=(eq_ref0 (pigeon_hole x)):(((eq hole) (pigeon_hole x)) (pigeon_hole x))
% Found (eq_ref0 (pigeon_hole x)) as proof of (((eq hole) (pigeon_hole x)) (pigeon_hole x0))
% Found ((eq_ref hole) (pigeon_hole x)) as proof of (((eq hole) (pigeon_hole x)) (pigeon_hole x0))
% Found ((eq_ref hole) (pigeon_hole x)) as proof of (((eq hole) (pigeon_hole x)) (pigeon_hole x0))
% Found ((eq_ref hole) (pigeon_hole x)) as proof of (((eq hole) (pigeon_hole x)) (pigeon_hole x0))
% Found choice_operator:=(fun (A:Type) (a:A)=> ((((classical_choice (A->Prop)) A) (fun (x3:(A->Prop))=> x3)) a)):(forall (A:Type), (A->((ex ((A->Prop)->A)) (fun (co:((A->Prop)->A))=> (forall (P:(A->Prop)), (((ex A) (fun (x:A)=> (P x)))->(P (co P))))))))
% Instantiate: b:=(forall (A:Type), (A->((ex ((A->Prop)->A)) (fun (co:((A->Prop)->A))=> (forall (P:(A->Prop)), (((ex A) (fun (x:A)=> (P x)))->(P (co P)))))))):Prop
% Found choice_operator as proof of b
% Found choice_operator:=(fun (A:Type) (a:A)=> ((((classical_choice (A->Prop)) A) (fun (x3:(A->Prop))=> x3)) a)):(forall (A:Type), (A->((ex ((A->Prop)->A)) (fun (co:((A->Prop)->A))=> (forall (P:(A->Prop)), (((ex A) (fun (x:A)=> (P x)))->(P (co P))))))))
% Instantiate: b:=(forall (A:Type), (A->((ex ((A->Prop)->A)) (fun (co:((A->Prop)->A))=> (forall (P:(A->Prop)), (((ex A) (fun (x:A)=> (P x)))->(P (co P)))))))):Prop
% Found choice_operator as proof of b
% Found choice_operator:=(fun (A:Type) (a:A)=> ((((classical_choice (A->Prop)) A) (fun (x3:(A->Prop))=> x3)) a)):(forall (A:Type), (A->((ex ((A->Prop)->A)) (fun (co:((A->Prop)->A))=> (forall (P:(A->Prop)), (((ex A) (fun (x:A)=> (P x)))->(P (co P))))))))
% Instantiate: b:=(forall (A:Type), (A->((ex ((A->Prop)->A)) (fun (co:((A->Prop)->A))=> (forall (P:(A->Prop)), (((ex A) (fun (x:A)=> (P x)))->(P (co P)))))))):Prop
% Found choice_operator as proof of b
% Found choice_operator:=(fun (A:Type) (a:A)=> ((((classical_choice (A->Prop)) A) (fun (x3:(A->Prop))=> x3)) a)):(forall (A:Type), (A->((ex ((A->Prop)->A)) (fu
% EOF
%------------------------------------------------------------------------------