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 P being set
for C, I, J being Element of A st C is_terminating_wrt f,P & I is_terminating_wrt f,P & J is_terminating_wrt f,P & P is_invariant_wrt C,f holds
if-then-else C,I,J is_terminating_wrt f,P
let S be non empty set ; :: thesis: for T being Subset of S
for f being ExecutionFunction of A,S,T
for P being set
for C, I, J being Element of A st C is_terminating_wrt f,P & I is_terminating_wrt f,P & J is_terminating_wrt f,P & P is_invariant_wrt C,f holds
if-then-else C,I,J is_terminating_wrt f,P
let T be Subset of S; :: thesis: for f being ExecutionFunction of A,S,T
for P being set
for C, I, J being Element of A st C is_terminating_wrt f,P & I is_terminating_wrt f,P & J is_terminating_wrt f,P & P is_invariant_wrt C,f holds
if-then-else C,I,J is_terminating_wrt f,P
let f be ExecutionFunction of A,S,T; :: thesis: for P being set
for C, I, J being Element of A st C is_terminating_wrt f,P & I is_terminating_wrt f,P & J is_terminating_wrt f,P & P is_invariant_wrt C,f holds
if-then-else C,I,J is_terminating_wrt f,P
let P be set ; :: thesis: for C, I, J being Element of A st C is_terminating_wrt f,P & I is_terminating_wrt f,P & J is_terminating_wrt f,P & P is_invariant_wrt C,f holds
if-then-else C,I,J is_terminating_wrt f,P
let C, I, J be Element of A; :: thesis: ( C is_terminating_wrt f,P & I is_terminating_wrt f,P & J is_terminating_wrt f,P & P is_invariant_wrt C,f implies if-then-else C,I,J is_terminating_wrt f,P )
assume that
A1: for s being Element of S st s in P holds
[s,C] in TerminatingPrograms A,S,T,f and
A2: for s being Element of S st s in P holds
[s,I] in TerminatingPrograms A,S,T,f and
A3: for s being Element of S st s in P holds
[s,J] in TerminatingPrograms A,S,T,f and
A4: for s being Element of S st s in P holds
f . s,C in P ; :: according to AOFA_000:def 38,AOFA_000:def 39 :: thesis: if-then-else C,I,J is_terminating_wrt f,P
let s be Element of S; :: according to AOFA_000:def 38 :: thesis: ( s in P implies [s,(if-then-else C,I,J)] in TerminatingPrograms A,S,T,f )
assume s in P ; :: thesis: [s,(if-then-else C,I,J)] in TerminatingPrograms A,S,T,f
then A5: ( ( f . s,C in T or f . s,C nin T ) & [s,C] in TerminatingPrograms A,S,T,f & f . s,C in P ) by A1, A4;
then ( [(f . s,C),I] in TerminatingPrograms A,S,T,f & [(f . s,C),J] in TerminatingPrograms A,S,T,f ) by A2, A3;
hence [s,(if-then-else C,I,J)] in TerminatingPrograms A,S,T,f by A5, Def35; :: thesis: verum