TSTP Solution File: SEU886^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEU886^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n115.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 13:33:20 EDT 2014
% Result : Theorem 32.93s
% Output : Proof 32.93s
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----ERROR: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEU886^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n115.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 11:40:46 CDT 2014
% % CPUTime : 32.93
% 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 0x1fce9e0>, <kernel.Type object at 0x1fce518>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (<kernel.Constant object at 0x2423098>, <kernel.Type object at 0x1fce7e8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x1fced40>, <kernel.DependentProduct object at 0x1fce368>) of role type named f
% Using role type
% Declaring f:(b->a)
% FOF formula (<kernel.Constant object at 0x1fce518>, <kernel.DependentProduct object at 0x1fcec20>) of role type named y
% Using role type
% Declaring y:(b->Prop)
% FOF formula (<kernel.Constant object at 0x1fce9e0>, <kernel.DependentProduct object at 0x1fce758>) of role type named x
% Using role type
% Declaring x:(b->Prop)
% FOF formula (forall (Xx0:a), (((ex b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))))->((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))))) of role conjecture named cX5203_pme
% Conjecture to prove = (forall (Xx0:a), (((ex b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))))->((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))))):Prop
% Parameter b_DUMMY:b.
% Parameter a_DUMMY:a.
% We need to prove ['(forall (Xx0:a), (((ex b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))))->((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))))']
% Parameter b:Type.
% Parameter a:Type.
% Parameter f:(b->a).
% Parameter y:(b->Prop).
% Parameter x:(b->Prop).
% Trying to prove (forall (Xx0:a), (((ex b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))))->((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq (b->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (b->Prop)) a0) (fun (x:b)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b0
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) b0)
% Found x0:((ex b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))))
% Instantiate: a0:=(fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))):(b->Prop)
% Found x0 as proof of ((ex b) a0)
% Found x0:((ex b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))))
% Instantiate: a0:=(fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))):(b->Prop)
% Found x0 as proof of ((ex b) a0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (b->Prop)) a0) (fun (x:b)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found ((eta_expansion_dep0 (fun (x3:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (b->Prop)) a0) (fun (x:b)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found ((eta_expansion_dep0 (fun (x3:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b0
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) b0)
% Found x02:(((eq a) Xx0) (f x00))
% Instantiate: x2:=x00:b
% Found x02 as proof of (((eq a) Xx0) (f x2))
% Found x02:(((eq a) Xx0) (f x00))
% Instantiate: x03:=x00:b
% Found x02 as proof of (((eq a) Xx0) (f x03))
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b0
% Found x0:((ex b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))))
% Instantiate: a0:=(fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))):(b->Prop)
% Found x0 as proof of ((ex b) a0)
% Found x0:((ex b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))))
% Instantiate: a0:=(fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))):(b->Prop)
% Found x0 as proof of ((ex b) a0)
% Found x0:((ex b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))):(b->Prop)
% Found x0 as proof of (P b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) (fun (x0:b)=> ((and (x x0)) (((eq a) Xx0) (f x0)))))
% Found (eta_expansion00 (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) b0)
% Found x0:((ex b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))))
% Instantiate: b0:=(fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))):(b->Prop)
% Found x0 as proof of (P b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) (fun (x:b)=> ((and (y x)) (((eq a) Xx0) (f x)))))
% Found (eta_expansion00 (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) b0)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (b->Prop)) a0) (fun (x:b)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (b->Prop)) a0) (fun (x:b)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found ((eta_expansion_dep0 (fun (x3:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b0
% Found x0:((ex b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))))
% Instantiate: f0:=(fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))):(b->Prop)
% Found x0 as proof of (P f0)
% Found x0:((ex b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))))
% Instantiate: f0:=(fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))):(b->Prop)
% Found x0 as proof of (P f0)
% Found x0:((ex b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))))
% Instantiate: f0:=(fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))):(b->Prop)
% Found x0 as proof of (P f0)
% Found x0:((ex b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))))
% Instantiate: f0:=(fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))):(b->Prop)
% Found x0 as proof of (P f0)
% Found x0:((ex b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))))
% Instantiate: a0:=(fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))):(b->Prop)
% Found x0 as proof of ((ex b) a0)
% Found x0:((ex b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))))
% Instantiate: a0:=(fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))):(b->Prop)
% Found x0 as proof of ((ex b) a0)
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and (y x1)) (((eq a) Xx0) (f x1))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and (y x1)) (((eq a) Xx0) (f x1))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and (y x1)) (((eq a) Xx0) (f x1))))
% Found (fun (x1:b)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) ((and (y x1)) (((eq a) Xx0) (f x1))))
% Found (fun (x1:b)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:b), (((eq Prop) (f0 x)) ((and (y x)) (((eq a) Xx0) (f x)))))
% Found eq_ref00:=(eq_ref0 (f0 x00)):(((eq Prop) (f0 x00)) (f0 x00))
% Found (eq_ref0 (f0 x00)) as proof of (((eq Prop) (f0 x00)) ((and (x x00)) (((eq a) Xx0) (f x00))))
% Found ((eq_ref Prop) (f0 x00)) as proof of (((eq Prop) (f0 x00)) ((and (x x00)) (((eq a) Xx0) (f x00))))
% Found ((eq_ref Prop) (f0 x00)) as proof of (((eq Prop) (f0 x00)) ((and (x x00)) (((eq a) Xx0) (f x00))))
% Found (fun (x00:b)=> ((eq_ref Prop) (f0 x00))) as proof of (((eq Prop) (f0 x00)) ((and (x x00)) (((eq a) Xx0) (f x00))))
% Found (fun (x00:b)=> ((eq_ref Prop) (f0 x00))) as proof of (forall (x0:b), (((eq Prop) (f0 x0)) ((and (x x0)) (((eq a) Xx0) (f x0)))))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and (y x1)) (((eq a) Xx0) (f x1))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and (y x1)) (((eq a) Xx0) (f x1))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and (y x1)) (((eq a) Xx0) (f x1))))
% Found (fun (x1:b)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) ((and (y x1)) (((eq a) Xx0) (f x1))))
% Found (fun (x1:b)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:b), (((eq Prop) (f0 x)) ((and (y x)) (((eq a) Xx0) (f x)))))
% Found eq_ref00:=(eq_ref0 (f0 x00)):(((eq Prop) (f0 x00)) (f0 x00))
% Found (eq_ref0 (f0 x00)) as proof of (((eq Prop) (f0 x00)) ((and (x x00)) (((eq a) Xx0) (f x00))))
% Found ((eq_ref Prop) (f0 x00)) as proof of (((eq Prop) (f0 x00)) ((and (x x00)) (((eq a) Xx0) (f x00))))
% Found ((eq_ref Prop) (f0 x00)) as proof of (((eq Prop) (f0 x00)) ((and (x x00)) (((eq a) Xx0) (f x00))))
% Found (fun (x00:b)=> ((eq_ref Prop) (f0 x00))) as proof of (((eq Prop) (f0 x00)) ((and (x x00)) (((eq a) Xx0) (f x00))))
% Found (fun (x00:b)=> ((eq_ref Prop) (f0 x00))) as proof of (forall (x0:b), (((eq Prop) (f0 x0)) ((and (x x0)) (((eq a) Xx0) (f x0)))))
% Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))
% Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) b0)
% Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) b0)
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b0
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b0
% Found x02:(((eq a) Xx0) (f x00))
% Instantiate: x2:=x00:b
% Found x02 as proof of (((eq a) Xx0) (f x2))
% Found x02:(((eq a) Xx0) (f x00))
% Instantiate: x03:=x00:b
% Found x02 as proof of (((eq a) Xx0) (f x03))
% Found x03:(x x00)
% Instantiate: x05:=x00:b
% Found x03 as proof of (x x05)
% Found x02:(((eq a) Xx0) (f x00))
% Instantiate: x05:=x00:b
% Found x02 as proof of (((eq a) Xx0) (f x05))
% Found ((conj10 x03) x02) as proof of ((and (x x05)) (((eq a) Xx0) (f x05)))
% Found (((conj1 (((eq a) Xx0) (f x05))) x03) x02) as proof of ((and (x x05)) (((eq a) Xx0) (f x05)))
% Found ((((conj (x x05)) (((eq a) Xx0) (f x05))) x03) x02) as proof of ((and (x x05)) (((eq a) Xx0) (f x05)))
% Found ((((conj (x x05)) (((eq a) Xx0) (f x05))) x03) x02) as proof of ((and (x x05)) (((eq a) Xx0) (f x05)))
% Found (ex_intro000 ((((conj (x x05)) (((eq a) Xx0) (f x05))) x03) x02)) as proof of ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))
% Found ((ex_intro00 x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02)) as proof of ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))
% Found (((ex_intro0 (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02)) as proof of ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))
% Found ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02)) as proof of ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))
% Found ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02)) as proof of ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))
% Found x04:(y x00)
% Instantiate: x2:=x00:b
% Found x04 as proof of (y x2)
% Found x02:(((eq a) Xx0) (f x00))
% Instantiate: x2:=x00:b
% Found x02 as proof of (((eq a) Xx0) (f x2))
% Found ((conj10 x04) x02) as proof of ((and (y x2)) (((eq a) Xx0) (f x2)))
% Found (((conj1 (((eq a) Xx0) (f x2))) x04) x02) as proof of ((and (y x2)) (((eq a) Xx0) (f x2)))
% Found ((((conj (y x2)) (((eq a) Xx0) (f x2))) x04) x02) as proof of ((and (y x2)) (((eq a) Xx0) (f x2)))
% Found ((((conj (y x2)) (((eq a) Xx0) (f x2))) x04) x02) as proof of ((and (y x2)) (((eq a) Xx0) (f x2)))
% Found (ex_intro000 ((((conj (y x2)) (((eq a) Xx0) (f x2))) x04) x02)) as proof of ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found ((ex_intro00 x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02)) as proof of ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found (((ex_intro0 (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02)) as proof of ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02)) as proof of ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02)) as proof of ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))
% Found ((conj00 ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02))) ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02))) as proof of ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))
% Found (((conj0 ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02))) ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02))) as proof of ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))
% Found ((((conj ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02))) ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02))) as proof of ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))
% Found (fun (x04:(y x00))=> ((((conj ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02))) ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02)))) as proof of ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))
% Found (fun (x03:(x x00)) (x04:(y x00))=> ((((conj ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02))) ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02)))) as proof of ((y x00)->((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))))
% Found (fun (x03:(x x00)) (x04:(y x00))=> ((((conj ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02))) ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02)))) as proof of ((x x00)->((y x00)->((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))))
% Found (and_rect10 (fun (x03:(x x00)) (x04:(y x00))=> ((((conj ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02))) ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02))))) as proof of ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))
% Found ((and_rect1 ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x03:(x x00)) (x04:(y x00))=> ((((conj ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02))) ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02))))) as proof of ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))
% Found (((fun (P:Type) (x2:((x x00)->((y x00)->P)))=> (((((and_rect (x x00)) (y x00)) P) x2) x01)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x03:(x x00)) (x04:(y x00))=> ((((conj ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02))) ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02))))) as proof of ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))
% Found (fun (x02:(((eq a) Xx0) (f x00)))=> (((fun (P:Type) (x2:((x x00)->((y x00)->P)))=> (((((and_rect (x x00)) (y x00)) P) x2) x01)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x03:(x x00)) (x04:(y x00))=> ((((conj ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02))) ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02)))))) as proof of ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))
% Found (fun (x01:((and (x x00)) (y x00))) (x02:(((eq a) Xx0) (f x00)))=> (((fun (P:Type) (x2:((x x00)->((y x00)->P)))=> (((((and_rect (x x00)) (y x00)) P) x2) x01)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x03:(x x00)) (x04:(y x00))=> ((((conj ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02))) ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02)))))) as proof of ((((eq a) Xx0) (f x00))->((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))))
% Found (fun (x01:((and (x x00)) (y x00))) (x02:(((eq a) Xx0) (f x00)))=> (((fun (P:Type) (x2:((x x00)->((y x00)->P)))=> (((((and_rect (x x00)) (y x00)) P) x2) x01)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x03:(x x00)) (x04:(y x00))=> ((((conj ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02))) ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02)))))) as proof of (((and (x x00)) (y x00))->((((eq a) Xx0) (f x00))->((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))))
% Found (and_rect00 (fun (x01:((and (x x00)) (y x00))) (x02:(((eq a) Xx0) (f x00)))=> (((fun (P:Type) (x2:((x x00)->((y x00)->P)))=> (((((and_rect (x x00)) (y x00)) P) x2) x01)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x03:(x x00)) (x04:(y x00))=> ((((conj ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02))) ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02))))))) as proof of ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))
% Found ((and_rect0 ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x01:((and (x x00)) (y x00))) (x02:(((eq a) Xx0) (f x00)))=> (((fun (P:Type) (x2:((x x00)->((y x00)->P)))=> (((((and_rect (x x00)) (y x00)) P) x2) x01)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x03:(x x00)) (x04:(y x00))=> ((((conj ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02))) ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02))))))) as proof of ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))
% Found (((fun (P:Type) (x2:(((and (x x00)) (y x00))->((((eq a) Xx0) (f x00))->P)))=> (((((and_rect ((and (x x00)) (y x00))) (((eq a) Xx0) (f x00))) P) x2) x1)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x01:((and (x x00)) (y x00))) (x02:(((eq a) Xx0) (f x00)))=> (((fun (P:Type) (x2:((x x00)->((y x00)->P)))=> (((((and_rect (x x00)) (y x00)) P) x2) x01)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x03:(x x00)) (x04:(y x00))=> ((((conj ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02))) ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02))))))) as proof of ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))
% Found (fun (x1:((and ((and (x x00)) (y x00))) (((eq a) Xx0) (f x00))))=> (((fun (P:Type) (x2:(((and (x x00)) (y x00))->((((eq a) Xx0) (f x00))->P)))=> (((((and_rect ((and (x x00)) (y x00))) (((eq a) Xx0) (f x00))) P) x2) x1)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x01:((and (x x00)) (y x00))) (x02:(((eq a) Xx0) (f x00)))=> (((fun (P:Type) (x2:((x x00)->((y x00)->P)))=> (((((and_rect (x x00)) (y x00)) P) x2) x01)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x03:(x x00)) (x04:(y x00))=> ((((conj ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02))) ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02)))))))) as proof of ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))
% Found (fun (x00:b) (x1:((and ((and (x x00)) (y x00))) (((eq a) Xx0) (f x00))))=> (((fun (P:Type) (x2:(((and (x x00)) (y x00))->((((eq a) Xx0) (f x00))->P)))=> (((((and_rect ((and (x x00)) (y x00))) (((eq a) Xx0) (f x00))) P) x2) x1)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x01:((and (x x00)) (y x00))) (x02:(((eq a) Xx0) (f x00)))=> (((fun (P:Type) (x2:((x x00)->((y x00)->P)))=> (((((and_rect (x x00)) (y x00)) P) x2) x01)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x03:(x x00)) (x04:(y x00))=> ((((conj ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02))) ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02)))))))) as proof of (((and ((and (x x00)) (y x00))) (((eq a) Xx0) (f x00)))->((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))))
% Found (fun (x00:b) (x1:((and ((and (x x00)) (y x00))) (((eq a) Xx0) (f x00))))=> (((fun (P:Type) (x2:(((and (x x00)) (y x00))->((((eq a) Xx0) (f x00))->P)))=> (((((and_rect ((and (x x00)) (y x00))) (((eq a) Xx0) (f x00))) P) x2) x1)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x01:((and (x x00)) (y x00))) (x02:(((eq a) Xx0) (f x00)))=> (((fun (P:Type) (x2:((x x00)->((y x00)->P)))=> (((((and_rect (x x00)) (y x00)) P) x2) x01)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x03:(x x00)) (x04:(y x00))=> ((((conj ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02))) ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02)))))))) as proof of (forall (x0:b), (((and ((and (x x0)) (y x0))) (((eq a) Xx0) (f x0)))->((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))))
% Found (ex_ind00 (fun (x00:b) (x1:((and ((and (x x00)) (y x00))) (((eq a) Xx0) (f x00))))=> (((fun (P:Type) (x2:(((and (x x00)) (y x00))->((((eq a) Xx0) (f x00))->P)))=> (((((and_rect ((and (x x00)) (y x00))) (((eq a) Xx0) (f x00))) P) x2) x1)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x01:((and (x x00)) (y x00))) (x02:(((eq a) Xx0) (f x00)))=> (((fun (P:Type) (x2:((x x00)->((y x00)->P)))=> (((((and_rect (x x00)) (y x00)) P) x2) x01)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x03:(x x00)) (x04:(y x00))=> ((((conj ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02))) ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02))))))))) as proof of ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))
% Found ((ex_ind0 ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x00:b) (x1:((and ((and (x x00)) (y x00))) (((eq a) Xx0) (f x00))))=> (((fun (P:Type) (x2:(((and (x x00)) (y x00))->((((eq a) Xx0) (f x00))->P)))=> (((((and_rect ((and (x x00)) (y x00))) (((eq a) Xx0) (f x00))) P) x2) x1)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x01:((and (x x00)) (y x00))) (x02:(((eq a) Xx0) (f x00)))=> (((fun (P:Type) (x2:((x x00)->((y x00)->P)))=> (((((and_rect (x x00)) (y x00)) P) x2) x01)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x03:(x x00)) (x04:(y x00))=> ((((conj ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02))) ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02))))))))) as proof of ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))
% Found (((fun (P:Prop) (x1:(forall (x0:b), (((and ((and (x x0)) (y x0))) (((eq a) Xx0) (f x0)))->P)))=> (((((ex_ind b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt))))) P) x1) x0)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x00:b) (x1:((and ((and (x x00)) (y x00))) (((eq a) Xx0) (f x00))))=> (((fun (P:Type) (x2:(((and (x x00)) (y x00))->((((eq a) Xx0) (f x00))->P)))=> (((((and_rect ((and (x x00)) (y x00))) (((eq a) Xx0) (f x00))) P) x2) x1)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x01:((and (x x00)) (y x00))) (x02:(((eq a) Xx0) (f x00)))=> (((fun (P:Type) (x2:((x x00)->((y x00)->P)))=> (((((and_rect (x x00)) (y x00)) P) x2) x01)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x03:(x x00)) (x04:(y x00))=> ((((conj ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02))) ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02))))))))) as proof of ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))
% Found (fun (x0:((ex b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt))))))=> (((fun (P:Prop) (x1:(forall (x0:b), (((and ((and (x x0)) (y x0))) (((eq a) Xx0) (f x0)))->P)))=> (((((ex_ind b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt))))) P) x1) x0)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x00:b) (x1:((and ((and (x x00)) (y x00))) (((eq a) Xx0) (f x00))))=> (((fun (P:Type) (x2:(((and (x x00)) (y x00))->((((eq a) Xx0) (f x00))->P)))=> (((((and_rect ((and (x x00)) (y x00))) (((eq a) Xx0) (f x00))) P) x2) x1)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x01:((and (x x00)) (y x00))) (x02:(((eq a) Xx0) (f x00)))=> (((fun (P:Type) (x2:((x x00)->((y x00)->P)))=> (((((and_rect (x x00)) (y x00)) P) x2) x01)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x03:(x x00)) (x04:(y x00))=> ((((conj ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02))) ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02)))))))))) as proof of ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))
% Found (fun (Xx0:a) (x0:((ex b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt))))))=> (((fun (P:Prop) (x1:(forall (x0:b), (((and ((and (x x0)) (y x0))) (((eq a) Xx0) (f x0)))->P)))=> (((((ex_ind b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt))))) P) x1) x0)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x00:b) (x1:((and ((and (x x00)) (y x00))) (((eq a) Xx0) (f x00))))=> (((fun (P:Type) (x2:(((and (x x00)) (y x00))->((((eq a) Xx0) (f x00))->P)))=> (((((and_rect ((and (x x00)) (y x00))) (((eq a) Xx0) (f x00))) P) x2) x1)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x01:((and (x x00)) (y x00))) (x02:(((eq a) Xx0) (f x00)))=> (((fun (P:Type) (x2:((x x00)->((y x00)->P)))=> (((((and_rect (x x00)) (y x00)) P) x2) x01)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x03:(x x00)) (x04:(y x00))=> ((((conj ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02))) ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02)))))))))) as proof of (((ex b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))))->((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))))
% Found (fun (Xx0:a) (x0:((ex b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt))))))=> (((fun (P:Prop) (x1:(forall (x0:b), (((and ((and (x x0)) (y x0))) (((eq a) Xx0) (f x0)))->P)))=> (((((ex_ind b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt))))) P) x1) x0)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x00:b) (x1:((and ((and (x x00)) (y x00))) (((eq a) Xx0) (f x00))))=> (((fun (P:Type) (x2:(((and (x x00)) (y x00))->((((eq a) Xx0) (f x00))->P)))=> (((((and_rect ((and (x x00)) (y x00))) (((eq a) Xx0) (f x00))) P) x2) x1)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x01:((and (x x00)) (y x00))) (x02:(((eq a) Xx0) (f x00)))=> (((fun (P:Type) (x2:((x x00)->((y x00)->P)))=> (((((and_rect (x x00)) (y x00)) P) x2) x01)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x03:(x x00)) (x04:(y x00))=> ((((conj ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02))) ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02)))))))))) as proof of (forall (Xx0:a), (((ex b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt)))))->((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))))
% Got proof (fun (Xx0:a) (x0:((ex b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt))))))=> (((fun (P:Prop) (x1:(forall (x0:b), (((and ((and (x x0)) (y x0))) (((eq a) Xx0) (f x0)))->P)))=> (((((ex_ind b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt))))) P) x1) x0)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x00:b) (x1:((and ((and (x x00)) (y x00))) (((eq a) Xx0) (f x00))))=> (((fun (P:Type) (x2:(((and (x x00)) (y x00))->((((eq a) Xx0) (f x00))->P)))=> (((((and_rect ((and (x x00)) (y x00))) (((eq a) Xx0) (f x00))) P) x2) x1)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x01:((and (x x00)) (y x00))) (x02:(((eq a) Xx0) (f x00)))=> (((fun (P:Type) (x2:((x x00)->((y x00)->P)))=> (((((and_rect (x x00)) (y x00)) P) x2) x01)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x03:(x x00)) (x04:(y x00))=> ((((conj ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02))) ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02))))))))))
% Time elapsed = 32.323098s
% node=3434 cost=1058.000000 depth=33
% ::::::::::::::::::::::
% % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% (fun (Xx0:a) (x0:((ex b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt))))))=> (((fun (P:Prop) (x1:(forall (x0:b), (((and ((and (x x0)) (y x0))) (((eq a) Xx0) (f x0)))->P)))=> (((((ex_ind b) (fun (Xt:b)=> ((and ((and (x Xt)) (y Xt))) (((eq a) Xx0) (f Xt))))) P) x1) x0)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x00:b) (x1:((and ((and (x x00)) (y x00))) (((eq a) Xx0) (f x00))))=> (((fun (P:Type) (x2:(((and (x x00)) (y x00))->((((eq a) Xx0) (f x00))->P)))=> (((((and_rect ((and (x x00)) (y x00))) (((eq a) Xx0) (f x00))) P) x2) x1)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x01:((and (x x00)) (y x00))) (x02:(((eq a) Xx0) (f x00)))=> (((fun (P:Type) (x2:((x x00)->((y x00)->P)))=> (((((and_rect (x x00)) (y x00)) P) x2) x01)) ((and ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))))) (fun (x03:(x x00)) (x04:(y x00))=> ((((conj ((ex b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt)))))) ((ex b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (x Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (x x00)) (((eq a) Xx0) (f x00))) x03) x02))) ((((ex_intro b) (fun (Xt:b)=> ((and (y Xt)) (((eq a) Xx0) (f Xt))))) x00) ((((conj (y x00)) (((eq a) Xx0) (f x00))) x04) x02))))))))))
% % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% EOF
%------------------------------------------------------------------------------