let A be preIfWhileAlgebra; :: thesis: for S being non empty set
for T being Subset of S
for f being ExecutionFunction of A,S,T
for C, I being Element of A st C is_terminating_wrt f & I is_terminating_wrt f & ( for s being Element of S holds f iteration_terminates_for I \; C,s ) holds
while (C,I) is_terminating_wrt f
let S be non empty set ; :: thesis: for T being Subset of S
for f being ExecutionFunction of A,S,T
for C, I being Element of A st C is_terminating_wrt f & I is_terminating_wrt f & ( for s being Element of S holds f iteration_terminates_for I \; C,s ) holds
while (C,I) is_terminating_wrt f
let T be Subset of S; :: thesis: for f being ExecutionFunction of A,S,T
for C, I being Element of A st C is_terminating_wrt f & I is_terminating_wrt f & ( for s being Element of S holds f iteration_terminates_for I \; C,s ) holds
while (C,I) is_terminating_wrt f
let f be ExecutionFunction of A,S,T; :: thesis: for C, I being Element of A st C is_terminating_wrt f & I is_terminating_wrt f & ( for s being Element of S holds f iteration_terminates_for I \; C,s ) holds
while (C,I) is_terminating_wrt f
let C, I be Element of A; :: thesis: ( C is_terminating_wrt f & I is_terminating_wrt f & ( for s being Element of S holds f iteration_terminates_for I \; C,s ) implies while (C,I) is_terminating_wrt f )
assume that
A1: C is_terminating_wrt f and
A2: I is_terminating_wrt f and
A3: for s being Element of S holds f iteration_terminates_for I \; C,s ; :: thesis: while (C,I) is_terminating_wrt f
let s be Element of S; :: according to AOFA_000:def 37 :: thesis: [s,(while (C,I))] in TerminatingPrograms (A,S,T,f)
f iteration_terminates_for I \; C,f . (s,C) by A3;
hence [s,(while (C,I))] in TerminatingPrograms (A,S,T,f) by A1, A2, Th114; :: thesis: verum