%------------------------------------------------------------------------------ % File : cocATP---0.2.0 % Problem : SEU630^2 : TPTP v6.1.0. Released v3.7.0. % Transfm : none % Format : tptp:raw % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % Computer : n185.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 13:32:39 EDT 2014 % Result : Theorem 2.37s % Output : Proof 2.37s % Verified : % SZS Type : None (Parsing solution fails) % Syntax : Number of formulae : 0 % Comments : %------------------------------------------------------------------------------ %----ERROR: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % % Problem : SEU630^2 : TPTP v6.1.0. Released v3.7.0. % % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % % Computer : n185.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 10:57:36 CDT 2014 % % CPUTime : 2.37 % 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 0x16840e0>, <kernel.DependentProduct object at 0x16a3b90>) of role type named in_type % Using role type % Declaring in:(fofType->(fofType->Prop)) % FOF formula (<kernel.Constant object at 0x16ffc68>, <kernel.Single object at 0x118d128>) of role type named emptyset_type % Using role type % Declaring emptyset:fofType % FOF formula (<kernel.Constant object at 0x16840e0>, <kernel.DependentProduct object at 0x16a3dd0>) of role type named setadjoin_type % Using role type % Declaring setadjoin:(fofType->(fofType->fofType)) % FOF formula (<kernel.Constant object at 0x118d128>, <kernel.DependentProduct object at 0x16a37a0>) of role type named powerset_type % Using role type % Declaring powerset:(fofType->fofType) % FOF formula (<kernel.Constant object at 0x118d128>, <kernel.DependentProduct object at 0x16a35a8>) of role type named dsetconstr_type % Using role type % Declaring dsetconstr:(fofType->((fofType->Prop)->fofType)) % FOF formula (<kernel.Constant object at 0x16a3ab8>, <kernel.Sort object at 0x118c5a8>) of role type named dsetconstrI_type % Using role type % Declaring dsetconstrI:Prop % FOF formula (((eq Prop) dsetconstrI) (forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) A)->((Xphi Xx)->((in Xx) ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy)))))))) of role definition named dsetconstrI % A new definition: (((eq Prop) dsetconstrI) (forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) A)->((Xphi Xx)->((in Xx) ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy)))))))) % Defined: dsetconstrI:=(forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) A)->((Xphi Xx)->((in Xx) ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy))))))) % FOF formula (<kernel.Constant object at 0x1191908>, <kernel.DependentProduct object at 0x1191e60>) of role type named binunion_type % Using role type % Declaring binunion:(fofType->(fofType->fofType)) % FOF formula (<kernel.Constant object at 0x1191050>, <kernel.DependentProduct object at 0x16a3dd0>) of role type named kpair_type % Using role type % Declaring kpair:(fofType->(fofType->fofType)) % FOF formula (((eq (fofType->(fofType->fofType))) kpair) (fun (Xx:fofType) (Xy:fofType)=> ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset)))) of role definition named kpair % A new definition: (((eq (fofType->(fofType->fofType))) kpair) (fun (Xx:fofType) (Xy:fofType)=> ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset)))) % Defined: kpair:=(fun (Xx:fofType) (Xy:fofType)=> ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))) % FOF formula (<kernel.Constant object at 0x1191e60>, <kernel.DependentProduct object at 0x16a37a0>) of role type named cartprod_type % Using role type % Declaring cartprod:(fofType->(fofType->fofType)) % FOF formula (((eq (fofType->(fofType->fofType))) cartprod) (fun (A:fofType) (B:fofType)=> ((dsetconstr (powerset (powerset ((binunion A) B)))) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) Xx) ((kpair Xy) Xz)))))))))))) of role definition named cartprod % A new definition: (((eq (fofType->(fofType->fofType))) cartprod) (fun (A:fofType) (B:fofType)=> ((dsetconstr (powerset (powerset ((binunion A) B)))) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) Xx) ((kpair Xy) Xz)))))))))))) % Defined: cartprod:=(fun (A:fofType) (B:fofType)=> ((dsetconstr (powerset (powerset ((binunion A) B)))) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) Xx) ((kpair Xy) Xz))))))))))) % FOF formula (<kernel.Constant object at 0x1191e60>, <kernel.Sort object at 0x118c5a8>) of role type named ubforcartprodlem3_type % Using role type % Declaring ubforcartprodlem3:Prop % FOF formula (((eq Prop) ubforcartprodlem3) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B))))))))) of role definition named ubforcartprodlem3 % A new definition: (((eq Prop) ubforcartprodlem3) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B))))))))) % Defined: ubforcartprodlem3:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B)))))))) % FOF formula (dsetconstrI->(ubforcartprodlem3->(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B)))))))) of role conjecture named cartprodpairin % Conjecture to prove = (dsetconstrI->(ubforcartprodlem3->(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B)))))))):Prop % We need to prove ['(dsetconstrI->(ubforcartprodlem3->(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B))))))))'] % Parameter fofType:Type. % Parameter in:(fofType->(fofType->Prop)). % Parameter emptyset:fofType. % Parameter setadjoin:(fofType->(fofType->fofType)). % Parameter powerset:(fofType->fofType). % Parameter dsetconstr:(fofType->((fofType->Prop)->fofType)). % Definition dsetconstrI:=(forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) A)->((Xphi Xx)->((in Xx) ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy))))))):Prop. % Parameter binunion:(fofType->(fofType->fofType)). % Definition kpair:=(fun (Xx:fofType) (Xy:fofType)=> ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))):(fofType->(fofType->fofType)). % Definition cartprod:=(fun (A:fofType) (B:fofType)=> ((dsetconstr (powerset (powerset ((binunion A) B)))) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) Xx) ((kpair Xy) Xz))))))))))):(fofType->(fofType->fofType)). % Definition ubforcartprodlem3:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B)))))))):Prop. % Trying to prove (dsetconstrI->(ubforcartprodlem3->(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B)))))))) % Found x0000000:=(x000000 x2):((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B)))) % Found (x000000 x2) as proof of ((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B)))) % Found ((x00000 Xy) x2) as proof of ((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B)))) % Found (((x0000 x1) Xy) x2) as proof of ((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B)))) % Found ((((x000 Xx) x1) Xy) x2) as proof of ((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B)))) % Found (((((x00 B) Xx) x1) Xy) x2) as proof of ((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B)))) % Found ((((((x0 A) B) Xx) x1) Xy) x2) as proof of ((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B)))) % Found ((((((x0 A) B) Xx) x1) Xy) x2) as proof of ((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B)))) % Found x1:((in Xx) A) % Instantiate: x4:=Xx:fofType % Found x1 as proof of ((in x4) A) % Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))):(((eq (fofType->Prop)) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) (fun (x:fofType)=> ((and ((in x) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x) Xz)))))))) % Found (eta_expansion_dep00 (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) b) % Found ((eta_expansion_dep0 (fun (x5:fofType)=> Prop)) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) b) % Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) b) % Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) b) % Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) b) % Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4)) % Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))))) % Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))))) % Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x) Xz)))))))) % Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4)) % Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))))) % Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))))) % Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x) Xz)))))))) % Found x2:((in Xy) B) % Instantiate: x5:=Xy:fofType % Found x2 as proof of ((in x5) B) % Found eq_ref00:=(eq_ref0 ((kpair Xx) Xy)):(((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy)) % Found (eq_ref0 ((kpair Xx) Xy)) as proof of (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) x5)) % Found ((eq_ref fofType) ((kpair Xx) Xy)) as proof of (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) x5)) % Found ((eq_ref fofType) ((kpair Xx) Xy)) as proof of (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) x5)) % Found ((eq_ref fofType) ((kpair Xx) Xy)) as proof of (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) x5)) % Found ((conj10 x2) ((eq_ref fofType) ((kpair Xx) Xy))) as proof of ((and ((in x5) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) x5))) % Found (((conj1 (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) x5))) x2) ((eq_ref fofType) ((kpair Xx) Xy))) as proof of ((and ((in x5) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) x5))) % Found ((((conj ((in x5) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) x5))) x2) ((eq_ref fofType) ((kpair Xx) Xy))) as proof of ((and ((in x5) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) x5))) % Found ((((conj ((in x5) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) x5))) x2) ((eq_ref fofType) ((kpair Xx) Xy))) as proof of ((and ((in x5) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) x5))) % Found (ex_intro010 ((((conj ((in x5) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) x5))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))) as proof of ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))) % Found ((ex_intro01 Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))) as proof of ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))) % Found (((ex_intro0 (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))) as proof of ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))) % Found (((ex_intro0 (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))) as proof of ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))) % Found ((conj00 x1) (((ex_intro0 (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy))))) as proof of ((and ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz)))))) % Found (((conj0 ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz)))))) x1) (((ex_intro0 (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy))))) as proof of ((and ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz)))))) % Found ((((conj ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz)))))) x1) (((ex_intro0 (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy))))) as proof of ((and ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz)))))) % Found ((((conj ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz)))))) x1) (((ex_intro0 (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy))))) as proof of ((and ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz)))))) % Found (ex_intro000 ((((conj ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz)))))) x1) (((ex_intro0 (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))) as proof of ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) % Found ((ex_intro00 Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) (((ex_intro0 (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))) as proof of ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) % Found (((ex_intro0 (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) (((ex_intro0 (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))) as proof of ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) % Found ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))) as proof of ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) % Found ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))) as proof of ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) % Found ((x300 ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy))))))) as proof of ((in ((kpair Xx) Xy)) ((cartprod A) B)) % Found (((x30 ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy))))))) as proof of ((in ((kpair Xx) Xy)) ((cartprod A) B)) % Found ((((x3 (fun (x7:fofType)=> ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) x7) ((kpair Xy0) Xz)))))))))) ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy))))))) as proof of ((in ((kpair Xx) Xy)) ((cartprod A) B)) % Found (((((x (powerset (powerset ((binunion A) B)))) (fun (x7:fofType)=> ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) x7) ((kpair Xy0) Xz)))))))))) ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy))))))) as proof of ((in ((kpair Xx) Xy)) ((cartprod A) B)) % Found (fun (x2:((in Xy) B))=> (((((x (powerset (powerset ((binunion A) B)))) (fun (x7:fofType)=> ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) x7) ((kpair Xy0) Xz)))))))))) ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))))) as proof of ((in ((kpair Xx) Xy)) ((cartprod A) B)) % Found (fun (Xy:fofType) (x2:((in Xy) B))=> (((((x (powerset (powerset ((binunion A) B)))) (fun (x7:fofType)=> ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) x7) ((kpair Xy0) Xz)))))))))) ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))))) as proof of (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B))) % Found (fun (x1:((in Xx) A)) (Xy:fofType) (x2:((in Xy) B))=> (((((x (powerset (powerset ((binunion A) B)))) (fun (x7:fofType)=> ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) x7) ((kpair Xy0) Xz)))))))))) ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))))) as proof of (forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B)))) % Found (fun (Xx:fofType) (x1:((in Xx) A)) (Xy:fofType) (x2:((in Xy) B))=> (((((x (powerset (powerset ((binunion A) B)))) (fun (x7:fofType)=> ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) x7) ((kpair Xy0) Xz)))))))))) ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))))) as proof of (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B))))) % Found (fun (B:fofType) (Xx:fofType) (x1:((in Xx) A)) (Xy:fofType) (x2:((in Xy) B))=> (((((x (powerset (powerset ((binunion A) B)))) (fun (x7:fofType)=> ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) x7) ((kpair Xy0) Xz)))))))))) ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))))) as proof of (forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B)))))) % Found (fun (A:fofType) (B:fofType) (Xx:fofType) (x1:((in Xx) A)) (Xy:fofType) (x2:((in Xy) B))=> (((((x (powerset (powerset ((binunion A) B)))) (fun (x7:fofType)=> ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) x7) ((kpair Xy0) Xz)))))))))) ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))))) as proof of (forall (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B)))))) % Found (fun (x0:ubforcartprodlem3) (A:fofType) (B:fofType) (Xx:fofType) (x1:((in Xx) A)) (Xy:fofType) (x2:((in Xy) B))=> (((((x (powerset (powerset ((binunion A) B)))) (fun (x7:fofType)=> ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) x7) ((kpair Xy0) Xz)))))))))) ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))))) as proof of (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B)))))) % Found (fun (x:dsetconstrI) (x0:ubforcartprodlem3) (A:fofType) (B:fofType) (Xx:fofType) (x1:((in Xx) A)) (Xy:fofType) (x2:((in Xy) B))=> (((((x (powerset (powerset ((binunion A) B)))) (fun (x7:fofType)=> ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) x7) ((kpair Xy0) Xz)))))))))) ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))))) as proof of (ubforcartprodlem3->(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B))))))) % Found (fun (x:dsetconstrI) (x0:ubforcartprodlem3) (A:fofType) (B:fofType) (Xx:fofType) (x1:((in Xx) A)) (Xy:fofType) (x2:((in Xy) B))=> (((((x (powerset (powerset ((binunion A) B)))) (fun (x7:fofType)=> ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) x7) ((kpair Xy0) Xz)))))))))) ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))))) as proof of (dsetconstrI->(ubforcartprodlem3->(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B)))))))) % Got proof (fun (x:dsetconstrI) (x0:ubforcartprodlem3) (A:fofType) (B:fofType) (Xx:fofType) (x1:((in Xx) A)) (Xy:fofType) (x2:((in Xy) B))=> (((((x (powerset (powerset ((binunion A) B)))) (fun (x7:fofType)=> ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) x7) ((kpair Xy0) Xz)))))))))) ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))))) % Time elapsed = 2.022636s % node=291 cost=1508.000000 depth=33 % :::::::::::::::::::::: % % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p % % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p % (fun (x:dsetconstrI) (x0:ubforcartprodlem3) (A:fofType) (B:fofType) (Xx:fofType) (x1:((in Xx) A)) (Xy:fofType) (x2:((in Xy) B))=> (((((x (powerset (powerset ((binunion A) B)))) (fun (x7:fofType)=> ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) x7) ((kpair Xy0) Xz)))))))))) ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))))) % % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p % EOF %------------------------------------------------------------------------------