cocATP---0.2.0.THM-Prf.s

%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SWW469^1 : TPTP v6.1.0. Released v5.3.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n184.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 13:37:19 EDT 2014

% Result   : Theorem 206.15s
% Output   : Proof 206.15s
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----ERROR: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SWW469^1 : TPTP v6.1.0. Released v5.3.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n184.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 09:11:51 CDT 2014
% % CPUTime  : 206.15 
% 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 0x227f908>, <kernel.Type object at 0x227f7a0>) of role type named ty_ty_tc__Com__Ostate
% Using role type
% Declaring state:Type
% FOF formula (<kernel.Constant object at 0x227f6c8>, <kernel.Sort object at 0x21455a8>) of role type named sy_c_HOL_Oinduct__false
% Using role type
% Declaring induct_false:Prop
% FOF formula (<kernel.Constant object at 0x227f908>, <kernel.Sort object at 0x21455a8>) of role type named sy_c_HOL_Oinduct__true
% Using role type
% Declaring induct_true:Prop
% FOF formula (<kernel.Constant object at 0x227ffc8>, <kernel.Sort object at 0x21455a8>) of role type named sy_c_Hoare__Mirabelle__ghhkfsbqqq_Ostate__not__singleton
% Using role type
% Declaring hoare_1821564147gleton:Prop
% FOF formula ((iff hoare_1821564147gleton) ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))) of role axiom named fact_0_state__not__singleton__def
% A new axiom: ((iff hoare_1821564147gleton) ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))
% FOF formula (induct_false->False) of role axiom named fact_1_induct__false__def
% A new axiom: (induct_false->False)
% FOF formula induct_true of role axiom named fact_2_induct__trueI
% A new axiom: induct_true
% FOF formula induct_true of role axiom named fact_3_induct__true__def
% A new axiom: induct_true
% FOF formula hoare_1821564147gleton of role hypothesis named conj_0
% A new axiom: hoare_1821564147gleton
% FOF formula (forall (T:state), ((forall (S:state), (((eq state) S) T))->False)) of role conjecture named conj_1
% Conjecture to prove = (forall (T:state), ((forall (S:state), (((eq state) S) T))->False)):Prop
% Parameter state_DUMMY:state.
% We need to prove ['(forall (T:state), ((forall (S:state), (((eq state) S) T))->False))']
% Parameter state:Type.
% Parameter induct_false:Prop.
% Parameter induct_true:Prop.
% Parameter hoare_1821564147gleton:Prop.
% Axiom fact_0_state__not__singleton__def:((iff hoare_1821564147gleton) ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))).
% Axiom fact_1_induct__false__def:(induct_false->False).
% Axiom fact_2_induct__trueI:induct_true.
% Axiom fact_3_induct__true__def:induct_true.
% Axiom conj_0:hoare_1821564147gleton.
% Trying to prove (forall (T:state), ((forall (S:state), (((eq state) S) T))->False))
% Found fact_1_induct__false__def0:False
% Found (fun (x:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((forall (S:state), (((eq state) S) T))->False)
% Found fact_1_induct__false__def0:False
% Found (fun (x:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (T:state) (x:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((forall (S:state), (((eq state) S) T))->False)
% Found (fun (T:state) (x:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (forall (T:state), ((forall (S:state), (((eq state) S) T))->False))
% Found fact_1_induct__false__def0:False
% Found (fun (x1:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x1:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found fact_1_induct__false__def0:False
% Found (fun (x:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found fact_1_induct__false__def0:False
% Found (fun (x1:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x0:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x1:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found (fun (x0:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x1:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False))
% Found fact_1_induct__false__def0:False
% Found (fun (x:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (T:state) (x:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (T:state) (x:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (forall (T:state), (not (forall (S:state), (((eq state) S) T))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found (x0 conj_0) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found (x0 conj_0) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found (x0 conj_0) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found (x0 conj_0) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found (x0 conj_0) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found (x0 conj_0) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found fact_1_induct__false__def0:False
% Found (fun (x1:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x1:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found x1:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found (x conj_0) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found (x conj_0) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x0:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x0:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)->(not (forall (S:state), (((eq state) S) T))))
% Found fact_1_induct__false__def0:False
% Found (fun (x1:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x0:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x1:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found (fun (x0:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x1:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False))
% Found x2:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found (x conj_0) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found (x conj_0) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found (x0 conj_0) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found (x0 conj_0) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x0:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0)))))))) (x0:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)->(not (forall (S:state), (((eq state) S) T))))
% Found (fun (x:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0)))))))) (x0:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0)))))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)->(not (forall (S:state), (((eq state) S) T)))))
% Found x2:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found (x conj_0) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found (x conj_0) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found (x0 conj_0) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found (x0 conj_0) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found (x0 conj_0) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found (x0 conj_0) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T:state)=> (not (((eq state) x2) T))))->False)
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:((ex state) (fun (T:state)=> (not (((eq state) x1) T))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found (fun (x2:((ex state) (fun (T:state)=> (not (((eq state) x1) T))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T:state)=> (not (((eq state) x1) T))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False))
% Found x1:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x1 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:state) (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T:state)=> (not (((eq state) x2) T))))->False)
% Found (fun (x2:state) (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> fact_1_induct__false__def0) as proof of (forall (x:state), (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->False))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found x1:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x1 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x0:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x0 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:((ex state) (fun (T:state)=> (not (((eq state) x1) T))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found (fun (x1:state) (x2:((ex state) (fun (T:state)=> (not (((eq state) x1) T))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T:state)=> (not (((eq state) x1) T))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False))
% Found (fun (x1:state) (x2:((ex state) (fun (T:state)=> (not (((eq state) x1) T))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of (forall (x:state), (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x2:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x2 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x1:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x1 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T:state)=> (not (((eq state) x2) T))))->False)
% Found x0:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x0 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found (fun (x2:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x2:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x2 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x1:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x1 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:((ex state) (fun (T:state)=> (not (((eq state) x1) T))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found (fun (x2:((ex state) (fun (T:state)=> (not (((eq state) x1) T))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T:state)=> (not (((eq state) x1) T))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x2:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x2 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:state) (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T:state)=> (not (((eq state) x2) T))))->False)
% Found (fun (x2:state) (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> fact_1_induct__false__def0) as proof of (forall (x:state), (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->False))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found (fun (x1:((ex state) (fun (T:state)=> (not (((eq state) x0) T))))) (x2:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False))
% Found (fun (x1:((ex state) (fun (T:state)=> (not (((eq state) x0) T))))) (x2:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T:state)=> (not (((eq state) x0) T))))->((hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)))
% Found fact_1_induct__false__def0:False
% Found (fun (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found fact_1_induct__false__def0:False
% Found (fun (x4:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x4:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x2:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x2 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x0:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x0 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:((ex state) (fun (T:state)=> (not (((eq state) x1) T))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found (fun (x1:state) (x2:((ex state) (fun (T:state)=> (not (((eq state) x1) T))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T:state)=> (not (((eq state) x1) T))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False))
% Found (fun (x1:state) (x2:((ex state) (fun (T:state)=> (not (((eq state) x1) T))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of (forall (x:state), (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)))
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found fact_1_induct__false__def0:False
% Found (fun (x4:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x4:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of (P0 x1)
% Found ((eq_ref0 x1) P) as proof of (P0 x1)
% Found (((eq_ref state) x1) P) as proof of (P0 x1)
% Found (((eq_ref state) x1) P) as proof of (P0 x1)
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of (P0 x2)
% Found ((eq_ref0 x2) P) as proof of (P0 x2)
% Found (((eq_ref state) x2) P) as proof of (P0 x2)
% Found (((eq_ref state) x2) P) as proof of (P0 x2)
% Found fact_1_induct__false__def0:False
% Found (fun (x4:((ex state) (fun (T0:state)=> (not (((eq state) x3) T0)))))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x4:((ex state) (fun (T0:state)=> (not (((eq state) x3) T0)))))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T0:state)=> (not (((eq state) x3) T0))))->False)
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (T:state) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (T:state) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (forall (T:state), (not (forall (S:state), (((eq state) S) T))))
% Found x1:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x1 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x4:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x4:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x4:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)->(not (forall (S:state), (((eq state) S) T))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T:state)=> (not (((eq state) x2) T))))->False)
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found (fun (x1:((ex state) (fun (T:state)=> (not (((eq state) x0) T))))) (x2:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False))
% Found (fun (x0:state) (x1:((ex state) (fun (T:state)=> (not (((eq state) x0) T))))) (x2:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T:state)=> (not (((eq state) x0) T))))->((hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)))
% Found (fun (x0:state) (x1:((ex state) (fun (T:state)=> (not (((eq state) x0) T))))) (x2:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of (forall (x:state), (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->((hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x2:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x2 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x2:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x2 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x0:((ex state) (fun (T0:state)=> (not (((eq state) x) T0))))) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x0:((ex state) (fun (T0:state)=> (not (((eq state) x) T0))))) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T0:state)=> (not (((eq state) x) T0))))->(not (forall (S:state), (((eq state) S) T))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x4:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x3:((ex state) (fun (T0:state)=> (not (((eq state) x2) T0))))) (x4:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x3:((ex state) (fun (T0:state)=> (not (((eq state) x2) T0))))) (x4:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T0:state)=> (not (((eq state) x2) T0))))->(not (forall (S:state), (((eq state) S) T))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:((ex state) (fun (T:state)=> (not (((eq state) x1) T))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found (fun (x2:((ex state) (fun (T:state)=> (not (((eq state) x1) T))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T:state)=> (not (((eq state) x1) T))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found fact_1_induct__false__def0:False
% Found (fun (x4:((ex state) (fun (T0:state)=> (not (((eq state) x3) T0)))))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x3:state) (x4:((ex state) (fun (T0:state)=> (not (((eq state) x3) T0)))))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T0:state)=> (not (((eq state) x3) T0))))->False)
% Found (fun (x3:state) (x4:((ex state) (fun (T0:state)=> (not (((eq state) x3) T0)))))=> fact_1_induct__false__def0) as proof of (forall (x:state), (((ex state) (fun (T0:state)=> (not (((eq state) x) T0))))->False))
% Found x2:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x2 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found eq_ref000:=(eq_ref00 P):((P x)->(P x))
% Found (eq_ref00 P) as proof of (P0 x)
% Found ((eq_ref0 x) P) as proof of (P0 x)
% Found (((eq_ref state) x) P) as proof of (P0 x)
% Found (((eq_ref state) x) P) as proof of (P0 x)
% Found fact_1_induct__false__def0:False
% Found (fun (x5:(not (((eq state) x1) x4)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x5:(not (((eq state) x1) x4)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x1) x4))->False)
% Found fact_1_induct__false__def0:False
% Found (fun (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (T:state) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x0:((ex state) (fun (T:state)=> (not (((eq state) x) T))))) (T:state) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (forall (T:state), (not (forall (S:state), (((eq state) S) T))))
% Found (fun (x0:((ex state) (fun (T:state)=> (not (((eq state) x) T))))) (T:state) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->(forall (T:state), (not (forall (S:state), (((eq state) S) T)))))
% Found fact_1_induct__false__def0:False
% Found (fun (x4:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x4:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x2:((ex state) (fun (T0:state)=> (not (((eq state) x1) T0))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x4:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)->(not (forall (S:state), (((eq state) S) T))))
% Found (fun (x2:((ex state) (fun (T0:state)=> (not (((eq state) x1) T0))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x4:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T0:state)=> (not (((eq state) x1) T0))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)->(not (forall (S:state), (((eq state) S) T)))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:state) (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T:state)=> (not (((eq state) x2) T))))->False)
% Found (fun (x2:state) (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> fact_1_induct__false__def0) as proof of (forall (x:state), (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->False))
% Found eq_ref000:=(eq_ref00 P):((P x)->(P x))
% Found (eq_ref00 P) as proof of (P0 x)
% Found ((eq_ref0 x) P) as proof of (P0 x)
% Found (((eq_ref state) x) P) as proof of (P0 x)
% Found (((eq_ref state) x) P) as proof of (P0 x)
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x5:(not (((eq state) x2) x4)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x5:(not (((eq state) x2) x4)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x2) x4))->False)
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x0:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x0 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found eq_ref000:=(eq_ref00 P):((P x)->(P x))
% Found (eq_ref00 P) as proof of (P0 x)
% Found ((eq_ref0 x) P) as proof of (P0 x)
% Found (((eq_ref state) x) P) as proof of (P0 x)
% Found (((eq_ref state) x) P) as proof of (P0 x)
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found eq_ref000:=(eq_ref00 P):((P x)->(P x))
% Found (eq_ref00 P) as proof of (P0 x)
% Found ((eq_ref0 x) P) as proof of (P0 x)
% Found (((eq_ref state) x) P) as proof of (P0 x)
% Found (((eq_ref state) x) P) as proof of (P0 x)
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x0:((ex state) (fun (T0:state)=> (not (((eq state) x) T0))))) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x:state) (x0:((ex state) (fun (T0:state)=> (not (((eq state) x) T0))))) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T0:state)=> (not (((eq state) x) T0))))->(not (forall (S:state), (((eq state) S) T))))
% Found (fun (x:state) (x0:((ex state) (fun (T0:state)=> (not (((eq state) x) T0))))) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (forall (x:state), (((ex state) (fun (T0:state)=> (not (((eq state) x) T0))))->(not (forall (S:state), (((eq state) S) T)))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x1:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x1 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x4:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x3:((ex state) (fun (T0:state)=> (not (((eq state) x2) T0))))) (x4:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x2:state) (x3:((ex state) (fun (T0:state)=> (not (((eq state) x2) T0))))) (x4:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T0:state)=> (not (((eq state) x2) T0))))->(not (forall (S:state), (((eq state) S) T))))
% Found (fun (x2:state) (x3:((ex state) (fun (T0:state)=> (not (((eq state) x2) T0))))) (x4:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (forall (x:state), (((ex state) (fun (T0:state)=> (not (((eq state) x) T0))))->(not (forall (S:state), (((eq state) S) T)))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:((ex state) (fun (T:state)=> (not (((eq state) x1) T))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found (fun (x1:state) (x2:((ex state) (fun (T:state)=> (not (((eq state) x1) T))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T:state)=> (not (((eq state) x1) T))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False))
% Found (fun (x1:state) (x2:((ex state) (fun (T:state)=> (not (((eq state) x1) T))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of (forall (x:state), (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x2:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)->(not (forall (S:state), (((eq state) S) T))))
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref state) x0) P) as proof of (P0 x0)
% Found (((eq_ref state) x0) P) as proof of (P0 x0)
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of (P0 x1)
% Found ((eq_ref0 x1) P) as proof of (P0 x1)
% Found (((eq_ref state) x1) P) as proof of (P0 x1)
% Found (((eq_ref state) x1) P) as proof of (P0 x1)
% Found x0:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x0 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x5:(not (((eq state) x1) x4)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x4:state) (x5:(not (((eq state) x1) x4)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x1) x4))->False)
% Found (fun (x4:state) (x5:(not (((eq state) x1) x4)))=> fact_1_induct__false__def0) as proof of (forall (x:state), ((not (((eq state) x1) x))->False))
% Found fact_1_induct__false__def0:False
% Found (fun (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (T:state) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x0:((ex state) (fun (T:state)=> (not (((eq state) x) T))))) (T:state) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (forall (T:state), (not (forall (S:state), (((eq state) S) T))))
% Found (fun (x:state) (x0:((ex state) (fun (T:state)=> (not (((eq state) x) T))))) (T:state) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->(forall (T:state), (not (forall (S:state), (((eq state) S) T)))))
% Found (fun (x:state) (x0:((ex state) (fun (T:state)=> (not (((eq state) x) T))))) (T:state) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (forall (x:state), (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->(forall (T:state), (not (forall (S:state), (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of (P0 x2)
% Found ((eq_ref0 x2) P) as proof of (P0 x2)
% Found (((eq_ref state) x2) P) as proof of (P0 x2)
% Found (((eq_ref state) x2) P) as proof of (P0 x2)
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x5:(not (((eq state) x2) x4)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x2) x4))->False)
% Found (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> fact_1_induct__false__def0) as proof of (forall (x:state), ((not (((eq state) x2) x))->False))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T:state)=> (not (((eq state) x2) T))))->False)
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x2:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)->(not (forall (S:state), (((eq state) S) T))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found (fun (x2:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x1:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x1 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x0:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x0 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found eq_ref00:=(eq_ref0 b):(((eq state) b) b)
% Found (eq_ref0 b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found x6:=(x x1):(((eq state) x1) T)
% Found (x x1) as proof of (((eq state) x1) b)
% Found (x x1) as proof of (((eq state) x1) b)
% Found (x x1) as proof of (((eq state) x1) b)
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x6:=(x x2):(((eq state) x2) T)
% Found (x x2) as proof of (((eq state) x2) b)
% Found (x x2) as proof of (((eq state) x2) b)
% Found (x x2) as proof of (((eq state) x2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq state) b) b)
% Found (eq_ref0 b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x2:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x2 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x1:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x1 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:((ex state) (fun (T0:state)=> (not (((eq state) x1) T0))))) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x2:((ex state) (fun (T0:state)=> (not (((eq state) x1) T0))))) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T0:state)=> (not (((eq state) x1) T0))))->(not (forall (S:state), (((eq state) S) T))))
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x1:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0)))))))) (x2:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)->(not (forall (S:state), (((eq state) S) T))))
% Found (fun (x1:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0)))))))) (x2:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0)))))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)->(not (forall (S:state), (((eq state) S) T)))))
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:((ex state) (fun (T:state)=> (not (((eq state) x1) T))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found (fun (x2:((ex state) (fun (T:state)=> (not (((eq state) x1) T))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T:state)=> (not (((eq state) x1) T))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x0:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x0 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x5:(not (((eq state) x0) x4)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x5:(not (((eq state) x0) x4)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x0) x4))->False)
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:state) (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T:state)=> (not (((eq state) x2) T))))->False)
% Found (fun (x2:state) (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> fact_1_induct__false__def0) as proof of (forall (x:state), (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->False))
% Found x1:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x1 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x0:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x0 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x1:((ex state) (fun (T0:state)=> (not (((eq state) x0) T0))))) (x2:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)->(not (forall (S:state), (((eq state) S) T))))
% Found (fun (x1:((ex state) (fun (T0:state)=> (not (((eq state) x0) T0))))) (x2:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T0:state)=> (not (((eq state) x0) T0))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)->(not (forall (S:state), (((eq state) S) T)))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x5:(not (((eq state) x1) x4)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x5:(not (((eq state) x1) x4)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x1) x4))->False)
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found (fun (x1:((ex state) (fun (T:state)=> (not (((eq state) x0) T))))) (x2:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False))
% Found (fun (x1:((ex state) (fun (T:state)=> (not (((eq state) x0) T))))) (x2:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T:state)=> (not (((eq state) x0) T))))->((hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)))
% Found fact_1_induct__false__def0:False
% Found (fun (x5:(not (((eq state) x2) x4)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x5:(not (((eq state) x2) x4)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x2) x4))->False)
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x2:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x2 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x1:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x1 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x2:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:((ex state) (fun (T0:state)=> (not (((eq state) x1) T0))))) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x1:state) (x2:((ex state) (fun (T0:state)=> (not (((eq state) x1) T0))))) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T0:state)=> (not (((eq state) x1) T0))))->(not (forall (S:state), (((eq state) S) T))))
% Found (fun (x1:state) (x2:((ex state) (fun (T0:state)=> (not (((eq state) x1) T0))))) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (forall (x:state), (((ex state) (fun (T0:state)=> (not (((eq state) x) T0))))->(not (forall (S:state), (((eq state) S) T)))))
% Found fact_1_induct__false__def0:False
% Found (fun (x2:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x1:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0)))))))) (x2:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)->(not (forall (S:state), (((eq state) S) T))))
% Found (fun (x0:((ex state) (fun (T0:state)=> (not (((eq state) x) T0))))) (x1:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0)))))))) (x2:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0)))))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)->(not (forall (S:state), (((eq state) S) T)))))
% Found (fun (x0:((ex state) (fun (T0:state)=> (not (((eq state) x) T0))))) (x1:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0)))))))) (x2:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T0:state)=> (not (((eq state) x) T0))))->((hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0)))))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)->(not (forall (S:state), (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:((ex state) (fun (T:state)=> (not (((eq state) x1) T))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found (fun (x1:state) (x2:((ex state) (fun (T:state)=> (not (((eq state) x1) T))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T:state)=> (not (((eq state) x1) T))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False))
% Found (fun (x1:state) (x2:((ex state) (fun (T:state)=> (not (((eq state) x1) T))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of (forall (x:state), (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x0:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x0 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of (P0 x1)
% Found ((eq_ref0 x1) P) as proof of (P0 x1)
% Found (((eq_ref state) x1) P) as proof of (P0 x1)
% Found (((eq_ref state) x1) P) as proof of (P0 x1)
% Found fact_1_induct__false__def0:False
% Found (fun (x10:(not (((eq state) x) x00)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x10:(not (((eq state) x) x00)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x) x00))->False)
% Found eq_ref00:=(eq_ref0 x):(((eq state) x) x)
% Found (eq_ref0 x) as proof of (((eq state) x) b)
% Found ((eq_ref state) x) as proof of (((eq state) x) b)
% Found ((eq_ref state) x) as proof of (((eq state) x) b)
% Found ((eq_ref state) x) as proof of (((eq state) x) b)
% Found eq_ref00:=(eq_ref0 b):(((eq state) b) b)
% Found (eq_ref0 b) as proof of (((eq state) b) x00)
% Found ((eq_ref state) b) as proof of (((eq state) b) x00)
% Found ((eq_ref state) b) as proof of (((eq state) b) x00)
% Found ((eq_ref state) b) as proof of (((eq state) b) x00)
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x10:(not (((eq state) x) x00)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x10:(not (((eq state) x) x00)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x) x00))->False)
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x1:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x1 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found eq_ref00:=(eq_ref0 x):(((eq state) x) x)
% Found (eq_ref0 x) as proof of (((eq state) x) b)
% Found ((eq_ref state) x) as proof of (((eq state) x) b)
% Found ((eq_ref state) x) as proof of (((eq state) x) b)
% Found ((eq_ref state) x) as proof of (((eq state) x) b)
% Found eq_ref00:=(eq_ref0 b):(((eq state) b) b)
% Found (eq_ref0 b) as proof of (((eq state) b) x00)
% Found ((eq_ref state) b) as proof of (((eq state) b) x00)
% Found ((eq_ref state) b) as proof of (((eq state) b) x00)
% Found ((eq_ref state) b) as proof of (((eq state) b) x00)
% Found fact_1_induct__false__def0:False
% Found (fun (x5:(not (((eq state) x0) x4)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x4:state) (x5:(not (((eq state) x0) x4)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x0) x4))->False)
% Found (fun (x4:state) (x5:(not (((eq state) x0) x4)))=> fact_1_induct__false__def0) as proof of (forall (x:state), ((not (((eq state) x0) x))->False))
% Found x0:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x0 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of (P0 x2)
% Found ((eq_ref0 x2) P) as proof of (P0 x2)
% Found (((eq_ref state) x2) P) as proof of (P0 x2)
% Found (((eq_ref state) x2) P) as proof of (P0 x2)
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of (P0 x1)
% Found ((eq_ref0 x1) P) as proof of (P0 x1)
% Found (((eq_ref state) x1) P) as proof of (P0 x1)
% Found (((eq_ref state) x1) P) as proof of (P0 x1)
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found eq_ref00:=(eq_ref0 x):(((eq state) x) x)
% Found (eq_ref0 x) as proof of (((eq state) x) b)
% Found ((eq_ref state) x) as proof of (((eq state) x) b)
% Found ((eq_ref state) x) as proof of (((eq state) x) b)
% Found ((eq_ref state) x) as proof of (((eq state) x) b)
% Found eq_ref00:=(eq_ref0 b):(((eq state) b) b)
% Found (eq_ref0 b) as proof of (((eq state) b) x00)
% Found ((eq_ref state) b) as proof of (((eq state) b) x00)
% Found ((eq_ref state) b) as proof of (((eq state) b) x00)
% Found ((eq_ref state) b) as proof of (((eq state) b) x00)
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x1:((ex state) (fun (T0:state)=> (not (((eq state) x0) T0))))) (x2:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)->(not (forall (S:state), (((eq state) S) T))))
% Found (fun (x0:state) (x1:((ex state) (fun (T0:state)=> (not (((eq state) x0) T0))))) (x2:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T0:state)=> (not (((eq state) x0) T0))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)->(not (forall (S:state), (((eq state) S) T)))))
% Found (fun (x0:state) (x1:((ex state) (fun (T0:state)=> (not (((eq state) x0) T0))))) (x2:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (forall (x:state), (((ex state) (fun (T0:state)=> (not (((eq state) x) T0))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)->(not (forall (S:state), (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x1:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found (x conj_0) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found (x conj_0) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x5:(not (((eq state) x1) x4)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x4:state) (x5:(not (((eq state) x1) x4)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x1) x4))->False)
% Found (fun (x4:state) (x5:(not (((eq state) x1) x4)))=> fact_1_induct__false__def0) as proof of (forall (x:state), ((not (((eq state) x1) x))->False))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x5:(not (((eq state) x2) x4)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x2) x4))->False)
% Found (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> fact_1_induct__false__def0) as proof of (forall (x:state), ((not (((eq state) x2) x))->False))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found eq_ref00:=(eq_ref0 b):(((eq state) b) b)
% Found (eq_ref0 b) as proof of (((eq state) b) x00)
% Found ((eq_ref state) b) as proof of (((eq state) b) x00)
% Found ((eq_ref state) b) as proof of (((eq state) b) x00)
% Found ((eq_ref state) b) as proof of (((eq state) b) x00)
% Found eq_ref00:=(eq_ref0 x):(((eq state) x) x)
% Found (eq_ref0 x) as proof of (((eq state) x) b)
% Found ((eq_ref state) x) as proof of (((eq state) x) b)
% Found ((eq_ref state) x) as proof of (((eq state) x) b)
% Found ((eq_ref state) x) as proof of (((eq state) x) b)
% Found eq_ref000:=(eq_ref00 P):((P x3)->(P x3))
% Found (eq_ref00 P) as proof of (P0 x3)
% Found ((eq_ref0 x3) P) as proof of (P0 x3)
% Found (((eq_ref state) x3) P) as proof of (P0 x3)
% Found (((eq_ref state) x3) P) as proof of (P0 x3)
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found (fun (x1:((ex state) (fun (T:state)=> (not (((eq state) x0) T))))) (x2:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False))
% Found (fun (x0:state) (x1:((ex state) (fun (T:state)=> (not (((eq state) x0) T))))) (x2:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T:state)=> (not (((eq state) x0) T))))->((hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)))
% Found (fun (x0:state) (x1:((ex state) (fun (T:state)=> (not (((eq state) x0) T))))) (x2:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of (forall (x:state), (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->((hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x2:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x2 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of (P0 x1)
% Found ((eq_ref0 x1) P) as proof of (P0 x1)
% Found (((eq_ref state) x1) P) as proof of (P0 x1)
% Found (((eq_ref state) x1) P) as proof of (P0 x1)
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x1:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x1 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of (P0 x2)
% Found ((eq_ref0 x2) P) as proof of (P0 x2)
% Found (((eq_ref state) x2) P) as proof of (P0 x2)
% Found (((eq_ref state) x2) P) as proof of (P0 x2)
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of (P0 x2)
% Found ((eq_ref0 x2) P) as proof of (P0 x2)
% Found (((eq_ref state) x2) P) as proof of (P0 x2)
% Found (((eq_ref state) x2) P) as proof of (P0 x2)
% Found fact_1_induct__false__def0:False
% Found (fun (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((forall (S:state), (((eq state) S) T))->False)
% Found x6:=(x x0):(((eq state) x0) T)
% Found (x x0) as proof of (((eq state) x0) b)
% Found (x x0) as proof of (((eq state) x0) b)
% Found (x x0) as proof of (((eq state) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq state) b) b)
% Found (eq_ref0 b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x2:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x1:(not (((eq state) x) x00))) (x2:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x1:(not (((eq state) x) x00))) (x2:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x) x00))->(not (forall (S:state), (((eq state) S) T))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x2:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x1:(not (((eq state) x) x00))) (x2:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x1:(not (((eq state) x) x00))) (x2:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x) x00))->(not (forall (S:state), (((eq state) S) T))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x0:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x0 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found eq_ref00:=(eq_ref0 b):(((eq state) b) b)
% Found (eq_ref0 b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found eq_ref00:=(eq_ref0 x1):(((eq state) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq state) x1) b)
% Found ((eq_ref state) x1) as proof of (((eq state) x1) b)
% Found ((eq_ref state) x1) as proof of (((eq state) x1) b)
% Found ((eq_ref state) x1) as proof of (((eq state) x1) b)
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found eq_ref00:=(eq_ref0 x2):(((eq state) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq state) x2) b)
% Found ((eq_ref state) x2) as proof of (((eq state) x2) b)
% Found ((eq_ref state) x2) as proof of (((eq state) x2) b)
% Found ((eq_ref state) x2) as proof of (((eq state) x2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq state) b) b)
% Found (eq_ref0 b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x10:(not (((eq state) x) x00)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x00:state) (x10:(not (((eq state) x) x00)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x) x00))->False)
% Found (fun (x00:state) (x10:(not (((eq state) x) x00)))=> fact_1_induct__false__def0) as proof of (forall (x0:state), ((not (((eq state) x) x0))->False))
% Found x2:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x2 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x6:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x6:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found fact_1_induct__false__def0:False
% Found (fun (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((forall (S:state), (((eq state) S) T))->False)
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x2:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x2 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x1:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x1 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x10:(not (((eq state) x) x00)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x00:state) (x10:(not (((eq state) x) x00)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x) x00))->False)
% Found (fun (x00:state) (x10:(not (((eq state) x) x00)))=> fact_1_induct__false__def0) as proof of (forall (x0:state), ((not (((eq state) x) x0))->False))
% Found x1:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found (x conj_0) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found (x conj_0) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x6:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x6:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x6:(not (((eq state) x1) x5)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x6:(not (((eq state) x1) x5)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x1) x5))->False)
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x6:(not (((eq state) x3) x5)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x6:(not (((eq state) x3) x5)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x3) x5))->False)
% Found fact_1_induct__false__def0:False
% Found (fun (x5:(not (((eq state) x1) x4)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x5:(not (((eq state) x1) x4)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x1) x4))->False)
% Found fact_1_induct__false__def0:False
% Found (fun (x6:(not (((eq state) x2) x5)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x6:(not (((eq state) x2) x5)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x2) x5))->False)
% Found x00:=(x0 x10):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found (x0 x10) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found (x0 x10) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (T:state) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((forall (S:state), (((eq state) S) T))->False)
% Found (fun (T:state) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (forall (T:state), ((forall (S:state), (((eq state) S) T))->False))
% Found fact_1_induct__false__def0:False
% Found (fun (x5:(not (((eq state) x2) x4)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x5:(not (((eq state) x2) x4)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x2) x4))->False)
% Found fact_1_induct__false__def0:False
% Found (fun (x2:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x1:(not (((eq state) x) x00))) (x2:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x00:state) (x1:(not (((eq state) x) x00))) (x2:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x) x00))->(not (forall (S:state), (((eq state) S) T))))
% Found (fun (x00:state) (x1:(not (((eq state) x) x00))) (x2:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (forall (x0:state), ((not (((eq state) x) x0))->(not (forall (S:state), (((eq state) S) T)))))
% Found x0:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x0 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x0:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x0 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x2:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x1:(not (((eq state) x) x00))) (x2:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x00:state) (x1:(not (((eq state) x) x00))) (x2:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x) x00))->(not (forall (S:state), (((eq state) S) T))))
% Found (fun (x00:state) (x1:(not (((eq state) x) x00))) (x2:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (forall (x0:state), ((not (((eq state) x) x0))->(not (forall (S:state), (((eq state) S) T)))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x2:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x2 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x2:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x2 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x0:((ex state) (fun (T0:state)=> (not (((eq state) x) T0))))) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((forall (S:state), (((eq state) S) T))->False)
% Found (fun (x0:((ex state) (fun (T0:state)=> (not (((eq state) x) T0))))) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T0:state)=> (not (((eq state) x) T0))))->((forall (S:state), (((eq state) S) T))->False))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x1:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x1 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x6:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x5:(not (((eq state) x1) x4))) (x6:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x5:(not (((eq state) x1) x4))) (x6:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x1) x4))->(not (forall (S:state), (((eq state) S) T))))
% Found x1:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x1 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x0:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x0 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x0:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x0 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x6:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x5:(not (((eq state) x2) x4))) (x6:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x5:(not (((eq state) x2) x4))) (x6:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x2) x4))->(not (forall (S:state), (((eq state) S) T))))
% Found x1:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x1 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found eq_ref000:=(eq_ref00 P):((P x)->(P x))
% Found (eq_ref00 P) as proof of (P0 x)
% Found ((eq_ref0 x) P) as proof of (P0 x)
% Found (((eq_ref state) x) P) as proof of (P0 x)
% Found (((eq_ref state) x) P) as proof of (P0 x)
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found x2:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x2 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x1:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x1 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x0:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x0 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x6:(not (((eq state) x1) x5)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x5:state) (x6:(not (((eq state) x1) x5)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x1) x5))->False)
% Found (fun (x5:state) (x6:(not (((eq state) x1) x5)))=> fact_1_induct__false__def0) as proof of (forall (x:state), ((not (((eq state) x1) x))->False))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref state) x0) P) as proof of (P0 x0)
% Found (((eq_ref state) x0) P) as proof of (P0 x0)
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found eq_ref000:=(eq_ref00 P):((P x)->(P x))
% Found (eq_ref00 P) as proof of (P0 x)
% Found ((eq_ref0 x) P) as proof of (P0 x)
% Found (((eq_ref state) x) P) as proof of (P0 x)
% Found (((eq_ref state) x) P) as proof of (P0 x)
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found eq_ref000:=(eq_ref00 P):((P x)->(P x))
% Found (eq_ref00 P) as proof of (P0 x)
% Found ((eq_ref0 x) P) as proof of (P0 x)
% Found (((eq_ref state) x) P) as proof of (P0 x)
% Found (((eq_ref state) x) P) as proof of (P0 x)
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x6:(not (((eq state) x3) x5)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x5:state) (x6:(not (((eq state) x3) x5)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x3) x5))->False)
% Found (fun (x5:state) (x6:(not (((eq state) x3) x5)))=> fact_1_induct__false__def0) as proof of (forall (x:state), ((not (((eq state) x3) x))->False))
% Found x2:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x2 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of (P0 x1)
% Found ((eq_ref0 x1) P) as proof of (P0 x1)
% Found (((eq_ref state) x1) P) as proof of (P0 x1)
% Found (((eq_ref state) x1) P) as proof of (P0 x1)
% Found eq_ref000:=(eq_ref00 P):((P x)->(P x))
% Found (eq_ref00 P) as proof of (P0 x)
% Found ((eq_ref0 x) P) as proof of (P0 x)
% Found (((eq_ref state) x) P) as proof of (P0 x)
% Found (((eq_ref state) x) P) as proof of (P0 x)
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x1:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x1 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x5:(not (((eq state) x1) x4)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x4:state) (x5:(not (((eq state) x1) x4)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x1) x4))->False)
% Found (fun (x4:state) (x5:(not (((eq state) x1) x4)))=> fact_1_induct__false__def0) as proof of (forall (x:state), ((not (((eq state) x1) x))->False))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref state) x0) P) as proof of (P0 x0)
% Found (((eq_ref state) x0) P) as proof of (P0 x0)
% Found x0:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x0 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x6:(not (((eq state) x2) x5)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x5:state) (x6:(not (((eq state) x2) x5)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x2) x5))->False)
% Found (fun (x5:state) (x6:(not (((eq state) x2) x5)))=> fact_1_induct__false__def0) as proof of (forall (x:state), ((not (((eq state) x2) x))->False))
% Found fact_1_induct__false__def0:False
% Found (fun (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (T:state) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((forall (S:state), (((eq state) S) T))->False)
% Found (fun (x0:((ex state) (fun (T:state)=> (not (((eq state) x) T))))) (T:state) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (forall (T:state), ((forall (S:state), (((eq state) S) T))->False))
% Found (fun (x0:((ex state) (fun (T:state)=> (not (((eq state) x) T))))) (T:state) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->(forall (T:state), ((forall (S:state), (((eq state) S) T))->False)))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref state) x0) P) as proof of (P0 x0)
% Found (((eq_ref state) x0) P) as proof of (P0 x0)
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found fact_1_induct__false__def0:False
% Found (fun (x5:(not (((eq state) x2) x4)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x2) x4))->False)
% Found (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> fact_1_induct__false__def0) as proof of (forall (x:state), ((not (((eq state) x2) x))->False))
% Found x00:=(x0 x10):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found (x0 x10) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found (x0 x10) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x00:=(x0 x10):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found (x0 x10) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found (x0 x10) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found eq_ref00:=(eq_ref0 b):(((eq state) b) b)
% Found (eq_ref0 b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found eq_ref00:=(eq_ref0 x1):(((eq state) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq state) x1) b)
% Found ((eq_ref state) x1) as proof of (((eq state) x1) b)
% Found ((eq_ref state) x1) as proof of (((eq state) x1) b)
% Found ((eq_ref state) x1) as proof of (((eq state) x1) b)
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x2:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x2 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of (P0 x1)
% Found ((eq_ref0 x1) P) as proof of (P0 x1)
% Found (((eq_ref state) x1) P) as proof of (P0 x1)
% Found (((eq_ref state) x1) P) as proof of (P0 x1)
% Found eq_ref000:=(eq_ref00 P):((P x1)->(P x1))
% Found (eq_ref00 P) as proof of (P0 x1)
% Found ((eq_ref0 x1) P) as proof of (P0 x1)
% Found (((eq_ref state) x1) P) as proof of (P0 x1)
% Found (((eq_ref state) x1) P) as proof of (P0 x1)
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found x2:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x2 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found eq_ref000:=(eq_ref00 P):((P x2)->(P x2))
% Found (eq_ref00 P) as proof of (P0 x2)
% Found ((eq_ref0 x2) P) as proof of (P0 x2)
% Found (((eq_ref state) x2) P) as proof of (P0 x2)
% Found (((eq_ref state) x2) P) as proof of (P0 x2)
% Found x1:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x1 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found eq_ref00:=(eq_ref0 x2):(((eq state) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq state) x2) b)
% Found ((eq_ref state) x2) as proof of (((eq state) x2) b)
% Found ((eq_ref state) x2) as proof of (((eq state) x2) b)
% Found ((eq_ref state) x2) as proof of (((eq state) x2) b)
% Found eq_ref00:=(eq_ref0 b):(((eq state) b) b)
% Found (eq_ref0 b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found fact_1_induct__false__def0:False
% Found (fun (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x0:((ex state) (fun (T0:state)=> (not (((eq state) x) T0))))) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((forall (S:state), (((eq state) S) T))->False)
% Found (fun (x:state) (x0:((ex state) (fun (T0:state)=> (not (((eq state) x) T0))))) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T0:state)=> (not (((eq state) x) T0))))->((forall (S:state), (((eq state) S) T))->False))
% Found (fun (x:state) (x0:((ex state) (fun (T0:state)=> (not (((eq state) x) T0))))) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (forall (x:state), (((ex state) (fun (T0:state)=> (not (((eq state) x) T0))))->((forall (S:state), (((eq state) S) T))->False)))
% Found eq_ref00:=(eq_ref0 x1):(((eq state) x1) x1)
% Found (eq_ref0 x1) as proof of (((eq state) x1) b)
% Found ((eq_ref state) x1) as proof of (((eq state) x1) b)
% Found ((eq_ref state) x1) as proof of (((eq state) x1) b)
% Found ((eq_ref state) x1) as proof of (((eq state) x1) b)
% Found eq_ref00:=(eq_ref0 b):(((eq state) b) b)
% Found (eq_ref0 b) as proof of (((eq state) b) x5)
% Found ((eq_ref state) b) as proof of (((eq state) b) x5)
% Found ((eq_ref state) b) as proof of (((eq state) b) x5)
% Found ((eq_ref state) b) as proof of (((eq state) b) x5)
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x6:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x5:(not (((eq state) x1) x4))) (x6:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x4:state) (x5:(not (((eq state) x1) x4))) (x6:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x1) x4))->(not (forall (S:state), (((eq state) S) T))))
% Found (fun (x4:state) (x5:(not (((eq state) x1) x4))) (x6:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (forall (x:state), ((not (((eq state) x1) x))->(not (forall (S:state), (((eq state) S) T)))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found eq_ref00:=(eq_ref0 x3):(((eq state) x3) x3)
% Found (eq_ref0 x3) as proof of (((eq state) x3) b)
% Found ((eq_ref state) x3) as proof of (((eq state) x3) b)
% Found ((eq_ref state) x3) as proof of (((eq state) x3) b)
% Found ((eq_ref state) x3) as proof of (((eq state) x3) b)
% Found eq_ref00:=(eq_ref0 b):(((eq state) b) b)
% Found (eq_ref0 b) as proof of (((eq state) b) x5)
% Found ((eq_ref state) b) as proof of (((eq state) b) x5)
% Found ((eq_ref state) b) as proof of (((eq state) b) x5)
% Found ((eq_ref state) b) as proof of (((eq state) b) x5)
% Found x0:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x0 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x6:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x5:(not (((eq state) x2) x4))) (x6:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x4:state) (x5:(not (((eq state) x2) x4))) (x6:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x2) x4))->(not (forall (S:state), (((eq state) S) T))))
% Found (fun (x4:state) (x5:(not (((eq state) x2) x4))) (x6:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (forall (x:state), ((not (((eq state) x2) x))->(not (forall (S:state), (((eq state) S) T)))))
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x2:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)->(not (forall (S:state), (((eq state) S) T))))
% Found eq_ref00:=(eq_ref0 b):(((eq state) b) b)
% Found (eq_ref0 b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found x6:=(x x1):(((eq state) x1) T)
% Found (x x1) as proof of (((eq state) x1) b)
% Found (x x1) as proof of (((eq state) x1) b)
% Found (x x1) as proof of (((eq state) x1) b)
% Found eq_ref00:=(eq_ref0 b):(((eq state) b) b)
% Found (eq_ref0 b) as proof of (((eq state) b) x5)
% Found ((eq_ref state) b) as proof of (((eq state) b) x5)
% Found ((eq_ref state) b) as proof of (((eq state) b) x5)
% Found ((eq_ref state) b) as proof of (((eq state) b) x5)
% Found x40:=(x4 x2):(((eq state) x2) T)
% Found (x4 x2) as proof of (((eq state) x2) b)
% Found (x4 x2) as proof of (((eq state) x2) b)
% Found (x4 x2) as proof of (((eq state) x2) b)
% Found x00:=(x0 x10):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found (x0 x10) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found (x0 x10) as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found eq_ref00:=(eq_ref0 b):(((eq state) b) b)
% Found (eq_ref0 b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found ((eq_ref state) b) as proof of (((eq state) b) x4)
% Found x6:=(x x2):(((eq state) x2) T)
% Found (x x2) as proof of (((eq state) x2) b)
% Found (x x2) as proof of (((eq state) x2) b)
% Found (x x2) as proof of (((eq state) x2) b)
% Found fact_1_induct__false__def0:False
% Found (fun (x5:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x5:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found (fun (x2:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x3:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> fact_1_induct__false__def0) as proof of ((hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x1:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x1 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x10:(not (((eq state) x) x00)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x10:(not (((eq state) x) x00)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x) x00))->False)
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x5:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x5:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found x0:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x0 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x2:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (not (forall (S:state), (((eq state) S) T)))
% Found (fun (x2:(((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)) (x3:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))->hoare_1821564147gleton)->(not (forall (S:state), (((eq state) S) T))))
% Found x2:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x2 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x5:(not (((eq state) x0) x4)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x5:(not (((eq state) x0) x4)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x0) x4))->False)
% Found fact_1_induct__false__def0:False
% Found (fun (x5:(not (((eq state) x0) x4)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x5:(not (((eq state) x0) x4)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x0) x4))->False)
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x1:=(x conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found x1 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T0:state)=> (not (((eq state) S) T0))))))
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found fact_1_induct__false__def0:False
% Found (fun (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (T:state) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of ((forall (S:state), (((eq state) S) T))->False)
% Found (fun (x0:((ex state) (fun (T:state)=> (not (((eq state) x) T))))) (T:state) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (forall (T:state), ((forall (S:state), (((eq state) S) T))->False))
% Found (fun (x:state) (x0:((ex state) (fun (T:state)=> (not (((eq state) x) T))))) (T:state) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->(forall (T:state), ((forall (S:state), (((eq state) S) T))->False)))
% Found (fun (x:state) (x0:((ex state) (fun (T:state)=> (not (((eq state) x) T))))) (T:state) (x1:(forall (S:state), (((eq state) S) T)))=> fact_1_induct__false__def0) as proof of (forall (x:state), (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->(forall (T:state), ((forall (S:state), (((eq state) S) T))->False))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00:=(x0 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found x00 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x5:(not (((eq state) x1) x4)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x5:(not (((eq state) x1) x4)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x1) x4))->False)
% Found fact_0_state__not__singleton__def__proj10:=(fact_0_state__not__singleton__def__proj1 conj_0):((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_0_state__not__singleton__def__proj10 as proof of ((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))
% Found fact_1_induct__false__def0:False
% Found (fun (x5:(not (((eq state) x1) x4)))=> fact_1_induct__false__def0) as proof of False
% Found (fun (x5:(not (((eq state) x1) x4)))=> fact_1_induct__false__def0) as proof of ((not (((eq state) x1) x4))->False)
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found conj_0:hoare_1821564147gleton
% Found conj_0 as proof of hoare_1821564147gleton
% Found x6:=(x x4):(((eq state) x4) T)
% Found (x x4) as proof of (((eq state) x4) b)
% Found (x x4) as proof of (((eq state) x4) b)
% Found (eq_sym000 (x x4)) as proof of (((eq state) b) x4)
% Found ((eq_sym00 b) (x x4)) as proof of (((eq state) b) x4)
% Found (((eq_sym0 x4) b) (x x4)) as proof of (((eq state) b) x4)
% Found ((((eq_sym state) x4) b) (x x4)) as proof of (((eq state) b) x4)
% Found ((((eq_sym state) x4) b) (x x4)) as proof of (((eq state) b) x4)
% Found ((eq_trans0000 (x x2)) ((((eq_sym state) x4) b) (x x4))) as proof of (((eq state) x2) x4)
% Found (((eq_trans000 T) (x x2)) ((((eq_sym state) x4) T) (x x4))) as proof of (((eq state) x2) x4)
% Found ((((fun (b:state)=> ((eq_trans00 b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4))) as proof of (((eq state) x2) x4)
% Found ((((fun (b:state)=> (((eq_trans0 x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4))) as proof of (((eq state) x2) x4)
% Found ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4))) as proof of (((eq state) x2) x4)
% Found ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4))) as proof of (((eq state) x2) x4)
% Found (x5 ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4)))) as proof of False
% Found (fun (x5:(not (((eq state) x2) x4)))=> (x5 ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4))))) as proof of False
% Found (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> (x5 ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4))))) as proof of ((not (((eq state) x2) x4))->False)
% Found (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> (x5 ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4))))) as proof of (forall (x:state), ((not (((eq state) x2) x))->False))
% Found (ex_ind10 (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> (x5 ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4)))))) as proof of False
% Found ((ex_ind1 False) (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> (x5 ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4)))))) as proof of False
% Found (((fun (P:Prop) (x4:(forall (x:state), ((not (((eq state) x2) x))->P)))=> (((((ex_ind state) (fun (T:state)=> (not (((eq state) x2) T)))) P) x4) x3)) False) (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> (x5 ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4)))))) as proof of False
% Found (fun (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> (((fun (P:Prop) (x4:(forall (x:state), ((not (((eq state) x2) x))->P)))=> (((((ex_ind state) (fun (T:state)=> (not (((eq state) x2) T)))) P) x4) x3)) False) (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> (x5 ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4))))))) as proof of False
% Found (fun (x2:state) (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> (((fun (P:Prop) (x4:(forall (x:state), ((not (((eq state) x2) x))->P)))=> (((((ex_ind state) (fun (T:state)=> (not (((eq state) x2) T)))) P) x4) x3)) False) (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> (x5 ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4))))))) as proof of (((ex state) (fun (T:state)=> (not (((eq state) x2) T))))->False)
% Found (fun (x2:state) (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> (((fun (P:Prop) (x4:(forall (x:state), ((not (((eq state) x2) x))->P)))=> (((((ex_ind state) (fun (T:state)=> (not (((eq state) x2) T)))) P) x4) x3)) False) (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> (x5 ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4))))))) as proof of (forall (x:state), (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->False))
% Found (ex_ind00 (fun (x2:state) (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> (((fun (P:Prop) (x4:(forall (x:state), ((not (((eq state) x2) x))->P)))=> (((((ex_ind state) (fun (T:state)=> (not (((eq state) x2) T)))) P) x4) x3)) False) (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> (x5 ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4)))))))) as proof of False
% Found ((ex_ind0 False) (fun (x2:state) (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> (((fun (P:Prop) (x4:(forall (x:state), ((not (((eq state) x2) x))->P)))=> (((((ex_ind state) (fun (T:state)=> (not (((eq state) x2) T)))) P) x4) x3)) False) (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> (x5 ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4)))))))) as proof of False
% Found (((fun (P:Prop) (x2:(forall (x:state), (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->P)))=> (((((ex_ind state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))) P) x2) x00)) False) (fun (x2:state) (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> (((fun (P:Prop) (x4:(forall (x:state), ((not (((eq state) x2) x))->P)))=> (((((ex_ind state) (fun (T:state)=> (not (((eq state) x2) T)))) P) x4) x3)) False) (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> (x5 ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4)))))))) as proof of False
% Found (((fun (P:Prop) (x2:(forall (x:state), (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->P)))=> (((((ex_ind state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))) P) x2) (x0 conj_0))) False) (fun (x2:state) (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> (((fun (P:Prop) (x4:(forall (x:state), ((not (((eq state) x2) x))->P)))=> (((((ex_ind state) (fun (T:state)=> (not (((eq state) x2) T)))) P) x4) x3)) False) (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> (x5 ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4)))))))) as proof of False
% Found (fun (x1:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> (((fun (P:Prop) (x2:(forall (x:state), (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->P)))=> (((((ex_ind state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))) P) x2) (x0 conj_0))) False) (fun (x2:state) (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> (((fun (P:Prop) (x4:(forall (x:state), ((not (((eq state) x2) x))->P)))=> (((((ex_ind state) (fun (T:state)=> (not (((eq state) x2) T)))) P) x4) x3)) False) (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> (x5 ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4))))))))) as proof of False
% Found (fun (x0:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x1:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> (((fun (P:Prop) (x2:(forall (x:state), (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->P)))=> (((((ex_ind state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))) P) x2) (x0 conj_0))) False) (fun (x2:state) (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> (((fun (P:Prop) (x4:(forall (x:state), ((not (((eq state) x2) x))->P)))=> (((((ex_ind state) (fun (T:state)=> (not (((eq state) x2) T)))) P) x4) x3)) False) (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> (x5 ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4))))))))) as proof of ((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False)
% Found (fun (x0:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x1:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> (((fun (P:Prop) (x2:(forall (x:state), (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->P)))=> (((((ex_ind state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))) P) x2) (x0 conj_0))) False) (fun (x2:state) (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> (((fun (P:Prop) (x4:(forall (x:state), ((not (((eq state) x2) x))->P)))=> (((((ex_ind state) (fun (T:state)=> (not (((eq state) x2) T)))) P) x4) x3)) False) (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> (x5 ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4))))))))) as proof of ((hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->False))
% Found (and_rect00 (fun (x0:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x1:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> (((fun (P:Prop) (x2:(forall (x:state), (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->P)))=> (((((ex_ind state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))) P) x2) (x0 conj_0))) False) (fun (x2:state) (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> (((fun (P:Prop) (x4:(forall (x:state), ((not (((eq state) x2) x))->P)))=> (((((ex_ind state) (fun (T:state)=> (not (((eq state) x2) T)))) P) x4) x3)) False) (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> (x5 ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4)))))))))) as proof of False
% Found ((and_rect0 False) (fun (x0:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x1:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> (((fun (P:Prop) (x2:(forall (x:state), (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->P)))=> (((((ex_ind state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))) P) x2) (x0 conj_0))) False) (fun (x2:state) (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> (((fun (P:Prop) (x4:(forall (x:state), ((not (((eq state) x2) x))->P)))=> (((((ex_ind state) (fun (T:state)=> (not (((eq state) x2) T)))) P) x4) x3)) False) (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> (x5 ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4)))))))))) as proof of False
% Found (((fun (P:Type) (x0:((hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->P)))=> (((((and_rect (hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)) P) x0) fact_0_state__not__singleton__def)) False) (fun (x0:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x1:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> (((fun (P:Prop) (x2:(forall (x:state), (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->P)))=> (((((ex_ind state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))) P) x2) (x0 conj_0))) False) (fun (x2:state) (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> (((fun (P:Prop) (x4:(forall (x:state), ((not (((eq state) x2) x))->P)))=> (((((ex_ind state) (fun (T:state)=> (not (((eq state) x2) T)))) P) x4) x3)) False) (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> (x5 ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4)))))))))) as proof of False
% Found (fun (x:(forall (S:state), (((eq state) S) T)))=> (((fun (P:Type) (x0:((hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->P)))=> (((((and_rect (hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)) P) x0) fact_0_state__not__singleton__def)) False) (fun (x0:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x1:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> (((fun (P:Prop) (x2:(forall (x:state), (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->P)))=> (((((ex_ind state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))) P) x2) (x0 conj_0))) False) (fun (x2:state) (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> (((fun (P:Prop) (x4:(forall (x:state), ((not (((eq state) x2) x))->P)))=> (((((ex_ind state) (fun (T:state)=> (not (((eq state) x2) T)))) P) x4) x3)) False) (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> (x5 ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4))))))))))) as proof of False
% Found (fun (T:state) (x:(forall (S:state), (((eq state) S) T)))=> (((fun (P:Type) (x0:((hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->P)))=> (((((and_rect (hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)) P) x0) fact_0_state__not__singleton__def)) False) (fun (x0:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x1:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> (((fun (P:Prop) (x2:(forall (x:state), (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->P)))=> (((((ex_ind state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))) P) x2) (x0 conj_0))) False) (fun (x2:state) (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> (((fun (P:Prop) (x4:(forall (x:state), ((not (((eq state) x2) x))->P)))=> (((((ex_ind state) (fun (T:state)=> (not (((eq state) x2) T)))) P) x4) x3)) False) (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> (x5 ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4))))))))))) as proof of ((forall (S:state), (((eq state) S) T))->False)
% Found (fun (T:state) (x:(forall (S:state), (((eq state) S) T)))=> (((fun (P:Type) (x0:((hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->P)))=> (((((and_rect (hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)) P) x0) fact_0_state__not__singleton__def)) False) (fun (x0:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x1:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> (((fun (P:Prop) (x2:(forall (x:state), (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->P)))=> (((((ex_ind state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))) P) x2) (x0 conj_0))) False) (fun (x2:state) (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> (((fun (P:Prop) (x4:(forall (x:state), ((not (((eq state) x2) x))->P)))=> (((((ex_ind state) (fun (T:state)=> (not (((eq state) x2) T)))) P) x4) x3)) False) (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> (x5 ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4))))))))))) as proof of (forall (T:state), ((forall (S:state), (((eq state) S) T))->False))
% Got proof (fun (T:state) (x:(forall (S:state), (((eq state) S) T)))=> (((fun (P:Type) (x0:((hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->P)))=> (((((and_rect (hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)) P) x0) fact_0_state__not__singleton__def)) False) (fun (x0:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x1:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> (((fun (P:Prop) (x2:(forall (x:state), (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->P)))=> (((((ex_ind state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))) P) x2) (x0 conj_0))) False) (fun (x2:state) (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> (((fun (P:Prop) (x4:(forall (x:state), ((not (((eq state) x2) x))->P)))=> (((((ex_ind state) (fun (T:state)=> (not (((eq state) x2) T)))) P) x4) x3)) False) (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> (x5 ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4)))))))))))
% Time elapsed = 204.442571s
% node=31981 cost=1187.000000 depth=35
% ::::::::::::::::::::::
% % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% (fun (T:state) (x:(forall (S:state), (((eq state) S) T)))=> (((fun (P:Type) (x0:((hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))->((((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)->P)))=> (((((and_rect (hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton)) P) x0) fact_0_state__not__singleton__def)) False) (fun (x0:(hoare_1821564147gleton->((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))))) (x1:(((ex state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T))))))->hoare_1821564147gleton))=> (((fun (P:Prop) (x2:(forall (x:state), (((ex state) (fun (T:state)=> (not (((eq state) x) T))))->P)))=> (((((ex_ind state) (fun (S:state)=> ((ex state) (fun (T:state)=> (not (((eq state) S) T)))))) P) x2) (x0 conj_0))) False) (fun (x2:state) (x3:((ex state) (fun (T:state)=> (not (((eq state) x2) T)))))=> (((fun (P:Prop) (x4:(forall (x:state), ((not (((eq state) x2) x))->P)))=> (((((ex_ind state) (fun (T:state)=> (not (((eq state) x2) T)))) P) x4) x3)) False) (fun (x4:state) (x5:(not (((eq state) x2) x4)))=> (x5 ((((fun (b:state)=> ((((eq_trans state) x2) b) x4)) T) (x x2)) ((((eq_sym state) x4) T) (x x4)))))))))))
% % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% EOF
%------------------------------------------------------------------------------