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 holds
( [s,C] in TerminatingPrograms A,S,T,f & ex r being non empty FinSequence of S st
( r . 1 = f . s,C & r . (len r) nin T & ( 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) ) ) ) )
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 holds
( [s,C] in TerminatingPrograms A,S,T,f & ex r being non empty FinSequence of S st
( r . 1 = f . s,C & r . (len r) nin T & ( 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) ) ) ) )
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 holds
( [s,C] in TerminatingPrograms A,S,T,f & ex r being non empty FinSequence of S st
( r . 1 = f . s,C & r . (len r) nin T & ( 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) ) ) ) )
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 holds
( [s,C] in TerminatingPrograms A,S,T,f & ex r being non empty FinSequence of S st
( r . 1 = f . s,C & r . (len r) nin T & ( 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) ) ) ) )
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 holds
( [s,C] in TerminatingPrograms A,S,T,f & ex r being non empty FinSequence of S st
( r . 1 = f . s,C & r . (len r) nin T & ( 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) ) ) ) )
let f be ExecutionFunction of A,S,T; :: thesis: ( A is free & [s,(while C,I)] in TerminatingPrograms A,S,T,f implies ( [s,C] in TerminatingPrograms A,S,T,f & ex r being non empty FinSequence of S st
( r . 1 = f . s,C & r . (len r) nin T & ( 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) ) ) ) ) )
set TP = TerminatingPrograms A,S,T,f;
set rr = s;
set IJ = while C,I;
assume that
A1: A is free and
A2: [s,(while C,I)] in TerminatingPrograms A,S,T,f ; :: thesis: ( [s,C] in TerminatingPrograms A,S,T,f & ex r being non empty FinSequence of S st
( r . 1 = f . s,C & r . (len r) nin T & ( 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) ) ) ) )
reconsider P = (TerminatingPrograms A,S,T,f) \ {[s,(while C,I)]} as Subset of [:S,the carrier of A:] ;
A3: [:S,(ElementaryInstructions A):] c= P

proof

let x, y be set ; :: according to RELAT_1:def 3 :: thesis: ( [x,y] nin [:S,(ElementaryInstructions A):] or not [x,y] nin P )
assume A4: [x,y] in [:S,(ElementaryInstructions A):] ; :: thesis: not [x,y] nin P
then y in ElementaryInstructions A by ZFMISC_1:106;
then A5: y <> while C,I by Th52;
A6: [:S,(ElementaryInstructions A):] c= TerminatingPrograms A,S,T,f by Def35;
[x,y] <> [s,(while C,I)] by A5, ZFMISC_1:33;
then [x,y] nin {[s,(while C,I)]} by TARSKI:def 1;
hence not [x,y] nin P by A4, A6, XBOOLE_0:def 5; :: thesis: verum

end;

A7: [:S,{(EmptyIns A)}:] c= P

proof

let x, y be set ; :: according to RELAT_1:def 3 :: thesis: ( [x,y] nin [:S,{(EmptyIns A)}:] or not [x,y] nin P )
assume A8: [x,y] in [:S,{(EmptyIns A)}:] ; :: thesis: not [x,y] nin P
then y in {(EmptyIns A)} by ZFMISC_1:106;
then y = EmptyIns A by TARSKI:def 1;
then A9: y <> while C,I by A1, Th72;
A10: [:S,{(EmptyIns A)}:] c= TerminatingPrograms A,S,T,f by Def35;
[x,y] <> [s,(while C,I)] by A9, ZFMISC_1:33;
then [x,y] nin {[s,(while C,I)]} by TARSKI:def 1;
hence not [x,y] nin P by A8, A10, XBOOLE_0:def 5; :: thesis: verum

end;

A11: now

let s be Element of S; :: thesis: for C, I, J being Element of A holds
( ( [s,I] in P & [(f . s,I),J] in P implies [s,(I \; J)] in P ) & ( [s,C] in P & [(f . s,C),I] in P & f . s,C in T implies [s,(if-then-else C,I,J)] in P ) & ( [s,C] in P & [(f . s,C),J] in P & f . s,C nin T implies [s,(if-then-else C,I,J)] in P ) )
let C, I, J be Element of A; :: thesis: ( ( [s,I] in P & [(f . s,I),J] in P implies [s,(I \; J)] in P ) & ( [s,C] in P & [(f . s,C),I] in P & f . s,C in T implies [s,(if-then-else C,I,J)] in P ) & ( [s,C] in P & [(f . s,C),J] in P & f . s,C nin T implies [s,(if-then-else C,I,J)] in P ) )

hereby :: thesis: ( ( [s,C] in P & [(f . s,C),I] in P & f . s,C in T implies [s,(if-then-else C,I,J)] in P ) & ( [s,C] in P & [(f . s,C),J] in P & f . s,C nin T implies [s,(if-then-else C,I,J)] in P ) )

assume that
A12: [s,I] in P and
A13: [(f . s,I),J] in P ; :: thesis: [s,(I \; J)] in P
A14: [s,I] in TerminatingPrograms A,S,T,f by A12, ZFMISC_1:64;
A15: [(f . s,I),J] in TerminatingPrograms A,S,T,f by A13, ZFMISC_1:64;
A16: while C,I <> I \; J by A1, Th73;
A17: [s,(I \; J)] in TerminatingPrograms A,S,T,f by A14, A15, Def35;
[s,(while C,I)] <> [s,(I \; J)] by A16, ZFMISC_1:33;
hence [s,(I \; J)] in P by A17, ZFMISC_1:64; :: thesis: verum

end;

hereby :: thesis: ( [s,C] in P & [(f . s,C),J] in P & f . s,C nin T implies [s,(if-then-else C,I,J)] in P )

assume that
A18: [s,C] in P and
A19: [(f . s,C),I] in P and
A20: f . s,C in T ; :: thesis: [s,(if-then-else C,I,J)] in P
A21: [s,C] in TerminatingPrograms A,S,T,f by A18, ZFMISC_1:64;
A22: [(f . s,C),I] in TerminatingPrograms A,S,T,f by A19, ZFMISC_1:64;
A23: while C,I <> if-then-else C,I,J by A1, Th74;
A24: [s,(if-then-else C,I,J)] in TerminatingPrograms A,S,T,f by A20, A21, A22, Def35;
[s,(while C,I)] <> [s,(if-then-else C,I,J)] by A23, ZFMISC_1:33;
hence [s,(if-then-else C,I,J)] in P by A24, ZFMISC_1:64; :: thesis: verum

end;

hereby :: thesis: verum

assume that
A25: [s,C] in P and
A26: [(f . s,C),J] in P and
A27: f . s,C nin T ; :: thesis: [s,(if-then-else C,I,J)] in P
A28: [s,C] in TerminatingPrograms A,S,T,f by A25, ZFMISC_1:64;
A29: [(f . s,C),J] in TerminatingPrograms A,S,T,f by A26, ZFMISC_1:64;
A30: while C,I <> if-then-else C,I,J by A1, Th74;
A31: [s,(if-then-else C,I,J)] in TerminatingPrograms A,S,T,f by A27, A28, A29, Def35;
[s,(while C,I)] <> [s,(if-then-else C,I,J)] by A30, ZFMISC_1:33;
hence [s,(if-then-else C,I,J)] in P by A31, ZFMISC_1:64; :: thesis: verum

end;

end;

hereby :: thesis: ex r being non empty FinSequence of S st
( r . 1 = f . s,C & r . (len r) nin T & ( 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) ) ) )

assume A32: [s,C] nin TerminatingPrograms A,S,T,f ; :: thesis: contradiction

now

let s be Element of S; :: thesis: for C9, I9, J being Element of A st [s,C9] in P & ex r being non empty FinSequence of S st
( r . 1 = f . s,C9 & r . (len r) nin T & ( for i being Nat st 1 <= i & i < len r holds
( r . i in T & [(r . i),(I9 \; C9)] in P & r . (i + 1) = f . (r . i),(I9 \; C9) ) ) ) holds
[s,(while C9,I9)] in P
let C9, I9, J be Element of A; :: thesis: ( [s,C9] in P & ex r being non empty FinSequence of S st
( r . 1 = f . s,C9 & r . (len r) nin T & ( for i being Nat st 1 <= i & i < len r holds
( r . i in T & [(r . i),(I9 \; C9)] in P & r . (i + 1) = f . (r . i),(I9 \; C9) ) ) ) implies [s,(while C9,I9)] in P )
assume A33: [s,C9] in P ; :: thesis: ( ex r being non empty FinSequence of S st
( r . 1 = f . s,C9 & r . (len r) nin T & ( for i being Nat st 1 <= i & i < len r holds
( r . i in T & [(r . i),(I9 \; C9)] in P & r . (i + 1) = f . (r . i),(I9 \; C9) ) ) ) implies [s,(while C9,I9)] in P )
given r being non empty FinSequence of S such that A34: r . 1 = f . s,C9 and
A35: r . (len r) nin T and
A36: for i being Nat st 1 <= i & i < len r holds
( r . i in T & [(r . i),(I9 \; C9)] in P & r . (i + 1) = f . (r . i),(I9 \; C9) ) ; :: thesis: [s,(while C9,I9)] in P

A37: now

let i be Nat; :: thesis: ( 1 <= i & i < len r implies ( r . i in T & [(r . i),(I9 \; C9)] in TerminatingPrograms A,S,T,f & r . (i + 1) = f . (r . i),(I9 \; C9) ) )
assume that
A38: 1 <= i and
A39: i < len r ; :: thesis: ( r . i in T & [(r . i),(I9 \; C9)] in TerminatingPrograms A,S,T,f & r . (i + 1) = f . (r . i),(I9 \; C9) )
[(r . i),(I9 \; C9)] in P by A36, A38, A39;
hence ( r . i in T & [(r . i),(I9 \; C9)] in TerminatingPrograms A,S,T,f & r . (i + 1) = f . (r . i),(I9 \; C9) ) by A36, A38, A39, ZFMISC_1:64; :: thesis: verum

end;

A40: [s,C9] in TerminatingPrograms A,S,T,f by A33, ZFMISC_1:64;
then A41: ( while C9,I9 <> while C,I or s <> s ) by A1, A32, Th75;
A42: [s,(while C9,I9)] in TerminatingPrograms A,S,T,f by A34, A35, A37, A40, Def35;
[s,(while C9,I9)] <> [s,(while C,I)] by A41, ZFMISC_1:33;
hence [s,(while C9,I9)] in P by A42, ZFMISC_1:64; :: thesis: verum

end;

then for s being Element of S
for C, I, J being Element of A holds
( ( [s,I] in P & [(f . s,I),J] in P implies [s,(I \; J)] in P ) & ( [s,C] in P & [(f . s,C),I] in P & f . s,C in T implies [s,(if-then-else C,I,J)] in P ) & ( [s,C] in P & [(f . s,C),J] in P & f . s,C nin T implies [s,(if-then-else C,I,J)] in P ) & ( [s,C] in P & ex r being non empty FinSequence of S st
( r . 1 = f . s,C & r . (len r) nin T & ( for i being Nat st 1 <= i & i < len r holds
( r . i in T & [(r . i),(I \; C)] in P & r . (i + 1) = f . (r . i),(I \; C) ) ) ) implies [s,(while C,I)] in P ) ) by A11;
then TerminatingPrograms A,S,T,f c= P by A3, A7, Def35;
hence contradiction by A2, ZFMISC_1:64; :: thesis: verum

end;

assume A43: for r being non empty FinSequence of S holds
( not r . 1 = f . s,C or not r . (len r) nin T or ex i being Nat st
( 1 <= i & i < len r & not ( r . i in T & [(r . i),(I \; C)] in TerminatingPrograms A,S,T,f & r . (i + 1) = f . (r . i),(I \; C) ) ) ) ; :: thesis: contradiction

now

let s be Element of S; :: thesis: for C9, I9, J being Element of A st [s,C9] in P & ex r being non empty FinSequence of S st
( r . 1 = f . s,C9 & r . (len r) nin T & ( for i being Nat st 1 <= i & i < len r holds
( r . i in T & [(r . i),(I9 \; C9)] in P & r . (i + 1) = f . (r . i),(I9 \; C9) ) ) ) holds
[s,(while C9,I9)] in P
let C9, I9, J be Element of A; :: thesis: ( [s,C9] in P & ex r being non empty FinSequence of S st
( r . 1 = f . s,C9 & r . (len r) nin T & ( for i being Nat st 1 <= i & i < len r holds
( r . i in T & [(r . i),(I9 \; C9)] in P & r . (i + 1) = f . (r . i),(I9 \; C9) ) ) ) implies [s,(while C9,I9)] in P )
assume A44: [s,C9] in P ; :: thesis: ( ex r being non empty FinSequence of S st
( r . 1 = f . s,C9 & r . (len r) nin T & ( for i being Nat st 1 <= i & i < len r holds
( r . i in T & [(r . i),(I9 \; C9)] in P & r . (i + 1) = f . (r . i),(I9 \; C9) ) ) ) implies [s,(while C9,I9)] in P )
given r being non empty FinSequence of S such that A45: r . 1 = f . s,C9 and
A46: r . (len r) nin T and
A47: for i being Nat st 1 <= i & i < len r holds
( r . i in T & [(r . i),(I9 \; C9)] in P & r . (i + 1) = f . (r . i),(I9 \; C9) ) ; :: thesis: [s,(while C9,I9)] in P

A48: now

let i be Nat; :: thesis: ( 1 <= i & i < len r implies ( r . i in T & [(r . i),(I9 \; C9)] in TerminatingPrograms A,S,T,f & r . (i + 1) = f . (r . i),(I9 \; C9) ) )
assume that
A49: 1 <= i and
A50: i < len r ; :: thesis: ( r . i in T & [(r . i),(I9 \; C9)] in TerminatingPrograms A,S,T,f & r . (i + 1) = f . (r . i),(I9 \; C9) )
[(r . i),(I9 \; C9)] in P by A47, A49, A50;
hence ( r . i in T & [(r . i),(I9 \; C9)] in TerminatingPrograms A,S,T,f & r . (i + 1) = f . (r . i),(I9 \; C9) ) by A47, A49, A50, ZFMISC_1:64; :: thesis: verum

end;

A51: [s,C9] in TerminatingPrograms A,S,T,f by A44, ZFMISC_1:64;
( I <> I9 or C <> C9 or s <> s ) by A43, A45, A46, A48;
then A52: ( while C9,I9 <> while C,I or s <> s ) by A1, Th75;
A53: [s,(while C9,I9)] in TerminatingPrograms A,S,T,f by A45, A46, A48, A51, Def35;
[s,(while C9,I9)] <> [s,(while C,I)] by A52, ZFMISC_1:33;
hence [s,(while C9,I9)] in P by A53, ZFMISC_1:64; :: thesis: verum

end;

then for s being Element of S
for C, I, J being Element of A holds
( ( [s,I] in P & [(f . s,I),J] in P implies [s,(I \; J)] in P ) & ( [s,C] in P & [(f . s,C),I] in P & f . s,C in T implies [s,(if-then-else C,I,J)] in P ) & ( [s,C] in P & [(f . s,C),J] in P & f . s,C nin T implies [s,(if-then-else C,I,J)] in P ) & ( [s,C] in P & ex r being non empty FinSequence of S st
( r . 1 = f . s,C & r . (len r) nin T & ( for i being Nat st 1 <= i & i < len r holds
( r . i in T & [(r . i),(I \; C)] in P & r . (i + 1) = f . (r . i),(I \; C) ) ) ) implies [s,(while C,I)] in P ) ) by A11;
then TerminatingPrograms A,S,T,f c= P by A3, A7, Def35;
hence contradiction by A2, ZFMISC_1:64; :: thesis: verum