let P be Instruction-Sequence of SCM+FSA; :: thesis: for I being really-closed parahalting MacroInstruction of SCM+FSA
for a being read-write Int-Location
for s being State of SCM+FSA st ex f being Function of (product (the_Values_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
let I be really-closed parahalting MacroInstruction of SCM+FSA ; :: thesis: for a being read-write Int-Location
for s being State of SCM+FSA st ex f being Function of (product (the_Values_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
let a be read-write Int-Location; :: thesis: for s being State of SCM+FSA st ex f being Function of (product (the_Values_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
let s be State of SCM+FSA; :: thesis: ( ex f being Function of (product (the_Values_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 )
A1: for k being Nat holds I is_halting_on (StepWhile>0 (a,I,P,s)) . k,P +* (while>0 (a,I)) by SCMFSA7B:19;
assume ex f being Function of (product (the_Values_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
hence while>0 (a,I) is_halting_on s,P by A1, Th19; :: thesis: verum