%------------------------------------------------------------------------------ % 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 %------------------------------------------------------------------------------