let s be State of SCM+FSA; :: thesis: for p being Instruction-Sequence of SCM+FSA
for a being read-write Int-Location
for J being good really-closed MacroInstruction of SCM+FSA st not J destroys a & ProperTimesBody a,J,s,p & 0 <= s . a holds
for k being Nat st k >= s . a holds
( ((StepTimes (a,J,p,s)) . k) . a = 0 & ((StepTimes (a,J,p,s)) . k) . (intloc 0) = 1 )
let p be Instruction-Sequence of SCM+FSA; :: thesis: for a being read-write Int-Location
for J being good really-closed MacroInstruction of SCM+FSA st not J destroys a & ProperTimesBody a,J,s,p & 0 <= s . a holds
for k being Nat st k >= s . a holds
( ((StepTimes (a,J,p,s)) . k) . a = 0 & ((StepTimes (a,J,p,s)) . k) . (intloc 0) = 1 )
let a be read-write Int-Location; :: thesis: for J being good really-closed MacroInstruction of SCM+FSA st not J destroys a & ProperTimesBody a,J,s,p & 0 <= s . a holds
for k being Nat st k >= s . a holds
( ((StepTimes (a,J,p,s)) . k) . a = 0 & ((StepTimes (a,J,p,s)) . k) . (intloc 0) = 1 )
let J be good really-closed MacroInstruction of SCM+FSA ; :: thesis: ( not J destroys a & ProperTimesBody a,J,s,p & 0 <= s . a implies for k being Nat st k >= s . a holds
( ((StepTimes (a,J,p,s)) . k) . a = 0 & ((StepTimes (a,J,p,s)) . k) . (intloc 0) = 1 ) )
set I = J;
assume that
A1: not J destroys a and
A2: ProperTimesBody a,J,s,p and
A3: 0 <= s . a ; :: thesis: for k being Nat st k >= s . a holds
( ((StepTimes (a,J,p,s)) . k) . a = 0 & ((StepTimes (a,J,p,s)) . k) . (intloc 0) = 1 )
set ST = StepTimes (a,J,p,s);
set SW = StepWhile>0 (a,(J ";" (SubFrom (a,(intloc 0)))),p,(Initialized s));
defpred S1[ Nat] means ( $1 >= s . a implies ( ((StepTimes (a,J,p,s)) . $1) . a = 0 & ((StepTimes (a,J,p,s)) . $1) . (intloc 0) = 1 ) );
A4: for k being Nat st S1[k] holds
S1[k + 1]
proof
reconsider sa = s . a as Element of NAT by A3, INT_1:3;
let k be Nat; :: thesis: ( S1[k] implies S1[k + 1] )
assume that
A5: ( k >= s . a implies ( ((StepTimes (a,J,p,s)) . k) . a = 0 & ((StepTimes (a,J,p,s)) . k) . (intloc 0) = 1 ) ) and
A6: k + 1 >= s . a ; :: thesis: ( ((StepTimes (a,J,p,s)) . (k + 1)) . a = 0 & ((StepTimes (a,J,p,s)) . (k + 1)) . (intloc 0) = 1 )
per cases ( k + 1 = sa or k + 1 > sa ) by A6, XXREAL_0:1;
suppose A7: k + 1 = sa ; :: thesis: ( ((StepTimes (a,J,p,s)) . (k + 1)) . a = 0 & ((StepTimes (a,J,p,s)) . (k + 1)) . (intloc 0) = 1 )
then (((StepTimes (a,J,p,s)) . (k + 1)) . a) + (k + 1) = s . a by A2, Th52, A1;
hence ((StepTimes (a,J,p,s)) . (k + 1)) . a = 0 by A7; :: thesis: ((StepTimes (a,J,p,s)) . (k + 1)) . (intloc 0) = 1
thus ((StepTimes (a,J,p,s)) . (k + 1)) . (intloc 0) = 1 by A2, A7, Th51, A1; :: thesis: verum
end;
suppose A8: k + 1 > sa ; :: thesis: ( ((StepTimes (a,J,p,s)) . (k + 1)) . a = 0 & ((StepTimes (a,J,p,s)) . (k + 1)) . (intloc 0) = 1 )
then A9: DataPart ((StepWhile>0 (a,(J ";" (SubFrom (a,(intloc 0)))),p,(Initialized s))) . (k + 1)) = DataPart ((StepWhile>0 (a,(J ";" (SubFrom (a,(intloc 0)))),p,(Initialized s))) . k) by A5, NAT_1:13, Th31;
hence ((StepTimes (a,J,p,s)) . (k + 1)) . a = 0 by A5, A8, NAT_1:13, SCMFSA_M:2; :: thesis: ((StepTimes (a,J,p,s)) . (k + 1)) . (intloc 0) = 1
thus ((StepTimes (a,J,p,s)) . (k + 1)) . (intloc 0) = 1 by A5, A8, A9, NAT_1:13, SCMFSA_M:2; :: thesis: verum
end;
end;
end;
A10: S1[ 0 ]
proof
assume A11: 0 >= s . a ; :: thesis: ( ((StepTimes (a,J,p,s)) . 0) . a = 0 & ((StepTimes (a,J,p,s)) . 0) . (intloc 0) = 1 )
thus ((StepTimes (a,J,p,s)) . 0) . a = (((StepTimes (a,J,p,s)) . 0) . a) + 0
.= 0 by A2, A3, A11, Th52, A1 ; :: thesis: ((StepTimes (a,J,p,s)) . 0) . (intloc 0) = 1
thus ((StepTimes (a,J,p,s)) . 0) . (intloc 0) = 1 by A2, A3, Th51, A1; :: thesis: verum
end;
thus for k being Nat holds S1[k] from NAT_1:sch 2(A10, A4); :: thesis: verum