%------------------------------------------------------------------------------ % File : cocATP---0.2.0 % Problem : SEU893^5 : TPTP v6.1.0. Released v4.0.0. % Transfm : none % Format : tptp:raw % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % Computer : n102.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 214.21s % Output : Proof 214.21s % Verified : % SZS Type : None (Parsing solution fails) % Syntax : Number of formulae : 0 % Comments : %------------------------------------------------------------------------------ %----ERROR: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % % Problem : SEU893^5 : TPTP v6.1.0. Released v4.0.0. % % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % % Computer : n102.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 11:41:11 CDT 2014 % % CPUTime : 214.21 % 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 0x1287ea8>, <kernel.Type object at 0x12874d0>) of role type named b_type % Using role type % Declaring b:Type % FOF formula (<kernel.Constant object at 0x1473908>, <kernel.Type object at 0x1287c20>) of role type named a_type % Using role type % Declaring a:Type % FOF formula (<kernel.Constant object at 0x1287b90>, <kernel.DependentProduct object at 0x1287368>) of role type named cF % Using role type % Declaring cF:(b->a) % FOF formula (<kernel.Constant object at 0x12874d0>, <kernel.DependentProduct object at 0x1287cb0>) of role type named cS % Using role type % Declaring cS:(b->Prop) % FOF formula (<kernel.Constant object at 0x1287ea8>, <kernel.DependentProduct object at 0x1287758>) of role type named cR % Using role type % Declaring cR:(b->Prop) % FOF formula (forall (Xx:a), (((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))->((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))))) of role conjecture named cTHM34A_pme % Conjecture to prove = (forall (Xx:a), (((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))->((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))))):Prop % Parameter b_DUMMY:b. % Parameter a_DUMMY:a. % We need to prove ['(forall (Xx:a), (((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))->((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))))'] % Parameter b:Type. % Parameter a:Type. % Parameter cF:(b->a). % Parameter cS:(b->Prop). % Parameter cR:(b->Prop). % Trying to prove (forall (Xx:a), (((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))->((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))))) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) % Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found classic0:=(classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))):((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) (not ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))))) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % Found x4:(((eq a) Xx) (cF x1)) % Instantiate: x0:=x1:b % Found x4 as proof of (((eq a) Xx) (cF x0)) % Found x4:(((eq a) Xx) (cF x1)) % Instantiate: x0:=x1:b % Found x4 as proof of (((eq a) Xx) (cF x0)) % Found x4:(((eq a) Xx) (cF x0)) % Instantiate: x2:=x0:b % Found x4 as proof of (((eq a) Xx) (cF x2)) % Found x4:(((eq a) Xx) (cF x0)) % Instantiate: x2:=x0:b % Found x4 as proof of (((eq a) Xx) (cF x2)) % Found x4:(((eq a) Xx) (cF x0)) % Instantiate: x2:=x0:b % Found x4 as proof of (((eq a) Xx) (cF x2)) % Found x4:(((eq a) Xx) (cF x0)) % Instantiate: x2:=x0:b % Found x4 as proof of (((eq a) Xx) (cF x2)) % 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 (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found ((eta_expansion_dep0 (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % 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 (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P b0) % Found eta_expansion000:=(eta_expansion00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cR x)) (((eq a) Xx) (cF x))))) % Found (eta_expansion00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eta_expansion0 Prop) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P b0) % Found eta_expansion000:=(eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cS x)) (((eq a) Xx) (cF x))))) % Found (eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eta_expansion0 Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) % Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found classic0:=(classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))):((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) (not ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))))) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0)) % Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0)))) % Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0)))) % Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0)))) % Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0)))) % Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0)) % Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0)))) % Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0)))) % Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0)))) % Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0)))) % Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0)) % Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0)))) % Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0)))) % Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0)))) % Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0)))) % Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0)) % Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0)))) % Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0)))) % Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0)))) % Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0)))) % Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: a0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of ((ex b) a0) % Found (or_intror00 x) as proof of (P ((ex b) a0)) % Found ((or_intror0 ((ex b) a0)) x) as proof of (P ((ex b) a0)) % Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((ex b) a0)) % Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((ex b) a0)) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: a0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of ((ex b) a0) % Found (or_intror00 x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0))) % Found ((or_intror0 ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0))) % Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0))) % Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0))) % Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) % Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found classic0:=(classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))):((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) (not ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % 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 (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found ((eta_expansion0 Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF 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 (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found ((eta_expansion0 Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found eq_ref000:=(eq_ref00 cS):((cS x1)->(cS x1)) % Found (eq_ref00 cS) as proof of ((cS x1)->(cS x0)) % Found ((eq_ref0 x1) cS) as proof of ((cS x1)->(cS x0)) % Found (((eq_ref b) x1) cS) as proof of ((cS x1)->(cS x0)) % Found (((eq_ref b) x1) cS) as proof of ((cS x1)->(cS x0)) % Found eq_ref000:=(eq_ref00 cR):((cR x1)->(cR x1)) % Found (eq_ref00 cR) as proof of ((cR x1)->(cR x0)) % Found ((eq_ref0 x1) cR) as proof of ((cR x1)->(cR x0)) % Found (((eq_ref b) x1) cR) as proof of ((cR x1)->(cR x0)) % Found (((eq_ref b) x1) cR) as proof of ((cR x1)->(cR x0)) % Found eq_ref000:=(eq_ref00 cR):((cR x0)->(cR x0)) % Found (eq_ref00 cR) as proof of ((cR x0)->(cR x2)) % Found ((eq_ref0 x0) cR) as proof of ((cR x0)->(cR x2)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x2)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x2)) % Found eq_ref000:=(eq_ref00 cS):((cS x0)->(cS x0)) % Found (eq_ref00 cS) as proof of ((cS x0)->(cS x2)) % Found ((eq_ref0 x0) cS) as proof of ((cS x0)->(cS x2)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x2)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x2)) % Found eq_ref000:=(eq_ref00 cR):((cR x0)->(cR x0)) % Found (eq_ref00 cR) as proof of ((cR x0)->(cR x2)) % Found ((eq_ref0 x0) cR) as proof of ((cR x0)->(cR x2)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x2)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x2)) % Found eq_ref000:=(eq_ref00 cS):((cS x0)->(cS x0)) % Found (eq_ref00 cS) as proof of ((cS x0)->(cS x2)) % Found ((eq_ref0 x0) cS) as proof of ((cS x0)->(cS x2)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x2)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x2)) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P b0) % Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P b0) % Found eta_expansion000:=(eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cS x)) (((eq a) Xx) (cF x))))) % Found (eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eta_expansion0 Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % 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 (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found ((eta_expansion_dep0 (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF 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 (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found ((eta_expansion_dep0 (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found x40:=(x4 (fun (x5:a)=> (cS x1))):((cS x1)->(cS x1)) % Found (x4 (fun (x5:a)=> (cS x1))) as proof of ((cS x1)->(cS x0)) % Found (x4 (fun (x5:a)=> (cS x1))) as proof of ((cS x1)->(cS x0)) % Found x40:=(x4 (fun (x5:a)=> (cR x1))):((cR x1)->(cR x1)) % Found (x4 (fun (x5:a)=> (cR x1))) as proof of ((cR x1)->(cR x0)) % Found (x4 (fun (x5:a)=> (cR x1))) as proof of ((cR x1)->(cR x0)) % Found x40:=(x4 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found x4:(((eq a) Xx) (cF x1)) % Instantiate: x0:=x1:b % Found (fun (x4:(((eq a) Xx) (cF x1)))=> x4) as proof of (((eq a) Xx) (cF x0)) % Found (fun (x3:((or (cR x1)) (cS x1))) (x4:(((eq a) Xx) (cF x1)))=> x4) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0))) % Found (fun (x3:((or (cR x1)) (cS x1))) (x4:(((eq a) Xx) (cF x1)))=> x4) as proof of (((or (cR x1)) (cS x1))->((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0)))) % Found (and_rect00 (fun (x3:((or (cR x1)) (cS x1))) (x4:(((eq a) Xx) (cF x1)))=> x4)) as proof of (((eq a) Xx) (cF x0)) % Found ((and_rect0 (((eq a) Xx) (cF x0))) (fun (x3:((or (cR x1)) (cS x1))) (x4:(((eq a) Xx) (cF x1)))=> x4)) as proof of (((eq a) Xx) (cF x0)) % Found (((fun (P:Type) (x3:(((or (cR x1)) (cS x1))->((((eq a) Xx) (cF x1))->P)))=> (((((and_rect ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))) P) x3) x2)) (((eq a) Xx) (cF x0))) (fun (x3:((or (cR x1)) (cS x1))) (x4:(((eq a) Xx) (cF x1)))=> x4)) as proof of (((eq a) Xx) (cF x0)) % Found (((fun (P:Type) (x3:(((or (cR x1)) (cS x1))->((((eq a) Xx) (cF x1))->P)))=> (((((and_rect ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))) P) x3) x2)) (((eq a) Xx) (cF x0))) (fun (x3:((or (cR x1)) (cS x1))) (x4:(((eq a) Xx) (cF x1)))=> x4)) as proof of (((eq a) Xx) (cF x0)) % Found x40:=(x4 (fun (x5:a)=> (cR x1))):((cR x1)->(cR x1)) % Found (x4 (fun (x5:a)=> (cR x1))) as proof of ((cR x1)->(cR x0)) % Found (x4 (fun (x5:a)=> (cR x1))) as proof of ((cR x1)->(cR x0)) % Found x4:(((eq a) Xx) (cF x1)) % Instantiate: x0:=x1:b % Found (fun (x4:(((eq a) Xx) (cF x1)))=> x4) as proof of (((eq a) Xx) (cF x0)) % Found (fun (x3:((or (cR x1)) (cS x1))) (x4:(((eq a) Xx) (cF x1)))=> x4) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0))) % Found (fun (x3:((or (cR x1)) (cS x1))) (x4:(((eq a) Xx) (cF x1)))=> x4) as proof of (((or (cR x1)) (cS x1))->((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0)))) % Found (and_rect00 (fun (x3:((or (cR x1)) (cS x1))) (x4:(((eq a) Xx) (cF x1)))=> x4)) as proof of (((eq a) Xx) (cF x0)) % Found ((and_rect0 (((eq a) Xx) (cF x0))) (fun (x3:((or (cR x1)) (cS x1))) (x4:(((eq a) Xx) (cF x1)))=> x4)) as proof of (((eq a) Xx) (cF x0)) % Found (((fun (P:Type) (x3:(((or (cR x1)) (cS x1))->((((eq a) Xx) (cF x1))->P)))=> (((((and_rect ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))) P) x3) x2)) (((eq a) Xx) (cF x0))) (fun (x3:((or (cR x1)) (cS x1))) (x4:(((eq a) Xx) (cF x1)))=> x4)) as proof of (((eq a) Xx) (cF x0)) % Found (((fun (P:Type) (x3:(((or (cR x1)) (cS x1))->((((eq a) Xx) (cF x1))->P)))=> (((((and_rect ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))) P) x3) x2)) (((eq a) Xx) (cF x0))) (fun (x3:((or (cR x1)) (cS x1))) (x4:(((eq a) Xx) (cF x1)))=> x4)) as proof of (((eq a) Xx) (cF x0)) % Found x40:=(x4 (fun (x5:a)=> (cS x1))):((cS x1)->(cS x1)) % Found (x4 (fun (x5:a)=> (cS x1))) as proof of ((cS x1)->(cS x0)) % Found (x4 (fun (x5:a)=> (cS x1))) as proof of ((cS x1)->(cS x0)) % Found x40:=(x4 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P b0) % Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P b0) % Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found x40:=(x4 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found x40:=(x4 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found x4:(((eq a) Xx) (cF x0)) % Instantiate: x2:=x0:b % Found (fun (x4:(((eq a) Xx) (cF x0)))=> x4) as proof of (((eq a) Xx) (cF x2)) % Found (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4) as proof of (((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2)))) % Found (and_rect00 (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4)) as proof of (((eq a) Xx) (cF x2)) % Found ((and_rect0 (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4)) as proof of (((eq a) Xx) (cF x2)) % Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4)) as proof of (((eq a) Xx) (cF x2)) % Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4)) as proof of (((eq a) Xx) (cF x2)) % Found x40:=(x4 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found eq_substitution00000:=(eq_substitution0000 (fun (x6:a)=> x6)):((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x0))) % Found (eq_substitution0000 (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found ((eq_substitution000 (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (((eq_substitution00 Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found ((((eq_substitution0 a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6))) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6))) as proof of (((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2)))) % Found (and_rect00 (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found ((and_rect0 (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found x40:=(x4 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found eq_substitution00000:=(eq_substitution0000 (fun (x6:a)=> x6)):((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x0))) % Found (eq_substitution0000 (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found ((eq_substitution000 (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (((eq_substitution00 Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found ((((eq_substitution0 a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6))) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6))) as proof of (((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2)))) % Found (and_rect00 (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found ((and_rect0 (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found x40:=(x4 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found x30:=(x3 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0)) % Found (x3 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x4)) % Found (x3 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x4)) % Found x30:=(x3 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0)) % Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4)) % Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4)) % Found eq_substitution00000:=(eq_substitution0000 (fun (x6:a)=> x6)):((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x0))) % Found (eq_substitution0000 (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found ((eq_substitution000 (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (((eq_substitution00 Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found ((((eq_substitution0 a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6))) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6))) as proof of (((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2)))) % Found (and_rect00 (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found ((and_rect0 (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found x40:=(x4 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found x30:=(x3 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0)) % Found (x3 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x4)) % Found (x3 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x4)) % Found x30:=(x3 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0)) % Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4)) % Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4)) % Found x30:=(x3 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0)) % Found (x3 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x4)) % Found (x3 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x4)) % Found x30:=(x3 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0)) % Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4)) % Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4)) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2)) % Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2)) % Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2)) % Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2)) % Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P b0) % Found eta_expansion000:=(eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cS x)) (((eq a) Xx) (cF x))))) % Found (eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eta_expansion0 Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P b0) % Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: a0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of ((ex b) a0) % Found (or_intror00 x) as proof of (P ((ex b) a0)) % Found ((or_intror0 ((ex b) a0)) x) as proof of (P ((ex b) a0)) % Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((ex b) a0)) % Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((ex b) a0)) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: a0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of ((ex b) a0) % Found (or_intror00 x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0))) % Found ((or_intror0 ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0))) % Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0))) % Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0))) % Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) % Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found classic0:=(classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))):((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) (not ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))))) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % Found x4:(((eq a) Xx) (cF x1)) % Instantiate: x0:=x1:b % Found x4 as proof of (((eq a) Xx) (cF x0)) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0)) % Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0)))) % Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0)))) % Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0)))) % Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0)))) % Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x))))) % Found x4:(((eq a) Xx) (cF x1)) % Instantiate: x0:=x1:b % Found x4 as proof of (((eq a) Xx) (cF x0)) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0)) % Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0)))) % Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0)))) % Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0)))) % Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0)))) % Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0)) % Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0)))) % Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0)))) % Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0)))) % Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cR x0)) (((eq a) Xx) (cF x0)))) % Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0)) % Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0)))) % Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0)))) % Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0)))) % Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF x0)))) % Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x))))) % Found x4:(((eq a) Xx) (cF x0)) % Instantiate: x2:=x0:b % Found x4 as proof of (((eq a) Xx) (cF x2)) % Found x4:(((eq a) Xx) (cF x0)) % Instantiate: x2:=x0:b % Found x4 as proof of (((eq a) Xx) (cF x2)) % Found x4:(((eq a) Xx) (cF x0)) % Instantiate: x2:=x0:b % Found x4 as proof of (((eq a) Xx) (cF x2)) % Found x4:(((eq a) Xx) (cF x0)) % Instantiate: x2:=x0:b % Found x4 as proof of (((eq a) Xx) (cF x2)) % Found x4:(((eq a) Xx) (cF x0)) % Instantiate: x2:=x0:b % Found x4 as proof of (((eq a) Xx) (cF x2)) % Found x4:(((eq a) Xx) (cF x0)) % Instantiate: x2:=x0:b % Found x4 as proof of (((eq a) Xx) (cF x2)) % Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) % Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found classic0:=(classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))):((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) (not ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: a0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of ((ex b) a0) % Found (or_intror00 x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0))) % Found ((or_intror0 ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0))) % Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0))) % Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: a0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of ((ex b) a0) % Found (or_intror00 x) as proof of (P ((ex b) a0)) % Found ((or_intror0 ((ex b) a0)) x) as proof of (P ((ex b) a0)) % Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((ex b) a0)) % Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((ex b) a0)) % Found x5:(cS x1) % Instantiate: x0:=x1:b % Found (fun (x5:(cS x1))=> x5) as proof of (cS x0) % Found (fun (x5:(cS x1))=> x5) as proof of ((cS x1)->(cS x0)) % Found eq_ref000:=(eq_ref00 cR):((cR x1)->(cR x1)) % Found (eq_ref00 cR) as proof of ((cR x1)->(cR x0)) % Found ((eq_ref0 x1) cR) as proof of ((cR x1)->(cR x0)) % Found (((eq_ref b) x1) cR) as proof of ((cR x1)->(cR x0)) % Found (((eq_ref b) x1) cR) as proof of ((cR x1)->(cR x0)) % Found (((eq_ref b) x1) cR) as proof of ((cR x1)->(cR x0)) % Found x5:(cR x0) % Instantiate: x2:=x0:b % Found (fun (x5:(cR x0))=> x5) as proof of (cR x2) % Found (fun (x5:(cR x0))=> x5) as proof of ((cR x0)->(cR x2)) % Found eq_ref000:=(eq_ref00 cS):((cS x0)->(cS x0)) % Found (eq_ref00 cS) as proof of ((cS x0)->(cS x2)) % Found ((eq_ref0 x0) cS) as proof of ((cS x0)->(cS x2)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x2)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x2)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x2)) % Found eq_ref000:=(eq_ref00 cS):((cS x1)->(cS x1)) % Found (eq_ref00 cS) as proof of ((cS x1)->(cS x0)) % Found ((eq_ref0 x1) cS) as proof of ((cS x1)->(cS x0)) % Found (((eq_ref b) x1) cS) as proof of ((cS x1)->(cS x0)) % Found (((eq_ref b) x1) cS) as proof of ((cS x1)->(cS x0)) % Found (((eq_ref b) x1) cS) as proof of ((cS x1)->(cS x0)) % Found x40:=(x4 (fun (x5:a)=> (cR x1))):((cR x1)->(cR x1)) % Found (x4 (fun (x5:a)=> (cR x1))) as proof of ((cR x1)->(cR x0)) % Found (x4 (fun (x5:a)=> (cR x1))) as proof of ((cR x1)->(cR x0)) % Found (x4 (fun (x5:a)=> (cR x1))) as proof of ((cR x1)->(cR x0)) % Found eq_ref000:=(eq_ref00 cR):((cR x0)->(cR x0)) % Found (eq_ref00 cR) as proof of ((cR x0)->(cR x2)) % Found ((eq_ref0 x0) cR) as proof of ((cR x0)->(cR x2)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x2)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x2)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x2)) % Found eq_ref000:=(eq_ref00 cS):((cS x0)->(cS x0)) % Found (eq_ref00 cS) as proof of ((cS x0)->(cS x2)) % Found ((eq_ref0 x0) cS) as proof of ((cS x0)->(cS x2)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x2)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x2)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x2)) % Found x40:=(x4 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found x40:=(x4 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found x5:(cS x0) % Instantiate: x2:=x0:b % Found (fun (x5:(cS x0))=> x5) as proof of (cS x2) % Found (fun (x5:(cS x0))=> x5) as proof of ((cS x0)->(cS x2)) % Found x40:=(x4 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2)) % Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2)) % Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x))))) % Found eq_ref000:=(eq_ref00 cS):((cS x0)->(cS x0)) % Found (eq_ref00 cS) as proof of ((cS x0)->(cS x4)) % Found ((eq_ref0 x0) cS) as proof of ((cS x0)->(cS x4)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x4)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x4)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x4)) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2)) % Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2)) % Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x))))) % Found x5:(cR x0) % Instantiate: x4:=x0:b % Found (fun (x5:(cR x0))=> x5) as proof of (cR x4) % Found (fun (x5:(cR x0))=> x5) as proof of ((cR x0)->(cR x4)) % Found eq_ref000:=(eq_ref00 cS):((cS x0)->(cS x0)) % Found (eq_ref00 cS) as proof of ((cS x0)->(cS x4)) % Found ((eq_ref0 x0) cS) as proof of ((cS x0)->(cS x4)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x4)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x4)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x4)) % Found eq_ref000:=(eq_ref00 cR):((cR x0)->(cR x0)) % Found (eq_ref00 cR) as proof of ((cR x0)->(cR x4)) % Found ((eq_ref0 x0) cR) as proof of ((cR x0)->(cR x4)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x4)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x4)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x4)) % Found x30:=(x3 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0)) % Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4)) % Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4)) % Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4)) % Found eq_ref000:=(eq_ref00 cR):((cR x0)->(cR x0)) % Found (eq_ref00 cR) as proof of ((cR x0)->(cR x4)) % Found ((eq_ref0 x0) cR) as proof of ((cR x0)->(cR x4)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x4)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x4)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x4)) % Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) % Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found classic0:=(classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))):((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) (not ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % 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 (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found ((eta_expansion_dep0 (fun (x5:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF 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 (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found ((eta_expansion0 Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF 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 (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found ((eta_expansion0 Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % 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 (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P b0) % Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P b0) % Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cR x)) (((eq a) Xx) (cF x))))) % Found (eta_expansion_dep00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eta_expansion_dep0 (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P b0) % Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P b0) % Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF 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 (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found ((eta_expansion0 Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF 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 (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found ((eta_expansion_dep0 (fun (x3:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found x4:(((eq a) Xx) (cF x1)) % Instantiate: x0:=x1:b % Found x4 as proof of (((eq a) Xx) (cF x0)) % Found x4:(((eq a) Xx) (cF x1)) % Instantiate: x0:=x1:b % Found x4 as proof of (((eq a) Xx) (cF x0)) % Found x4:(((eq a) Xx) (cF x1)) % Instantiate: x0:=x1:b % Found x4 as proof of (((eq a) Xx) (cF x0)) % Found x5:(cR x1) % Instantiate: x0:=x1:b % Found x5 as proof of (cR x0) % Found ((conj00 x5) x4) as proof of ((and (cR x0)) (((eq a) Xx) (cF x0))) % Found (((conj0 (((eq a) Xx) (cF x0))) x5) x4) as proof of ((and (cR x0)) (((eq a) Xx) (cF x0))) % Found ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x5) x4) as proof of ((and (cR x0)) (((eq a) Xx) (cF x0))) % Found (fun (x5:(cR x1))=> ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x5) x4)) as proof of ((and (cR x0)) (((eq a) Xx) (cF x0))) % Found (fun (x5:(cR x1))=> ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x5) x4)) as proof of ((cR x1)->((and (cR x0)) (((eq a) Xx) (cF x0)))) % Found x4:(((eq a) Xx) (cF x1)) % Found x4 as proof of (((eq a) Xx) (cF x0)) % Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx) % Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx) % Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found x5:(cS x1) % Instantiate: x0:=x1:b % Found x5 as proof of (cS x0) % Found x4:(((eq a) Xx) (cF x1)) % Instantiate: x0:=x1:b % Found x4 as proof of (((eq a) Xx) (cF x0)) % Found ((conj00 x5) x4) as proof of ((and (cS x0)) (((eq a) Xx) (cF x0))) % Found (((conj0 (((eq a) Xx) (cF x0))) x5) x4) as proof of ((and (cS x0)) (((eq a) Xx) (cF x0))) % Found ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x5) x4) as proof of ((and (cS x0)) (((eq a) Xx) (cF x0))) % Found (fun (x5:(cS x1))=> ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x5) x4)) as proof of ((and (cS x0)) (((eq a) Xx) (cF x0))) % Found (fun (x5:(cS x1))=> ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x5) x4)) as proof of ((cS x1)->((and (cS x0)) (((eq a) Xx) (cF x0)))) % Found x4:(((eq a) Xx) (cF x1)) % Found x4 as proof of (((eq a) Xx) (cF x0)) % Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx) % Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx) % Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found x4:(((eq a) Xx) (cF x0)) % Instantiate: x2:=x0:b % Found x4 as proof of (((eq a) Xx) (cF x2)) % Found x5:(cS x0) % Instantiate: x2:=x0:b % Found x5 as proof of (cS x2) % Found ((conj00 x5) x4) as proof of ((and (cS x2)) (((eq a) Xx) (cF x2))) % Found (((conj0 (((eq a) Xx) (cF x2))) x5) x4) as proof of ((and (cS x2)) (((eq a) Xx) (cF x2))) % Found ((((conj (cS x2)) (((eq a) Xx) (cF x2))) x5) x4) as proof of ((and (cS x2)) (((eq a) Xx) (cF x2))) % Found (fun (x5:(cS x0))=> ((((conj (cS x2)) (((eq a) Xx) (cF x2))) x5) x4)) as proof of ((and (cS x2)) (((eq a) Xx) (cF x2))) % Found (fun (x5:(cS x0))=> ((((conj (cS x2)) (((eq a) Xx) (cF x2))) x5) x4)) as proof of ((cS x0)->((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found x4:(((eq a) Xx) (cF x0)) % Instantiate: x2:=x0:b % Found x4 as proof of (((eq a) Xx) (cF x2)) % Found x4:(((eq a) Xx) (cF x0)) % Found x4 as proof of (((eq a) Xx) (cF x2)) % Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx) % Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx) % Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found eq_substitution00000:=(eq_substitution0000 (fun (x6:a)=> x6)):((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x1))) % Found (eq_substitution0000 (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0))) % Found ((eq_substitution000 (cF x1)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0))) % Found (((eq_substitution00 Xx) (cF x1)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0))) % Found ((((eq_substitution0 a) Xx) (cF x1)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0))) % Found (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0))) % Found (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0))) % Found (fun (x3:((or (cR x1)) (cS x1)))=> (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6))) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0))) % Found (fun (x3:((or (cR x1)) (cS x1)))=> (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6))) as proof of (((or (cR x1)) (cS x1))->((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0)))) % Found (and_rect00 (fun (x3:((or (cR x1)) (cS x1)))=> (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x0)) % Found ((and_rect0 (((eq a) Xx) (cF x0))) (fun (x3:((or (cR x1)) (cS x1)))=> (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x0)) % Found (((fun (P:Type) (x3:(((or (cR x1)) (cS x1))->((((eq a) Xx) (cF x1))->P)))=> (((((and_rect ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))) P) x3) x2)) (((eq a) Xx) (cF x0))) (fun (x3:((or (cR x1)) (cS x1)))=> (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x0)) % Found (fun (x2:((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))=> (((fun (P:Type) (x3:(((or (cR x1)) (cS x1))->((((eq a) Xx) (cF x1))->P)))=> (((((and_rect ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))) P) x3) x2)) (((eq a) Xx) (cF x0))) (fun (x3:((or (cR x1)) (cS x1)))=> (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6))))) as proof of (((eq a) Xx) (cF x0)) % Found eq_substitution00000:=(eq_substitution0000 (fun (x6:a)=> x6)):((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x1))) % Found (eq_substitution0000 (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0))) % Found ((eq_substitution000 (cF x1)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0))) % Found (((eq_substitution00 Xx) (cF x1)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0))) % Found ((((eq_substitution0 a) Xx) (cF x1)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0))) % Found (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0))) % Found (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0))) % Found (fun (x3:((or (cR x1)) (cS x1)))=> (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6))) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0))) % Found (fun (x3:((or (cR x1)) (cS x1)))=> (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6))) as proof of (((or (cR x1)) (cS x1))->((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0)))) % Found (and_rect00 (fun (x3:((or (cR x1)) (cS x1)))=> (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x0)) % Found ((and_rect0 (((eq a) Xx) (cF x0))) (fun (x3:((or (cR x1)) (cS x1)))=> (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x0)) % Found (((fun (P:Type) (x3:(((or (cR x1)) (cS x1))->((((eq a) Xx) (cF x1))->P)))=> (((((and_rect ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))) P) x3) x2)) (((eq a) Xx) (cF x0))) (fun (x3:((or (cR x1)) (cS x1)))=> (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x0)) % Found (fun (x2:((and ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))))=> (((fun (P:Type) (x3:(((or (cR x1)) (cS x1))->((((eq a) Xx) (cF x1))->P)))=> (((((and_rect ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))) P) x3) x2)) (((eq a) Xx) (cF x0))) (fun (x3:((or (cR x1)) (cS x1)))=> (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6))))) as proof of (((eq a) Xx) (cF x0)) % Found x4:(((eq a) Xx) (cF x0)) % Instantiate: x2:=x0:b % Found x4 as proof of (((eq a) Xx) (cF x2)) % Found x5:(cR x0) % Instantiate: x2:=x0:b % Found x5 as proof of (cR x2) % Found x4:(((eq a) Xx) (cF x0)) % Instantiate: x2:=x0:b % Found x4 as proof of (((eq a) Xx) (cF x2)) % Found ((conj00 x5) x4) as proof of ((and (cR x2)) (((eq a) Xx) (cF x2))) % Found (((conj0 (((eq a) Xx) (cF x2))) x5) x4) as proof of ((and (cR x2)) (((eq a) Xx) (cF x2))) % Found ((((conj (cR x2)) (((eq a) Xx) (cF x2))) x5) x4) as proof of ((and (cR x2)) (((eq a) Xx) (cF x2))) % Found (fun (x5:(cR x0))=> ((((conj (cR x2)) (((eq a) Xx) (cF x2))) x5) x4)) as proof of ((and (cR x2)) (((eq a) Xx) (cF x2))) % Found (fun (x5:(cR x0))=> ((((conj (cR x2)) (((eq a) Xx) (cF x2))) x5) x4)) as proof of ((cR x0)->((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found x4:(((eq a) Xx) (cF x0)) % Found x4 as proof of (((eq a) Xx) (cF x2)) % Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx) % Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx) % Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found x4:(((eq a) Xx) (cF x0)) % Instantiate: x2:=x0:b % Found x4 as proof of (((eq a) Xx) (cF x2)) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found x5:(cR x0) % Instantiate: x4:=x0:b % Found x5 as proof of (cR x4) % Found ((conj00 x5) x3) as proof of ((and (cR x4)) (((eq a) Xx) (cF x4))) % Found (((conj0 (((eq a) Xx) (cF x4))) x5) x3) as proof of ((and (cR x4)) (((eq a) Xx) (cF x4))) % Found ((((conj (cR x4)) (((eq a) Xx) (cF x4))) x5) x3) as proof of ((and (cR x4)) (((eq a) Xx) (cF x4))) % Found (fun (x5:(cR x0))=> ((((conj (cR x4)) (((eq a) Xx) (cF x4))) x5) x3)) as proof of ((and (cR x4)) (((eq a) Xx) (cF x4))) % Found (fun (x5:(cR x0))=> ((((conj (cR x4)) (((eq a) Xx) (cF x4))) x5) x3)) as proof of ((cR x0)->((and (cR x4)) (((eq a) Xx) (cF x4)))) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found x3:(((eq a) Xx) (cF x0)) % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx) % Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx) % Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found x5:(cR x0) % Instantiate: x2:=x0:b % Found x5 as proof of (cR x2) % Found x4:(((eq a) Xx) (cF x0)) % Instantiate: x2:=x0:b % Found x4 as proof of (((eq a) Xx) (cF x2)) % Found ((conj00 x5) x4) as proof of ((and (cR x2)) (((eq a) Xx) (cF x2))) % Found (((conj0 (((eq a) Xx) (cF x2))) x5) x4) as proof of ((and (cR x2)) (((eq a) Xx) (cF x2))) % Found ((((conj (cR x2)) (((eq a) Xx) (cF x2))) x5) x4) as proof of ((and (cR x2)) (((eq a) Xx) (cF x2))) % Found (fun (x5:(cR x0))=> ((((conj (cR x2)) (((eq a) Xx) (cF x2))) x5) x4)) as proof of ((and (cR x2)) (((eq a) Xx) (cF x2))) % Found (fun (x5:(cR x0))=> ((((conj (cR x2)) (((eq a) Xx) (cF x2))) x5) x4)) as proof of ((cR x0)->((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found x4:(((eq a) Xx) (cF x0)) % Found x4 as proof of (((eq a) Xx) (cF x2)) % Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx) % Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx) % Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found x5:(cS x0) % Instantiate: x4:=x0:b % Found x5 as proof of (cS x4) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found ((conj00 x5) x3) as proof of ((and (cS x4)) (((eq a) Xx) (cF x4))) % Found (((conj0 (((eq a) Xx) (cF x4))) x5) x3) as proof of ((and (cS x4)) (((eq a) Xx) (cF x4))) % Found ((((conj (cS x4)) (((eq a) Xx) (cF x4))) x5) x3) as proof of ((and (cS x4)) (((eq a) Xx) (cF x4))) % Found (fun (x5:(cS x0))=> ((((conj (cS x4)) (((eq a) Xx) (cF x4))) x5) x3)) as proof of ((and (cS x4)) (((eq a) Xx) (cF x4))) % Found (fun (x5:(cS x0))=> ((((conj (cS x4)) (((eq a) Xx) (cF x4))) x5) x3)) as proof of ((cS x0)->((and (cS x4)) (((eq a) Xx) (cF x4)))) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found x3:(((eq a) Xx) (cF x0)) % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx) % Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx) % Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found x5:(cS x0) % Instantiate: x2:=x0:b % Found x5 as proof of (cS x2) % Found x4:(((eq a) Xx) (cF x0)) % Instantiate: x2:=x0:b % Found x4 as proof of (((eq a) Xx) (cF x2)) % Found ((conj00 x5) x4) as proof of ((and (cS x2)) (((eq a) Xx) (cF x2))) % Found (((conj0 (((eq a) Xx) (cF x2))) x5) x4) as proof of ((and (cS x2)) (((eq a) Xx) (cF x2))) % Found ((((conj (cS x2)) (((eq a) Xx) (cF x2))) x5) x4) as proof of ((and (cS x2)) (((eq a) Xx) (cF x2))) % Found (fun (x5:(cS x0))=> ((((conj (cS x2)) (((eq a) Xx) (cF x2))) x5) x4)) as proof of ((and (cS x2)) (((eq a) Xx) (cF x2))) % Found (fun (x5:(cS x0))=> ((((conj (cS x2)) (((eq a) Xx) (cF x2))) x5) x4)) as proof of ((cS x0)->((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found x4:(((eq a) Xx) (cF x0)) % Instantiate: x2:=x0:b % Found x4 as proof of (((eq a) Xx) (cF x2)) % Found x4:(((eq a) Xx) (cF x0)) % Found x4 as proof of (((eq a) Xx) (cF x2)) % Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx) % Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx) % Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found x5:(cR x0) % Instantiate: x4:=x0:b % Found x5 as proof of (cR x4) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found ((conj00 x5) x3) as proof of ((and (cR x4)) (((eq a) Xx) (cF x4))) % Found (((conj0 (((eq a) Xx) (cF x4))) x5) x3) as proof of ((and (cR x4)) (((eq a) Xx) (cF x4))) % Found ((((conj (cR x4)) (((eq a) Xx) (cF x4))) x5) x3) as proof of ((and (cR x4)) (((eq a) Xx) (cF x4))) % Found (fun (x5:(cR x0))=> ((((conj (cR x4)) (((eq a) Xx) (cF x4))) x5) x3)) as proof of ((and (cR x4)) (((eq a) Xx) (cF x4))) % Found (fun (x5:(cR x0))=> ((((conj (cR x4)) (((eq a) Xx) (cF x4))) x5) x3)) as proof of ((cR x0)->((and (cR x4)) (((eq a) Xx) (cF x4)))) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found x3:(((eq a) Xx) (cF x0)) % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx) % Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx) % Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found x5:(cS x0) % Instantiate: x4:=x0:b % Found x5 as proof of (cS x4) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found ((conj00 x5) x3) as proof of ((and (cS x4)) (((eq a) Xx) (cF x4))) % Found (((conj0 (((eq a) Xx) (cF x4))) x5) x3) as proof of ((and (cS x4)) (((eq a) Xx) (cF x4))) % Found ((((conj (cS x4)) (((eq a) Xx) (cF x4))) x5) x3) as proof of ((and (cS x4)) (((eq a) Xx) (cF x4))) % Found (fun (x5:(cS x0))=> ((((conj (cS x4)) (((eq a) Xx) (cF x4))) x5) x3)) as proof of ((and (cS x4)) (((eq a) Xx) (cF x4))) % Found (fun (x5:(cS x0))=> ((((conj (cS x4)) (((eq a) Xx) (cF x4))) x5) x3)) as proof of ((cS x0)->((and (cS x4)) (((eq a) Xx) (cF x4)))) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found x3:(((eq a) Xx) (cF x0)) % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx) % Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx) % Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found x5:(cR x0) % Instantiate: x4:=x0:b % Found x5 as proof of (cR x4) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found ((conj00 x5) x3) as proof of ((and (cR x4)) (((eq a) Xx) (cF x4))) % Found (((conj0 (((eq a) Xx) (cF x4))) x5) x3) as proof of ((and (cR x4)) (((eq a) Xx) (cF x4))) % Found ((((conj (cR x4)) (((eq a) Xx) (cF x4))) x5) x3) as proof of ((and (cR x4)) (((eq a) Xx) (cF x4))) % Found (fun (x5:(cR x0))=> ((((conj (cR x4)) (((eq a) Xx) (cF x4))) x5) x3)) as proof of ((and (cR x4)) (((eq a) Xx) (cF x4))) % Found (fun (x5:(cR x0))=> ((((conj (cR x4)) (((eq a) Xx) (cF x4))) x5) x3)) as proof of ((cR x0)->((and (cR x4)) (((eq a) Xx) (cF x4)))) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found x3:(((eq a) Xx) (cF x0)) % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx) % Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx) % Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found x5:(cS x0) % Instantiate: x4:=x0:b % Found x5 as proof of (cS x4) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x4:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found ((conj00 x5) x3) as proof of ((and (cS x4)) (((eq a) Xx) (cF x4))) % Found (((conj0 (((eq a) Xx) (cF x4))) x5) x3) as proof of ((and (cS x4)) (((eq a) Xx) (cF x4))) % Found ((((conj (cS x4)) (((eq a) Xx) (cF x4))) x5) x3) as proof of ((and (cS x4)) (((eq a) Xx) (cF x4))) % Found (fun (x5:(cS x0))=> ((((conj (cS x4)) (((eq a) Xx) (cF x4))) x5) x3)) as proof of ((and (cS x4)) (((eq a) Xx) (cF x4))) % Found (fun (x5:(cS x0))=> ((((conj (cS x4)) (((eq a) Xx) (cF x4))) x5) x3)) as proof of ((cS x0)->((and (cS x4)) (((eq a) Xx) (cF x4)))) % Found x3:(((eq a) Xx) (cF x0)) % Found x3 as proof of (((eq a) Xx) (cF x4)) % Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx) % Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx) % Found (eq_ref0 Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found ((eq_ref a) Xx) as proof of (((eq a) Xx) Xx) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P b0) % Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cR x)) (((eq a) Xx) (cF x))))) % Found (eta_expansion_dep00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eta_expansion_dep0 (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P b0) % Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P b0) % Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (eq_ref0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P b0) % Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cR x)) (((eq a) Xx) (cF x))))) % Found (eta_expansion_dep00 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eta_expansion_dep0 (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % 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 (cR x4)) (((eq a) Xx) (cF x4)))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cR x4)) (((eq a) Xx) (cF x4)))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cR x4)) (((eq a) Xx) (cF x4)))) % Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and (cR x4)) (((eq a) Xx) (cF x4)))) % Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x))))) % 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 (cS x4)) (((eq a) Xx) (cF x4)))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF x4)))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF x4)))) % Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF x4)))) % Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x))))) % 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 (cS x4)) (((eq a) Xx) (cF x4)))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF x4)))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF x4)))) % Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF x4)))) % Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x))))) % 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 (cR x4)) (((eq a) Xx) (cF x4)))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cR x4)) (((eq a) Xx) (cF x4)))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cR x4)) (((eq a) Xx) (cF x4)))) % Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and (cR x4)) (((eq a) Xx) (cF x4)))) % Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2)) % Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2)) % Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2)) % Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2)) % Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P b0) % Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P b0) % Found eta_expansion000:=(eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cS x)) (((eq a) Xx) (cF x))))) % Found (eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eta_expansion0 Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: a0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of ((ex b) a0) % Found (or_intror00 x) as proof of (P ((ex b) a0)) % Found ((or_intror0 ((ex b) a0)) x) as proof of (P ((ex b) a0)) % Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((ex b) a0)) % Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((ex b) a0)) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: a0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of ((ex b) a0) % Found (or_intror00 x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0))) % Found ((or_intror0 ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0))) % Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0))) % Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: a0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of ((ex b) a0) % Found (or_intror00 x) as proof of (P ((ex b) a0)) % Found ((or_intror0 ((ex b) a0)) x) as proof of (P ((ex b) a0)) % Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((ex b) a0)) % Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((ex b) a0)) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: a0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of ((ex b) a0) % Found (or_intror00 x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0))) % Found ((or_intror0 ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0))) % Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0))) % Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0))) % Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) % Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found classic0:=(classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))):((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) (not ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % 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 (cR x4)) (((eq a) Xx) (cF x4)))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cR x4)) (((eq a) Xx) (cF x4)))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cR x4)) (((eq a) Xx) (cF x4)))) % Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and (cR x4)) (((eq a) Xx) (cF x4)))) % Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x))))) % 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 (cR x4)) (((eq a) Xx) (cF x4)))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cR x4)) (((eq a) Xx) (cF x4)))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cR x4)) (((eq a) Xx) (cF x4)))) % Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and (cR x4)) (((eq a) Xx) (cF x4)))) % Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x))))) % 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 (cS x4)) (((eq a) Xx) (cF x4)))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF x4)))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF x4)))) % Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF x4)))) % Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x))))) % 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 (cS x4)) (((eq a) Xx) (cF x4)))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF x4)))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF x4)))) % Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF x4)))) % Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x))))) % 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 (cS x4)) (((eq a) Xx) (cF x4)))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF x4)))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF x4)))) % Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF x4)))) % Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x))))) % 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 (cS x4)) (((eq a) Xx) (cF x4)))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF x4)))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF x4)))) % Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF x4)))) % Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x))))) % 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 (cR x4)) (((eq a) Xx) (cF x4)))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cR x4)) (((eq a) Xx) (cF x4)))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cR x4)) (((eq a) Xx) (cF x4)))) % Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and (cR x4)) (((eq a) Xx) (cF x4)))) % Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x))))) % 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 (cR x4)) (((eq a) Xx) (cF x4)))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cR x4)) (((eq a) Xx) (cF x4)))) % Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cR x4)) (((eq a) Xx) (cF x4)))) % Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and (cR x4)) (((eq a) Xx) (cF x4)))) % Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2)) % Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2)) % Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2)) % Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2)) % Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P b0) % Found eta_expansion000:=(eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cS x)) (((eq a) Xx) (cF x))))) % Found (eta_expansion00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eta_expansion0 Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P b0) % Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found eq_ref000:=(eq_ref00 cS):((cS x1)->(cS x1)) % Found (eq_ref00 cS) as proof of ((cS x1)->(cS x0)) % Found ((eq_ref0 x1) cS) as proof of ((cS x1)->(cS x0)) % Found (((eq_ref b) x1) cS) as proof of ((cS x1)->(cS x0)) % Found (((eq_ref b) x1) cS) as proof of ((cS x1)->(cS x0)) % Found eq_ref000:=(eq_ref00 cR):((cR x1)->(cR x1)) % Found (eq_ref00 cR) as proof of ((cR x1)->(cR x0)) % Found ((eq_ref0 x1) cR) as proof of ((cR x1)->(cR x0)) % Found (((eq_ref b) x1) cR) as proof of ((cR x1)->(cR x0)) % Found (((eq_ref b) x1) cR) as proof of ((cR x1)->(cR x0)) % Found eq_ref000:=(eq_ref00 cS):((cS x0)->(cS x0)) % Found (eq_ref00 cS) as proof of ((cS x0)->(cS x2)) % Found ((eq_ref0 x0) cS) as proof of ((cS x0)->(cS x2)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x2)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x2)) % Found eq_ref000:=(eq_ref00 cR):((cR x0)->(cR x0)) % Found (eq_ref00 cR) as proof of ((cR x0)->(cR x2)) % Found ((eq_ref0 x0) cR) as proof of ((cR x0)->(cR x2)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x2)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x2)) % Found eq_ref000:=(eq_ref00 cS):((cS x0)->(cS x0)) % Found (eq_ref00 cS) as proof of ((cS x0)->(cS x2)) % Found ((eq_ref0 x0) cS) as proof of ((cS x0)->(cS x2)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x2)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x2)) % Found eq_ref000:=(eq_ref00 cR):((cR x0)->(cR x0)) % Found (eq_ref00 cR) as proof of ((cR x0)->(cR x2)) % Found ((eq_ref0 x0) cR) as proof of ((cR x0)->(cR x2)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x2)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x2)) % Found eq_ref000:=(eq_ref00 cS):((cS x0)->(cS x0)) % Found (eq_ref00 cS) as proof of ((cS x0)->(cS x2)) % Found ((eq_ref0 x0) cS) as proof of ((cS x0)->(cS x2)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x2)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x2)) % Found eq_ref000:=(eq_ref00 cR):((cR x0)->(cR x0)) % Found (eq_ref00 cR) as proof of ((cR x0)->(cR x2)) % Found ((eq_ref0 x0) cR) as proof of ((cR x0)->(cR x2)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x2)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x2)) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: a0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of ((ex b) a0) % Found (or_intror00 x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0))) % Found ((or_intror0 ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0))) % Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0))) % Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: a0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of ((ex b) a0) % Found (or_intror00 x) as proof of (P ((ex b) a0)) % Found ((or_intror0 ((ex b) a0)) x) as proof of (P ((ex b) a0)) % Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((ex b) a0)) % Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) a0)) x) as proof of (P ((ex b) a0)) % Found eq_substitution00000:=(eq_substitution0000 (fun (x6:a)=> x6)):((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x1))) % Found (eq_substitution0000 (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0))) % Found ((eq_substitution000 (cF x1)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0))) % Found (((eq_substitution00 Xx) (cF x1)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0))) % Found ((((eq_substitution0 a) Xx) (cF x1)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0))) % Found (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0))) % Found (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0))) % Found (fun (x3:((or (cR x1)) (cS x1)))=> (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6))) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0))) % Found (fun (x3:((or (cR x1)) (cS x1)))=> (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6))) as proof of (((or (cR x1)) (cS x1))->((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0)))) % Found (and_rect00 (fun (x3:((or (cR x1)) (cS x1)))=> (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x0)) % Found ((and_rect0 (((eq a) Xx) (cF x0))) (fun (x3:((or (cR x1)) (cS x1)))=> (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x0)) % Found (((fun (P:Type) (x3:(((or (cR x1)) (cS x1))->((((eq a) Xx) (cF x1))->P)))=> (((((and_rect ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))) P) x3) x2)) (((eq a) Xx) (cF x0))) (fun (x3:((or (cR x1)) (cS x1)))=> (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x0)) % Found (((fun (P:Type) (x3:(((or (cR x1)) (cS x1))->((((eq a) Xx) (cF x1))->P)))=> (((((and_rect ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))) P) x3) x2)) (((eq a) Xx) (cF x0))) (fun (x3:((or (cR x1)) (cS x1)))=> (((((eq_substitution a) a) Xx) (cF x1)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x0)) % Found x4:(((eq a) Xx) (cF x1)) % Instantiate: x0:=x1:b % Found (fun (x4:(((eq a) Xx) (cF x1)))=> x4) as proof of (((eq a) Xx) (cF x0)) % Found (fun (x3:((or (cR x1)) (cS x1))) (x4:(((eq a) Xx) (cF x1)))=> x4) as proof of ((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0))) % Found (fun (x3:((or (cR x1)) (cS x1))) (x4:(((eq a) Xx) (cF x1)))=> x4) as proof of (((or (cR x1)) (cS x1))->((((eq a) Xx) (cF x1))->(((eq a) Xx) (cF x0)))) % Found (and_rect00 (fun (x3:((or (cR x1)) (cS x1))) (x4:(((eq a) Xx) (cF x1)))=> x4)) as proof of (((eq a) Xx) (cF x0)) % Found ((and_rect0 (((eq a) Xx) (cF x0))) (fun (x3:((or (cR x1)) (cS x1))) (x4:(((eq a) Xx) (cF x1)))=> x4)) as proof of (((eq a) Xx) (cF x0)) % Found (((fun (P:Type) (x3:(((or (cR x1)) (cS x1))->((((eq a) Xx) (cF x1))->P)))=> (((((and_rect ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))) P) x3) x2)) (((eq a) Xx) (cF x0))) (fun (x3:((or (cR x1)) (cS x1))) (x4:(((eq a) Xx) (cF x1)))=> x4)) as proof of (((eq a) Xx) (cF x0)) % Found (((fun (P:Type) (x3:(((or (cR x1)) (cS x1))->((((eq a) Xx) (cF x1))->P)))=> (((((and_rect ((or (cR x1)) (cS x1))) (((eq a) Xx) (cF x1))) P) x3) x2)) (((eq a) Xx) (cF x0))) (fun (x3:((or (cR x1)) (cS x1))) (x4:(((eq a) Xx) (cF x1)))=> x4)) as proof of (((eq a) Xx) (cF x0)) % Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) % Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found classic0:=(classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))):((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) (not ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % Found eq_substitution00000:=(eq_substitution0000 (fun (x6:a)=> x6)):((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x0))) % Found (eq_substitution0000 (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found ((eq_substitution000 (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (((eq_substitution00 Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found ((((eq_substitution0 a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6))) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6))) as proof of (((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2)))) % Found (and_rect00 (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found ((and_rect0 (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found eq_substitution00000:=(eq_substitution0000 (fun (x6:a)=> x6)):((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x0))) % Found (eq_substitution0000 (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found ((eq_substitution000 (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (((eq_substitution00 Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found ((((eq_substitution0 a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6))) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6))) as proof of (((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2)))) % Found (and_rect00 (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found ((and_rect0 (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found eq_ref00:=(eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))):(((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) % Found (eq_ref0 ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found ((eq_ref Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (((eq Prop) ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) b0) % Found classic0:=(classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))):((or ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) (not ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % Found (classic ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) as proof of (P b0) % Found eq_substitution00000:=(eq_substitution0000 (fun (x6:a)=> x6)):((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x0))) % Found (eq_substitution0000 (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found ((eq_substitution000 (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (((eq_substitution00 Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found ((((eq_substitution0 a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6))) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6))) as proof of (((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2)))) % Found (and_rect00 (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found ((and_rect0 (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found x4:(((eq a) Xx) (cF x0)) % Instantiate: x2:=x0:b % Found (fun (x4:(((eq a) Xx) (cF x0)))=> x4) as proof of (((eq a) Xx) (cF x2)) % Found (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4) as proof of (((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2)))) % Found (and_rect00 (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4)) as proof of (((eq a) Xx) (cF x2)) % Found ((and_rect0 (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4)) as proof of (((eq a) Xx) (cF x2)) % Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4)) as proof of (((eq a) Xx) (cF x2)) % Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4)) as proof of (((eq a) Xx) (cF x2)) % Found x4:(((eq a) Xx) (cF x0)) % Instantiate: x2:=x0:b % Found (fun (x4:(((eq a) Xx) (cF x0)))=> x4) as proof of (((eq a) Xx) (cF x2)) % Found (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4) as proof of (((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2)))) % Found (and_rect00 (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4)) as proof of (((eq a) Xx) (cF x2)) % Found ((and_rect0 (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4)) as proof of (((eq a) Xx) (cF x2)) % Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4)) as proof of (((eq a) Xx) (cF x2)) % Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0))) (x4:(((eq a) Xx) (cF x0)))=> x4)) as proof of (((eq a) Xx) (cF x2)) % Found eq_substitution00000:=(eq_substitution0000 (fun (x6:a)=> x6)):((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x0))) % Found (eq_substitution0000 (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found ((eq_substitution000 (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (((eq_substitution00 Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found ((((eq_substitution0 a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6))) as proof of ((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2))) % Found (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6))) as proof of (((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->(((eq a) Xx) (cF x2)))) % Found (and_rect00 (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found ((and_rect0 (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found (((fun (P:Type) (x3:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:((or (cR x0)) (cS x0)))=> (((((eq_substitution a) a) Xx) (cF x0)) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2)) % Found x40:=(x4 (fun (x5:a)=> (cR x1))):((cR x1)->(cR x1)) % Found (x4 (fun (x5:a)=> (cR x1))) as proof of ((cR x1)->(cR x0)) % Found (x4 (fun (x5:a)=> (cR x1))) as proof of ((cR x1)->(cR x0)) % Found x40:=(x4 (fun (x5:a)=> (cS x1))):((cS x1)->(cS x1)) % Found (x4 (fun (x5:a)=> (cS x1))) as proof of ((cS x1)->(cS x0)) % Found (x4 (fun (x5:a)=> (cS x1))) as proof of ((cS x1)->(cS x0)) % Found x40:=(x4 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found x40:=(x4 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found x40:=(x4 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found x40:=(x4 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found x40:=(x4 (fun (x5:a)=> (cS x1))):((cS x1)->(cS x1)) % Found (x4 (fun (x5:a)=> (cS x1))) as proof of ((cS x1)->(cS x0)) % Found (x4 (fun (x5:a)=> (cS x1))) as proof of ((cS x1)->(cS x0)) % Found x40:=(x4 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found x40:=(x4 (fun (x5:a)=> (cR x1))):((cR x1)->(cR x1)) % Found (x4 (fun (x5:a)=> (cR x1))) as proof of ((cR x1)->(cR x0)) % Found (x4 (fun (x5:a)=> (cR x1))) as proof of ((cR x1)->(cR x0)) % Found x40:=(x4 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found x40:=(x4 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found x40:=(x4 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found x40:=(x4 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found x40:=(x4 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found x30:=(x3 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0)) % Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4)) % Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4)) % Found x30:=(x3 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0)) % Found (x3 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x4)) % Found (x3 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x4)) % Found x40:=(x4 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found x40:=(x4 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found x30:=(x3 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0)) % Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4)) % Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4)) % Found x30:=(x3 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0)) % Found (x3 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x4)) % Found (x3 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x4)) % Found x30:=(x3 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0)) % Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4)) % Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4)) % Found x30:=(x3 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0)) % Found (x3 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x4)) % Found (x3 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x4)) % Found x30:=(x3 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0)) % Found (x3 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x4)) % Found (x3 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x4)) % Found x30:=(x3 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0)) % Found (x3 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x4)) % Found (x3 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x4)) % Found x30:=(x3 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0)) % Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4)) % Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4)) % Found x30:=(x3 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0)) % Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4)) % Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4)) % Found x30:=(x3 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0)) % Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4)) % Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4)) % Found x30:=(x3 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0)) % Found (x3 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x4)) % Found (x3 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x4)) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2)) % Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2)) % Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF x))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2)) % Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: f:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P f) % Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2)) % Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cR x2)) (((eq a) Xx) (cF x2)))) % Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and (cR x)) (((eq a) Xx) (cF x))))) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x5:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x5)) % Found x4:(cS x0) % Instantiate: x5:=x0:b % Found x4 as proof of (cS x5) % Found ((conj00 x4) x3) as proof of ((and (cS x5)) (((eq a) Xx) (cF x5))) % Found (((conj0 (((eq a) Xx) (cF x5))) x4) x3) as proof of ((and (cS x5)) (((eq a) Xx) (cF x5))) % Found ((((conj (cS x5)) (((eq a) Xx) (cF x5))) x4) x3) as proof of ((and (cS x5)) (((eq a) Xx) (cF x5))) % Found ((((conj (cS x5)) (((eq a) Xx) (cF x5))) x4) x3) as proof of ((and (cS x5)) (((eq a) Xx) (cF x5))) % Found (ex_intro000 ((((conj (cS x5)) (((eq a) Xx) (cF x5))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found ((ex_intro00 x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (((ex_intro0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (fun (x4:(cS x0))=> ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3))) as proof of ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (fun (x4:(cS x0))=> ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3))) as proof of ((cS x0)->((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x5:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x5)) % Found x4:(cR x0) % Instantiate: x5:=x0:b % Found x4 as proof of (cR x5) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x5:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x5)) % Found ((conj00 x4) x3) as proof of ((and (cR x5)) (((eq a) Xx) (cF x5))) % Found (((conj0 (((eq a) Xx) (cF x5))) x4) x3) as proof of ((and (cR x5)) (((eq a) Xx) (cF x5))) % Found ((((conj (cR x5)) (((eq a) Xx) (cF x5))) x4) x3) as proof of ((and (cR x5)) (((eq a) Xx) (cF x5))) % Found ((((conj (cR x5)) (((eq a) Xx) (cF x5))) x4) x3) as proof of ((and (cR x5)) (((eq a) Xx) (cF x5))) % Found (ex_intro000 ((((conj (cR x5)) (((eq a) Xx) (cF x5))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found ((ex_intro00 x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((ex_intro0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (fun (x4:(cR x0))=> ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))) as proof of ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (fun (x4:(cR x0))=> ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))) as proof of ((cR x0)->((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x5:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x5)) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x5:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x5)) % Found x4:(cR x0) % Instantiate: x5:=x0:b % Found x4 as proof of (cR x5) % Found ((conj00 x4) x3) as proof of ((and (cR x5)) (((eq a) Xx) (cF x5))) % Found (((conj0 (((eq a) Xx) (cF x5))) x4) x3) as proof of ((and (cR x5)) (((eq a) Xx) (cF x5))) % Found ((((conj (cR x5)) (((eq a) Xx) (cF x5))) x4) x3) as proof of ((and (cR x5)) (((eq a) Xx) (cF x5))) % Found ((((conj (cR x5)) (((eq a) Xx) (cF x5))) x4) x3) as proof of ((and (cR x5)) (((eq a) Xx) (cF x5))) % Found (ex_intro000 ((((conj (cR x5)) (((eq a) Xx) (cF x5))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found ((ex_intro00 x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((ex_intro0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (fun (x4:(cR x0))=> ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))) as proof of ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (fun (x4:(cR x0))=> ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))) as proof of ((cR x0)->((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x5:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x5)) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x5:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x5)) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x5:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x5)) % Found x4:(cS x0) % Instantiate: x5:=x0:b % Found x4 as proof of (cS x5) % Found ((conj00 x4) x3) as proof of ((and (cS x5)) (((eq a) Xx) (cF x5))) % Found (((conj0 (((eq a) Xx) (cF x5))) x4) x3) as proof of ((and (cS x5)) (((eq a) Xx) (cF x5))) % Found ((((conj (cS x5)) (((eq a) Xx) (cF x5))) x4) x3) as proof of ((and (cS x5)) (((eq a) Xx) (cF x5))) % Found ((((conj (cS x5)) (((eq a) Xx) (cF x5))) x4) x3) as proof of ((and (cS x5)) (((eq a) Xx) (cF x5))) % Found (ex_intro000 ((((conj (cS x5)) (((eq a) Xx) (cF x5))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found ((ex_intro00 x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (((ex_intro0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (fun (x4:(cS x0))=> ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3))) as proof of ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (fun (x4:(cS x0))=> ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3))) as proof of ((cS x0)->((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x5:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x5)) % Found x4:(cR x0) % Instantiate: x5:=x0:b % Found x4 as proof of (cR x5) % Found ((conj00 x4) x3) as proof of ((and (cR x5)) (((eq a) Xx) (cF x5))) % Found (((conj0 (((eq a) Xx) (cF x5))) x4) x3) as proof of ((and (cR x5)) (((eq a) Xx) (cF x5))) % Found ((((conj (cR x5)) (((eq a) Xx) (cF x5))) x4) x3) as proof of ((and (cR x5)) (((eq a) Xx) (cF x5))) % Found ((((conj (cR x5)) (((eq a) Xx) (cF x5))) x4) x3) as proof of ((and (cR x5)) (((eq a) Xx) (cF x5))) % Found (ex_intro000 ((((conj (cR x5)) (((eq a) Xx) (cF x5))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found ((ex_intro00 x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((ex_intro0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (fun (x4:(cR x0))=> ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))) as proof of ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (fun (x4:(cR x0))=> ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))) as proof of ((cR x0)->((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x5:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x5)) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x5:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x5)) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x5:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x5)) % Found x4:(cS x0) % Instantiate: x5:=x0:b % Found x4 as proof of (cS x5) % Found ((conj00 x4) x3) as proof of ((and (cS x5)) (((eq a) Xx) (cF x5))) % Found (((conj0 (((eq a) Xx) (cF x5))) x4) x3) as proof of ((and (cS x5)) (((eq a) Xx) (cF x5))) % Found ((((conj (cS x5)) (((eq a) Xx) (cF x5))) x4) x3) as proof of ((and (cS x5)) (((eq a) Xx) (cF x5))) % Found ((((conj (cS x5)) (((eq a) Xx) (cF x5))) x4) x3) as proof of ((and (cS x5)) (((eq a) Xx) (cF x5))) % Found (ex_intro000 ((((conj (cS x5)) (((eq a) Xx) (cF x5))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found ((ex_intro00 x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (((ex_intro0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (fun (x4:(cS x0))=> ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3))) as proof of ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (fun (x4:(cS x0))=> ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3))) as proof of ((cS x0)->((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) % Found x5:(cR x1) % Instantiate: x0:=x1:b % Found (fun (x5:(cR x1))=> x5) as proof of (cR x0) % Found (fun (x5:(cR x1))=> x5) as proof of ((cR x1)->(cR x0)) % Found eq_ref000:=(eq_ref00 cS):((cS x1)->(cS x1)) % Found (eq_ref00 cS) as proof of ((cS x1)->(cS x0)) % Found ((eq_ref0 x1) cS) as proof of ((cS x1)->(cS x0)) % Found (((eq_ref b) x1) cS) as proof of ((cS x1)->(cS x0)) % Found (((eq_ref b) x1) cS) as proof of ((cS x1)->(cS x0)) % Found (((eq_ref b) x1) cS) as proof of ((cS x1)->(cS x0)) % Found x5:(cS x0) % Instantiate: x2:=x0:b % Found (fun (x5:(cS x0))=> x5) as proof of (cS x2) % Found (fun (x5:(cS x0))=> x5) as proof of ((cS x0)->(cS x2)) % Found eq_ref000:=(eq_ref00 cR):((cR x0)->(cR x0)) % Found (eq_ref00 cR) as proof of ((cR x0)->(cR x2)) % Found ((eq_ref0 x0) cR) as proof of ((cR x0)->(cR x2)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x2)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x2)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x2)) % Found x40:=(x4 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found x40:=(x4 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found eq_ref000:=(eq_ref00 cR):((cR x1)->(cR x1)) % Found (eq_ref00 cR) as proof of ((cR x1)->(cR x0)) % Found ((eq_ref0 x1) cR) as proof of ((cR x1)->(cR x0)) % Found (((eq_ref b) x1) cR) as proof of ((cR x1)->(cR x0)) % Found (((eq_ref b) x1) cR) as proof of ((cR x1)->(cR x0)) % Found (((eq_ref b) x1) cR) as proof of ((cR x1)->(cR x0)) % Found eq_ref000:=(eq_ref00 cS):((cS x1)->(cS x1)) % Found (eq_ref00 cS) as proof of ((cS x1)->(cS x0)) % Found ((eq_ref0 x1) cS) as proof of ((cS x1)->(cS x0)) % Found (((eq_ref b) x1) cS) as proof of ((cS x1)->(cS x0)) % Found (((eq_ref b) x1) cS) as proof of ((cS x1)->(cS x0)) % Found (((eq_ref b) x1) cS) as proof of ((cS x1)->(cS x0)) % Found x40:=(x4 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found x40:=(x4 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found (x4 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x2)) % Found eq_ref000:=(eq_ref00 cR):((cR x0)->(cR x0)) % Found (eq_ref00 cR) as proof of ((cR x0)->(cR x2)) % Found ((eq_ref0 x0) cR) as proof of ((cR x0)->(cR x2)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x2)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x2)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x2)) % Found x5:(cR x0) % Instantiate: x2:=x0:b % Found (fun (x5:(cR x0))=> x5) as proof of (cR x2) % Found (fun (x5:(cR x0))=> x5) as proof of ((cR x0)->(cR x2)) % Found eq_ref000:=(eq_ref00 cS):((cS x0)->(cS x0)) % Found (eq_ref00 cS) as proof of ((cS x0)->(cS x2)) % Found ((eq_ref0 x0) cS) as proof of ((cS x0)->(cS x2)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x2)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x2)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x2)) % Found x5:(cS x0) % Instantiate: x2:=x0:b % Found (fun (x5:(cS x0))=> x5) as proof of (cS x2) % Found (fun (x5:(cS x0))=> x5) as proof of ((cS x0)->(cS x2)) % Found eq_ref000:=(eq_ref00 cS):((cS x0)->(cS x0)) % Found (eq_ref00 cS) as proof of ((cS x0)->(cS x2)) % Found ((eq_ref0 x0) cS) as proof of ((cS x0)->(cS x2)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x2)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x2)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x2)) % Found x30:=(x3 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0)) % Found (x3 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x4)) % Found (x3 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x4)) % Found (x3 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x4)) % Found eq_ref000:=(eq_ref00 cR):((cR x0)->(cR x0)) % Found (eq_ref00 cR) as proof of ((cR x0)->(cR x4)) % Found ((eq_ref0 x0) cR) as proof of ((cR x0)->(cR x4)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x4)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x4)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x4)) % Found x5:(cS x0) % Instantiate: x4:=x0:b % Found (fun (x5:(cS x0))=> x5) as proof of (cS x4) % Found (fun (x5:(cS x0))=> x5) as proof of ((cS x0)->(cS x4)) % Found eq_ref000:=(eq_ref00 cS):((cS x0)->(cS x0)) % Found (eq_ref00 cS) as proof of ((cS x0)->(cS x4)) % Found ((eq_ref0 x0) cS) as proof of ((cS x0)->(cS x4)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x4)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x4)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x4)) % Found x40:=(x4 (fun (x5:a)=> (cR x0))):((cR x0)->(cR x0)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found (x4 (fun (x5:a)=> (cR x0))) as proof of ((cR x0)->(cR x2)) % Found eq_ref000:=(eq_ref00 cS):((cS x0)->(cS x0)) % Found (eq_ref00 cS) as proof of ((cS x0)->(cS x4)) % Found ((eq_ref0 x0) cS) as proof of ((cS x0)->(cS x4)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x4)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x4)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x4)) % Found x5:(cR x0) % Instantiate: x4:=x0:b % Found (fun (x5:(cR x0))=> x5) as proof of (cR x4) % Found (fun (x5:(cR x0))=> x5) as proof of ((cR x0)->(cR x4)) % Found eq_ref000:=(eq_ref00 cS):((cS x0)->(cS x0)) % Found (eq_ref00 cS) as proof of ((cS x0)->(cS x4)) % Found ((eq_ref0 x0) cS) as proof of ((cS x0)->(cS x4)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x4)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x4)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x4)) % Found x30:=(x3 (fun (x5:a)=> (cS x0))):((cS x0)->(cS x0)) % Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4)) % Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4)) % Found (x3 (fun (x5:a)=> (cS x0))) as proof of ((cS x0)->(cS x4)) % Found x5:(cR x0) % Instantiate: x4:=x0:b % Found (fun (x5:(cR x0))=> x5) as proof of (cR x4) % Found (fun (x5:(cR x0))=> x5) as proof of ((cR x0)->(cR x4)) % Found eq_ref000:=(eq_ref00 cR):((cR x0)->(cR x0)) % Found (eq_ref00 cR) as proof of ((cR x0)->(cR x4)) % Found ((eq_ref0 x0) cR) as proof of ((cR x0)->(cR x4)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x4)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x4)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x4)) % Found eq_ref000:=(eq_ref00 cR):((cR x0)->(cR x0)) % Found (eq_ref00 cR) as proof of ((cR x0)->(cR x4)) % Found ((eq_ref0 x0) cR) as proof of ((cR x0)->(cR x4)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x4)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x4)) % Found (((eq_ref b) x0) cR) as proof of ((cR x0)->(cR x4)) % Found eq_ref000:=(eq_ref00 cS):((cS x0)->(cS x0)) % Found (eq_ref00 cS) as proof of ((cS x0)->(cS x4)) % Found ((eq_ref0 x0) cS) as proof of ((cS x0)->(cS x4)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x4)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x4)) % Found (((eq_ref b) x0) cS) as proof of ((cS x0)->(cS x4)) % 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 (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found ((eta_expansion_dep0 (fun (x5:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF 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 (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found ((eta_expansion_dep0 (fun (x5:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P b0) % Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and (cS x)) (((eq a) Xx) (cF x))))) % Found (eta_expansion_dep00 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eta_expansion_dep0 (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) % Instantiate: b0:=(fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))):(b->Prop) % Found x as proof of (P b0) % Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (eq_ref0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) b0) % Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF 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 (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found ((eta_expansion0 Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF 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 (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found ((eta_expansion0 Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((eta_expansion b) Prop) a0) as proof of (((eq (b->Prop)) a0) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x5:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x5)) % Found x4:(cS x0) % Instantiate: x5:=x0:b % Found x4 as proof of (cS x5) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x5:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x5)) % Found ((conj00 x4) x3) as proof of ((and (cS x5)) (((eq a) Xx) (cF x5))) % Found (((conj0 (((eq a) Xx) (cF x5))) x4) x3) as proof of ((and (cS x5)) (((eq a) Xx) (cF x5))) % Found ((((conj (cS x5)) (((eq a) Xx) (cF x5))) x4) x3) as proof of ((and (cS x5)) (((eq a) Xx) (cF x5))) % Found ((((conj (cS x5)) (((eq a) Xx) (cF x5))) x4) x3) as proof of ((and (cS x5)) (((eq a) Xx) (cF x5))) % Found (ex_intro000 ((((conj (cS x5)) (((eq a) Xx) (cF x5))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found ((ex_intro00 x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (((ex_intro0 (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) % Found (or_intror00 ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3))) as proof of ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) % Found ((or_intror0 ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3))) as proof of ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) % Found (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3))) as proof of ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) % Found (fun (x4:(cS x0))=> (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3)))) as proof of ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) % Found (fun (x4:(cS x0))=> (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3)))) as proof of ((cS x0)->((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) % Found x3:(((eq a) Xx) (cF x0)) % Instantiate: x5:=x0:b % Found x3 as proof of (((eq a) Xx) (cF x5)) % Found x4:(cR x0) % Instantiate: x5:=x0:b % Found x4 as proof of (cR x5) % Found ((conj00 x4) x3) as proof of ((and (cR x5)) (((eq a) Xx) (cF x5))) % Found (((conj0 (((eq a) Xx) (cF x5))) x4) x3) as proof of ((and (cR x5)) (((eq a) Xx) (cF x5))) % Found ((((conj (cR x5)) (((eq a) Xx) (cF x5))) x4) x3) as proof of ((and (cR x5)) (((eq a) Xx) (cF x5))) % Found ((((conj (cR x5)) (((eq a) Xx) (cF x5))) x4) x3) as proof of ((and (cR x5)) (((eq a) Xx) (cF x5))) % Found (ex_intro000 ((((conj (cR x5)) (((eq a) Xx) (cF x5))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found ((ex_intro00 x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (((ex_intro0 (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) % Found (or_introl00 ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))) as proof of ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) % Found ((or_introl0 ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))) as proof of ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) % Found (((or_introl ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))) as proof of ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) % Found (fun (x4:(cR x0))=> (((or_introl ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3)))) as proof of ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) % Found (fun (x4:(cR x0))=> (((or_introl ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3)))) as proof of ((cR x0)->((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) % Found ((or_ind00 (fun (x4:(cR x0))=> (((or_introl ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))))) (fun (x4:(cS x0))=> (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3))))) as proof of ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) % Found (((or_ind0 ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x4:(cR x0))=> (((or_introl ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))))) (fun (x4:(cS x0))=> (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3))))) as proof of ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) % Found ((((fun (P:Prop) (x4:((cR x0)->P)) (x5:((cS x0)->P))=> ((((((or_ind (cR x0)) (cS x0)) P) x4) x5) x2)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x4:(cR x0))=> (((or_introl ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))))) (fun (x4:(cS x0))=> (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3))))) as proof of ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) % Found (fun (x3:(((eq a) Xx) (cF x0)))=> ((((fun (P:Prop) (x4:((cR x0)->P)) (x5:((cS x0)->P))=> ((((((or_ind (cR x0)) (cS x0)) P) x4) x5) x2)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x4:(cR x0))=> (((or_introl ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))))) (fun (x4:(cS x0))=> (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3)))))) as proof of ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) % Found (fun (x2:((or (cR x0)) (cS x0))) (x3:(((eq a) Xx) (cF x0)))=> ((((fun (P:Prop) (x4:((cR x0)->P)) (x5:((cS x0)->P))=> ((((((or_ind (cR x0)) (cS x0)) P) x4) x5) x2)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x4:(cR x0))=> (((or_introl ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))))) (fun (x4:(cS x0))=> (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3)))))) as proof of ((((eq a) Xx) (cF x0))->((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) % Found (fun (x2:((or (cR x0)) (cS x0))) (x3:(((eq a) Xx) (cF x0)))=> ((((fun (P:Prop) (x4:((cR x0)->P)) (x5:((cS x0)->P))=> ((((((or_ind (cR x0)) (cS x0)) P) x4) x5) x2)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x4:(cR x0))=> (((or_introl ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))))) (fun (x4:(cS x0))=> (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3)))))) as proof of (((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))))) % Found (and_rect00 (fun (x2:((or (cR x0)) (cS x0))) (x3:(((eq a) Xx) (cF x0)))=> ((((fun (P:Prop) (x4:((cR x0)->P)) (x5:((cS x0)->P))=> ((((((or_ind (cR x0)) (cS x0)) P) x4) x5) x2)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x4:(cR x0))=> (((or_introl ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))))) (fun (x4:(cS x0))=> (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3))))))) as proof of ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) % Found ((and_rect0 ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x2:((or (cR x0)) (cS x0))) (x3:(((eq a) Xx) (cF x0)))=> ((((fun (P:Prop) (x4:((cR x0)->P)) (x5:((cS x0)->P))=> ((((((or_ind (cR x0)) (cS x0)) P) x4) x5) x2)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x4:(cR x0))=> (((or_introl ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))))) (fun (x4:(cS x0))=> (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3))))))) as proof of ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) % Found (((fun (P:Type) (x2:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x2:((or (cR x0)) (cS x0))) (x3:(((eq a) Xx) (cF x0)))=> ((((fun (P:Prop) (x4:((cR x0)->P)) (x5:((cS x0)->P))=> ((((((or_ind (cR x0)) (cS x0)) P) x4) x5) x2)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x4:(cR x0))=> (((or_introl ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))))) (fun (x4:(cS x0))=> (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3))))))) as proof of ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) % Found (fun (x1:((and ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))=> (((fun (P:Type) (x2:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x2:((or (cR x0)) (cS x0))) (x3:(((eq a) Xx) (cF x0)))=> ((((fun (P:Prop) (x4:((cR x0)->P)) (x5:((cS x0)->P))=> ((((((or_ind (cR x0)) (cS x0)) P) x4) x5) x2)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x4:(cR x0))=> (((or_introl ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))))) (fun (x4:(cS x0))=> (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3)))))))) as proof of ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) % Found (fun (x0:b) (x1:((and ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))=> (((fun (P:Type) (x2:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x2:((or (cR x0)) (cS x0))) (x3:(((eq a) Xx) (cF x0)))=> ((((fun (P:Prop) (x4:((cR x0)->P)) (x5:((cS x0)->P))=> ((((((or_ind (cR x0)) (cS x0)) P) x4) x5) x2)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x4:(cR x0))=> (((or_introl ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))))) (fun (x4:(cS x0))=> (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3)))))))) as proof of (((and ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0)))->((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) % Found (fun (x0:b) (x1:((and ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))=> (((fun (P:Type) (x2:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x2:((or (cR x0)) (cS x0))) (x3:(((eq a) Xx) (cF x0)))=> ((((fun (P:Prop) (x4:((cR x0)->P)) (x5:((cS x0)->P))=> ((((((or_ind (cR x0)) (cS x0)) P) x4) x5) x2)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x4:(cR x0))=> (((or_introl ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))))) (fun (x4:(cS x0))=> (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3)))))))) as proof of (forall (x:b), (((and ((or (cR x)) (cS x))) (((eq a) Xx) (cF x)))->((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))))) % Found (ex_ind00 (fun (x0:b) (x1:((and ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))=> (((fun (P:Type) (x2:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x2:((or (cR x0)) (cS x0))) (x3:(((eq a) Xx) (cF x0)))=> ((((fun (P:Prop) (x4:((cR x0)->P)) (x5:((cS x0)->P))=> ((((((or_ind (cR x0)) (cS x0)) P) x4) x5) x2)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x4:(cR x0))=> (((or_introl ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))))) (fun (x4:(cS x0))=> (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3))))))))) as proof of ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) % Found ((ex_ind0 ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x0:b) (x1:((and ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))=> (((fun (P:Type) (x2:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x2:((or (cR x0)) (cS x0))) (x3:(((eq a) Xx) (cF x0)))=> ((((fun (P:Prop) (x4:((cR x0)->P)) (x5:((cS x0)->P))=> ((((((or_ind (cR x0)) (cS x0)) P) x4) x5) x2)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x4:(cR x0))=> (((or_introl ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))))) (fun (x4:(cS x0))=> (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3))))))))) as proof of ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) % Found (((fun (P:Prop) (x0:(forall (x:b), (((and ((or (cR x)) (cS x))) (((eq a) Xx) (cF x)))->P)))=> (((((ex_ind b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) P) x0) x)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x0:b) (x1:((and ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))=> (((fun (P:Type) (x2:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x2:((or (cR x0)) (cS x0))) (x3:(((eq a) Xx) (cF x0)))=> ((((fun (P:Prop) (x4:((cR x0)->P)) (x5:((cS x0)->P))=> ((((((or_ind (cR x0)) (cS x0)) P) x4) x5) x2)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x4:(cR x0))=> (((or_introl ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))))) (fun (x4:(cS x0))=> (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3))))))))) as proof of ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) % Found (fun (x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))))=> (((fun (P:Prop) (x0:(forall (x:b), (((and ((or (cR x)) (cS x))) (((eq a) Xx) (cF x)))->P)))=> (((((ex_ind b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) P) x0) x)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x0:b) (x1:((and ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))=> (((fun (P:Type) (x2:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x2:((or (cR x0)) (cS x0))) (x3:(((eq a) Xx) (cF x0)))=> ((((fun (P:Prop) (x4:((cR x0)->P)) (x5:((cS x0)->P))=> ((((((or_ind (cR x0)) (cS x0)) P) x4) x5) x2)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x4:(cR x0))=> (((or_introl ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))))) (fun (x4:(cS x0))=> (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3)))))))))) as proof of ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) % Found (fun (Xx:a) (x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))))=> (((fun (P:Prop) (x0:(forall (x:b), (((and ((or (cR x)) (cS x))) (((eq a) Xx) (cF x)))->P)))=> (((((ex_ind b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) P) x0) x)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x0:b) (x1:((and ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))=> (((fun (P:Type) (x2:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x2:((or (cR x0)) (cS x0))) (x3:(((eq a) Xx) (cF x0)))=> ((((fun (P:Prop) (x4:((cR x0)->P)) (x5:((cS x0)->P))=> ((((((or_ind (cR x0)) (cS x0)) P) x4) x5) x2)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x4:(cR x0))=> (((or_introl ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))))) (fun (x4:(cS x0))=> (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3)))))))))) as proof of (((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))->((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) % Found (fun (Xx:a) (x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))))=> (((fun (P:Prop) (x0:(forall (x:b), (((and ((or (cR x)) (cS x))) (((eq a) Xx) (cF x)))->P)))=> (((((ex_ind b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) P) x0) x)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x0:b) (x1:((and ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))=> (((fun (P:Type) (x2:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x2:((or (cR x0)) (cS x0))) (x3:(((eq a) Xx) (cF x0)))=> ((((fun (P:Prop) (x4:((cR x0)->P)) (x5:((cS x0)->P))=> ((((((or_ind (cR x0)) (cS x0)) P) x4) x5) x2)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x4:(cR x0))=> (((or_introl ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))))) (fun (x4:(cS x0))=> (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3)))))))))) as proof of (forall (Xx:a), (((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt)))))->((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))))) % Got proof (fun (Xx:a) (x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))))=> (((fun (P:Prop) (x0:(forall (x:b), (((and ((or (cR x)) (cS x))) (((eq a) Xx) (cF x)))->P)))=> (((((ex_ind b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) P) x0) x)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x0:b) (x1:((and ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))=> (((fun (P:Type) (x2:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x2:((or (cR x0)) (cS x0))) (x3:(((eq a) Xx) (cF x0)))=> ((((fun (P:Prop) (x4:((cR x0)->P)) (x5:((cS x0)->P))=> ((((((or_ind (cR x0)) (cS x0)) P) x4) x5) x2)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x4:(cR x0))=> (((or_introl ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))))) (fun (x4:(cS x0))=> (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3)))))))))) % Time elapsed = 212.084098s % node=22959 cost=1032.000000 depth=32 % :::::::::::::::::::::: % % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p % % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p % (fun (Xx:a) (x:((ex b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))))=> (((fun (P:Prop) (x0:(forall (x:b), (((and ((or (cR x)) (cS x))) (((eq a) Xx) (cF x)))->P)))=> (((((ex_ind b) (fun (Xt:b)=> ((and ((or (cR Xt)) (cS Xt))) (((eq a) Xx) (cF Xt))))) P) x0) x)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x0:b) (x1:((and ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))))=> (((fun (P:Type) (x2:(((or (cR x0)) (cS x0))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((or (cR x0)) (cS x0))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x2:((or (cR x0)) (cS x0))) (x3:(((eq a) Xx) (cF x0)))=> ((((fun (P:Prop) (x4:((cR x0)->P)) (x5:((cS x0)->P))=> ((((((or_ind (cR x0)) (cS x0)) P) x4) x5) x2)) ((or ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))))) (fun (x4:(cR x0))=> (((or_introl ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cR x0)) (((eq a) Xx) (cF x0))) x4) x3))))) (fun (x4:(cS x0))=> (((or_intror ((ex b) (fun (Xt:b)=> ((and (cR Xt)) (((eq a) Xx) (cF Xt)))))) ((ex b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt)))))) ((((ex_intro b) (fun (Xt:b)=> ((and (cS Xt)) (((eq a) Xx) (cF Xt))))) x0) ((((conj (cS x0)) (((eq a) Xx) (cF x0))) x4) x3)))))))))) % % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p % EOF %------------------------------------------------------------------------------