let s1, s2 be State of SCM+FSA; :: thesis: for I being Program of SCM+FSA
for a being Int-Location st not I refers a & ( for b being Int-Location st a <> b holds
s1 . b = s2 . b ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) & I is_closed_on s1 holds
for k being Element of NAT holds
( ( for b being Int-Location st a <> b holds
(Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k)) . b = (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k)) . b ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k)) . f = (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k)) . f ) & IC (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k)) = IC (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k)) & CurInstr ((ProgramPart (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))),(Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))) = CurInstr ((ProgramPart (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k))),(Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k))) )
let I be Program of SCM+FSA; :: thesis: for a being Int-Location st not I refers a & ( for b being Int-Location st a <> b holds
s1 . b = s2 . b ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) & I is_closed_on s1 holds
for k being Element of NAT holds
( ( for b being Int-Location st a <> b holds
(Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k)) . b = (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k)) . b ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k)) . f = (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k)) . f ) & IC (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k)) = IC (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k)) & CurInstr ((ProgramPart (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))),(Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))) = CurInstr ((ProgramPart (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k))),(Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k))) )
let a be Int-Location ; :: thesis: ( not I refers a & ( for b being Int-Location st a <> b holds
s1 . b = s2 . b ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) & I is_closed_on s1 implies for k being Element of NAT holds
( ( for b being Int-Location st a <> b holds
(Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k)) . b = (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k)) . b ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k)) . f = (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k)) . f ) & IC (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k)) = IC (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k)) & CurInstr ((ProgramPart (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))),(Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))) = CurInstr ((ProgramPart (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k))),(Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k))) ) )
assume A1: not I refers a ; :: thesis: ( ex b being Int-Location st
( a <> b & not s1 . b = s2 . b ) or ex f being FinSeq-Location st not s1 . f = s2 . f or not I is_closed_on s1 or for k being Element of NAT holds
( ( for b being Int-Location st a <> b holds
(Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k)) . b = (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k)) . b ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k)) . f = (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k)) . f ) & IC (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k)) = IC (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k)) & CurInstr ((ProgramPart (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))),(Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))) = CurInstr ((ProgramPart (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k))),(Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k))) ) )
set S2 = s2 +* (I +* (Start-At (0,SCM+FSA)));
set S1 = s1 +* (I +* (Start-At (0,SCM+FSA)));
defpred S1[ State of SCM+FSA, State of SCM+FSA] means ( ( for b being Int-Location st a <> b holds
$1 . b = $2 . b ) & ( for f being FinSeq-Location holds $1 . f = $2 . f ) );
assume that
A2: for b being Int-Location st a <> b holds
s1 . b = s2 . b and
A3: for f being FinSeq-Location holds s1 . f = s2 . f ; :: thesis: ( not I is_closed_on s1 or for k being Element of NAT holds
( ( for b being Int-Location st a <> b holds
(Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k)) . b = (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k)) . b ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k)) . f = (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k)) . f ) & IC (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k)) = IC (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k)) & CurInstr ((ProgramPart (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))),(Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))) = CurInstr ((ProgramPart (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k))),(Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k))) ) )
A4: IC SCM+FSA in dom (I +* (Start-At (0,SCM+FSA))) by COMPOS_1:141;
A5: now
let f be FinSeq-Location ; :: thesis: (s1 +* (I +* (Start-At (0,SCM+FSA)))) . f = (s2 +* (I +* (Start-At (0,SCM+FSA)))) . f
A6: not f in dom (I +* (Start-At (0,SCM+FSA))) by SCMFSA6B:13;
hence (s1 +* (I +* (Start-At (0,SCM+FSA)))) . f = s1 . f by FUNCT_4:12
.= s2 . f by A3
.= (s2 +* (I +* (Start-At (0,SCM+FSA)))) . f by A6, FUNCT_4:12 ;
:: thesis: verum
end;
A7: Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),0) = s2 +* (I +* (Start-At (0,SCM+FSA))) by EXTPRO_1:3;
defpred S2[ Nat] means ( S1[ Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),$1), Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),$1)] & IC (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),$1)) = IC (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),$1)) & CurInstr ((ProgramPart (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),$1))),(Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),$1))) = CurInstr ((ProgramPart (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),$1))),(Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),$1))) );
A8: I c= I +* (Start-At (0,SCM+FSA)) by SCMFSA8A:9;
A9: IC (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),0)) = (s1 +* (I +* (Start-At (0,SCM+FSA)))) . (IC SCM+FSA) by EXTPRO_1:3
.= (I +* (Start-At (0,SCM+FSA))) . (IC SCM+FSA) by A4, FUNCT_4:14
.= (s2 +* (I +* (Start-At (0,SCM+FSA)))) . (IC SCM+FSA) by A4, FUNCT_4:14
.= IC (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),0)) by EXTPRO_1:3 ;
I +* (Start-At (0,SCM+FSA)) c= s1 +* (I +* (Start-At (0,SCM+FSA))) by FUNCT_4:26;
then A10: I c= s1 +* (I +* (Start-At (0,SCM+FSA))) by A8, XBOOLE_1:1;
A11: now
let b be Int-Location ; :: thesis: ( a <> b implies (s1 +* (I +* (Start-At (0,SCM+FSA)))) . b = (s2 +* (I +* (Start-At (0,SCM+FSA)))) . b )
assume A12: a <> b ; :: thesis: (s1 +* (I +* (Start-At (0,SCM+FSA)))) . b = (s2 +* (I +* (Start-At (0,SCM+FSA)))) . b
A13: not b in dom (I +* (Start-At (0,SCM+FSA))) by SCMFSA6B:12;
hence (s1 +* (I +* (Start-At (0,SCM+FSA)))) . b = s1 . b by FUNCT_4:12
.= s2 . b by A2, A12
.= (s2 +* (I +* (Start-At (0,SCM+FSA)))) . b by A13, FUNCT_4:12 ;
:: thesis: verum
end;
A14: Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),0) = s1 +* (I +* (Start-At (0,SCM+FSA))) by EXTPRO_1:3;
assume A15: I is_closed_on s1 ; :: thesis: for k being Element of NAT holds
( ( for b being Int-Location st a <> b holds
(Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k)) . b = (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k)) . b ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k)) . f = (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k)) . f ) & IC (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k)) = IC (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k)) & CurInstr ((ProgramPart (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))),(Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))) = CurInstr ((ProgramPart (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k))),(Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k))) )
A16: IC (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),0)) in dom I by A15, SCMFSA7B:def 7;
I +* (Start-At (0,SCM+FSA)) c= s2 +* (I +* (Start-At (0,SCM+FSA))) by FUNCT_4:26;
then A17: I c= s2 +* (I +* (Start-At (0,SCM+FSA))) by A8, XBOOLE_1:1;
A18: for k being Element of NAT st S2[k] holds
S2[k + 1]
proof
let k be Element of NAT ; :: thesis: ( S2[k] implies S2[k + 1] )
T: ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k)) by AMI_1:123;
A19: Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1)) = Following ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))),(Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k)))),(Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))) by T ;
A20: IC (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k)) in dom I by A15, SCMFSA7B:def 7;
A21: ProgramPart I = I by RELAT_1:209;
then A22: I c= Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1)) by A17, AMI_1:99;
T: ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k)) by AMI_1:123;
A23: Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1)) = Following ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k))),(Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k)))),(Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k))) by T ;
Y: (ProgramPart (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))) /. (IC (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))) = (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k)) . (IC (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))) by COMPOS_1:38;
I c= Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k) by A10, A21, AMI_1:99;
then CurInstr ((ProgramPart (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))),(Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))) = I . (IC (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))) by A20, Y, GRFUNC_1:8;
then CurInstr ((ProgramPart (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))),(Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))) in rng I by A20, FUNCT_1:def 5;
then A24: not CurInstr ((ProgramPart (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))),(Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))) refers a by A1, SCMFSA7B:def 2;
assume A25: S2[k] ; :: thesis: S2[k + 1]
hence S1[ Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1)), Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1))] by A19, A23, A24, Th37; :: thesis: ( IC (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1))) = IC (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1))) & CurInstr ((ProgramPart (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1)))),(Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1)))) = CurInstr ((ProgramPart (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1)))),(Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1)))) )
thus A26: IC (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1))) = IC (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1))) by A25, A19, A23, A24, Th37; :: thesis: CurInstr ((ProgramPart (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1)))),(Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1)))) = CurInstr ((ProgramPart (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1)))),(Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1))))
A27: IC (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1))) in dom I by A15, SCMFSA7B:def 7;
Y: (ProgramPart (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1)))) /. (IC (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1)))) = (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1))) . (IC (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1)))) by COMPOS_1:38;
Z: (ProgramPart (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1)))) /. (IC (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1)))) = (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1))) . (IC (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1)))) by COMPOS_1:38;
I c= Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1)) by A10, A21, AMI_1:99;
hence CurInstr ((ProgramPart (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1)))),(Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1)))) = I . (IC (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1)))) by A27, Y, GRFUNC_1:8
.= CurInstr ((ProgramPart (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1)))),(Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),(k + 1)))) by A22, A26, A27, Z, GRFUNC_1:8 ;
:: thesis: verum
end;
V: (ProgramPart (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),0))) /. (IC (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),0))) = (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),0)) . (IC (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),0))) by COMPOS_1:38;
U: (ProgramPart (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),0))) /. (IC (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),0))) = (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),0)) . (IC (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),0))) by COMPOS_1:38;
CurInstr ((ProgramPart (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),0))),(Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),0))) = (s1 +* (I +* (Start-At (0,SCM+FSA)))) . (IC (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),0))) by V, EXTPRO_1:3
.= I . (IC (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),0))) by A10, A16, GRFUNC_1:8
.= (s2 +* (I +* (Start-At (0,SCM+FSA)))) . (IC (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),0))) by A17, A9, A16, GRFUNC_1:8
.= CurInstr ((ProgramPart (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),0))),(Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),0))) by U, EXTPRO_1:3 ;
then A28: S2[ 0 ] by A11, A5, A14, A7, A9;
for k being Element of NAT holds S2[k] from NAT_1:sch 1(A28, A18);
 hence for k being Element of NAT holds
( ( for b being Int-Location st a <> b holds
(Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k)) . b = (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k)) . b ) & ( for f being FinSeq-Location holds (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k)) . f = (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k)) . f ) & IC (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k)) = IC (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k)) & CurInstr ((ProgramPart (Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))),(Comput ((ProgramPart (s1 +* (I +* (Start-At (0,SCM+FSA))))),(s1 +* (I +* (Start-At (0,SCM+FSA)))),k))) = CurInstr ((ProgramPart (Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k))),(Comput ((ProgramPart (s2 +* (I +* (Start-At (0,SCM+FSA))))),(s2 +* (I +* (Start-At (0,SCM+FSA)))),k))) ) ; :: thesis: verum