let A be preIfWhileAlgebra; :: thesis: for I being Element of A
for S being non empty set
for T being Subset of S
for s being Element of S
for f being ExecutionFunction of A,S,T st s nin T holds
( f iteration_terminates_for I,s & iteration-degree I,s,f = 0 )
let I be Element of A; :: thesis: for S being non empty set
for T being Subset of S
for s being Element of S
for f being ExecutionFunction of A,S,T st s nin T holds
( f iteration_terminates_for I,s & iteration-degree I,s,f = 0 )
let S be non empty set ; :: thesis: for T being Subset of S
for s being Element of S
for f being ExecutionFunction of A,S,T st s nin T holds
( f iteration_terminates_for I,s & iteration-degree I,s,f = 0 )
let T be Subset of S; :: thesis: for s being Element of S
for f being ExecutionFunction of A,S,T st s nin T holds
( f iteration_terminates_for I,s & iteration-degree I,s,f = 0 )
let s be Element of S; :: thesis: for f being ExecutionFunction of A,S,T st s nin T holds
( f iteration_terminates_for I,s & iteration-degree I,s,f = 0 )
let f be ExecutionFunction of A,S,T; :: thesis: ( s nin T implies ( f iteration_terminates_for I,s & iteration-degree I,s,f = 0 ) )
assume A1: s nin T ; :: thesis: ( f iteration_terminates_for I,s & iteration-degree I,s,f = 0 )
set r = <*s*>;
A2: ( <*s*> . 1 = s & len <*s*> = 1 ) by FINSEQ_1:57;
A3: for i being Nat st 1 <= i & i < len <*s*> holds
( <*s*> . i in T & <*s*> . (i + 1) = f . (<*s*> . i),I ) by FINSEQ_1:57;
A4: 1 - 1 = 0. ;
thus f iteration_terminates_for I,s :: thesis: iteration-degree I,s,f = 0

proof

take r = <*s*>; :: according to AOFA_000:def 33 :: thesis: ( r . 1 = s & r . (len r) nin T & ( for i being Nat st 1 <= i & i < len r holds
( r . i in T & r . (i + 1) = f . (r . i),I ) ) )
thus ( r . 1 = s & r . (len r) nin T & ( for i being Nat st 1 <= i & i < len r holds
( r . i in T & r . (i + 1) = f . (r . i),I ) ) ) by A1, A2; :: thesis: verum

end;

hence iteration-degree I,s,f = 0 by A1, A3, A4, Def34, A2; :: thesis: verum