let P be Instruction-Sequence of SCM+FSA; :: thesis: for I being parahalting Program of
for a being read-write Int-Location
for s being State of SCM+FSA st ex f being Function of (product the Object-Kind of SCM+FSA),NAT st
for k being Nat holds
( ( f . ((StepWhile=0 (a,I,P,s)) . (k + 1)) < f . ((StepWhile=0 (a,I,P,s)) . k) or f . ((StepWhile=0 (a,I,P,s)) . k) = 0 ) & ( f . ((StepWhile=0 (a,I,P,s)) . k) = 0 implies ((StepWhile=0 (a,I,P,s)) . k) . a <> 0 ) & ( ((StepWhile=0 (a,I,P,s)) . k) . a <> 0 implies f . ((StepWhile=0 (a,I,P,s)) . k) = 0 ) ) holds
( while=0 (a,I) is_halting_on s,P & while=0 (a,I) is_closed_on s,P )
let I be parahalting Program of ; :: thesis: for a being read-write Int-Location
for s being State of SCM+FSA st ex f being Function of (product the Object-Kind of SCM+FSA),NAT st
for k being Nat holds
( ( f . ((StepWhile=0 (a,I,P,s)) . (k + 1)) < f . ((StepWhile=0 (a,I,P,s)) . k) or f . ((StepWhile=0 (a,I,P,s)) . k) = 0 ) & ( f . ((StepWhile=0 (a,I,P,s)) . k) = 0 implies ((StepWhile=0 (a,I,P,s)) . k) . a <> 0 ) & ( ((StepWhile=0 (a,I,P,s)) . k) . a <> 0 implies f . ((StepWhile=0 (a,I,P,s)) . k) = 0 ) ) holds
( while=0 (a,I) is_halting_on s,P & while=0 (a,I) is_closed_on s,P )
let a be read-write Int-Location ; :: thesis: for s being State of SCM+FSA st ex f being Function of (product the Object-Kind of SCM+FSA),NAT st
for k being Nat holds
( ( f . ((StepWhile=0 (a,I,P,s)) . (k + 1)) < f . ((StepWhile=0 (a,I,P,s)) . k) or f . ((StepWhile=0 (a,I,P,s)) . k) = 0 ) & ( f . ((StepWhile=0 (a,I,P,s)) . k) = 0 implies ((StepWhile=0 (a,I,P,s)) . k) . a <> 0 ) & ( ((StepWhile=0 (a,I,P,s)) . k) . a <> 0 implies f . ((StepWhile=0 (a,I,P,s)) . k) = 0 ) ) holds
( while=0 (a,I) is_halting_on s,P & while=0 (a,I) is_closed_on s,P )
let s be State of SCM+FSA; :: thesis: ( ex f being Function of (product the Object-Kind of SCM+FSA),NAT st
for k being Nat holds
( ( f . ((StepWhile=0 (a,I,P,s)) . (k + 1)) < f . ((StepWhile=0 (a,I,P,s)) . k) or f . ((StepWhile=0 (a,I,P,s)) . k) = 0 ) & ( f . ((StepWhile=0 (a,I,P,s)) . k) = 0 implies ((StepWhile=0 (a,I,P,s)) . k) . a <> 0 ) & ( ((StepWhile=0 (a,I,P,s)) . k) . a <> 0 implies f . ((StepWhile=0 (a,I,P,s)) . k) = 0 ) ) implies ( while=0 (a,I) is_halting_on s,P & while=0 (a,I) is_closed_on s,P ) )
A1: for k being Nat holds
( I is_closed_on (StepWhile=0 (a,I,P,s)) . k,P +* (while=0 (a,I)) & I is_halting_on (StepWhile=0 (a,I,P,s)) . k,P +* (while=0 (a,I)) ) by SCMFSA7B:18, SCMFSA7B:19;
assume ex f being Function of (product the Object-Kind of SCM+FSA),NAT st
for k being Nat holds
( ( f . ((StepWhile=0 (a,I,P,s)) . (k + 1)) < f . ((StepWhile=0 (a,I,P,s)) . k) or f . ((StepWhile=0 (a,I,P,s)) . k) = 0 ) & ( f . ((StepWhile=0 (a,I,P,s)) . k) = 0 implies ((StepWhile=0 (a,I,P,s)) . k) . a <> 0 ) & ( ((StepWhile=0 (a,I,P,s)) . k) . a <> 0 implies f . ((StepWhile=0 (a,I,P,s)) . k) = 0 ) ) ; :: thesis: ( while=0 (a,I) is_halting_on s,P & while=0 (a,I) is_closed_on s,P )
hence ( while=0 (a,I) is_halting_on s,P & while=0 (a,I) is_closed_on s,P ) by A1, Th32; :: thesis: verum