let A be preIfWhileAlgebra; :: thesis: for C, 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 A is free & [s,(while C,I)] in TerminatingPrograms A,S,T,f & f . s,C in T holds
[(f . s,C),I] in TerminatingPrograms A,S,T,f
let C, 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 A is free & [s,(while C,I)] in TerminatingPrograms A,S,T,f & f . s,C in T holds
[(f . s,C),I] in TerminatingPrograms A,S,T,f
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 A is free & [s,(while C,I)] in TerminatingPrograms A,S,T,f & f . s,C in T holds
[(f . s,C),I] in TerminatingPrograms A,S,T,f
let T be Subset of S; :: thesis: for s being Element of S
for f being ExecutionFunction of A,S,T st A is free & [s,(while C,I)] in TerminatingPrograms A,S,T,f & f . s,C in T holds
[(f . s,C),I] in TerminatingPrograms A,S,T,f
let s be Element of S; :: thesis: for f being ExecutionFunction of A,S,T st A is free & [s,(while C,I)] in TerminatingPrograms A,S,T,f & f . s,C in T holds
[(f . s,C),I] in TerminatingPrograms A,S,T,f
let f be ExecutionFunction of A,S,T; :: thesis: ( A is free & [s,(while C,I)] in TerminatingPrograms A,S,T,f & f . s,C in T implies [(f . s,C),I] in TerminatingPrograms A,S,T,f )
set TP = TerminatingPrograms A,S,T,f;
set IJ = while C,I;
assume A1: ( A is free & [s,(while C,I)] in TerminatingPrograms A,S,T,f & f . s,C in T ) ; :: thesis: [(f . s,C),I] in TerminatingPrograms A,S,T,f
then consider r being non empty FinSequence of S such that
A2: ( r . 1 = f . s,C & r . (len r) nin T ) and
A3: for i being Nat st 1 <= i & i < len r holds
( r . i in T & [(r . i),(I \; C)] in TerminatingPrograms A,S,T,f & r . (i + 1) = f . (r . i),(I \; C) ) by Th99;
len r >= 1 by NAT_1:14;
then 1 < len r by A1, A2, XXREAL_0:1;
then [(r . 1),(I \; C)] in TerminatingPrograms A,S,T,f by A3;
hence [(f . s,C),I] in TerminatingPrograms A,S,T,f by A1, A2, Th97; :: thesis: verum