scpqsort.miz

begin

 reserve x for Int_position,

n,p0 for Nat;

 registration

 let I,J be shiftable Program of SCMPDS , a be Int_position, k1 be Integer;

 cluster ( if>0 (a,k1,I,J)) -> shiftable;

 correctness

 proof

 set i = ((a,k1) <=0_goto (( card I) + 2)), G = ( Goto (( card J) + 1));

 ((I ';' G) ';' J) = ((I ';' ( Load ( goto (( card J) + 1)))) ';' J) by SCMPDS_6:def 1;

 then

 reconsider IJ = ((I ';' G) ';' J) as shiftable Program of SCMPDS ;

 ( if>0 (a,k1,I,J)) = (((i ';' I) ';' G) ';' J) by SCMPDS_6:def 5

 .= ((i ';' (I ';' G)) ';' J) by SCMPDS_4: 14

 .= (i ';' IJ) by SCMPDS_4: 14

 .= (( Load i) ';' IJ) by SCMPDS_4:def 2;

 hence thesis;

 end;

 end

 reserve P,Q,V for Instruction-Sequence of SCMPDS ;

 theorem :: SCPQSORT:1



 Th1: for s be 0 -started State of SCMPDS , I be halt-free shiftable Program of SCMPDS , J be shiftable Program of SCMPDS , a,b be Int_position, k1 be Integer st (s . ( DataLoc ((s . a),k1))) > 0 & I is_closed_on (s,P) & I is_halting_on (s,P) holds (( IExec (( if>0 (a,k1,I,J)),P,s)) . b) = (( IExec (I,P,s)) . b)

 proof

 let s be 0 -started State of SCMPDS , I be halt-free shiftable Program of SCMPDS , J be shiftable Program of SCMPDS , a,b be Int_position, k1 be Integer;

 assume that

 A1: (s . ( DataLoc ((s . a),k1))) > 0 and

 A2: I is_closed_on (s,P) and

 A3: I is_halting_on (s,P);



 A4: not b in ( dom ( Start-At (((( card I) + ( card J)) + 2), SCMPDS ))) by SCMPDS_4: 18;

 ( IExec (( if>0 (a,k1,I,J)),P,s)) = (( IExec (I,P,s)) +* ( Start-At (((( card I) + ( card J)) + 2), SCMPDS ))) by A1, A2, A3, SCMPDS_6: 70;

 hence thesis by A4, FUNCT_4: 11;

 end;

 set A = NAT , D = SCM-Data-Loc ;



 Lm1: for a be Int_position, i be Integer, I be Program of SCMPDS holds ( while>0 (a,i,I)) = (((a,i) <=0_goto (( card I) + 2)) ';' (I ';' ( goto ( - (( card I) + 1)))))

 proof

 let a be Int_position, i be Integer, I be Program of SCMPDS ;

 set i1 = ((a,i) <=0_goto (( card I) + 2)), i2 = ( goto ( - (( card I) + 1)));



 thus ( while>0 (a,i,I)) = ((i1 ';' I) ';' i2) by SCMPDS_8:def 3

 .= (i1 ';' (I ';' i2)) by SCMPDS_4: 15;

 end;



 Lm2: for I be Program of SCMPDS , a be Int_position, i be Integer holds ( Shift (I,1)) c= ( while>0 (a,i,I))

 proof

 let I be Program of SCMPDS , a be Int_position, i be Integer;

 set i1 = ((a,i) <=0_goto (( card I) + 2)), i2 = ( goto ( - (( card I) + 1)));



 A1: ( while>0 (a,i,I)) = ((i1 ';' I) ';' i2) by SCMPDS_8:def 3

 .= ((i1 ';' I) ';' ( Load i2)) by SCMPDS_4:def 3

 .= ((( Load i1) ';' I) ';' ( Load i2)) by SCMPDS_4:def 2;

 ( card ( Load i1)) = 1 by COMPOS_1: 54;

 hence thesis by A1, SCMPDS_7: 3;

 end;

 theorem :: SCPQSORT:2



 Th2: for s,sm be State of SCMPDS , I be halt-free shiftable Program of SCMPDS , a be Int_position, i be Integer, m be Nat st I is_closed_on (s,P) & I is_halting_on (s,P) & (s . ( DataLoc ((s . a),i))) > 0 & m = (( LifeSpan ((P +* ( stop I)),( Initialize s))) + 2) & sm = ( Comput ((P +* ( stop ( while>0 (a,i,I)))),( Initialize s),m)) holds ( DataPart sm) = ( DataPart ( IExec (I,P,( Initialize s)))) & ( Initialize sm) = sm

 proof

 let s,sm be State of SCMPDS , I be halt-free shiftable Program of SCMPDS , a be Int_position, i be Integer, m be Nat;

 set b = ( DataLoc ((s . a),i));

 set WH = ( while>0 (a,i,I)), iWH = ( stop WH), IsI = ( stop I);

 set i1 = ((a,i) <=0_goto (( card I) + 2)), i2 = ( goto ( - (( card I) + 1)));

 set s2 = ( Initialize s), P2 = (P +* ( stop I)), s3 = ( Initialize s), P3 = (P +* ( stop WH)), s4 = ( Comput (P3,s3,1)), P4 = P3;

 assume that

 A1: I is_closed_on (s,P) and

 A2: I is_halting_on (s,P) and

 A3: (s . b) > 0 and

 A4: m = (( LifeSpan (P2,s2)) + 2) and

 A5: sm = ( Comput (P3,s3,m));



 A6: IsI c= P2 by FUNCT_4: 25;



 A7: iWH c= P3 by FUNCT_4: 25;

 WH c= iWH by AFINSQ_1: 74;

 then

 A8: WH c= P3 by A7, XBOOLE_1: 1;

 ( Shift (I,1)) c= WH by Lm2;

 then

 A9: ( Shift (I,1)) c= P4 by A8, XBOOLE_1: 1;



 A10: WH = (i1 ';' (I ';' i2)) by Lm1;



 A11: ( Comput (P3,s3,( 0 + 1))) = ( Following (P3,( Comput (P3,s3, 0 )))) by EXTPRO_1: 3

 .= ( Exec (i1,s3)) by A10, SCMPDS_6: 11;

 set m2 = ( LifeSpan (P2,s2)), s5 = ( Comput (P4,s4,m2)), P5 = P4, l1 = (( card I) + 1);



 A12: ( IC s3) = 0 by MEMSTR_0:def 11;



 A13: not b in ( dom ( Start-At ( 0 , SCMPDS ))) by SCMPDS_4: 18;

 not a in ( dom ( Start-At ( 0 , SCMPDS ))) by SCMPDS_4: 18;



 then

 A14: (s3 . ( DataLoc ((s3 . a),i))) = (s3 . b) by FUNCT_4: 11

 .= (s . b) by A13, FUNCT_4: 11;

 set m3 = (m2 + 1);

 set s6 = ( Comput (P3,s3,m3)), P6 = P3;



 A15: s6 = s5 by EXTPRO_1: 4;

 set s7 = ( Comput (P3,s3,(m3 + 1)));

 (( card I) + 1) < (( card I) + 2) by XREAL_1: 6;

 then

 A16: l1 in ( dom WH) by SCMPDS_8: 18;



 A17: I is_closed_on (s2,P2) by A1, SCMPDS_6: 24;

 for x holds (s2 . x) = (s4 . x) by A11, SCMPDS_2: 56;

 then

 A18: ( DataPart s2) = ( DataPart s4) by SCMPDS_4: 8;



 A19: P2 halts_on s2 by A2, SCMPDS_6:def 3;

 (P2 +* ( stop I)) halts_on ( Initialize s2) by A2, SCMPDS_6:def 3;

 then

 A20: I is_halting_on (s2,P2) by SCMPDS_6:def 3;



 A21: ( IC s4) = ( 0 + 1) by A3, A12, A11, A14, SCMPDS_2: 56;

 then ( DataPart ( Comput (P2,s2,m2))) = ( DataPart s5) by A6, A20, A17, A18, A9, SCMPDS_7: 18;



 then

 A22: ( DataPart s5) = ( DataPart ( Result (P2,s2))) by A19, EXTPRO_1: 23

 .= ( DataPart ( IExec (I,P,( Initialize s)))) by SCMPDS_4:def 5;



 A23: ( IC s5) = l1 by A6, A20, A17, A21, A18, A9, SCMPDS_7: 18;



 then

 A24: ( CurInstr (P6,s6)) = (P5 . l1) by A15, PBOOLE: 143

 .= (WH . l1) by A16, A8, GRFUNC_1: 2

 .= i2 by SCMPDS_8: 19;



 A25: s7 = ( Following (P3,s6)) by EXTPRO_1: 3

 .= ( Exec (i2,s6)) by A24;

 now

 let x;



 thus (sm . x) = (s6 . x) by A4, A5, A25, SCMPDS_2: 54

 .= (s5 . x) by EXTPRO_1: 4;

 end;

 hence ( DataPart sm) = ( DataPart ( IExec (I,P,( Initialize s)))) by A22, SCMPDS_4: 8;

 ( IC s7) = ( ICplusConst (s6,( 0 - (( card I) + 1)))) by A25, SCMPDS_2: 54

 .= 0 by A23, A15, SCMPDS_7: 1;

 hence thesis by A4, A5, MEMSTR_0: 46;

 end;

 theorem :: SCPQSORT:3



 Th3: for s be State of SCMPDS , I be Program of SCMPDS st for t be 0 -started State of SCMPDS , Q st ( DataPart t) = ( DataPart s) holds I is_halting_on (t,Q) holds I is_closed_on (s,P)

 proof

 let s be State of SCMPDS , I be Program of SCMPDS ;

 assume

 A1: for t be 0 -started State of SCMPDS , Q st ( DataPart t) = ( DataPart s) holds I is_halting_on (t,Q);

 set pI = ( stop I), sI = ( Initialize s), PI = (P +* pI);

 defpred X[ Nat] means not ( IC ( Comput (PI,sI,$1))) in ( dom pI);



 A2: for a be Int_position holds (s . a) = (sI . a)

 proof

 let a be Int_position;

 not a in ( dom ( Start-At ( 0 , SCMPDS ))) by SCMPDS_4: 18;

 hence thesis by FUNCT_4: 11;

 end;

 assume not I is_closed_on (s,P);

 then ex k be Nat st X[k] by SCMPDS_6:def 2;

 then

 A3: ex k be Nat st X[k];

 consider n be Nat such that

 A4: X[n] and

 A5: for m be Nat st X[m] holds n <= m from NAT_1:sch 5( A3);

 reconsider n as Nat;

 set s2 = ( Comput (PI,sI,n)), P2 = PI, Ig = ((( IC s2),(( IC s2) + 1)) --> (( goto 1),( goto ( - 1))));

 reconsider P0 = (PI +* Ig) as Instruction-Sequence of SCMPDS ;

 reconsider P1 = (P2 +* Ig) as Instruction-Sequence of SCMPDS ;

 reconsider P3 = (PI +* (( IC s2),( goto 1))) as Instruction-Sequence of SCMPDS ;

 reconsider P4 = (P3 +* ((( IC s2) + 1),( goto ( - 1)))) as Instruction-Sequence of SCMPDS ;



 A6: P0 = P4 by FUNCT_7: 139;



 A7: for m be Nat st m < n holds ( IC ( Comput (PI,sI,m))) in ( dom pI) by A5;



 A8: ( stop I) c= PI by FUNCT_4: 25;

 pI c= P3 by A4, FUNCT_4: 25, FUNCT_7: 89;

 then

 A9: ( stop I) c= P0 by A6, A4, AFINSQ_1: 73, FUNCT_7: 89;

 then

 A10: ( Comput (P0,sI,n)) = s2 by A8, A7, SCMPDS_4: 21;

 ( DataPart sI) = ( DataPart s) by A2, SCMPDS_4: 8;

 then I is_halting_on (sI,P0) by A1;

 then

 A11: (P0 +* pI) halts_on ( Initialize sI) by SCMPDS_6:def 3;



 A12: not P0 halts_on s2 by SCMPDS_4: 20;

 (P0 +* pI) = P0 by A9, FUNCT_4: 98;

 hence contradiction by A12, A10, A11, EXTPRO_1: 22;

 end;

 theorem :: SCPQSORT:4



 Th4: for i1,i2,i3,i4 be Instruction of SCMPDS holds ( card (((i1 ';' i2) ';' i3) ';' i4)) = 4

 proof

 let x1,x2,x3,x4 be Instruction of SCMPDS ;



 thus ( card (((x1 ';' x2) ';' x3) ';' x4)) = (( card ((x1 ';' x2) ';' x3)) + 1) by SCMP_GCD: 4

 .= ((( card (x1 ';' x2)) + 1) + 1) by SCMP_GCD: 4

 .= ((2 + 1) + 1) by SCMP_GCD: 5

 .= 4;

 end;

 theorem :: SCPQSORT:5



 Th5: for s be 0 -started State of SCMPDS , I be halt-free shiftable Program of SCMPDS , a,x,y be Int_position, i,c be Integer st (s . x) >= (c + (s . ( DataLoc ((s . a),i)))) & (for t be 0 -started State of SCMPDS , Q st (t . x) >= (c + (t . ( DataLoc ((s . a),i)))) & (t . y) = (s . y) & (t . a) = (s . a) & (t . ( DataLoc ((s . a),i))) > 0 holds (( IExec (I,Q,t)) . a) = (t . a) & I is_closed_on (t,Q) & I is_halting_on (t,Q) & (( IExec (I,Q,t)) . ( DataLoc ((s . a),i))) < (t . ( DataLoc ((s . a),i))) & (( IExec (I,Q,t)) . x) >= (c + (( IExec (I,Q,t)) . ( DataLoc ((s . a),i)))) & (( IExec (I,Q,t)) . y) = (t . y)) holds ( while>0 (a,i,I)) is_closed_on (s,P) & ( while>0 (a,i,I)) is_halting_on (s,P) & ((s . ( DataLoc ((s . a),i))) > 0 implies ( IExec (( while>0 (a,i,I)),P,s)) = ( IExec (( while>0 (a,i,I)),P,( Initialize ( IExec (I,P,s))))))

 proof

 let s be 0 -started State of SCMPDS , I be halt-free shiftable Program of SCMPDS , a,x1,y1 be Int_position, i,c be Integer;

 set b = ( DataLoc ((s . a),i));

 assume (s . x1) >= (c + (s . b));

 then

 A1: for x st x in {x1} holds (s . x) >= (c + (s . b)) by TARSKI:def 1;

 assume

 A2: for t be 0 -started State of SCMPDS , Q st (t . x1) >= (c + (t . b)) & (t . y1) = (s . y1) & (t . a) = (s . a) & (t . b) > 0 holds (( IExec (I,Q,t)) . a) = (t . a) & I is_closed_on (t,Q) & I is_halting_on (t,Q) & (( IExec (I,Q,t)) . b) < (t . b) & (( IExec (I,Q,t)) . x1) >= (c + (( IExec (I,Q,t)) . b)) & (( IExec (I,Q,t)) . y1) = (t . y1);

 now

 let t be 0 -started State of SCMPDS , Q;

 assume that

 A3: for x st x in {x1} holds (t . x) >= (c + (t . b)) and

 A4: for x st x in {y1} holds (t . x) = (s . x) and

 A5: (t . a) = (s . a) and

 A6: (t . b) > 0 ;

 y1 in {y1} by TARSKI:def 1;

 then

 A7: (t . y1) = (s . y1) by A4;

 x1 in {x1} by TARSKI:def 1;

 then

 A8: (t . x1) >= (c + (t . b)) by A3;

 hence (( IExec (I,Q,t)) . a) = (t . a) & I is_closed_on (t,Q) & I is_halting_on (t,Q) & (( IExec (I,Q,t)) . b) < (t . b) by A2, A5, A6, A7;

 hereby

 let x;

 assume

 A9: x in {x1};

 (( IExec (I,Q,t)) . x1) >= (c + (( IExec (I,Q,t)) . b)) by A2, A5, A6, A8, A7;

 hence (( IExec (I,Q,t)) . x) >= (c + (( IExec (I,Q,t)) . b)) by A9, TARSKI:def 1;

 end;

 let x;

 assume

 A10: x in {y1};



 hence (( IExec (I,Q,t)) . x) = (( IExec (I,Q,t)) . y1) by TARSKI:def 1

 .= (t . y1) by A2, A5, A6, A8, A7

 .= (t . x) by A10, TARSKI:def 1;

 end;

 hence thesis by A1, SCMPDS_8: 27;

 end;

 theorem :: SCPQSORT:6



 Th6: for s be 0 -started State of SCMPDS , I be halt-free shiftable Program of SCMPDS , a,x,y be Int_position, i,c be Integer st (s . x) >= c & (for t be 0 -started State of SCMPDS , Q st (t . x) >= c & (t . y) = (s . y) & (t . a) = (s . a) & (t . ( DataLoc ((s . a),i))) > 0 holds (( IExec (I,Q,t)) . a) = (t . a) & I is_closed_on (t,Q) & I is_halting_on (t,Q) & (( IExec (I,Q,t)) . ( DataLoc ((s . a),i))) < (t . ( DataLoc ((s . a),i))) & (( IExec (I,Q,t)) . x) >= c & (( IExec (I,Q,t)) . y) = (t . y)) holds ( while>0 (a,i,I)) is_closed_on (s,P) & ( while>0 (a,i,I)) is_halting_on (s,P) & ((s . ( DataLoc ((s . a),i))) > 0 implies ( IExec (( while>0 (a,i,I)),P,s)) = ( IExec (( while>0 (a,i,I)),P,( Initialize ( IExec (I,P,s))))))

 proof

 let s be 0 -started State of SCMPDS , I be halt-free shiftable Program of SCMPDS , a,x,y be Int_position, i,c be Integer;

 set b = ( DataLoc ((s . a),i));

 defpred P[ set] means ex t be State of SCMPDS st t = $1 & (t . x) >= c & (t . y) = (s . y);

 consider f be Function of ( product ( the_Values_of SCMPDS )), NAT such that

 A1: for s be State of SCMPDS holds ((s . b) <= 0 implies (f . s) = 0 ) & ((s . b) > 0 implies (f . s) = (s . b)) by SCMPDS_8: 5;

 deffunc F( State of SCMPDS ) = (f . $1);



 A2: for t be 0 -started State of SCMPDS st P[t] & F(t) = 0 holds (t . b) <= 0 by A1;

 assume

 A3: (s . x) >= c;



 A4: P[s] by A3;

 assume

 A5: for t be 0 -started State of SCMPDS , Q st (t . x) >= c & (t . y) = (s . y) & (t . a) = (s . a) & (t . b) > 0 holds (( IExec (I,Q,t)) . a) = (t . a) & I is_closed_on (t,Q) & I is_halting_on (t,Q) & (( IExec (I,Q,t)) . b) < (t . b) & (( IExec (I,Q,t)) . x) >= c & (( IExec (I,Q,t)) . y) = (t . y);

 A6:

 now

 let t be 0 -started State of SCMPDS , Q;

 assume that

 A7: P[t] and

 A8: (t . a) = (s . a) and

 A9: (t . b) > 0 ;

 set It = ( IExec (I,Q,t)), t2 = ( Initialize It);

 thus (( IExec (I,Q,t)) . a) = (t . a) & I is_closed_on (t,Q) & I is_halting_on (t,Q) by A5, A8, A9, A7;

 thus F(t2) < F(t)

 proof



 A10: F(t) = (t . b) by A1, A9;

 assume

 A11: F(t2) >= F(t);

 then (t2 . b) > 0 by A1, A9, A10;



 then F(t2) = (t2 . b) by A1

 .= (It . b) by SCMPDS_5: 15;

 hence contradiction by A5, A8, A9, A7, A11, A10;

 end;

 thus P[( Initialize It)]

 proof

 take v = ( Initialize It);

 thus v = ( Initialize It);

 (It . x) >= c by A5, A8, A9, A7;

 hence (v . x) >= c by SCMPDS_5: 15;

 (It . y) = (t . y) by A5, A8, A9, A7;

 hence thesis by A7, SCMPDS_5: 15;

 end;

 end;

 ( while>0 (a,i,I)) is_closed_on (s,P) & ( while>0 (a,i,I)) is_halting_on (s,P) from SCMPDS_8:sch 3( A2, A4, A6);

 hence ( while>0 (a,i,I)) is_closed_on (s,P) & ( while>0 (a,i,I)) is_halting_on (s,P);

 assume

 A12: (s . b) > 0 ;

 ( IExec (( while>0 (a,i,I)),P,s)) = ( IExec (( while>0 (a,i,I)),P,( Initialize ( IExec (I,P,s))))) from SCMPDS_8:sch 4( A12, A2, A4, A6);

 hence thesis;

 end;

 theorem :: SCPQSORT:7



 Th7: for s be 0 -started State of SCMPDS , I be halt-free shiftable Program of SCMPDS , a,x1,x2,x3,x4 be Int_position, i,c,md be Integer st (s . x4) = (((s . x3) - c) + (s . x1)) & md <= ((s . x3) - c) & (for t be 0 -started State of SCMPDS , Q st (t . x4) = (((t . x3) - c) + (t . x1)) & md <= ((t . x3) - c) & (t . x2) = (s . x2) & (t . a) = (s . a) & (t . ( DataLoc ((s . a),i))) > 0 holds (( IExec (I,Q,t)) . a) = (t . a) & I is_closed_on (t,Q) & I is_halting_on (t,Q) & (( IExec (I,Q,t)) . ( DataLoc ((s . a),i))) < (t . ( DataLoc ((s . a),i))) & (( IExec (I,Q,t)) . x4) = (((( IExec (I,Q,t)) . x3) - c) + (( IExec (I,Q,t)) . x1)) & md <= ((( IExec (I,Q,t)) . x3) - c) & (( IExec (I,Q,t)) . x2) = (t . x2)) holds ( while>0 (a,i,I)) is_closed_on (s,P) & ( while>0 (a,i,I)) is_halting_on (s,P) & ((s . ( DataLoc ((s . a),i))) > 0 implies ( IExec (( while>0 (a,i,I)),P,s)) = ( IExec (( while>0 (a,i,I)),P,( Initialize ( IExec (I,P,s))))))

 proof

 let s be 0 -started State of SCMPDS , I be halt-free shiftable Program of SCMPDS , a,x1,x2,x3,x4 be Int_position, i,c,md be Integer;

 set b = ( DataLoc ((s . a),i));

 defpred P[ set] means ex t be State of SCMPDS st t = $1 & (t . x4) = (((t . x3) - c) + (t . x1)) & md <= ((t . x3) - c) & (t . x2) = (s . x2);

 assume that

 A1: (s . x4) = (((s . x3) - c) + (s . x1)) and

 A2: md <= ((s . x3) - c);

 consider f be Function of ( product ( the_Values_of SCMPDS )), NAT such that

 A3: for s be State of SCMPDS holds ((s . b) <= 0 implies (f . s) = 0 ) & ((s . b) > 0 implies (f . s) = (s . b)) by SCMPDS_8: 5;

 deffunc F( State of SCMPDS ) = (f . $1);



 A4: for t be 0 -started State of SCMPDS st P[t] & F(t) = 0 holds (t . b) <= 0 by A3;

 assume

 A5: for t be 0 -started State of SCMPDS , Q st (t . x4) = (((t . x3) - c) + (t . x1)) & md <= ((t . x3) - c) & (t . x2) = (s . x2) & (t . a) = (s . a) & (t . ( DataLoc ((s . a),i))) > 0 holds (( IExec (I,Q,t)) . a) = (t . a) & I is_closed_on (t,Q) & I is_halting_on (t,Q) & (( IExec (I,Q,t)) . ( DataLoc ((s . a),i))) < (t . ( DataLoc ((s . a),i))) & (( IExec (I,Q,t)) . x4) = (((( IExec (I,Q,t)) . x3) - c) + (( IExec (I,Q,t)) . x1)) & md <= ((( IExec (I,Q,t)) . x3) - c) & (( IExec (I,Q,t)) . x2) = (t . x2);

 A6:

 now

 let t be 0 -started State of SCMPDS , Q;

 assume that

 A7: P[t] and

 A8: (t . a) = (s . a) and

 A9: (t . b) > 0 ;

 set It = ( IExec (I,Q,t)), t2 = ( Initialize It);

 consider v be State of SCMPDS such that

 A10: v = t and

 A11: (v . x4) = (((v . x3) - c) + (v . x1)) and

 A12: md <= ((v . x3) - c) and

 A13: (v . x2) = (s . x2) by A7;



 A14: (t . x2) = (s . x2) by A10, A13;



 A15: (t . x4) = (((t . x3) - c) + (t . x1)) by A10, A11;



 A16: md <= ((t . x3) - c) by A10, A12;

 thus (( IExec (I,Q,t)) . a) = (t . a) & I is_closed_on (t,Q) & I is_halting_on (t,Q) by A5, A8, A9, A15, A13, A10, A12;

 thus F(t2) < F(t)

 proof



 A17: F(t) = (t . b) by A3, A9;

 assume

 A18: F(t2) >= F(t);

 then (t2 . b) > 0 by A3, A9, A17;



 then F(t2) = (t2 . b) by A3

 .= (It . b) by SCMPDS_5: 15;

 hence contradiction by A5, A8, A9, A15, A16, A10, A13, A18, A17;

 end;

 thus P[( Initialize It)]

 proof

 take v = ( Initialize It);

 thus v = ( Initialize It);

 (It . x4) = (((It . x3) - c) + (It . x1)) by A5, A8, A9, A15, A16, A14;

 then (v . x4) = (((It . x3) - c) + (It . x1)) by SCMPDS_5: 15;

 then (v . x4) = (((v . x3) - c) + (It . x1)) by SCMPDS_5: 15;

 hence (v . x4) = (((v . x3) - c) + (v . x1)) by SCMPDS_5: 15;

 md <= ((It . x3) - c) by A5, A8, A9, A15, A16, A14;

 hence md <= ((v . x3) - c) by SCMPDS_5: 15;

 (It . x2) = (t . x2) by A5, A8, A9, A15, A16, A10, A13;

 hence thesis by A10, A13, SCMPDS_5: 15;

 end;

 end;



 A19: P[s] by A1, A2;

 ( while>0 (a,i,I)) is_closed_on (s,P) & ( while>0 (a,i,I)) is_halting_on (s,P) from SCMPDS_8:sch 3( A4, A19, A6);

 hence ( while>0 (a,i,I)) is_closed_on (s,P) & ( while>0 (a,i,I)) is_halting_on (s,P);

 assume

 A20: (s . b) > 0 ;

 ( IExec (( while>0 (a,i,I)),P,s)) = ( IExec (( while>0 (a,i,I)),P,( Initialize ( IExec (I,P,s))))) from SCMPDS_8:sch 4( A20, A4, A19, A6);

 hence thesis;

 end;

 theorem :: SCPQSORT:8



 Th8: for f be FinSequence of INT , m,k1,k,n be Nat st k1 = (k - 1) & f is_non_decreasing_on (m,k1) & f is_non_decreasing_on ((k + 1),n) & (for i be Nat st m <= i & i < k holds (f . i) <= (f . k)) & (for i be Nat st k < i & i <= n holds (f . k) <= (f . i)) holds f is_non_decreasing_on (m,n)

 proof

 let f be FinSequence of INT , m,k1,k,n be Nat;

 assume that

 A1: k1 = (k - 1);

 assume

 A2: f is_non_decreasing_on (m,k1);

 assume

 A3: f is_non_decreasing_on ((k + 1),n);

 assume

 A4: for i be Nat st m <= i & i < k holds (f . i) <= (f . k);

 assume

 A5: for i be Nat st k < i & i <= n holds (f . k) <= (f . i);

 now

 let i,j be Nat;

 assume that

 A6: m <= i and

 A7: i <= j and

 A8: j <= n;

 per cases by XXREAL_0: 1;

 suppose j < k;

 then (j + 1) <= k by INT_1: 7;

 then j <= k1 by A1, XREAL_1: 19;

 hence (f . i) <= (f . j) by A2, A6, A7, FINSEQ_6:def 9;

 end;

 suppose

 A9: j = k;

 per cases ;

 suppose i = j;

 hence (f . i) <= (f . j);

 end;

 suppose i <> j;

 then i < j by A7, XXREAL_0: 1;

 hence (f . i) <= (f . j) by A4, A6, A9;

 end;

 end;

 suppose

 A10: j > k;

 per cases by XXREAL_0: 1;

 suppose i < k;

 then

 A11: (f . k) >= (f . i) by A4, A6;

 (f . j) >= (f . k) by A5, A8, A10;

 hence (f . i) <= (f . j) by A11, XXREAL_0: 2;

 end;

 suppose

 A12: i = k;

 per cases ;

 suppose i = j;

 hence (f . i) <= (f . j);

 end;

 suppose i <> j;

 then i < j by A7, XXREAL_0: 1;

 hence (f . i) <= (f . j) by A5, A8, A12;

 end;

 end;

 suppose i > k;

 then i >= (k + 1) by INT_1: 7;

 hence (f . i) <= (f . j) by A3, A7, A8, FINSEQ_6:def 9;

 end;

 end;

 end;

 hence thesis by FINSEQ_6:def 9;

 end;



 Lm3: for s be 0 -started State of SCMPDS , I be halt-free shiftable Program of SCMPDS , a be Int_position, i,c be Integer, f,g be FinSequence of INT , m,n,m1 be Nat st (s . a) = c & ( len f) = n & ( len g) = n & f is_FinSequence_on (s,m) & g is_FinSequence_on (( IExec (( while>0 (a,i,I)),P,s)),m) & 1 = (s . ( DataLoc (c,i))) & m1 = ((m + n) + 1) & (m + 1) = (s . ( intpos m1)) & (m + n) = (s . ( intpos (m1 + 1))) & (for t be 0 -started State of SCMPDS , Q holds for f1,f2 be FinSequence of INT , k1,k2,y1,yn be Nat st (t . a) = c & ((2 * k1) + 1) = (t . ( DataLoc (c,i))) & k2 = (((m + n) + (2 * k1)) + 1) & (m + y1) = (t . ( intpos k2)) & (m + yn) = (t . ( intpos (k2 + 1))) & (1 <= y1 & yn <= n or y1 >= yn) holds I is_closed_on (t,Q) & I is_halting_on (t,Q) & (( IExec (I,Q,t)) . a) = (t . a) & (for j be Nat st (1 <= j & j < ((2 * k1) + 1)) holds (( IExec (I,Q,t)) . ( intpos ((m + n) + j))) = (t . ( intpos ((m + n) + j)))) & (y1 >= yn implies (( IExec (I,Q,t)) . ( DataLoc (c,i))) = ((2 * k1) - 1) & (for j be Nat st (1 <= j & j <= n) holds (( IExec (I,Q,t)) . ( intpos (m + j))) = (t . ( intpos (m + j))))) & (y1 < yn implies (( IExec (I,Q,t)) . ( DataLoc (c,i))) = ((2 * k1) + 3) & (for j be Nat st 1 <= j & j < y1 or yn < j & j <= n holds (( IExec (I,Q,t)) . ( intpos (m + j))) = (t . ( intpos (m + j)))) & (ex ym be Nat st y1 <= ym & ym <= yn & (m + y1) = (( IExec (I,Q,t)) . ( intpos k2)) & ((m + ym) - 1) = (( IExec (I,Q,t)) . ( intpos (k2 + 1))) & ((m + ym) + 1) = (( IExec (I,Q,t)) . ( intpos (k2 + 2))) & (m + yn) = (( IExec (I,Q,t)) . ( intpos (k2 + 3))) & (for j be Nat st y1 <= j & j < ym holds (( IExec (I,Q,t)) . ( intpos (m + j))) <= (( IExec (I,Q,t)) . ( intpos (m + ym)))) & (for j be Nat st ym < j & j <= yn holds (( IExec (I,Q,t)) . ( intpos (m + j))) >= (( IExec (I,Q,t)) . ( intpos (m + ym)))))) & (f1 is_FinSequence_on (t,m) & f2 is_FinSequence_on (( IExec (I,Q,t)),m) & ( len f1) = n & ( len f2) = n implies (f1,f2) are_fiberwise_equipotent )) holds ( while>0 (a,i,I)) is_halting_on (s,P) & (f,g) are_fiberwise_equipotent & g is_non_decreasing_on (1,n)

 proof

 let s be 0 -started State of SCMPDS , I be halt-free shiftable Program of SCMPDS , a be Int_position, i,c be Integer, f,g be FinSequence of INT , m,n,m1 be Nat;



 A1: ( Initialize s) = s by MEMSTR_0: 44;

 set b = ( DataLoc (c,i));

 assume that

 A2: (s . a) = c and

 A3: ( len f) = n and

 A4: ( len g) = n;

 set WH = ( while>0 (a,i,I)), sWH = ( stop WH);

 assume

 A5: f is_FinSequence_on (s,m);

 defpred P[ Nat] means for t be 0 -started State of SCMPDS , Q holds for f1 be FinSequence of INT , k1,k2,y1,yn be Nat st (t . a) = c & ((2 * k1) + 1) = (t . b) & k2 = (((m + n) + (2 * k1)) + 1) & (1 <= y1 & yn <= n or y1 >= yn) & (m + y1) = (t . ( intpos k2)) & (m + yn) = (t . ( intpos (k2 + 1))) & (yn - y1) <= $1 & f1 is_FinSequence_on (t,m) & ( len f1) = n holds ex k be Nat, f2 be FinSequence of INT st ( Initialize ( Comput ((Q +* sWH),t,k))) = ( Comput ((Q +* sWH),t,k)) & f2 is_FinSequence_on (( Comput ((Q +* sWH),t,k)),m) & ( len f2) = n & (f1,f2) are_fiberwise_equipotent & f2 is_non_decreasing_on (y1,yn) & (for j be Nat st y1 < yn & (1 <= j & j < y1 or yn < j & j <= n) holds (f2 . j) = (t . ( intpos (m + j)))) & (for j be Nat st y1 >= yn & 1 <= j & j <= n holds (f2 . j) = (t . ( intpos (m + j)))) & (for j be Nat st 1 <= j & j < ((2 * k1) + 1) holds (( Comput ((Q +* sWH),t,k)) . ( intpos ((m + n) + j))) = (t . ( intpos ((m + n) + j)))) & (( Comput ((Q +* sWH),t,k)) . b) = ((t . b) - 2) & (( Comput ((Q +* sWH),t,k)) . a) = c;

 assume

 A6: g is_FinSequence_on (( IExec (( while>0 (a,i,I)),P,s)),m);

 assume

 A7: 1 = (s . b);

 assume that

 A8: m1 = ((m + n) + 1) and

 A9: (m + 1) = (s . ( intpos m1)) and

 A10: (m + n) = (s . ( intpos (m1 + 1)));



 A11: m1 = (((m + n) + (2 * 0 )) + 1) by A8;

 assume

 A12: for t be 0 -started State of SCMPDS , Q holds for f1,f2 be FinSequence of INT , k1,k2,y1,yn be Nat st (t . a) = c & ((2 * k1) + 1) = (t . b) & k2 = (((m + n) + (2 * k1)) + 1) & (m + y1) = (t . ( intpos k2)) & (m + yn) = (t . ( intpos (k2 + 1))) & (1 <= y1 & yn <= n or y1 >= yn) holds I is_closed_on (t,Q) & I is_halting_on (t,Q) & (( IExec (I,Q,t)) . a) = (t . a) & (for j be Nat st (1 <= j & j < ((2 * k1) + 1)) holds (( IExec (I,Q,t)) . ( intpos ((m + n) + j))) = (t . ( intpos ((m + n) + j)))) & (y1 >= yn implies (( IExec (I,Q,t)) . b) = ((2 * k1) - 1) & for j be Nat st 1 <= j & j <= n holds (( IExec (I,Q,t)) . ( intpos (m + j))) = (t . ( intpos (m + j)))) & (y1 < yn implies (( IExec (I,Q,t)) . b) = ((2 * k1) + 3) & (for j be Nat st 1 <= j & j < y1 or yn < j & j <= n holds (( IExec (I,Q,t)) . ( intpos (m + j))) = (t . ( intpos (m + j)))) & ex ym be Nat st y1 <= ym & ym <= yn & (m + y1) = (( IExec (I,Q,t)) . ( intpos k2)) & ((m + ym) - 1) = (( IExec (I,Q,t)) . ( intpos (k2 + 1))) & ((m + ym) + 1) = (( IExec (I,Q,t)) . ( intpos (k2 + 2))) & (m + yn) = (( IExec (I,Q,t)) . ( intpos (k2 + 3))) & (for j be Nat st y1 <= j & j < ym holds (( IExec (I,Q,t)) . ( intpos (m + j))) <= (( IExec (I,Q,t)) . ( intpos (m + ym)))) & for j be Nat st ym < j & j <= yn holds (( IExec (I,Q,t)) . ( intpos (m + j))) >= (( IExec (I,Q,t)) . ( intpos (m + ym)))) & (f1 is_FinSequence_on (t,m) & f2 is_FinSequence_on (( IExec (I,Q,t)),m) & ( len f1) = n & ( len f2) = n implies (f1,f2) are_fiberwise_equipotent );



 A13: P[ 0 ]

 proof

 let t be 0 -started State of SCMPDS , Q;

 let f1 be FinSequence of INT , k1,k2,y1,yn be Nat;



 A14: ( Initialize t) = t by MEMSTR_0: 44;

 assume that

 A15: (t . a) = c and

 A16: ((2 * k1) + 1) = (t . b) and

 A17: k2 = (((m + n) + (2 * k1)) + 1) and

 A18: 1 <= y1 & yn <= n or y1 >= yn and

 A19: (m + y1) = (t . ( intpos k2)) and

 A20: (m + yn) = (t . ( intpos (k2 + 1))) and

 A21: (yn - y1) <= 0 and

 A22: f1 is_FinSequence_on (t,m) and

 A23: ( len f1) = n;



 A24: I is_halting_on (t,Q) by A12, A15, A16, A17, A18, A19, A20;

 take k = (( LifeSpan ((Q +* ( stop I)),t)) + 2);

 set tk = ( Comput ((Q +* sWH),t,k));



 A25: I is_closed_on (t,Q) by A12, A15, A16, A17, A18, A19, A20;

 then

 A26: ( DataPart tk) = ( DataPart ( IExec (I,Q,t))) by A15, A16, A24, Th2, A14;

 consider f2 be FinSequence of INT such that

 A27: ( len f2) = n and

 A28: for i be Nat st 1 <= i & i <= ( len f2) holds (f2 . i) = (( IExec (I,Q,t)) . ( intpos (m + i))) by SCPISORT: 1;

 take f2;

 thus ( Initialize tk) = tk by A15, A16, A25, A24, Th2, A14;

 now

 let i be Nat;

 assume that

 A29: 1 <= i and

 A30: i <= ( len f2);



 thus (f2 . i) = (( IExec (I,Q,t)) . ( intpos (m + i))) by A28, A29, A30

 .= (tk . ( intpos (m + i))) by A26, SCMPDS_4: 8;

 end;

 hence f2 is_FinSequence_on (tk,m);

 thus ( len f2) = n by A27;

 f2 is_FinSequence_on (( IExec (I,Q,t)),m) by A28;

 hence (f1,f2) are_fiberwise_equipotent by A12, A15, A16, A17, A18, A19, A20, A22, A23, A27;

 thus f2 is_non_decreasing_on (y1,yn) by A21, FINSEQ_6: 165, XREAL_1: 50;

 thus for j be Nat st y1 < yn & (1 <= j & j < y1 or yn < j & j <= n) holds (f2 . j) = (t . ( intpos (m + j))) by A21, XREAL_1: 50;

 hereby

 let j be Nat;

 assume that

 A31: y1 >= yn and

 A32: 1 <= j and

 A33: j <= n;



 thus (f2 . j) = (( IExec (I,Q,t)) . ( intpos (m + j))) by A27, A28, A32, A33

 .= (t . ( intpos (m + j))) by A12, A15, A16, A17, A19, A20, A31, A32, A33;

 end;

 hereby

 let j be Nat;

 assume that

 A34: 1 <= j and

 A35: j < ((2 * k1) + 1);



 thus (tk . ( intpos ((m + n) + j))) = (( IExec (I,Q,t)) . ( intpos ((m + n) + j))) by A26, SCMPDS_4: 8

 .= (t . ( intpos ((m + n) + j))) by A12, A15, A16, A17, A18, A19, A20, A34, A35;

 end;

 y1 >= yn implies (( IExec (I,Q,t)) . b) = ((2 * k1) - 1) & for j be Nat st 1 <= j & j <= n holds (( IExec (I,Q,t)) . ( intpos (m + j))) = (t . ( intpos (m + j))) by A12, A15, A16, A17, A19, A20;

 hence (tk . b) = ((t . b) - 2) by A16, A21, A26, SCMPDS_4: 8, XREAL_1: 50;

 (( IExec (I,Q,t)) . a) = (t . a) by A12, A15, A16, A17, A18, A19, A20;

 hence thesis by A15, A26, SCMPDS_4: 8;

 end;

 A36:

 now

 let k be Nat;

 assume

 A37: P[k];

 P[(k + 1)]

 proof

 let t be 0 -started State of SCMPDS , Q;

 let f1 be FinSequence of INT , k1,k2,y1,yn be Nat;



 A38: ( Initialize t) = t by MEMSTR_0: 44;

 assume that

 A39: (t . a) = c and

 A40: ((2 * k1) + 1) = (t . b) and

 A41: k2 = (((m + n) + (2 * k1)) + 1) and

 A42: 1 <= y1 & yn <= n or y1 >= yn and

 A43: (m + y1) = (t . ( intpos k2)) and

 A44: (m + yn) = (t . ( intpos (k2 + 1))) and

 A45: (yn - y1) <= (k + 1) and

 A46: f1 is_FinSequence_on (t,m) and

 A47: ( len f1) = n;

 per cases ;

 suppose (yn - y1) <= 0 ;

 hence ex kk be Nat, f2 be FinSequence of INT st ( Initialize ( Comput ((Q +* sWH),t,kk))) = ( Comput ((Q +* sWH),t,kk)) & f2 is_FinSequence_on (( Comput ((Q +* sWH),t,kk)),m) & ( len f2) = n & (f1,f2) are_fiberwise_equipotent & f2 is_non_decreasing_on (y1,yn) & (for j be Nat st y1 < yn & (1 <= j & j < y1 or yn < j & j <= n) holds (f2 . j) = (t . ( intpos (m + j)))) & (for j be Nat st y1 >= yn & 1 <= j & j <= n holds (f2 . j) = (t . ( intpos (m + j)))) & (for j be Nat st (1 <= j & j < ((2 * k1) + 1)) holds (( Comput ((Q +* sWH),t,kk)) . ( intpos ((m + n) + j))) = (t . ( intpos ((m + n) + j)))) & (( Comput ((Q +* sWH),t,kk)) . b) = ((t . b) - 2) & (( Comput ((Q +* sWH),t,kk)) . a) = c by A13, A39, A40, A41, A42, A43, A44, A46, A47;

 end;

 suppose

 A48: (yn - y1) > 0 ;

 set m1 = (( LifeSpan ((Q +* ( stop I)),t)) + 2);

 set t1 = ( Comput ((Q +* sWH),t,m1)), Q1 = (Q +* sWH);



 A49: I is_halting_on (t,Q) by A12, A39, A40, A41, A42, A43, A44;

 (y1 - 1) >= 0 by A42, A48, XREAL_1: 47, XREAL_1: 48;

 then

 reconsider y0 = (y1 - 1) as Element of NAT by INT_1: 3;

 set jj = ((2 * k1) + 1);



 A50: ((yn - y1) - 1) <= ((k + 1) - 1) by A45, XREAL_1: 9;



 A51: yn <= (y1 + (k + 1)) by A45, XREAL_1: 20;

 consider f2 be FinSequence of INT such that

 A52: ( len f2) = n and

 A53: for i be Nat st 1 <= i & i <= ( len f2) holds (f2 . i) = (t1 . ( intpos (m + i))) by SCPISORT: 1;

 set It = ( IExec (I,Q,t));



 A54: y1 < yn implies (It . b) = ((2 * k1) + 3) & (for j be Nat st 1 <= j & j < y1 or yn < j & j <= n holds (( IExec (I,Q,t)) . ( intpos (m + j))) = (t . ( intpos (m + j)))) & ex ym be Nat st y1 <= ym & ym <= yn & (m + y1) = (It . ( intpos k2)) & ((m + ym) - 1) = (It . ( intpos (k2 + 1))) & ((m + ym) + 1) = (It . ( intpos (k2 + 2))) & (m + yn) = (It . ( intpos (k2 + 3))) & (for j be Nat st y1 <= j & j < ym holds (( IExec (I,Q,t)) . ( intpos (m + j))) <= (( IExec (I,Q,t)) . ( intpos (m + ym)))) & for j be Nat st ym < j & j <= yn holds (( IExec (I,Q,t)) . ( intpos (m + j))) >= (( IExec (I,Q,t)) . ( intpos (m + ym))) by A12, A39, A40, A41, A42, A43, A44;

 then

 consider ym be Nat such that

 A55: y1 <= ym and

 A56: ym <= yn and

 A57: (m + y1) = (It . ( intpos k2)) and

 A58: ((m + ym) - 1) = (It . ( intpos (k2 + 1))) and

 A59: ((m + ym) + 1) = (It . ( intpos (k2 + 2))) and

 A60: (m + yn) = (It . ( intpos (k2 + 3))) and

 A61: for j be Nat st y1 <= j & j < ym holds (( IExec (I,Q,t)) . ( intpos (m + j))) <= (( IExec (I,Q,t)) . ( intpos (m + ym))) and

 A62: for j be Nat st ym < j & j <= yn holds (( IExec (I,Q,t)) . ( intpos (m + j))) >= (( IExec (I,Q,t)) . ( intpos (m + ym))) by A48, XREAL_1: 47;



 A63: I is_closed_on (t,Q) by A12, A39, A40, A41, A42, A43, A44;

 then

 A64: ( DataPart t1) = ( DataPart ( IExec (I,Q,t))) by A39, A40, A49, Th2, A38;

 then

 A65: (t1 . b) = ((2 * k1) + 3) by A48, A54, SCMPDS_4: 8, XREAL_1: 47;



 A66: ym >= 1 by A42, A48, A55, XREAL_1: 47, XXREAL_0: 2;

 then

 reconsider yc = (ym - 1) as Element of NAT by INT_1: 3, XREAL_1: 48;



 A67: yc <= yn by A56, XREAL_1: 146, XXREAL_0: 2;

 then

 A68: yc <= n by A42, A48, XREAL_1: 47, XXREAL_0: 2;



 A69: (t1 . b) = ((2 * (k1 + 1)) + 1) by A48, A54, A64, SCMPDS_4: 8, XREAL_1: 47;

 (( IExec (I,Q,t)) . a) = (t . a) by A12, A39, A40, A41, A42, A43, A44;

 then

 A70: (t1 . a) = c by A39, A64, SCMPDS_4: 8;

 set k3 = (((m + n) + (2 * (k1 + 1))) + 1), yd = (ym + 1);



 A71: yd > ym by XREAL_1: 29;

 then

 A72: yd >= y1 by A55, XXREAL_0: 2;

 then

 A73: yd >= 1 by A42, A48, XREAL_1: 47, XXREAL_0: 2;



 A74: (t1 . ( intpos (k3 + 1))) = (m + yn) by A41, A64, A60, SCMPDS_4: 8;

 (ym + (1 + k)) >= (y1 + (1 + k)) by A55, XREAL_1: 6;

 then yn <= (yd + k) by A51, XXREAL_0: 2;

 then

 A75: (yn - yd) <= k by XREAL_1: 20;



 A76: (t1 . ( intpos k3)) = (m + yd) by A41, A64, A59, SCMPDS_4: 8;



 A77: (( Initialize t1) . a) = (t1 . a) by SCMPDS_5: 15;



 A78: (( Initialize t1) . b) = (t1 . b) by SCMPDS_5: 15;



 A79: (( Initialize t1) . ( intpos k3)) = (t1 . ( intpos k3)) by SCMPDS_5: 15;



 A80: (( Initialize t1) . ( intpos (k3 + 1))) = (t1 . ( intpos (k3 + 1))) by SCMPDS_5: 15;

 f2 is_FinSequence_on (( Initialize t1),m)

 proof

 let i be Nat;

 assume 1 <= i & i <= ( len f2);

 then (f2 . i) = (t1 . ( intpos (m + i))) by A53;

 hence thesis by SCMPDS_5: 15;

 end;

 then

 consider kl be Nat, f3 be FinSequence of INT such that

 A81: ( Initialize ( Comput ((Q1 +* sWH),( Initialize t1),kl))) = ( Comput ((Q1 +* sWH),( Initialize t1),kl)) and

 A82: f3 is_FinSequence_on (( Comput ((Q1 +* sWH),( Initialize t1),kl)),m) and

 A83: ( len f3) = n and

 A84: (f2,f3) are_fiberwise_equipotent and

 A85: f3 is_non_decreasing_on (yd,yn) and

 A86: for j be Nat st yd < yn & (1 <= j & j < yd or yn < j & j <= n) holds (f3 . j) = (( Initialize t1) . ( intpos (m + j))) and

 A87: for j be Nat st yd >= yn & 1 <= j & j <= n holds (f3 . j) = (( Initialize t1) . ( intpos (m + j))) and

 A88: for j be Nat st 1 <= j & j < ((2 * (k1 + 1)) + 1) holds (( Comput ((Q1 +* sWH),( Initialize t1),kl)) . ( intpos ((m + n) + j))) = (( Initialize t1) . ( intpos ((m + n) + j))) and

 A89: (( Comput ((Q1 +* sWH),( Initialize t1),kl)) . b) = ((( Initialize t1) . b) - 2) and

 A90: (( Comput ((Q1 +* sWH),( Initialize t1),kl)) . a) = c by A37, A73, A75, A52, A42, A48, A70, A77, A69, A78, A76, A79, A74, A80, XREAL_1: 47;



 A91: ( Initialize ( Comput ((Q1 +* sWH),( Initialize t1),kl))) = ( Comput ((Q1 +* sWH),( Initialize t1),kl)) by A81;



 A92: f3 is_FinSequence_on (( Comput ((Q1 +* sWH),( Initialize t1),kl)),m) by A82;



 A93: ( len f3) = n by A83;



 A94: (f2,f3) are_fiberwise_equipotent by A84;



 A95: f3 is_non_decreasing_on (yd,yn) by A85;



 A96: for j be Nat st yd < yn & (1 <= j & j < yd or yn < j & j <= n) holds (f3 . j) = (t1 . ( intpos (m + j)))

 proof

 let j be Nat;

 assume yd < yn & (1 <= j & j < yd or yn < j & j <= n);

 then (f3 . j) = (( Initialize t1) . ( intpos (m + j))) by A86;

 hence thesis by SCMPDS_5: 15;

 end;



 A97: for j be Nat st yd >= yn & 1 <= j & j <= n holds (f3 . j) = (t1 . ( intpos (m + j)))

 proof

 let j be Nat;

 assume yd >= yn & 1 <= j & j <= n;

 then (f3 . j) = (( Initialize t1) . ( intpos (m + j))) by A87;

 hence thesis by SCMPDS_5: 15;

 end;



 A98: for j be Nat st 1 <= j & j < ((2 * (k1 + 1)) + 1) holds (( Comput ((Q1 +* sWH),( Initialize t1),kl)) . ( intpos ((m + n) + j))) = (t1 . ( intpos ((m + n) + j)))

 proof

 let j be Nat;

 assume 1 <= j & j < ((2 * (k1 + 1)) + 1);

 then (( Comput ((Q1 +* sWH),( Initialize t1),kl)) . ( intpos ((m + n) + j))) = (( Initialize t1) . ( intpos ((m + n) + j))) by A88;

 hence thesis by SCMPDS_5: 15;

 end;

 set t2 = ( Comput ((Q1 +* sWH),( Initialize t1),kl)), Q2 = (Q1 +* sWH);



 A99: (t2 . b) = ((t1 . b) - 2) by A89, A78;



 A100: (t2 . a) = c by A90;



 A101: ((2 * k1) + 3) = ((2 * (k1 + 1)) + 1);

 then jj < ((2 * (k1 + 1)) + 1) by XREAL_1: 6;



 then

 A102: (t2 . ( intpos k2)) = (t1 . ( intpos ((m + n) + jj))) by A41, A98, NAT_1: 11

 .= (m + y1) by A41, A64, A57, SCMPDS_4: 8;



 A103: ym <= n by A42, A48, A56, XREAL_1: 47, XXREAL_0: 2;

 A104:

 now

 per cases ;

 suppose yd < yn;

 hence (f3 . ym) = (t1 . ( intpos (m + ym))) by A71, A96, A66;

 end;

 suppose yd >= yn;

 hence (f3 . ym) = (t1 . ( intpos (m + ym))) by A97, A66, A103;

 end;

 end;

 A105:

 now

 let i be Nat;

 assume that

 A106: 1 <= i and

 A107: i <= ym;



 A108: i <= n by A103, A107, XXREAL_0: 2;



 A109: i < yd by A71, A107, XXREAL_0: 2;

 now

 per cases ;

 suppose yd < yn;

 hence (f3 . i) = (t1 . ( intpos (m + i))) by A96, A106, A109;

 end;

 suppose yd >= yn;

 hence (f3 . i) = (t1 . ( intpos (m + i))) by A97, A106, A108;

 end;

 end;

 hence (f3 . i) = (f2 . i) by A52, A53, A106, A108;

 end;

 yc <= (yn - 1) by A56, XREAL_1: 9;

 then (yc - y1) <= ((yn - 1) - y1) by XREAL_1: 9;

 then

 A110: (yc - y1) <= k by A50, XXREAL_0: 2;



 A111: yc < yd by A71, XREAL_1: 146, XXREAL_0: 2;

 set jj = ((2 * k1) + 2);

 jj >= 2 by NAT_1: 11;

 then

 A112: jj >= 1 by XXREAL_0: 2;

 jj < ((2 * (k1 + 1)) + 1) by A101, XREAL_1: 6;



 then

 A113: (t2 . ( intpos (k2 + 1))) = (t1 . ( intpos ((m + n) + jj))) by A41, A98, A112

 .= (m + yc) by A41, A64, A58, SCMPDS_4: 8;



 A114: 1 <= y1 & yc <= n or y1 >= yc by A48, A68, A42, XREAL_1: 47;



 A115: (( Initialize t2) . a) = (t2 . a) by SCMPDS_5: 15;



 A116: (( Initialize t2) . b) = (t2 . b) by SCMPDS_5: 15;



 A117: (( Initialize t2) . ( intpos k2)) = (t2 . ( intpos k2)) by SCMPDS_5: 15;



 A118: (( Initialize t2) . ( intpos (k2 + 1))) = (t2 . ( intpos (k2 + 1))) by SCMPDS_5: 15;

 f3 is_FinSequence_on (( Initialize t2),m)

 proof

 let i be Nat;

 assume 1 <= i & i <= ( len f3);

 then (f3 . i) = (t2 . ( intpos (m + i))) by A92;

 hence thesis by SCMPDS_5: 15;

 end;

 then

 consider km be Nat, f4 be FinSequence of INT such that

 A119: ( Initialize ( Comput ((Q2 +* sWH),( Initialize t2),km))) = ( Comput ((Q2 +* sWH),( Initialize t2),km)) and

 A120: f4 is_FinSequence_on (( Comput ((Q2 +* sWH),( Initialize t2),km)),m) and

 A121: ( len f4) = n and

 A122: (f3,f4) are_fiberwise_equipotent and

 A123: f4 is_non_decreasing_on (y1,yc) and

 A124: for j be Nat st y1 < yc & (1 <= j & j < y1 or yc < j & j <= n) holds (f4 . j) = (( Initialize t2) . ( intpos (m + j))) and

 A125: for j be Nat st y1 >= yc & 1 <= j & j <= n holds (f4 . j) = (( Initialize t2) . ( intpos (m + j))) and

 A126: for j be Nat st 1 <= j & j < ((2 * k1) + 1) holds (( Comput ((Q2 +* sWH),( Initialize t2),km)) . ( intpos ((m + n) + j))) = (( Initialize t2) . ( intpos ((m + n) + j))) and

 A127: (( Comput ((Q2 +* sWH),( Initialize t2),km)) . b) = ((( Initialize t2) . b) - 2) and

 A128: (( Comput ((Q2 +* sWH),( Initialize t2),km)) . a) = c by A37, A110, A93, A114, A41, A115, A100, A116, A65, A99, A117, A102, A118, A113;



 A129: ( Initialize ( Comput ((Q2 +* sWH),( Initialize t2),km))) = ( Comput ((Q2 +* sWH),( Initialize t2),km)) by A119;



 A130: f4 is_FinSequence_on (( Comput ((Q2 +* sWH),( Initialize t2),km)),m) by A120;



 A131: ( len f4) = n by A121;



 A132: (f3,f4) are_fiberwise_equipotent by A122;



 A133: f4 is_non_decreasing_on (y1,yc) by A123;



 A134: for j be Nat st y1 < yc & (1 <= j & j < y1 or yc < j & j <= n) holds (f4 . j) = (t2 . ( intpos (m + j)))

 proof

 let j be Nat;

 assume y1 < yc & (1 <= j & j < y1 or yc < j & j <= n);

 then (f4 . j) = (( Initialize t2) . ( intpos (m + j))) by A124;

 hence thesis by SCMPDS_5: 15;

 end;



 A135: for j be Nat st y1 >= yc & 1 <= j & j <= n holds (f4 . j) = (t2 . ( intpos (m + j)))

 proof

 let j be Nat;

 assume y1 >= yc & 1 <= j & j <= n;

 then (f4 . j) = (( Initialize t2) . ( intpos (m + j))) by A125;

 hence thesis by SCMPDS_5: 15;

 end;



 A136: for j be Nat st 1 <= j & j < ((2 * k1) + 1) holds (( Comput ((Q2 +* sWH),( Initialize t2),km)) . ( intpos ((m + n) + j))) = (t2 . ( intpos ((m + n) + j)))

 proof

 let j be Nat;

 assume 1 <= j & j < ((2 * k1) + 1);

 then (( Comput ((Q2 +* sWH),( Initialize t2),km)) . ( intpos ((m + n) + j))) = (( Initialize t2) . ( intpos ((m + n) + j))) by A126;

 hence thesis by SCMPDS_5: 15;

 end;



 A137: (( Comput ((Q2 +* sWH),( Initialize t2),km)) . b) = ((t2 . b) - 2) by A127, A116;



 A138: (( Comput ((Q2 +* sWH),( Initialize t2),km)) . a) = c by A128;

 A139:

 now

 let i be Nat;

 assume that

 A140: yc < i and

 A141: i <= ( len f4);



 A142: (1 + 0 ) <= i by A140, INT_1: 7;

 now

 per cases ;

 suppose y1 < yc;

 hence (f4 . i) = (t2 . ( intpos (m + i))) by A131, A134, A140, A141;

 end;

 suppose y1 >= yc;

 hence (f4 . i) = (t2 . ( intpos (m + i))) by A131, A135, A141, A142;

 end;

 end;

 hence (f4 . i) = (f3 . i) by A92, A93, A131, A141, A142;

 end;

 then (f4 . ym) = (f3 . ym) by A131, A103, XREAL_1: 146;

 then

 A143: (f4 . ym) = (( IExec (I,Q,t)) . ( intpos (m + ym))) by A64, A104, SCMPDS_4: 8;

 A144:

 now

 let i be Nat;

 assume that

 A145: yn < i and

 A146: i <= ( len f3);



 A147: (1 + 0 ) <= i by A145, INT_1: 7;

 now

 per cases ;

 suppose yd < yn;

 hence (f3 . i) = (t1 . ( intpos (m + i))) by A93, A96, A145, A146;

 end;

 suppose yd >= yn;

 hence (f3 . i) = (t1 . ( intpos (m + i))) by A93, A97, A146, A147;

 end;

 end;

 hence (f3 . i) = (f2 . i) by A52, A53, A93, A146, A147;

 end;

 A148:

 now

 let i be Nat;

 assume that

 A149: ym < i and

 A150: i <= yn;

 consider j be Nat such that

 A151: ym < j and

 A152: j <= yn and

 A153: (f3 . i) = (f2 . j) by A42, A48, A56, A93, A94, A105, A144, A149, A150, RFINSEQ: 32, XREAL_1: 47;



 A154: yc < i by A149, XREAL_1: 146, XXREAL_0: 2;



 A155: 1 <= j by A66, A151, XXREAL_0: 2;



 A156: j <= ( len f2) by A42, A48, A52, A152, XREAL_1: 47, XXREAL_0: 2;

 i <= ( len f4) by A42, A48, A131, A150, XREAL_1: 47, XXREAL_0: 2;



 then (f4 . i) = (f2 . j) by A139, A154, A153

 .= (t1 . ( intpos (m + j))) by A53, A155, A156

 .= (( IExec (I,Q,t)) . ( intpos (m + j))) by A64, SCMPDS_4: 8;

 hence (f4 . ym) <= (f4 . i) by A62, A143, A151, A152;

 end;



 A157: yn > y1 by A48, XREAL_1: 47;

 A158:

 now

 let i be Nat;

 assume that

 A159: 1 <= i and

 A160: i <= y0;

 (i - 1) < (y1 - 1) by A160, XREAL_1: 146, XXREAL_0: 2;

 then

 A161: i < y1 by XREAL_1: 9;

 y1 <= n by A42, A157, XXREAL_0: 2;

 then

 A162: i <= n by A161, XXREAL_0: 2;

 now

 per cases ;

 suppose y1 < yc;

 hence (f4 . i) = (t2 . ( intpos (m + i))) by A134, A159, A161;

 end;

 suppose y1 >= yc;

 hence (f4 . i) = (t2 . ( intpos (m + i))) by A135, A159, A162;

 end;

 end;

 hence (f4 . i) = (f3 . i) by A92, A93, A159, A162;

 end;



 A163: y0 <= yc by A55, XREAL_1: 9;

 A164:

 now

 let i be Nat;

 assume that

 A165: y1 <= i and

 A166: i < ym;

 (i + 1) <= ym by A166, INT_1: 7;

 then

 A167: i <= yc by XREAL_1: 19;

 y0 < i by A165, XREAL_1: 146, XXREAL_0: 2;

 then

 consider j be Nat such that

 A168: y0 < j and

 A169: j <= yc and

 A170: (f4 . i) = (f3 . j) by A68, A131, A132, A163, A158, A139, A167, RFINSEQ: 32;



 A171: (1 + 0 ) <= j by A168, INT_1: 7;



 A172: j <= n by A68, A169, XXREAL_0: 2;



 A173: j < yd by A111, A169, XXREAL_0: 2;

 now

 per cases ;

 suppose yd < yn;

 hence (f3 . j) = (t1 . ( intpos (m + j))) by A96, A171, A173;

 end;

 suppose yd >= yn;

 hence (f3 . j) = (t1 . ( intpos (m + j))) by A97, A171, A172;

 end;

 end;

 then

 A174: (f4 . i) = (( IExec (I,Q,t)) . ( intpos (m + j))) by A64, A170, SCMPDS_4: 8;



 A175: j < ym by A169, XREAL_1: 146, XXREAL_0: 2;

 ((y1 - 1) + 1) <= j by A168, INT_1: 7;

 hence (f4 . i) <= (f4 . ym) by A61, A143, A175, A174;

 end;

 take mm = (m1 + (kl + km));

 set tm = ( Comput ((Q +* sWH),t,mm));

 take f4;

 ( Initialize t1) = t1 by A39, A40, A63, A49, Th2, A38;



 then

 A176: tm = ( Comput ((Q +* sWH),( Initialize t1),(kl + km))) by EXTPRO_1: 4

 .= ( Comput ((Q2 +* sWH),( Initialize t2),km)) by A91, EXTPRO_1: 4;

 hence ( Initialize tm) = tm by A129;

 thus f4 is_FinSequence_on (tm,m) by A130, A176;

 thus ( len f4) = n by A131;

 now

 let i be Nat;

 assume that

 A177: 1 <= i and

 A178: i <= ( len f2);



 thus (f2 . i) = (t1 . ( intpos (m + i))) by A53, A177, A178

 .= (( IExec (I,Q,t)) . ( intpos (m + i))) by A64, SCMPDS_4: 8;

 end;

 then f2 is_FinSequence_on (( IExec (I,Q,t)),m);

 then (f1,f2) are_fiberwise_equipotent by A12, A39, A40, A41, A42, A43, A44, A46, A47, A52;

 then (f1,f3) are_fiberwise_equipotent by A94, CLASSES1: 76;

 hence (f1,f4) are_fiberwise_equipotent by A132, CLASSES1: 76;

 A179:

 now

 let j be Nat;

 assume that

 A180: yd <= j and

 A181: j <= yn;



 A182: 1 <= j by A73, A180, XXREAL_0: 2;



 A183: j <= n by A42, A48, A181, XREAL_1: 47, XXREAL_0: 2;



 A184: yc < j by A111, A180, XXREAL_0: 2;

 now

 per cases ;

 suppose y1 < yc;

 hence (f4 . j) = (t2 . ( intpos (m + j))) by A134, A183, A184;

 end;

 suppose y1 >= yc;

 hence (f4 . j) = (t2 . ( intpos (m + j))) by A135, A183, A182;

 end;

 end;

 hence (f4 . j) = (f3 . j) by A92, A93, A183, A182;

 end;

 now

 let i,j be Nat;

 assume that

 A185: yd <= i and

 A186: i <= j and

 A187: j <= yn;

 yd <= j by A185, A186, XXREAL_0: 2;

 then

 A188: (f4 . j) = (f3 . j) by A179, A187;

 i <= yn by A186, A187, XXREAL_0: 2;

 then (f4 . i) = (f3 . i) by A179, A185;

 hence (f4 . i) <= (f4 . j) by A95, A185, A186, A187, A188, FINSEQ_6:def 9;

 end;

 then f4 is_non_decreasing_on (yd,yn) by FINSEQ_6:def 9;

 hence f4 is_non_decreasing_on (y1,yn) by A133, A164, A148, Th8;

 thus for j be Nat st y1 < yn & (1 <= j & j < y1 or yn < j & j <= n) holds (f4 . j) = (t . ( intpos (m + j)))

 proof

 let j be Nat;

 assume that

 A189: y1 < yn and

 A190: 1 <= j & j < y1 or yn < j & j <= n;



 A191: 1 <= j & j <= n

 proof

 per cases by A190;

 suppose

 A192: 1 <= j & j < y1;

 then j < yn by A189, XXREAL_0: 2;

 hence thesis by A42, A48, A192, XREAL_1: 47, XXREAL_0: 2;

 end;

 suppose

 A193: yn < j & j <= n;

 then y1 < j by A189, XXREAL_0: 2;

 hence thesis by A42, A48, A193, XREAL_1: 47, XXREAL_0: 2;

 end;

 end;



 A194: 1 <= j & j < yd or yn < j & j <= n

 proof

 per cases by A190;

 suppose 1 <= j & j < y1;

 hence thesis by A72, XXREAL_0: 2;

 end;

 suppose yn < j & j <= n;

 hence thesis;

 end;

 end;

 A195:

 now

 per cases ;

 suppose yd < yn;

 hence (f3 . j) = (t1 . ( intpos (m + j))) by A96, A194;

 end;

 suppose yd >= yn;

 hence (f3 . j) = (t1 . ( intpos (m + j))) by A97, A191;

 end;

 end;



 A196: 1 <= j & j < y1 or yc < j & j <= n

 proof

 per cases by A190;

 suppose 1 <= j & j < y1;

 hence thesis;

 end;

 suppose yn < j & j <= n;

 hence thesis by A67, XXREAL_0: 2;

 end;

 end;

 now

 per cases ;

 suppose y1 < yc;

 hence (f4 . j) = (t2 . ( intpos (m + j))) by A134, A196;

 end;

 suppose y1 >= yc;

 hence (f4 . j) = (t2 . ( intpos (m + j))) by A135, A191;

 end;

 end;



 hence (f4 . j) = (f3 . j) by A92, A93, A191

 .= (( IExec (I,Q,t)) . ( intpos (m + j))) by A64, A195, SCMPDS_4: 8

 .= (t . ( intpos (m + j))) by A12, A39, A40, A41, A42, A43, A44, A189, A190;

 end;

 thus for j be Nat st y1 >= yn & 1 <= j & j <= n holds (f4 . j) = (t . ( intpos (m + j))) by A48, XREAL_1: 47;

 hereby

 let j be Nat;

 assume that

 A197: 1 <= j and

 A198: j < ((2 * k1) + 1);

 ((2 * k1) + 1) < ((2 * (k1 + 1)) + 1) by A101, XREAL_1: 6;

 then

 A199: j < ((2 * (k1 + 1)) + 1) by A198, XXREAL_0: 2;



 thus (tm . ( intpos ((m + n) + j))) = (t2 . ( intpos ((m + n) + j))) by A136, A176, A197, A198

 .= (t1 . ( intpos ((m + n) + j))) by A98, A197, A199

 .= (( IExec (I,Q,t)) . ( intpos ((m + n) + j))) by A64, SCMPDS_4: 8

 .= (t . ( intpos ((m + n) + j))) by A12, A39, A40, A41, A42, A43, A44, A197, A198;

 end;

 thus (tm . b) = ((t . b) - 2) by A40, A65, A99, A137, A176;

 thus (tm . a) = c by A138, A176;

 end;

 end;

 hence P[(k + 1)];

 end;



 A200: for k be Nat holds P[k] from NAT_1:sch 2( A13, A36);

 ex k be Nat, f2 be FinSequence of INT st ( Initialize ( Comput ((P +* sWH),s,k))) = ( Comput ((P +* sWH),s,k)) & f2 is_FinSequence_on (( Comput ((P +* sWH),s,k)),m) & ( len f2) = n & (f,f2) are_fiberwise_equipotent & f2 is_non_decreasing_on (1,n) & (for j be Nat st 1 < n & (1 <= j & j < 1 or n < j & j <= n) holds (f2 . j) = (s . ( intpos (m + j)))) & (for j be Nat st 1 >= n & 1 <= j & j <= n holds (f2 . j) = (s . ( intpos (m + j)))) & (for j be Nat st (1 <= j & j < ((2 * 0 ) + 1)) holds (( Comput ((P +* sWH),s,k)) . ( intpos ((m + n) + j))) = (s . ( intpos ((m + n) + j)))) & (( Comput ((P +* sWH),s,k)) . b) = ((s . b) - 2) & (( Comput ((P +* sWH),s,k)) . a) = c

 proof

 per cases ;

 suppose (n - 1) <= 0 ;

 hence thesis by A2, A3, A5, A7, A9, A10, A13, A11;

 end;

 suppose (n - 1) > 0 ;

 then

 reconsider nn = (n - 1) as Element of NAT by INT_1: 3;

 P[nn] by A200;

 hence thesis by A2, A3, A5, A7, A9, A10, A11;

 end;

 end;

 then

 consider k be Nat, f2 be FinSequence of INT such that

 A201: ( Initialize ( Comput ((P +* sWH),s,k))) = ( Comput ((P +* sWH),s,k)) and

 A202: f2 is_FinSequence_on (( Comput ((P +* sWH),s,k)),m) and

 A203: ( len f2) = n and

 A204: (f,f2) are_fiberwise_equipotent and

 A205: f2 is_non_decreasing_on (1,n) and

 A206: (( Comput ((P +* sWH),s,k)) . b) = ((s . b) - 2) and

 A207: (( Comput ((P +* sWH),s,k)) . a) = c;

 set sk = ( Comput ((P +* sWH),s,k)), s1 = ( Initialize sk), P1 = ((P +* sWH) +* sWH), s2 = ( Comput (P1,s1,1)), P2 = P1;



 A208: ( IC s1) = 0 by MEMSTR_0:def 11;

 set i1 = ((a,i) <=0_goto (( card I) + 2)), i2 = ( goto ( - (( card I) + 1)));



 A209: ( card WH) = (( card I) + 2) by SCMPDS_8: 17;

 then

 A210: (( card I) + 2) in ( dom sWH) by COMPOS_1: 64;



 A211: ( dom g) = ( Seg n) by A4, FINSEQ_1:def 3;

 sWH c= P2 by FUNCT_4: 25;



 then

 A212: (P2 . (( card I) + 2)) = (sWH . (( card I) + 2)) by A210, GRFUNC_1: 2

 .= ( halt SCMPDS ) by A209, COMPOS_1: 64;



 A213: WH = (i1 ';' (I ';' i2)) by Lm1;



 A214: ( Comput (P1,s1,( 0 + 1))) = ( Following (P1,( Comput (P1,s1, 0 )))) by EXTPRO_1: 3

 .= ( Exec (i1,s1)) by A213, SCMPDS_6: 11;

 ( IC s2) = ( ICplusConst (s1,(( card I) + 2))) by A7, A201, A206, A207, A214, SCMPDS_2: 56

 .= ( 0 + (( card I) + 2)) by A208, SCMPDS_6: 12;

 then

 A215: ( CurInstr (P2,s2)) = ( halt SCMPDS ) by A212, PBOOLE: 143;



 A216: s2 = ( Comput ((P +* sWH),s,(k + 1))) by A201, EXTPRO_1: 4;



 A217: (P +* sWH) halts_on s by A215, A216, EXTPRO_1: 29;

 hence WH is_halting_on (s,P) by A1, SCMPDS_6:def 3;



 A218: ( Result ((P +* sWH),s)) = s2 by A215, A216, A217, EXTPRO_1:def 9;

 now

 let i be Nat;

 reconsider a = i as Nat;

 set xi = ( intpos (m + a));

 assume

 A219: i in ( dom g);

 then

 A220: 1 <= i by A211, FINSEQ_1: 1;



 A221: i <= n by A211, A219, FINSEQ_1: 1;

 ( IExec (WH,P,s)) = s2 by A218, SCMPDS_4:def 5;



 hence (g . i) = (s2 . xi) by A4, A6, A220, A221

 .= (s1 . xi) by A214, SCMPDS_2: 56

 .= (f2 . i) by A201, A202, A203, A220, A221;

 end;

 hence thesis by A4, A203, A204, A205, FINSEQ_2: 9;

 end;



 Lm4: for s be 0 -started State of SCMPDS , I be halt-free shiftable Program of SCMPDS , a be Int_position, i,c be Integer, m,n,m1 be Nat st (s . a) = c & 1 = (s . ( DataLoc (c,i))) & m1 = ((m + n) + 1) & (m + 1) = (s . ( intpos m1)) & (m + n) = (s . ( intpos (m1 + 1))) & (for t be 0 -started State of SCMPDS , Q holds for f1,f2 be FinSequence of INT , k1,k2,y1,yn be Nat st (t . a) = c & ((2 * k1) + 1) = (t . ( DataLoc (c,i))) & k2 = (((m + n) + (2 * k1)) + 1) & (m + y1) = (t . ( intpos k2)) & (m + yn) = (t . ( intpos (k2 + 1))) & (1 <= y1 & yn <= n or y1 >= yn) holds I is_closed_on (t,Q) & I is_halting_on (t,Q) & (( IExec (I,Q,t)) . a) = (t . a) & (for j be Nat st (1 <= j & j < ((2 * k1) + 1)) holds (( IExec (I,Q,t)) . ( intpos ((m + n) + j))) = (t . ( intpos ((m + n) + j)))) & (y1 >= yn implies (( IExec (I,Q,t)) . ( DataLoc (c,i))) = ((2 * k1) - 1) & (for j be Nat st (1 <= j & j <= n) holds (( IExec (I,Q,t)) . ( intpos (m + j))) = (t . ( intpos (m + j))))) & (y1 < yn implies (( IExec (I,Q,t)) . ( DataLoc (c,i))) = ((2 * k1) + 3) & (for j be Nat st 1 <= j & j < y1 or yn < j & j <= n holds (( IExec (I,Q,t)) . ( intpos (m + j))) = (t . ( intpos (m + j)))) & (ex ym be Nat st y1 <= ym & ym <= yn & (m + y1) = (( IExec (I,Q,t)) . ( intpos k2)) & ((m + ym) - 1) = (( IExec (I,Q,t)) . ( intpos (k2 + 1))) & ((m + ym) + 1) = (( IExec (I,Q,t)) . ( intpos (k2 + 2))) & (m + yn) = (( IExec (I,Q,t)) . ( intpos (k2 + 3))) & (for j be Nat st y1 <= j & j < ym holds (( IExec (I,Q,t)) . ( intpos (m + j))) <= (( IExec (I,Q,t)) . ( intpos (m + ym)))) & (for j be Nat st ym < j & j <= yn holds (( IExec (I,Q,t)) . ( intpos (m + j))) >= (( IExec (I,Q,t)) . ( intpos (m + ym)))))) & (f1 is_FinSequence_on (t,m) & f2 is_FinSequence_on (( IExec (I,Q,t)),m) & ( len f1) = n & ( len f2) = n implies (f1,f2) are_fiberwise_equipotent )) holds ( while>0 (a,i,I)) is_halting_on (s,P)

 proof

 let s be 0 -started State of SCMPDS , I be halt-free shiftable Program of SCMPDS , a be Int_position, i,c be Integer, m,n,m1 be Nat;

 set b = ( DataLoc (c,i));

 assume

 A1: (s . a) = c;

 consider f be FinSequence of INT such that

 A2: ( len f) = n and

 A3: for i be Nat st 1 <= i & i <= ( len f) holds (f . i) = (s . ( intpos (m + i))) by SCPISORT: 1;



 A4: f is_FinSequence_on (s,m) by A3;

 set ss = ( IExec (( while>0 (a,i,I)),P,s));

 assume

 A5: 1 = (s . b);

 consider g be FinSequence of INT such that

 A6: ( len g) = n and

 A7: for i be Nat st 1 <= i & i <= ( len g) holds (g . i) = (ss . ( intpos (m + i))) by SCPISORT: 1;



 A8: g is_FinSequence_on (ss,m) by A7;

 assume that

 A9: m1 = ((m + n) + 1) and

 A10: (m + 1) = (s . ( intpos m1)) and

 A11: (m + n) = (s . ( intpos (m1 + 1)));

 assume for t be 0 -started State of SCMPDS , Q holds for f1,f2 be FinSequence of INT , k1,k2,y1,yn be Nat st (t . a) = c & ((2 * k1) + 1) = (t . b) & k2 = (((m + n) + (2 * k1)) + 1) & (m + y1) = (t . ( intpos k2)) & (m + yn) = (t . ( intpos (k2 + 1))) & (1 <= y1 & yn <= n or y1 >= yn) holds I is_closed_on (t,Q) & I is_halting_on (t,Q) & (( IExec (I,Q,t)) . a) = (t . a) & (for j be Nat st (1 <= j & j < ((2 * k1) + 1)) holds (( IExec (I,Q,t)) . ( intpos ((m + n) + j))) = (t . ( intpos ((m + n) + j)))) & (y1 >= yn implies (( IExec (I,Q,t)) . b) = ((2 * k1) - 1) & for j be Nat st 1 <= j & j <= n holds (( IExec (I,Q,t)) . ( intpos (m + j))) = (t . ( intpos (m + j)))) & (y1 < yn implies (( IExec (I,Q,t)) . b) = ((2 * k1) + 3) & (for j be Nat st 1 <= j & j < y1 or yn < j & j <= n holds (( IExec (I,Q,t)) . ( intpos (m + j))) = (t . ( intpos (m + j)))) & ex ym be Nat st y1 <= ym & ym <= yn & (m + y1) = (( IExec (I,Q,t)) . ( intpos k2)) & ((m + ym) - 1) = (( IExec (I,Q,t)) . ( intpos (k2 + 1))) & ((m + ym) + 1) = (( IExec (I,Q,t)) . ( intpos (k2 + 2))) & (m + yn) = (( IExec (I,Q,t)) . ( intpos (k2 + 3))) & (for j be Nat st y1 <= j & j < ym holds (( IExec (I,Q,t)) . ( intpos (m + j))) <= (( IExec (I,Q,t)) . ( intpos (m + ym)))) & for j be Nat st ym < j & j <= yn holds (( IExec (I,Q,t)) . ( intpos (m + j))) >= (( IExec (I,Q,t)) . ( intpos (m + ym)))) & (f1 is_FinSequence_on (t,m) & f2 is_FinSequence_on (( IExec (I,Q,t)),m) & ( len f1) = n & ( len f2) = n implies (f1,f2) are_fiberwise_equipotent );

 hence thesis by A1, A5, A9, A10, A11, A2, A4, A6, A8, Lm3;

 end;



 Lm5: for s be 0 -started State of SCMPDS , I be halt-free shiftable Program of SCMPDS , a be Int_position, i,c be Integer, m,n,m1 be Nat st (s . a) = c & 1 = (s . ( DataLoc (c,i))) & m1 = ((m + n) + 1) & (m + 1) = (s . ( intpos m1)) & (m + n) = (s . ( intpos (m1 + 1))) & (for t be 0 -started State of SCMPDS , Q holds for f1,f2 be FinSequence of INT , k1,k2,y1,yn be Nat st (t . a) = c & ((2 * k1) + 1) = (t . ( DataLoc (c,i))) & k2 = (((m + n) + (2 * k1)) + 1) & (m + y1) = (t . ( intpos k2)) & (m + yn) = (t . ( intpos (k2 + 1))) & (1 <= y1 & yn <= n or y1 >= yn) holds I is_closed_on (t,Q) & I is_halting_on (t,Q) & (( IExec (I,Q,t)) . a) = (t . a) & (for j be Nat st (1 <= j & j < ((2 * k1) + 1)) holds (( IExec (I,Q,t)) . ( intpos ((m + n) + j))) = (t . ( intpos ((m + n) + j)))) & (y1 >= yn implies (( IExec (I,Q,t)) . ( DataLoc (c,i))) = ((2 * k1) - 1) & (for j be Nat st (1 <= j & j <= n) holds (( IExec (I,Q,t)) . ( intpos (m + j))) = (t . ( intpos (m + j))))) & (y1 < yn implies (( IExec (I,Q,t)) . ( DataLoc (c,i))) = ((2 * k1) + 3) & (for j be Nat st 1 <= j & j < y1 or yn < j & j <= n holds (( IExec (I,Q,t)) . ( intpos (m + j))) = (t . ( intpos (m + j)))) & (ex ym be Nat st y1 <= ym & ym <= yn & (m + y1) = (( IExec (I,Q,t)) . ( intpos k2)) & ((m + ym) - 1) = (( IExec (I,Q,t)) . ( intpos (k2 + 1))) & ((m + ym) + 1) = (( IExec (I,Q,t)) . ( intpos (k2 + 2))) & (m + yn) = (( IExec (I,Q,t)) . ( intpos (k2 + 3))) & (for j be Nat st y1 <= j & j < ym holds (( IExec (I,Q,t)) . ( intpos (m + j))) <= (( IExec (I,Q,t)) . ( intpos (m + ym)))) & (for j be Nat st ym < j & j <= yn holds (( IExec (I,Q,t)) . ( intpos (m + j))) >= (( IExec (I,Q,t)) . ( intpos (m + ym)))))) & (f1 is_FinSequence_on (t,m) & f2 is_FinSequence_on (( IExec (I,Q,t)),m) & ( len f1) = n & ( len f2) = n implies (f1,f2) are_fiberwise_equipotent )) holds ( while>0 (a,i,I)) is_halting_on (s,P) & ( while>0 (a,i,I)) is_closed_on (s,P)

 proof

 let s be 0 -started State of SCMPDS , I be halt-free shiftable Program of SCMPDS , a be Int_position, i,c be Integer, m,n,m1 be Nat;

 set b = ( DataLoc (c,i));

 assume

 A1: (s . a) = c;

 assume

 A2: 1 = (s . b);

 assume that

 A3: m1 = ((m + n) + 1) and

 A4: (m + 1) = (s . ( intpos m1)) and

 A5: (m + n) = (s . ( intpos (m1 + 1)));

 assume

 A6: for t be 0 -started State of SCMPDS , Q holds for f1,f2 be FinSequence of INT , k1,k2,y1,yn be Nat st (t . a) = c & ((2 * k1) + 1) = (t . b) & k2 = (((m + n) + (2 * k1)) + 1) & (m + y1) = (t . ( intpos k2)) & (m + yn) = (t . ( intpos (k2 + 1))) & (1 <= y1 & yn <= n or y1 >= yn) holds I is_closed_on (t,Q) & I is_halting_on (t,Q) & (( IExec (I,Q,t)) . a) = (t . a) & (for j be Nat st (1 <= j & j < ((2 * k1) + 1)) holds (( IExec (I,Q,t)) . ( intpos ((m + n) + j))) = (t . ( intpos ((m + n) + j)))) & (y1 >= yn implies (( IExec (I,Q,t)) . b) = ((2 * k1) - 1) & for j be Nat st 1 <= j & j <= n holds (( IExec (I,Q,t)) . ( intpos (m + j))) = (t . ( intpos (m + j)))) & (y1 < yn implies (( IExec (I,Q,t)) . b) = ((2 * k1) + 3) & (for j be Nat st 1 <= j & j < y1 or yn < j & j <= n holds (( IExec (I,Q,t)) . ( intpos (m + j))) = (t . ( intpos (m + j)))) & ex ym be Nat st y1 <= ym & ym <= yn & (m + y1) = (( IExec (I,Q,t)) . ( intpos k2)) & ((m + ym) - 1) = (( IExec (I,Q,t)) . ( intpos (k2 + 1))) & ((m + ym) + 1) = (( IExec (I,Q,t)) . ( intpos (k2 + 2))) & (m + yn) = (( IExec (I,Q,t)) . ( intpos (k2 + 3))) & (for j be Nat st y1 <= j & j < ym holds (( IExec (I,Q,t)) . ( intpos (m + j))) <= (( IExec (I,Q,t)) . ( intpos (m + ym)))) & for j be Nat st ym < j & j <= yn holds (( IExec (I,Q,t)) . ( intpos (m + j))) >= (( IExec (I,Q,t)) . ( intpos (m + ym)))) & (f1 is_FinSequence_on (t,m) & f2 is_FinSequence_on (( IExec (I,Q,t)),m) & ( len f1) = n & ( len f2) = n implies (f1,f2) are_fiberwise_equipotent );

 A7:

 now

 let v be 0 -started State of SCMPDS ;

 let V;

 assume

 A8: ( DataPart v) = ( DataPart s);

 then

 A9: 1 = (v . b) by A2, SCMPDS_4: 8;



 A10: (m + n) = (v . ( intpos (m1 + 1))) by A5, A8, SCMPDS_4: 8;



 A11: (m + 1) = (v . ( intpos m1)) by A4, A8, SCMPDS_4: 8;

 (v . a) = c by A1, A8, SCMPDS_4: 8;

 hence ( while>0 (a,i,I)) is_halting_on (v,V) by A3, A6, A9, A11, A10, Lm4;

 end;

 hence ( while>0 (a,i,I)) is_halting_on (s,P);

 thus thesis by A7, Th3;

 end;

 begin

 definition

 :: SCPQSORT:def1

 func Partition -> Program of SCMPDS equals ((((((((( GBP ,5) := ( GBP ,4)) ';' ( SubFrom ( GBP ,5, GBP ,2))) ';' (( GBP ,3) := ( GBP ,2))) ';' ( AddTo ( GBP ,3,1))) ';' ( while>0 ( GBP ,5,((( while>0 ( GBP ,5,(((((( GBP ,7) := ( GBP ,5)) ';' ( AddTo ( GBP ,5,( - 1)))) ';' (( GBP ,6) := (( intpos 4), 0 ))) ';' ( SubFrom ( GBP ,6,( intpos 2), 0 ))) ';' ( if>0 ( GBP ,6,(( AddTo ( GBP ,4,( - 1))) ';' ( AddTo ( GBP ,7,( - 1)))),( Load (( GBP ,5) := 0 ))))))) ';' ( while>0 ( GBP ,7,(((((( GBP ,5) := ( GBP ,7)) ';' ( AddTo ( GBP ,7,( - 1)))) ';' (( GBP ,6) := (( intpos 2), 0 ))) ';' ( SubFrom ( GBP ,6,( intpos 3), 0 ))) ';' ( if>0 ( GBP ,6,(( AddTo ( GBP ,3,1)) ';' ( AddTo ( GBP ,5,( - 1)))),( Load (( GBP ,7) := 0 )))))))) ';' ( if>0 ( GBP ,5,((((((( GBP ,6) := (( intpos 4), 0 )) ';' ((( intpos 4), 0 ) := (( intpos 3), 0 ))) ';' ((( intpos 3), 0 ) := ( GBP ,6))) ';' ( AddTo ( GBP ,5,( - 2)))) ';' ( AddTo ( GBP ,3,1))) ';' ( AddTo ( GBP ,4,( - 1)))))))))) ';' (( GBP ,6) := (( intpos 4), 0 ))) ';' ((( intpos 4), 0 ) := (( intpos 2), 0 ))) ';' ((( intpos 2), 0 ) := ( GBP ,6)));

 coherence ;

 end

 begin

 definition

 let n,p0 be Nat;

 :: SCPQSORT:def2

 func QuickSort (n,p0) -> Program of SCMPDS equals ((((( GBP := 0 ) ';' ( SBP := 1)) ';' (( SBP ,(p0 + n)) := (p0 + 1))) ';' (( SBP ,((p0 + n) + 1)) := (p0 + n))) ';' ( while>0 ( GBP ,1,(((( GBP ,2) := ( SBP ,((p0 + n) + 1))) ';' ( SubFrom ( GBP ,2, SBP ,(p0 + n)))) ';' ( if>0 ( GBP ,2,((((( GBP ,2) := ( SBP ,(p0 + n))) ';' (( GBP ,4) := ( SBP ,((p0 + n) + 1)))) ';' Partition ) ';' ((((((( SBP ,((p0 + n) + 3)) := ( SBP ,((p0 + n) + 1))) ';' (( SBP ,((p0 + n) + 1)) := ( GBP ,4))) ';' (( SBP ,((p0 + n) + 2)) := ( GBP ,4))) ';' ( AddTo ( SBP ,((p0 + n) + 1),( - 1)))) ';' ( AddTo ( SBP ,((p0 + n) + 2),1))) ';' ( AddTo ( GBP ,1,2)))),( Load ( AddTo ( GBP ,1,( - 2))))))))));

 coherence ;

 end

 set i1 = (( GBP ,7) := ( GBP ,5)), i2 = ( AddTo ( GBP ,5,( - 1))), i3 = (( GBP ,6) := (( intpos 4), 0 )), i4 = ( SubFrom ( GBP ,6,( intpos 2), 0 )), i5 = ( AddTo ( GBP ,4,( - 1))), i6 = ( AddTo ( GBP ,7,( - 1))), i7 = ( Load (( GBP ,5) := 0 )), IF1 = ( if>0 ( GBP ,6,(i5 ';' i6),i7)), WB1 = ((((i1 ';' i2) ';' i3) ';' i4) ';' IF1), WH1 = ( while>0 ( GBP ,5,WB1)), j1 = (( GBP ,5) := ( GBP ,7)), j2 = ( AddTo ( GBP ,7,( - 1))), j3 = (( GBP ,6) := (( intpos 2), 0 )), j4 = ( SubFrom ( GBP ,6,( intpos 3), 0 )), j5 = ( AddTo ( GBP ,3,1)), j6 = ( AddTo ( GBP ,5,( - 1))), j7 = ( Load (( GBP ,7) := 0 )), IF2 = ( if>0 ( GBP ,6,(j5 ';' j6),j7)), WB2 = ((((j1 ';' j2) ';' j3) ';' j4) ';' IF2), WH2 = ( while>0 ( GBP ,7,WB2)), k1 = (( GBP ,5) := ( GBP ,4)), k2 = ( SubFrom ( GBP ,5, GBP ,2)), k3 = (( GBP ,3) := ( GBP ,2)), k4 = ( AddTo ( GBP ,3,1)), K4 = (((k1 ';' k2) ';' k3) ';' k4), k5 = (( GBP ,6) := (( intpos 4), 0 )), k6 = ((( intpos 4), 0 ) := (( intpos 3), 0 )), k7 = ((( intpos 3), 0 ) := ( GBP ,6)), k8 = ( AddTo ( GBP ,5,( - 2))), k9 = ( AddTo ( GBP ,3,1)), k0 = ( AddTo ( GBP ,4,( - 1))), IF3 = ( if>0 ( GBP ,5,(((((k5 ';' k6) ';' k7) ';' k8) ';' k9) ';' k0))), WB3 = ((WH1 ';' WH2) ';' IF3), WH3 = ( while>0 ( GBP ,5,WB3)), j8 = (( GBP ,6) := (( intpos 4), 0 )), j9 = ((( intpos 4), 0 ) := (( intpos 2), 0 )), j0 = ((( intpos 2), 0 ) := ( GBP ,6));

 set a1 = ( intpos 1), a2 = ( intpos 2), a3 = ( intpos 3), a4 = ( intpos 4), a5 = ( intpos 5), a6 = ( intpos 6), a7 = ( intpos 7);



 Lm6: ( card WB1) = 9

 proof



 thus ( card WB1) = (( card (((i1 ';' i2) ';' i3) ';' i4)) + ( card IF1)) by AFINSQ_1: 17

 .= (4 + ( card IF1)) by Th4

 .= (4 + ((( card (i5 ';' i6)) + ( card i7)) + 2)) by SCMPDS_6: 65

 .= (4 + ((2 + ( card i7)) + 2)) by SCMP_GCD: 5

 .= (4 + ((2 + 1) + 2)) by COMPOS_1: 54

 .= 9;

 end;



 Lm7: for s be 0 -started State of SCMPDS , md,me be Nat st (s . a2) = md & (s . a4) = me & md >= 8 & me >= 8 & (s . GBP ) = 0 holds (( IExec (WB1,P,s)) . GBP ) = 0 & (( IExec (WB1,P,s)) . a1) = (s . a1) & (( IExec (WB1,P,s)) . a2) = (s . a2) & (( IExec (WB1,P,s)) . a3) = (s . a3) & (for i be Nat st i >= 8 holds (( IExec (WB1,P,s)) . ( intpos i)) = (s . ( intpos i))) & ((s . ( intpos md)) < (s . ( intpos me)) implies (( IExec (WB1,P,s)) . a5) = ((s . a5) - 1) & (( IExec (WB1,P,s)) . a4) = ((s . a4) - 1) & (( IExec (WB1,P,s)) . a7) = ((s . a5) - 1)) & ((s . ( intpos md)) >= (s . ( intpos me)) implies (( IExec (WB1,P,s)) . a5) = 0 & (( IExec (WB1,P,s)) . a4) = (s . a4) & (( IExec (WB1,P,s)) . a7) = (s . a5))

 proof

 set a = GBP ;

 let s be 0 -started State of SCMPDS , md,me be Nat;

 assume that

 A1: (s . a2) = md and

 A2: (s . a4) = me and

 A3: md >= 8 and

 A4: me >= 8 and

 A5: (s . a) = 0 ;

 set t0 = s, Q0 = P, t1 = ( IExec ((((i1 ';' i2) ';' i3) ';' i4),P,s)), Q1 = P, t2 = ( IExec (((i1 ';' i2) ';' i3),P,s)), t3 = ( IExec ((i1 ';' i2),P,s)), Q2 = P, t4 = ( Exec (i1,t0)), Q4 = Q0;



 A6: (t0 . a) = 0 by A5;



 A7: ( DataLoc ((t0 . a),7)) = ( intpos ( 0 + 7)) by A5, SCMP_GCD: 1;

 then

 A8: (t4 . a) = 0 by A6, AMI_3: 10, SCMPDS_2: 47;

 then

 A9: ( DataLoc ((t4 . a),5)) = ( intpos ( 0 + 5)) by SCMP_GCD: 1;



 A10: (t4 . a4) = me by A7, A2, AMI_3: 10, SCMPDS_2: 47;



 A11: (t3 . a4) = (( Exec (i2,t4)) . a4) by SCMPDS_5: 42

 .= me by A10, A9, AMI_3: 10, SCMPDS_2: 48;

 then

 A12: ( DataLoc ((t3 . a4), 0 )) = ( intpos (me + 0 )) by SCMP_GCD: 1;



 A13: (t3 . a) = (( Exec (i2,t4)) . a) by SCMPDS_5: 42

 .= 0 by A8, A9, AMI_3: 10, SCMPDS_2: 48;

 then

 A14: ( DataLoc ((t3 . a),6)) = ( intpos ( 0 + 6)) by SCMP_GCD: 1;



 A15: (t4 . a2) = md by A7, A1, AMI_3: 10, SCMPDS_2: 47;



 A16: (t3 . a2) = (( Exec (i2,t4)) . a2) by SCMPDS_5: 42

 .= md by A15, A9, AMI_3: 10, SCMPDS_2: 48;



 A17: (t4 . a5) = (s . a5) by A7, AMI_3: 10, SCMPDS_2: 47;

 set t01 = ( Initialize ( Initialize t1)), ii7 = (( GBP ,5) := 0 ), t5 = ( Exec (i5,t01));



 A18: (t2 . a2) = (( Exec (i3,t3)) . a2) by SCMPDS_5: 41

 .= md by A16, A14, AMI_3: 10, SCMPDS_2: 47;



 A19: (t2 . a) = (( Exec (i3,t3)) . a) by SCMPDS_5: 41

 .= 0 by A13, A14, AMI_3: 10, SCMPDS_2: 47;

 then

 A20: ( DataLoc ((t2 . a),6)) = ( intpos ( 0 + 6)) by SCMP_GCD: 1;

 ( DataLoc ((t0 . a),5)) = ( intpos ( 0 + 5)) by A6, SCMP_GCD: 1;

 then

 A21: (t4 . a7) = (s . a5) by A7, SCMPDS_2: 47;



 A22: (t3 . a7) = (( Exec (i2,t4)) . a7) by SCMPDS_5: 42

 .= (s . a5) by A21, A9, AMI_3: 10, SCMPDS_2: 48;



 A23: (t2 . a7) = (( Exec (i3,t3)) . a7) by SCMPDS_5: 41

 .= (s . a5) by A22, A14, AMI_3: 10, SCMPDS_2: 47;

 (t1 . a7) = (( Exec (i4,t2)) . a7) by SCMPDS_5: 41

 .= (s . a5) by A23, A20, AMI_3: 10, SCMPDS_2: 50;

 then

 A24: (t01 . a7) = (s . a5) by SCMPDS_5: 15;

 (t1 . a2) = (( Exec (i4,t2)) . a2) by SCMPDS_5: 41

 .= md by A18, A20, AMI_3: 10, SCMPDS_2: 50;

 then

 A25: (t01 . a2) = (s . a2) by A1, SCMPDS_5: 15;

 A26:

 now

 let i be Nat;

 assume i >= 8;

 then i > 7 by XXREAL_0: 2;

 hence (t4 . ( intpos i)) = (s . ( intpos i)) by A7, AMI_3: 10, SCMPDS_2: 47;

 end;

 A27:

 now

 let i be Nat;

 assume

 A28: i >= 8;

 then

 A29: i > 5 by XXREAL_0: 2;



 thus (t3 . ( intpos i)) = (( Exec (i2,t4)) . ( intpos i)) by SCMPDS_5: 42

 .= (t4 . ( intpos i)) by A9, A29, AMI_3: 10, SCMPDS_2: 48

 .= (s . ( intpos i)) by A26, A28;

 end;

 A30:

 now

 let i be Nat;

 assume

 A31: i >= 8;

 then

 A32: i > 6 by XXREAL_0: 2;



 thus (t2 . ( intpos i)) = (( Exec (i3,t3)) . ( intpos i)) by SCMPDS_5: 41

 .= (t3 . ( intpos i)) by A14, A32, AMI_3: 10, SCMPDS_2: 47

 .= (s . ( intpos i)) by A27, A31;

 end;

 A33:

 now

 let i be Nat;

 assume

 A34: i >= 8;

 then

 A35: i > 6 by XXREAL_0: 2;



 thus (t1 . ( intpos i)) = (( Exec (i4,t2)) . ( intpos i)) by SCMPDS_5: 41

 .= (t2 . ( intpos i)) by A20, A35, AMI_3: 10, SCMPDS_2: 50

 .= (s . ( intpos i)) by A30, A34;

 end;



 A36: (t4 . a3) = (s . a3) by A7, AMI_3: 10, SCMPDS_2: 47;



 A37: (t3 . a3) = (( Exec (i2,t4)) . a3) by SCMPDS_5: 42

 .= (s . a3) by A36, A9, AMI_3: 10, SCMPDS_2: 48;



 A38: (t4 . a1) = (s . a1) by A7, AMI_3: 10, SCMPDS_2: 47;



 A39: (t3 . a1) = (( Exec (i2,t4)) . a1) by SCMPDS_5: 42

 .= (s . a1) by A38, A9, AMI_3: 10, SCMPDS_2: 48;



 A40: (t2 . a1) = (( Exec (i3,t3)) . a1) by SCMPDS_5: 41

 .= (s . a1) by A39, A14, AMI_3: 10, SCMPDS_2: 47;

 (t1 . a1) = (( Exec (i4,t2)) . a1) by SCMPDS_5: 41

 .= (s . a1) by A40, A20, AMI_3: 10, SCMPDS_2: 50;

 then

 A41: (t01 . a1) = (s . a1) by SCMPDS_5: 15;



 A42: (t2 . a3) = (( Exec (i3,t3)) . a3) by SCMPDS_5: 41

 .= (s . a3) by A37, A14, AMI_3: 10, SCMPDS_2: 47;

 (t1 . a3) = (( Exec (i4,t2)) . a3) by SCMPDS_5: 41

 .= (s . a3) by A42, A20, AMI_3: 10, SCMPDS_2: 50;

 then

 A43: (t01 . a3) = (s . a3) by SCMPDS_5: 15;



 A44: (t1 . a) = (( Exec (i4,t2)) . a) by SCMPDS_5: 41

 .= 0 by A19, A20, AMI_3: 10, SCMPDS_2: 50;

 then

 A45: (t01 . a) = 0 by SCMPDS_5: 15;

 then

 A46: ( DataLoc ((t01 . a),5)) = ( intpos ( 0 + 5)) by SCMP_GCD: 1;

 a3 <> ( DataLoc ((t01 . a),4)) by A45, AMI_3: 10, SCMP_GCD: 1;

 then

 A47: (t5 . a3) = (s . a3) by A43, SCMPDS_2: 48;

 a2 <> ( DataLoc ((t01 . a),4)) by A45, AMI_3: 10, SCMP_GCD: 1;

 then

 A48: (t5 . a2) = (s . a2) by A25, SCMPDS_2: 48;

 a <> ( DataLoc ((t01 . a),4)) by A45, AMI_3: 10, SCMP_GCD: 1;

 then

 A49: (t5 . a) = 0 by A45, SCMPDS_2: 48;

 then

 A50: ( DataLoc ((t5 . a),7)) = ( intpos ( 0 + 7)) by SCMP_GCD: 1;



 A51: (t2 . a6) = (( Exec (i3,t3)) . a6) by SCMPDS_5: 41

 .= (t3 . ( intpos me)) by A14, A12, SCMPDS_2: 47

 .= (s . ( intpos me)) by A4, A27;

 (t1 . a6) = (( Exec (i4,t2)) . a6) by SCMPDS_5: 41

 .= ((t2 . a6) - (t2 . ( DataLoc ((t2 . a2), 0 )))) by A20, SCMPDS_2: 50

 .= ((t2 . a6) - (t2 . ( intpos (md + 0 )))) by A18, SCMP_GCD: 1

 .= ((s . ( intpos me)) - (s . ( intpos md))) by A3, A51, A30;

 then

 A52: (t1 . ( DataLoc ((t1 . a),6))) = ((s . ( intpos me)) - (s . ( intpos md))) by A44, SCMP_GCD: 1;

 a1 <> ( DataLoc ((t01 . a),4)) by A45, AMI_3: 10, SCMP_GCD: 1;

 then

 A53: (t5 . a1) = (s . a1) by A41, SCMPDS_2: 48;

 A54:

 now

 per cases ;

 suppose

 A55: (t01 . ( DataLoc ((t01 . a),6))) <= 0 ;



 hence (( IExec (IF1,Q1,( Initialize t1))) . a) = (( IExec (i7,Q1,( Initialize t1))) . a) by SCMPDS_6: 74

 .= (( Exec (ii7,t01)) . a) by SCMPDS_5: 40

 .= 0 by A45, A46, AMI_3: 10, SCMPDS_2: 46;



 thus (( IExec (IF1,Q1,( Initialize t1))) . a1) = (( IExec (i7,Q1,( Initialize t1))) . a1) by A55, SCMPDS_6: 74

 .= (( Exec (ii7,t01)) . a1) by SCMPDS_5: 40

 .= (s . a1) by A41, A46, AMI_3: 10, SCMPDS_2: 46;



 thus (( IExec (IF1,Q1,( Initialize t1))) . a2) = (( IExec (i7,Q1,( Initialize t1))) . a2) by A55, SCMPDS_6: 74

 .= (( Exec (ii7,t01)) . a2) by SCMPDS_5: 40

 .= (s . a2) by A25, A46, AMI_3: 10, SCMPDS_2: 46;



 thus (( IExec (IF1,Q1,( Initialize t1))) . a3) = (( IExec (i7,Q1,( Initialize t1))) . a3) by A55, SCMPDS_6: 74

 .= (( Exec (ii7,t01)) . a3) by SCMPDS_5: 40

 .= (s . a3) by A43, A46, AMI_3: 10, SCMPDS_2: 46;

 let i be Nat;

 assume

 A56: i >= 8;

 then

 A57: i > 5 by XXREAL_0: 2;



 thus (( IExec (WB1,P,s)) . ( intpos i)) = (( IExec (IF1,Q1,( Initialize t1))) . ( intpos i)) by SCMPDS_5: 35

 .= (( IExec (i7,Q1,( Initialize t1))) . ( intpos i)) by A55, SCMPDS_6: 74

 .= (( Exec (ii7,t01)) . ( intpos i)) by SCMPDS_5: 40

 .= (t01 . ( intpos i)) by A46, A57, AMI_3: 10, SCMPDS_2: 46

 .= (t1 . ( intpos i)) by SCMPDS_5: 15

 .= (s . ( intpos i)) by A33, A56;

 end;

 suppose

 A58: (t01 . ( DataLoc ((t01 . a),6))) > 0 ;



 thus (( IExec (IF1,Q1,( Initialize t1))) . a) = (( IExec ((i5 ';' i6),Q1,( Initialize t1))) . a) by A58, SCMPDS_6: 73

 .= (( Exec (i6,t5)) . a) by SCMPDS_5: 42

 .= 0 by A49, A50, AMI_3: 10, SCMPDS_2: 48;



 thus (( IExec (IF1,Q1,( Initialize t1))) . a1) = (( IExec ((i5 ';' i6),Q1,( Initialize t1))) . a1) by A58, SCMPDS_6: 73

 .= (( Exec (i6,t5)) . a1) by SCMPDS_5: 42

 .= (s . a1) by A53, A50, AMI_3: 10, SCMPDS_2: 48;



 thus (( IExec (IF1,Q1,( Initialize t1))) . a2) = (( IExec ((i5 ';' i6),Q1,( Initialize t1))) . a2) by A58, SCMPDS_6: 73

 .= (( Exec (i6,t5)) . a2) by SCMPDS_5: 42

 .= (s . a2) by A48, A50, AMI_3: 10, SCMPDS_2: 48;



 thus (( IExec (IF1,Q1,( Initialize t1))) . a3) = (( IExec ((i5 ';' i6),Q1,( Initialize t1))) . a3) by A58, SCMPDS_6: 73

 .= (( Exec (i6,t5)) . a3) by SCMPDS_5: 42

 .= (s . a3) by A47, A50, AMI_3: 10, SCMPDS_2: 48;

 let i be Nat;

 assume

 A59: i >= 8;

 then

 A60: i > 7 by XXREAL_0: 2;

 i > 4 by A59, XXREAL_0: 2;

 then

 A61: ( intpos i) <> ( DataLoc ((t01 . a),4)) by A45, AMI_3: 10, SCMP_GCD: 1;



 thus (( IExec (WB1,P,s)) . ( intpos i)) = (( IExec (IF1,Q1,( Initialize t1))) . ( intpos i)) by SCMPDS_5: 35

 .= (( IExec ((i5 ';' i6),Q1,( Initialize t1))) . ( intpos i)) by A58, SCMPDS_6: 73

 .= (( Exec (i6,t5)) . ( intpos i)) by SCMPDS_5: 42

 .= (t5 . ( intpos i)) by A50, A60, AMI_3: 10, SCMPDS_2: 48

 .= (t01 . ( intpos i)) by A61, SCMPDS_2: 48

 .= (t1 . ( intpos i)) by SCMPDS_5: 15

 .= (s . ( intpos i)) by A33, A59;

 end;

 end;

 hence (( IExec (WB1,P,s)) . a) = 0 & (( IExec (WB1,P,s)) . a1) = (s . a1) & (( IExec (WB1,P,s)) . a2) = (s . a2) & (( IExec (WB1,P,s)) . a3) = (s . a3) by SCMPDS_5: 35;

 thus for i be Nat st i >= 8 holds (( IExec (WB1,P,s)) . ( intpos i)) = (s . ( intpos i)) by A54;



 A62: (t3 . a5) = (( Exec (i2,t4)) . a5) by SCMPDS_5: 42

 .= ((t4 . a5) + ( - 1)) by A9, SCMPDS_2: 48

 .= ((s . a5) - 1) by A17;



 A63: (t2 . a5) = (( Exec (i3,t3)) . a5) by SCMPDS_5: 41

 .= ((s . a5) - 1) by A62, A14, AMI_3: 10, SCMPDS_2: 47;

 (t1 . a5) = (( Exec (i4,t2)) . a5) by SCMPDS_5: 41

 .= ((s . a5) - 1) by A63, A20, AMI_3: 10, SCMPDS_2: 50;

 then

 A64: (t01 . a5) = ((s . a5) - 1) by SCMPDS_5: 15;



 A65: (t2 . a4) = (( Exec (i3,t3)) . a4) by SCMPDS_5: 41

 .= me by A11, A14, AMI_3: 10, SCMPDS_2: 47;

 (t1 . a4) = (( Exec (i4,t2)) . a4) by SCMPDS_5: 41

 .= me by A65, A20, AMI_3: 10, SCMPDS_2: 50;

 then

 A66: (t01 . a4) = (s . a4) by A2, SCMPDS_5: 15;



 A67: ( DataLoc ((t01 . a),4)) = ( intpos ( 0 + 4)) by A45, SCMP_GCD: 1;



 A68: (t01 . ( DataLoc ((t01 . GBP ),6))) = (t01 . ( DataLoc ((t1 . GBP ),6))) by SCMPDS_5: 15

 .= (t1 . ( DataLoc ((t1 . GBP ),6))) by SCMPDS_5: 15;

 hereby



 A69: a5 <> ( DataLoc ((t01 . a),4)) by A45, AMI_3: 10, SCMP_GCD: 1;

 assume (s . ( intpos md)) < (s . ( intpos me));

 then

 A70: 0 < ((s . ( intpos me)) - (s . ( intpos md))) by XREAL_1: 50;



 A71: (t01 . ( DataLoc ((t01 . GBP ),6))) = (t01 . ( DataLoc ((t1 . GBP ),6))) by SCMPDS_5: 15

 .= (t1 . ( DataLoc ((t1 . GBP ),6))) by SCMPDS_5: 15;



 thus (( IExec (WB1,P,s)) . a5) = (( IExec (IF1,Q1,t01)) . a5) by SCMPDS_5: 35

 .= (( IExec ((i5 ';' i6),Q1,t01)) . a5) by A52, A70, A71, SCMPDS_6: 73

 .= (( Exec (i6,t5)) . a5) by SCMPDS_5: 42

 .= (t5 . a5) by A50, AMI_3: 10, SCMPDS_2: 48

 .= ((s . a5) - 1) by A64, A69, SCMPDS_2: 48;



 thus (( IExec (WB1,P,s)) . a4) = (( IExec (IF1,Q1,( Initialize t01))) . a4) by SCMPDS_5: 35

 .= (( IExec ((i5 ';' i6),Q1,( Initialize t01))) . a4) by A52, A71, A70, SCMPDS_6: 73

 .= (( Exec (i6,t5)) . a4) by SCMPDS_5: 42

 .= (t5 . a4) by A50, AMI_3: 10, SCMPDS_2: 48

 .= ((t01 . a4) + ( - 1)) by A67, SCMPDS_2: 48

 .= ((s . a4) - 1) by A66;



 A72: a7 <> ( DataLoc ((t01 . a),4)) by A45, AMI_3: 10, SCMP_GCD: 1;



 thus (( IExec (WB1,P,s)) . a7) = (( IExec (IF1,Q1,( Initialize t1))) . a7) by SCMPDS_5: 35

 .= (( IExec ((i5 ';' i6),Q1,( Initialize t1))) . a7) by A52, A71, A70, SCMPDS_6: 73

 .= (( Exec (i6,t5)) . a7) by SCMPDS_5: 42

 .= ((t5 . a7) + ( - 1)) by A50, SCMPDS_2: 48

 .= ((t5 . a7) - 1)

 .= ((s . a5) - 1) by A24, A72, SCMPDS_2: 48;

 end;

 assume (s . ( intpos md)) >= (s . ( intpos me));

 then

 A73: ((s . ( intpos me)) - (s . ( intpos md))) <= 0 by XREAL_1: 47;



 thus (( IExec (WB1,P,s)) . a5) = (( IExec (IF1,Q1,( Initialize t1))) . a5) by SCMPDS_5: 35

 .= (( IExec (i7,Q1,( Initialize t1))) . a5) by A52, A68, A73, SCMPDS_6: 74

 .= (( Exec (ii7,t01)) . a5) by SCMPDS_5: 40

 .= 0 by A46, SCMPDS_2: 46;



 thus (( IExec (WB1,P,s)) . a4) = (( IExec (IF1,Q1,( Initialize t1))) . a4) by SCMPDS_5: 35

 .= (( IExec (i7,Q1,( Initialize t1))) . a4) by A52, A68, A73, SCMPDS_6: 74

 .= (( Exec (ii7,t01)) . a4) by SCMPDS_5: 40

 .= (s . a4) by A46, A66, AMI_3: 10, SCMPDS_2: 46;



 thus (( IExec (WB1,P,s)) . a7) = (( IExec (IF1,Q1,( Initialize t1))) . a7) by SCMPDS_5: 35

 .= (( IExec (i7,Q1,( Initialize t1))) . a7) by A52, A68, A73, SCMPDS_6: 74

 .= (( Exec (ii7,t01)) . a7) by SCMPDS_5: 40

 .= (s . a5) by A24, A46, AMI_3: 10, SCMPDS_2: 46;

 end;



 Lm8: for s be 0 -started State of SCMPDS , m4,md be Nat st (s . GBP ) = 0 & (s . a5) > 0 & (s . a4) = (m4 + (s . a5)) & m4 >= 8 & (s . a2) = md & md >= 8 holds (( IExec (WH1,P,s)) . GBP ) = 0 & (( IExec (WH1,P,s)) . a1) = (s . a1) & (( IExec (WH1,P,s)) . a5) = 0 & (( IExec (WH1,P,s)) . a2) = (s . a2) & (( IExec (WH1,P,s)) . a3) = (s . a3) & (for i be Nat st i >= 8 holds (( IExec (WH1,P,s)) . ( intpos i)) = (s . ( intpos i))) & ex mE be Nat st mE = (( IExec (WH1,P,s)) . a7) & (( IExec (WH1,P,s)) . a4) = (m4 + mE) & mE <= (s . a5) & (for i be Nat st (m4 + mE) = (( IExec (WH1,P,s)) . ( intpos (m4 + mE))))

 proof

 set a = GBP ;

 let s be 0 -started State of SCMPDS , m4,md be Nat;

 assume that

 A1: (s . GBP ) = 0 and

 A2: (s . a5) > 0 and

 A3: (s . a4) = (m4 + (s . a5)) and

 A4: m4 >= 8 and

 A5: (s . a2) = md and

 A6: md >= 8;

 (s . a5) >= (1 + 0 ) by A2, INT_1: 7;

 then

 reconsider m5 = ((s . a5) - 1) as Element of NAT by INT_1: 3, XREAL_1: 48;



 A7: (s . a5) = (m5 + 1);

 defpred P[ Nat] means for t be 0 -started State of SCMPDS , Q st (t . a) = 0 & (t . a5) = ($1 + 1) & (t . a4) = (m4 + (t . a5)) & (t . a2) = md holds (( IExec (WH1,Q,t)) . a) = 0 & (( IExec (WH1,Q,t)) . a1) = (t . a1) & (( IExec (WH1,Q,t)) . a5) = 0 & (( IExec (WH1,Q,t)) . a2) = (t . a2) & (( IExec (WH1,Q,t)) . a3) = (t . a3) & (for i be Nat st i >= 8 holds (( IExec (WH1,Q,t)) . ( intpos i)) = (t . ( intpos i))) & ex mE be Nat st mE = (( IExec (WH1,Q,t)) . a7) & (( IExec (WH1,Q,t)) . a4) = (m4 + mE) & mE <= (t . a5) & (for i be Nat st (m4 + mE) = (( IExec (WH1,Q,t)) . ( intpos (m4 + mE))));

 A8:

 now

 let k be Nat;

 assume

 A9: P[k];

 thus P[(k + 1)]

 proof

 set me = (m4 + ((k + 1) + 1));

 let t be 0 -started State of SCMPDS ;

 let Q;

 set b = ( DataLoc ((t . a),5));

 assume that

 A10: (t . a) = 0 and

 A11: (t . a5) = ((k + 1) + 1) and

 A12: (t . a4) = (m4 + (t . a5)) and

 A13: (t . a2) = md;

 set It = ( IExec (WB1,Q,t)), IT = Q, It0 = ( Initialize It);



 A14: me >= m4 by NAT_1: 11;

 then

 A15: me >= 8 by A4, XXREAL_0: 2;

 then

 A16: (It . a1) = (t . a1) by A6, A10, A11, A12, A13, Lm7;



 A17: (t . ( intpos md)) >= (t . ( intpos me)) implies (It . a5) = 0 & (It . a4) = (t . a4) & (It . a7) = (t . a5) by A6, A10, A11, A12, A13, A15, Lm7;



 A18: b = ( intpos ( 0 + 5)) by A10, SCMP_GCD: 1;

 A19:

 now

 let v be 0 -started State of SCMPDS ;

 let V;

 assume that

 A20: (v . a4) >= (m4 + (v . b)) and

 A21: (v . a2) = (t . a2) and

 A22: (v . a) = (t . a) and

 A23: (v . b) > 0 ;



 A24: (m4 + (v . b)) > (m4 + 0 ) by A23, XREAL_1: 6;

 then

 reconsider ME = (v . a4) as Element of NAT by A20, INT_1: 3;

 (v . a4) >= m4 by A20, A24, XXREAL_0: 2;

 then

 A25: ME >= 8 by A4, XXREAL_0: 2;

 then

 A26: (v . ( intpos md)) < (v . ( intpos ME)) implies (( IExec (WB1,V,v)) . a5) = ((v . a5) - 1) & (( IExec (WB1,V,v)) . a4) = ((v . a4) - 1) & (( IExec (WB1,V,v)) . a7) = ((v . a5) - 1) by A6, A10, A13, A21, A22, Lm7;

 thus (( IExec (WB1,V,v)) . a) = (v . a) by A6, A10, A13, A21, A22, A25, Lm7;

 thus WB1 is_closed_on (v,V) & WB1 is_halting_on (v,V) by SCMPDS_6: 20, SCMPDS_6: 21;



 A27: (v . ( intpos md)) >= (v . ( intpos ME)) implies (( IExec (WB1,V,v)) . a5) = 0 & (( IExec (WB1,V,v)) . a4) = (v . a4) & (( IExec (WB1,V,v)) . a7) = (v . a5) by A6, A10, A13, A21, A22, A25, Lm7;

 hereby

 per cases ;

 suppose

 A28: (v . ( intpos md)) < (v . ( intpos ME));

 hence (( IExec (WB1,V,v)) . b) < (v . b) by A18, A26, XREAL_1: 146;

 ((m4 + (v . b)) - 1) = (m4 + (( IExec (WB1,V,v)) . b)) by A18, A26, A28;

 hence (( IExec (WB1,V,v)) . a4) >= (m4 + (( IExec (WB1,V,v)) . b)) by A20, A26, A28, XREAL_1: 9;

 end;

 suppose

 A29: (v . ( intpos md)) >= (v . ( intpos ME));

 hence (( IExec (WB1,V,v)) . b) < (v . b) by A6, A10, A13, A18, A21, A22, A23, A25, Lm7;

 (m4 + (( IExec (WB1,V,v)) . b)) < (m4 + (v . b)) by A18, A23, A27, A29, XREAL_1: 6;

 hence (( IExec (WB1,V,v)) . a4) >= (m4 + (( IExec (WB1,V,v)) . b)) by A20, A27, A29, XXREAL_0: 2;

 end;

 end;

 thus (( IExec (WB1,V,v)) . a2) = (v . a2) by A6, A10, A13, A21, A22, A25, Lm7;

 end;



 A30: (It . a3) = (t . a3) by A6, A10, A11, A12, A13, A15, Lm7;



 A31: (t . ( intpos md)) < (t . ( intpos me)) implies (It . a5) = ((t . a5) - 1) & (It . a4) = ((t . a4) - 1) & (It . a7) = ((t . a5) - 1) by A6, A10, A11, A12, A13, A15, Lm7;



 A32: (It . a) = 0 by A6, A10, A11, A12, A13, A15, Lm7;

 then

 A33: (It0 . a) = 0 by SCMPDS_5: 15;



 A34: ( DataLoc ((It . a),5)) = ( intpos ( 0 + 5)) by A32, SCMP_GCD: 1;



 A35: (It . a2) = (t . a2) by A6, A10, A11, A12, A13, A15, Lm7;

 then

 A36: (It0 . a2) = (t . a2) by SCMPDS_5: 15;

 per cases ;

 suppose

 A37: (t . ( intpos md)) < (t . ( intpos me));

 then

 A38: (It . a4) = (m4 + (It . a5)) by A12, A31;



 A39: (It0 . a4) = (m4 + (It . a5)) by A12, A31, A37, SCMPDS_5: 15

 .= (m4 + (It0 . a5)) by SCMPDS_5: 15;



 A40: (It0 . a5) = (It . a5) by SCMPDS_5: 15;



 A41: (It0 . a1) = (It . a1) by SCMPDS_5: 15;



 A42: (It0 . a3) = (It . a3) by SCMPDS_5: 15;



 A43: (It0 . a4) = (It . a4) by SCMPDS_5: 15;

 (It . a5) = (k + 1) by A11, A31, A37;

 then

 consider mE be Nat such that

 A44: mE = (( IExec (WH1,IT,( Initialize It))) . a7) and

 A45: (( IExec (WH1,IT,( Initialize It))) . a4) = (m4 + mE) and

 A46: mE <= (It . a5) and

 A47: for i be Nat st (m4 + mE) < i & i <= (It . a4) holds (( IExec (WH1,IT,( Initialize It))) . ( intpos md)) < (( IExec (WH1,IT,( Initialize It))) . ( intpos i)) and

 A48: mE = 0 or (( IExec (WH1,IT,( Initialize It))) . ( intpos md)) >= (( IExec (WH1,IT,( Initialize It))) . ( intpos (m4 + mE))) by A9, A12, A13, A31, A17, A33, A36, A39, A40;



 A49: ( IExec (WH1,Q,t)) = ( IExec (WH1,IT,( Initialize It))) by A11, A12, A18, A19, Th5;

 hence (( IExec (WH1,Q,t)) . a) = 0 by A9, A11, A12, A13, A31, A17, A38, A40, A36, A33, A43;

 thus (( IExec (WH1,Q,t)) . a1) = (t . a1) by A9, A11, A12, A13, A16, A31, A17, A38, A49, A40, A36, A33, A41, A43;

 thus (( IExec (WH1,Q,t)) . a5) = 0 by A9, A11, A12, A13, A31, A17, A38, A49, A40, A36, A33, A43;

 thus (( IExec (WH1,Q,t)) . a2) = (t . a2) by A9, A11, A12, A13, A31, A17, A38, A49, A40, A36, A33, A43;

 thus (( IExec (WH1,Q,t)) . a3) = (t . a3) by A9, A11, A12, A13, A30, A31, A17, A38, A49, A40, A36, A33, A42, A43;

 A50:

 now

 let i be Nat;

 assume

 A51: i >= 8;



 hence (( IExec (WH1,Q,t)) . ( intpos i)) = (It0 . ( intpos i)) by A9, A11, A12, A13, A31, A17, A38, A49, A40, A36, A33, A43

 .= (It . ( intpos i)) by SCMPDS_5: 15

 .= (t . ( intpos i)) by A6, A10, A11, A12, A13, A15, A51, Lm7;

 end;

 hence for i be Nat st i >= 8 holds (( IExec (WH1,Q,t)) . ( intpos i)) = (t . ( intpos i));

 take mE;

 thus mE = (( IExec (WH1,Q,t)) . a7) by A11, A12, A18, A19, A44, Th5;

 thus (( IExec (WH1,Q,t)) . a4) = (m4 + mE) by A11, A12, A18, A19, A45, Th5;

 thus mE <= (t . a5) by A31, A37, A46, XREAL_1: 146, XXREAL_0: 2;



 A52: (It . a4) = ((t . a4) - 1) by A6, A10, A11, A12, A13, A15, A37, Lm7;

 hereby

 let i be Nat;

 assume that

 A53: (m4 + mE) < i and

 A54: i <= (t . a4);

 per cases ;

 suppose

 A55: i = (t . a4);

 (( IExec (WH1,Q,t)) . ( intpos me)) = (t . ( intpos me)) by A4, A14, A50, XXREAL_0: 2;

 hence (( IExec (WH1,Q,t)) . ( intpos md)) < (( IExec (WH1,Q,t)) . ( intpos i)) by A6, A11, A12, A37, A50, A55;

 end;

 suppose i <> (t . a4);

 then i < (t . a4) by A54, XXREAL_0: 1;

 then (i + 1) <= (t . a4) by INT_1: 7;

 then i <= (It . a4) by A52, XREAL_1: 19;

 hence (( IExec (WH1,Q,t)) . ( intpos md)) < (( IExec (WH1,Q,t)) . ( intpos i)) by A49, A47, A53;

 end;

 end;

 thus mE = 0 or (( IExec (WH1,Q,t)) . ( intpos md)) >= (( IExec (WH1,Q,t)) . ( intpos (m4 + mE))) by A49, A48;

 end;

 suppose

 A56: (t . ( intpos md)) >= (t . ( intpos me));

 A57:

 now



 A58: (( Initialize It) . GBP ) = (It . GBP ) by SCMPDS_5: 15;



 A59: (( Initialize It) . ( DataLoc ((It . GBP ),5))) = (It . ( DataLoc ((It . GBP ),5))) by SCMPDS_5: 15;

 let x;



 thus (( IExec (WH1,Q,t)) . x) = (( IExec (WH1,IT,( Initialize It))) . x) by A11, A12, A18, A19, Th5

 .= (( Initialize It) . x) by A17, A34, A56, A58, A59, SCMPDS_8: 23

 .= (It . x) by SCMPDS_5: 15;

 end;

 hence (( IExec (WH1,Q,t)) . a) = 0 by A32;

 thus (( IExec (WH1,Q,t)) . a1) = (t . a1) by A16, A57;

 thus (( IExec (WH1,Q,t)) . a5) = 0 by A17, A56, A57;

 thus (( IExec (WH1,Q,t)) . a2) = (t . a2) by A35, A57;

 thus (( IExec (WH1,Q,t)) . a3) = (t . a3) by A30, A57;

 A60:

 now

 let i be Nat;

 assume

 A61: i >= 8;



 thus (( IExec (WH1,Q,t)) . ( intpos i)) = (It . ( intpos i)) by A57

 .= (t . ( intpos i)) by A6, A10, A11, A12, A13, A15, A61, Lm7;

 end;

 hence for i be Nat st i >= 8 holds (( IExec (WH1,Q,t)) . ( intpos i)) = (t . ( intpos i));

 take ME = ((k + 1) + 1);

 thus (( IExec (WH1,Q,t)) . a7) = ME by A11, A17, A56, A57;

 thus (( IExec (WH1,Q,t)) . a4) = (m4 + ME) by A11, A12, A17, A56, A57;

 thus ME <= (t . a5) by A11;

 thus for i be Nat st (m4 + ME) < i & i <= (t . a4) & not (( IExec (WH1,Q,t)) . ( intpos md)) < (( IExec (WH1,Q,t)) . ( intpos i)) holds contradiction by A11, A12;

 (( IExec (WH1,Q,t)) . ( intpos me)) = (t . ( intpos me)) by A4, A14, A60, XXREAL_0: 2;

 hence ME = 0 or (( IExec (WH1,Q,t)) . ( intpos md)) >= (( IExec (WH1,Q,t)) . ( intpos (m4 + ME))) by A6, A56, A60;

 end;

 end;

 end;



 A62: P[ 0 ]

 proof

 set me = (m4 + 1);

 let t be 0 -started State of SCMPDS ;

 let Q;

 set It = ( IExec (WB1,Q,t)), IT = Q;

 set b = ( DataLoc ((t . a),5));

 assume that

 A63: (t . a) = 0 and

 A64: (t . a5) = ( 0 + 1) and

 A65: (t . a4) = (m4 + (t . a5)) and

 A66: (t . a2) = md;



 A67: me >= m4 by NAT_1: 11;

 then

 A68: me >= 8 by A4, XXREAL_0: 2;

 then

 A69: (t . ( intpos md)) < (t . ( intpos me)) implies (It . a5) = ((t . a5) - 1) & (It . a4) = ((t . a4) - 1) & (It . a7) = ((t . a5) - 1) by A6, A63, A64, A65, A66, Lm7;



 A70: (t . ( intpos md)) >= (t . ( intpos me)) implies (It . a5) = 0 & (It . a4) = (t . a4) & (It . a7) = (t . a5) by A6, A63, A64, A65, A66, A68, Lm7;



 A71: b = ( intpos ( 0 + 5)) by A63, SCMP_GCD: 1;

 A72:

 now

 let v be 0 -started State of SCMPDS , V;

 assume that

 A73: (v . a4) >= (m4 + (v . b)) and

 A74: (v . a2) = (t . a2) and

 A75: (v . a) = (t . a) and

 A76: (v . b) > 0 ;



 A77: (m4 + (v . b)) > (m4 + 0 ) by A76, XREAL_1: 6;

 then

 reconsider ME = (v . a4) as Element of NAT by A73, INT_1: 3;

 (v . a4) >= m4 by A73, A77, XXREAL_0: 2;

 then

 A78: ME >= 8 by A4, XXREAL_0: 2;

 then

 A79: (v . ( intpos md)) < (v . ( intpos ME)) implies (( IExec (WB1,V,v)) . a5) = ((v . a5) - 1) & (( IExec (WB1,V,v)) . a4) = ((v . a4) - 1) & (( IExec (WB1,V,v)) . a7) = ((v . a5) - 1) by A6, A63, A66, A74, A75, Lm7;

 thus (( IExec (WB1,V,v)) . a) = (v . a) by A6, A63, A66, A74, A75, A78, Lm7;

 thus WB1 is_closed_on (v,V) & WB1 is_halting_on (v,V) by SCMPDS_6: 20, SCMPDS_6: 21;



 A80: (v . ( intpos md)) >= (v . ( intpos ME)) implies (( IExec (WB1,V,v)) . a5) = 0 & (( IExec (WB1,V,v)) . a4) = (v . a4) & (( IExec (WB1,V,v)) . a7) = (v . a5) by A6, A63, A66, A74, A75, A78, Lm7;

 hereby

 per cases ;

 suppose

 A81: (v . ( intpos md)) < (v . ( intpos ME));

 hence (( IExec (WB1,V,v)) . b) < (v . b) by A71, A79, XREAL_1: 146;

 ((m4 + (v . b)) - 1) = (m4 + (( IExec (WB1,V,v)) . b)) by A71, A79, A81;

 hence (( IExec (WB1,V,v)) . a4) >= (m4 + (( IExec (WB1,V,v)) . b)) by A73, A79, A81, XREAL_1: 9;

 end;

 suppose

 A82: (v . ( intpos md)) >= (v . ( intpos ME));

 hence (( IExec (WB1,V,v)) . b) < (v . b) by A6, A63, A66, A71, A74, A75, A76, A78, Lm7;

 (m4 + (( IExec (WB1,V,v)) . b)) < (m4 + (v . b)) by A71, A76, A80, A82, XREAL_1: 6;

 hence (( IExec (WB1,V,v)) . a4) >= (m4 + (( IExec (WB1,V,v)) . b)) by A73, A80, A82, XXREAL_0: 2;

 end;

 end;

 thus (( IExec (WB1,V,v)) . a2) = (v . a2) by A6, A63, A66, A74, A75, A78, Lm7;

 end;



 A83: (It . a) = 0 by A6, A63, A64, A65, A66, A68, Lm7;

 then

 A84: (It . ( DataLoc ((It . a),5))) = 0 by A64, A69, A70, SCMP_GCD: 1;

 A85:

 now



 A86: (( Initialize It) . GBP ) = (It . GBP ) by SCMPDS_5: 15;



 A87: (( Initialize It) . ( DataLoc ((It . GBP ),5))) = (It . ( DataLoc ((It . GBP ),5))) by SCMPDS_5: 15;

 let x;



 thus (( IExec (WH1,Q,t)) . x) = (( IExec (WH1,Q,( Initialize It))) . x) by A64, A65, A71, A72, Th5

 .= (( Initialize It) . x) by A84, A86, A87, SCMPDS_8: 23

 .= (It . x) by SCMPDS_5: 15;

 end;

 hence (( IExec (WH1,Q,t)) . a) = 0 by A83;

 (It . a1) = (t . a1) by A6, A63, A64, A65, A66, A68, Lm7;

 hence (( IExec (WH1,Q,t)) . a1) = (t . a1) by A85;

 thus (( IExec (WH1,Q,t)) . a5) = 0 by A64, A69, A70, A85;

 (It . a2) = (t . a2) by A6, A63, A64, A65, A66, A68, Lm7;

 hence (( IExec (WH1,Q,t)) . a2) = (t . a2) by A85;

 (It . a3) = (t . a3) by A6, A63, A64, A65, A66, A68, Lm7;

 hence (( IExec (WH1,Q,t)) . a3) = (t . a3) by A85;

 A88:

 now

 let i be Nat;

 assume

 A89: i >= 8;



 thus (( IExec (WH1,Q,t)) . ( intpos i)) = (It . ( intpos i)) by A85

 .= (t . ( intpos i)) by A6, A63, A64, A65, A66, A68, A89, Lm7;

 end;

 hence for i be Nat st i >= 8 holds (( IExec (WH1,Q,t)) . ( intpos i)) = (t . ( intpos i));



 A90: (( IExec (WH1,Q,t)) . ( intpos me)) = (t . ( intpos me)) by A4, A67, A88, XXREAL_0: 2;

 per cases ;

 suppose

 A91: (t . ( intpos md)) < (t . ( intpos me));

 take ME = 0 ;

 thus (( IExec (WH1,Q,t)) . a7) = ME by A64, A69, A85, A91;

 thus (( IExec (WH1,Q,t)) . a4) = (m4 + ME) by A64, A65, A69, A85, A91;

 thus ME <= (t . a5) by A64;

 hereby

 let i be Nat;

 assume that

 A92: (m4 + ME) < i and

 A93: i <= (t . a4);

 (m4 + 1) <= i by A92, INT_1: 7;

 then i = (m4 + 1) by A64, A65, A93, XXREAL_0: 1;

 hence (( IExec (WH1,Q,t)) . ( intpos md)) < (( IExec (WH1,Q,t)) . ( intpos i)) by A6, A88, A90, A91;

 end;

 thus ME = 0 or (( IExec (WH1,Q,t)) . ( intpos md)) >= (( IExec (WH1,Q,t)) . ( intpos (m4 + ME)));

 end;

 suppose

 A94: (t . ( intpos md)) >= (t . ( intpos me));

 take ME = 1;

 thus (( IExec (WH1,Q,t)) . a7) = ME by A64, A70, A85, A94;

 thus (( IExec (WH1,Q,t)) . a4) = (m4 + ME) by A64, A65, A70, A85, A94;

 thus ME <= (t . a5) by A64;

 thus for i be Nat st (m4 + ME) = (( IExec (WH1,Q,t)) . ( intpos (m4 + ME))) by A6, A88, A90, A94;

 end;

 end;

 for k be Nat holds P[k] from NAT_1:sch 2( A62, A8);

 hence thesis by A1, A3, A5, A7;

 end;



 Lm9: for s be 0 -started State of SCMPDS , m4,md be Nat st (s . GBP ) = 0 & (s . a4) = (m4 + (s . a5)) & m4 >= 8 & (s . a2) = md & md >= 8 holds WH1 is_closed_on (s,P) & WH1 is_halting_on (s,P)

 proof

 set a = GBP ;

 let s be 0 -started State of SCMPDS , m4,md be Nat;

 assume that

 A1: (s . GBP ) = 0 and

 A2: (s . a4) = (m4 + (s . a5)) and

 A3: m4 >= 8 and

 A4: (s . a2) = md and

 A5: md >= 8;

 set b = ( DataLoc ((s . a),5));



 A6: b = ( intpos ( 0 + 5)) by A1, SCMP_GCD: 1;

 now

 let v be 0 -started State of SCMPDS , V;

 assume that

 A7: (v . a4) >= (m4 + (v . b)) and

 A8: (v . a2) = (s . a2) and

 A9: (v . a) = (s . a) and

 A10: (v . b) > 0 ;



 A11: (m4 + (v . b)) > (m4 + 0 ) by A10, XREAL_1: 6;

 then

 reconsider ME = (v . a4) as Element of NAT by A7, INT_1: 3;

 (v . a4) >= m4 by A7, A11, XXREAL_0: 2;

 then

 A12: ME >= 8 by A3, XXREAL_0: 2;

 then

 A13: (v . ( intpos md)) < (v . ( intpos ME)) implies (( IExec (WB1,V,v)) . a5) = ((v . a5) - 1) & (( IExec (WB1,V,v)) . a4) = ((v . a4) - 1) & (( IExec (WB1,V,v)) . a7) = ((v . a5) - 1) by A1, A4, A5, A8, A9, Lm7;

 thus (( IExec (WB1,V,v)) . a) = (v . a) by A1, A4, A5, A8, A9, A12, Lm7;

 thus WB1 is_closed_on (v,V) & WB1 is_halting_on (v,V) by SCMPDS_6: 20, SCMPDS_6: 21;



 A14: (v . ( intpos md)) >= (v . ( intpos ME)) implies (( IExec (WB1,V,v)) . a5) = 0 & (( IExec (WB1,V,v)) . a4) = (v . a4) & (( IExec (WB1,V,v)) . a7) = (v . a5) by A1, A4, A5, A8, A9, A12, Lm7;

 hereby

 per cases ;

 suppose

 A15: (v . ( intpos md)) < (v . ( intpos ME));

 hence (( IExec (WB1,V,v)) . b) < (v . b) by A6, A13, XREAL_1: 146;

 ((m4 + (v . b)) - 1) = (m4 + (( IExec (WB1,V,v)) . b)) by A6, A13, A15;

 hence (( IExec (WB1,V,v)) . a4) >= (m4 + (( IExec (WB1,V,v)) . b)) by A7, A13, A15, XREAL_1: 9;

 end;

 suppose

 A16: (v . ( intpos md)) >= (v . ( intpos ME));

 hence (( IExec (WB1,V,v)) . b) < (v . b) by A1, A4, A5, A6, A8, A9, A10, A12, Lm7;

 (m4 + (( IExec (WB1,V,v)) . b)) < (m4 + (v . b)) by A6, A10, A14, A16, XREAL_1: 6;

 hence (( IExec (WB1,V,v)) . a4) >= (m4 + (( IExec (WB1,V,v)) . b)) by A7, A14, A16, XXREAL_0: 2;

 end;

 end;

 thus (( IExec (WB1,V,v)) . a2) = (v . a2) by A1, A4, A5, A8, A9, A12, Lm7;

 end;

 hence thesis by A2, A6, Th5;

 end;



 Lm10: ( card WH1) = 11

 proof



 thus ( card WH1) = (9 + 2) by Lm6, SCMPDS_8: 17

 .= 11;

 end;



 Lm11: ( card WB2) = 9

 proof



 thus ( card WB2) = (( card (((j1 ';' j2) ';' j3) ';' j4)) + ( card IF2)) by AFINSQ_1: 17

 .= (4 + ( card IF2)) by Th4

 .= (4 + ((( card (j5 ';' j6)) + ( card j7)) + 2)) by SCMPDS_6: 65

 .= (4 + ((2 + ( card j7)) + 2)) by SCMP_GCD: 5

 .= (4 + ((2 + 1) + 2)) by COMPOS_1: 54

 .= 9;

 end;



 Lm12: ( card WH2) = 11

 proof



 thus ( card WH2) = (9 + 2) by Lm11, SCMPDS_8: 17

 .= 11;

 end;



 Lm13: for s be 0 -started State of SCMPDS , md,me be Nat st (s . a2) = md & (s . a3) = me & md >= 8 & me >= 8 & (s . GBP ) = 0 holds (( IExec (WB2,P,s)) . GBP ) = 0 & (( IExec (WB2,P,s)) . a1) = (s . a1) & (( IExec (WB2,P,s)) . a2) = (s . a2) & (( IExec (WB2,P,s)) . a4) = (s . a4) & (for i be Nat st i >= 8 holds (( IExec (WB2,P,s)) . ( intpos i)) = (s . ( intpos i))) & ((s . ( intpos md)) > (s . ( intpos me)) implies (( IExec (WB2,P,s)) . a7) = ((s . a7) - 1) & (( IExec (WB2,P,s)) . a3) = ((s . a3) + 1) & (( IExec (WB2,P,s)) . a5) = ((s . a7) - 1)) & ((s . ( intpos md)) <= (s . ( intpos me)) implies (( IExec (WB2,P,s)) . a7) = 0 & (( IExec (WB2,P,s)) . a3) = (s . a3) & (( IExec (WB2,P,s)) . a5) = (s . a7))

 proof

 set a = GBP ;

 let s be 0 -started State of SCMPDS , md,me be Nat;

 assume that

 A1: (s . a2) = md and

 A2: (s . a3) = me and

 A3: md >= 8 and

 A4: me >= 8 and

 A5: (s . a) = 0 ;

 set t0 = s, t1 = ( IExec ((((j1 ';' j2) ';' j3) ';' j4),P,s)), Q1 = P, t2 = ( IExec (((j1 ';' j2) ';' j3),P,s)), t3 = ( IExec ((j1 ';' j2),P,s)), t4 = ( Exec (j1,t0));



 A6: (t0 . a) = 0 by A5;



 A7: ( DataLoc ((t0 . a),5)) = ( intpos ( 0 + 5)) by A5, SCMP_GCD: 1;

 then

 A8: (t4 . a) = 0 by A5, AMI_3: 10, SCMPDS_2: 47;

 then

 A9: ( DataLoc ((t4 . a),7)) = ( intpos ( 0 + 7)) by SCMP_GCD: 1;



 A10: (t4 . a2) = md by A7, A1, AMI_3: 10, SCMPDS_2: 47;



 A11: (t3 . a2) = (( Exec (j2,t4)) . a2) by SCMPDS_5: 42

 .= md by A10, A9, AMI_3: 10, SCMPDS_2: 48;

 then

 A12: ( DataLoc ((t3 . a2), 0 )) = ( intpos (md + 0 )) by SCMP_GCD: 1;



 A13: (t3 . a) = (( Exec (j2,t4)) . a) by SCMPDS_5: 42

 .= 0 by A8, A9, AMI_3: 10, SCMPDS_2: 48;

 then

 A14: ( DataLoc ((t3 . a),6)) = ( intpos ( 0 + 6)) by SCMP_GCD: 1;



 A15: (t4 . a3) = me by A7, A2, AMI_3: 10, SCMPDS_2: 47;



 A16: (t3 . a3) = (( Exec (j2,t4)) . a3) by SCMPDS_5: 42

 .= me by A15, A9, AMI_3: 10, SCMPDS_2: 48;



 A17: (t4 . a7) = (s . a7) by A7, AMI_3: 10, SCMPDS_2: 47;

 set t01 = ( Initialize t1), jj7 = (( GBP ,7) := 0 ), t5 = ( Exec (j5,t01));



 A18: (t2 . a3) = (( Exec (j3,t3)) . a3) by SCMPDS_5: 41

 .= me by A16, A14, AMI_3: 10, SCMPDS_2: 47;



 A19: (t2 . a) = (( Exec (j3,t3)) . a) by SCMPDS_5: 41

 .= 0 by A13, A14, AMI_3: 10, SCMPDS_2: 47;

 then

 A20: ( DataLoc ((t2 . a),6)) = ( intpos ( 0 + 6)) by SCMP_GCD: 1;



 A21: (t4 . a4) = (s . a4) by A7, AMI_3: 10, SCMPDS_2: 47;



 A22: (t3 . a4) = (( Exec (j2,t4)) . a4) by SCMPDS_5: 42

 .= (s . a4) by A21, A9, AMI_3: 10, SCMPDS_2: 48;



 A23: (t4 . a1) = (s . a1) by A7, AMI_3: 10, SCMPDS_2: 47;



 A24: (t3 . a1) = (( Exec (j2,t4)) . a1) by SCMPDS_5: 42

 .= (s . a1) by A23, A9, AMI_3: 10, SCMPDS_2: 48;



 A25: (t2 . a1) = (( Exec (j3,t3)) . a1) by SCMPDS_5: 41

 .= (s . a1) by A24, A14, AMI_3: 10, SCMPDS_2: 47;

 (t1 . a1) = (( Exec (j4,t2)) . a1) by SCMPDS_5: 41

 .= (s . a1) by A25, A20, AMI_3: 10, SCMPDS_2: 50;

 then

 A26: (t01 . a1) = (s . a1) by SCMPDS_5: 15;



 A27: (t2 . a4) = (( Exec (j3,t3)) . a4) by SCMPDS_5: 41

 .= (s . a4) by A22, A14, AMI_3: 10, SCMPDS_2: 47;

 (t1 . a4) = (( Exec (j4,t2)) . a4) by SCMPDS_5: 41

 .= (s . a4) by A27, A20, AMI_3: 10, SCMPDS_2: 50;

 then

 A28: (t01 . a4) = (s . a4) by SCMPDS_5: 15;



 A29: (t1 . a) = (( Exec (j4,t2)) . a) by SCMPDS_5: 41

 .= 0 by A19, A20, AMI_3: 10, SCMPDS_2: 50;

 then

 A30: (t01 . a) = 0 by SCMPDS_5: 15;

 then

 A31: ( DataLoc ((t01 . a),7)) = ( intpos ( 0 + 7)) by SCMP_GCD: 1;

 ( DataLoc ((t0 . a),7)) = ( intpos ( 0 + 7)) by A6, SCMP_GCD: 1;

 then

 A32: (t4 . a5) = (s . a7) by A7, SCMPDS_2: 47;



 A33: (t3 . a5) = (( Exec (j2,t4)) . a5) by SCMPDS_5: 42

 .= (s . a7) by A32, A9, AMI_3: 10, SCMPDS_2: 48;



 A34: (t2 . a5) = (( Exec (j3,t3)) . a5) by SCMPDS_5: 41

 .= (s . a7) by A33, A14, AMI_3: 10, SCMPDS_2: 47;

 (t1 . a5) = (( Exec (j4,t2)) . a5) by SCMPDS_5: 41

 .= (s . a7) by A34, A20, AMI_3: 10, SCMPDS_2: 50;

 then

 A35: (t01 . a5) = (s . a7) by SCMPDS_5: 15;



 A36: (t2 . a2) = (( Exec (j3,t3)) . a2) by SCMPDS_5: 41

 .= md by A11, A14, AMI_3: 10, SCMPDS_2: 47;

 (t1 . a2) = (( Exec (j4,t2)) . a2) by SCMPDS_5: 41

 .= md by A36, A20, AMI_3: 10, SCMPDS_2: 50;

 then

 A37: (t01 . a2) = (s . a2) by A1, SCMPDS_5: 15;

 a4 <> ( DataLoc ((t01 . a),3)) by A30, AMI_3: 10, SCMP_GCD: 1;

 then

 A38: (t5 . a4) = (s . a4) by A28, SCMPDS_2: 48;

 a2 <> ( DataLoc ((t01 . a),3)) by A30, AMI_3: 10, SCMP_GCD: 1;

 then

 A39: (t5 . a2) = (s . a2) by A37, SCMPDS_2: 48;

 a <> ( DataLoc ((t01 . a),3)) by A30, AMI_3: 10, SCMP_GCD: 1;

 then

 A40: (t5 . a) = 0 by A30, SCMPDS_2: 48;

 then

 A41: ( DataLoc ((t5 . a),5)) = ( intpos ( 0 + 5)) by SCMP_GCD: 1;

 A42:

 now

 let i be Nat;

 assume i >= 8;

 then i > 5 by XXREAL_0: 2;

 hence (t4 . ( intpos i)) = (s . ( intpos i)) by A7, AMI_3: 10, SCMPDS_2: 47;

 end;

 A43:

 now

 let i be Nat;

 assume

 A44: i >= 8;

 then

 A45: i > 7 by XXREAL_0: 2;



 thus (t3 . ( intpos i)) = (( Exec (j2,t4)) . ( intpos i)) by SCMPDS_5: 42

 .= (t4 . ( intpos i)) by A9, A45, AMI_3: 10, SCMPDS_2: 48

 .= (s . ( intpos i)) by A42, A44;

 end;

 A46:

 now

 let i be Nat;

 assume

 A47: i >= 8;

 then

 A48: i > 6 by XXREAL_0: 2;



 thus (t2 . ( intpos i)) = (( Exec (j3,t3)) . ( intpos i)) by SCMPDS_5: 41

 .= (t3 . ( intpos i)) by A14, A48, AMI_3: 10, SCMPDS_2: 47

 .= (s . ( intpos i)) by A43, A47;

 end;

 A49:

 now

 let i be Nat;

 assume

 A50: i >= 8;

 then

 A51: i > 6 by XXREAL_0: 2;



 thus (t1 . ( intpos i)) = (( Exec (j4,t2)) . ( intpos i)) by SCMPDS_5: 41

 .= (t2 . ( intpos i)) by A20, A51, AMI_3: 10, SCMPDS_2: 50

 .= (s . ( intpos i)) by A46, A50;

 end;



 A52: (t2 . a6) = (( Exec (j3,t3)) . a6) by SCMPDS_5: 41

 .= (t3 . ( intpos md)) by A14, A12, SCMPDS_2: 47

 .= (s . ( intpos md)) by A3, A43;

 (t1 . a6) = (( Exec (j4,t2)) . a6) by SCMPDS_5: 41

 .= ((t2 . a6) - (t2 . ( DataLoc ((t2 . a3), 0 )))) by A20, SCMPDS_2: 50

 .= ((t2 . a6) - (t2 . ( intpos (me + 0 )))) by A18, SCMP_GCD: 1

 .= ((s . ( intpos md)) - (s . ( intpos me))) by A4, A52, A46;

 then

 A53: (t1 . ( DataLoc ((t1 . a),6))) = ((s . ( intpos md)) - (s . ( intpos me))) by A29, SCMP_GCD: 1;

 a1 <> ( DataLoc ((t01 . a),3)) by A30, AMI_3: 10, SCMP_GCD: 1;

 then

 A54: (t5 . a1) = (s . a1) by A26, SCMPDS_2: 48;

 A55:

 now

 per cases ;

 suppose

 A56: (t01 . ( DataLoc ((t01 . a),6))) <= 0 ;



 hence (( IExec (IF2,Q1,( Initialize t1))) . a) = (( IExec (j7,Q1,( Initialize t1))) . a) by SCMPDS_6: 74

 .= (( Exec (jj7,t01)) . a) by SCMPDS_5: 40

 .= 0 by A30, A31, AMI_3: 10, SCMPDS_2: 46;



 thus (( IExec (IF2,Q1,( Initialize t1))) . a1) = (( IExec (j7,Q1,( Initialize t1))) . a1) by A56, SCMPDS_6: 74

 .= (( Exec (jj7,t01)) . a1) by SCMPDS_5: 40

 .= (s . a1) by A26, A31, AMI_3: 10, SCMPDS_2: 46;



 thus (( IExec (IF2,Q1,( Initialize t1))) . a2) = (( IExec (j7,Q1,( Initialize t1))) . a2) by A56, SCMPDS_6: 74

 .= (( Exec (jj7,t01)) . a2) by SCMPDS_5: 40

 .= (s . a2) by A37, A31, AMI_3: 10, SCMPDS_2: 46;



 thus (( IExec (IF2,Q1,( Initialize t1))) . a4) = (( IExec (j7,Q1,( Initialize t1))) . a4) by A56, SCMPDS_6: 74

 .= (( Exec (jj7,t01)) . a4) by SCMPDS_5: 40

 .= (s . a4) by A28, A31, AMI_3: 10, SCMPDS_2: 46;

 let i be Nat;

 assume

 A57: i >= 8;

 then

 A58: i > 7 by XXREAL_0: 2;



 thus (( IExec (WB2,P,s)) . ( intpos i)) = (( IExec (IF2,Q1,( Initialize t1))) . ( intpos i)) by SCMPDS_5: 35

 .= (( IExec (j7,Q1,( Initialize t1))) . ( intpos i)) by A56, SCMPDS_6: 74

 .= (( Exec (jj7,t01)) . ( intpos i)) by SCMPDS_5: 40

 .= (t01 . ( intpos i)) by A31, A58, AMI_3: 10, SCMPDS_2: 46

 .= (t1 . ( intpos i)) by SCMPDS_5: 15

 .= (s . ( intpos i)) by A49, A57;

 end;

 suppose

 A59: (t01 . ( DataLoc ((t01 . a),6))) > 0 ;



 hence (( IExec (IF2,Q1,( Initialize t1))) . a) = (( IExec ((j5 ';' j6),Q1,( Initialize t1))) . a) by SCMPDS_6: 73

 .= (( Exec (j6,t5)) . a) by SCMPDS_5: 42

 .= 0 by A40, A41, AMI_3: 10, SCMPDS_2: 48;



 thus (( IExec (IF2,Q1,( Initialize t1))) . a1) = (( IExec ((j5 ';' j6),Q1,( Initialize t1))) . a1) by A59, SCMPDS_6: 73

 .= (( Exec (j6,t5)) . a1) by SCMPDS_5: 42

 .= (s . a1) by A54, A41, AMI_3: 10, SCMPDS_2: 48;



 thus (( IExec (IF2,Q1,( Initialize t1))) . a2) = (( IExec ((j5 ';' j6),Q1,( Initialize t1))) . a2) by A59, SCMPDS_6: 73

 .= (( Exec (j6,t5)) . a2) by SCMPDS_5: 42

 .= (s . a2) by A39, A41, AMI_3: 10, SCMPDS_2: 48;



 thus (( IExec (IF2,Q1,( Initialize t1))) . a4) = (( IExec ((j5 ';' j6),Q1,( Initialize t1))) . a4) by A59, SCMPDS_6: 73

 .= (( Exec (j6,t5)) . a4) by SCMPDS_5: 42

 .= (s . a4) by A38, A41, AMI_3: 10, SCMPDS_2: 48;

 let i be Nat;

 assume

 A60: i >= 8;

 then

 A61: i > 5 by XXREAL_0: 2;

 i > 3 by A60, XXREAL_0: 2;

 then

 A62: ( intpos i) <> ( DataLoc ((t01 . a),3)) by A30, AMI_3: 10, SCMP_GCD: 1;



 thus (( IExec (WB2,P,s)) . ( intpos i)) = (( IExec (IF2,Q1,( Initialize t1))) . ( intpos i)) by SCMPDS_5: 35

 .= (( IExec ((j5 ';' j6),Q1,( Initialize t1))) . ( intpos i)) by A59, SCMPDS_6: 73

 .= (( Exec (j6,t5)) . ( intpos i)) by SCMPDS_5: 42

 .= (t5 . ( intpos i)) by A41, A61, AMI_3: 10, SCMPDS_2: 48

 .= (t01 . ( intpos i)) by A62, SCMPDS_2: 48

 .= (t1 . ( intpos i)) by SCMPDS_5: 15

 .= (s . ( intpos i)) by A49, A60;

 end;

 end;

 hence (( IExec (WB2,P,s)) . a) = 0 & (( IExec (WB2,P,s)) . a1) = (s . a1) & (( IExec (WB2,P,s)) . a2) = (s . a2) & (( IExec (WB2,P,s)) . a4) = (s . a4) by SCMPDS_5: 35;

 thus for i be Nat st i >= 8 holds (( IExec (WB2,P,s)) . ( intpos i)) = (s . ( intpos i)) by A55;



 A63: (t3 . a7) = (( Exec (j2,t4)) . a7) by SCMPDS_5: 42

 .= ((t4 . a7) + ( - 1)) by A9, SCMPDS_2: 48

 .= ((s . a7) - 1) by A17;



 A64: (t2 . a7) = (( Exec (j3,t3)) . a7) by SCMPDS_5: 41

 .= ((s . a7) - 1) by A63, A14, AMI_3: 10, SCMPDS_2: 47;

 (t1 . a7) = (( Exec (j4,t2)) . a7) by SCMPDS_5: 41

 .= ((s . a7) - 1) by A64, A20, AMI_3: 10, SCMPDS_2: 50;

 then

 A65: (t01 . a7) = ((s . a7) - 1) by SCMPDS_5: 15;

 (t1 . a3) = (( Exec (j4,t2)) . a3) by SCMPDS_5: 41

 .= me by A18, A20, AMI_3: 10, SCMPDS_2: 50;

 then

 A66: (t01 . a3) = (s . a3) by A2, SCMPDS_5: 15;



 A67: ( DataLoc ((t01 . a),3)) = ( intpos ( 0 + 3)) by A30, SCMP_GCD: 1;



 A68: (t01 . ( DataLoc ((t01 . a),6))) = (t1 . ( DataLoc ((t01 . a),6))) by SCMPDS_5: 15

 .= (t1 . ( DataLoc ((t1 . a),6))) by SCMPDS_5: 15;

 hereby



 A69: a7 <> ( DataLoc ((t01 . a),3)) by A30, AMI_3: 10, SCMP_GCD: 1;

 assume (s . ( intpos md)) > (s . ( intpos me));

 then

 A70: ((s . ( intpos md)) - (s . ( intpos me))) > 0 by XREAL_1: 50;



 thus (( IExec (WB2,P,s)) . a7) = (( IExec (IF2,Q1,( Initialize t1))) . a7) by SCMPDS_5: 35

 .= (( IExec ((j5 ';' j6),Q1,( Initialize t1))) . a7) by A53, A68, A70, SCMPDS_6: 73

 .= (( Exec (j6,t5)) . a7) by SCMPDS_5: 42

 .= (t5 . a7) by A41, AMI_3: 10, SCMPDS_2: 48

 .= ((s . a7) - 1) by A65, A69, SCMPDS_2: 48;



 thus (( IExec (WB2,P,s)) . a3) = (( IExec (IF2,Q1,( Initialize t1))) . a3) by SCMPDS_5: 35

 .= (( IExec ((j5 ';' j6),Q1,( Initialize t1))) . a3) by A53, A68, A70, SCMPDS_6: 73

 .= (( Exec (j6,t5)) . a3) by SCMPDS_5: 42

 .= (t5 . a3) by A41, AMI_3: 10, SCMPDS_2: 48

 .= ((s . a3) + 1) by A67, A66, SCMPDS_2: 48;



 A71: a5 <> ( DataLoc ((t01 . a),3)) by A30, AMI_3: 10, SCMP_GCD: 1;



 thus (( IExec (WB2,P,s)) . a5) = (( IExec (IF2,Q1,( Initialize t1))) . a5) by SCMPDS_5: 35

 .= (( IExec ((j5 ';' j6),Q1,( Initialize t1))) . a5) by A53, A68, A70, SCMPDS_6: 73

 .= (( Exec (j6,t5)) . a5) by SCMPDS_5: 42

 .= ((t5 . a5) + ( - 1)) by A41, SCMPDS_2: 48

 .= ((t5 . a5) - 1)

 .= ((s . a7) - 1) by A35, A71, SCMPDS_2: 48;

 end;

 hereby

 assume (s . ( intpos md)) <= (s . ( intpos me));

 then

 A72: 0 >= ((s . ( intpos md)) - (s . ( intpos me))) by XREAL_1: 47;



 thus (( IExec (WB2,P,s)) . a7) = (( IExec (IF2,Q1,( Initialize t1))) . a7) by SCMPDS_5: 35

 .= (( IExec (j7,Q1,( Initialize t1))) . a7) by A53, A68, A72, SCMPDS_6: 74

 .= (( Exec (jj7,t01)) . a7) by SCMPDS_5: 40

 .= 0 by A31, SCMPDS_2: 46;



 thus (( IExec (WB2,P,s)) . a3) = (( IExec (IF2,Q1,( Initialize t1))) . a3) by SCMPDS_5: 35

 .= (( IExec (j7,Q1,( Initialize t1))) . a3) by A53, A68, A72, SCMPDS_6: 74

 .= (( Exec (jj7,t01)) . a3) by SCMPDS_5: 40

 .= (s . a3) by A31, A66, AMI_3: 10, SCMPDS_2: 46;



 thus (( IExec (WB2,P,s)) . a5) = (( IExec (IF2,Q1,( Initialize t1))) . a5) by SCMPDS_5: 35

 .= (( IExec (j7,Q1,( Initialize t1))) . a5) by A53, A68, A72, SCMPDS_6: 74

 .= (( Exec (jj7,t01)) . a5) by SCMPDS_5: 40

 .= (s . a7) by A35, A31, AMI_3: 10, SCMPDS_2: 46;

 end;

 end;



 Lm14: for s be 0 -started State of SCMPDS , m3,md be Nat st (s . GBP ) = 0 & (s . a7) > 0 & ((s . a3) + (s . a7)) = m3 & (s . a3) >= 8 & (s . a2) = md & md >= 8 holds (( IExec (WH2,P,s)) . GBP ) = 0 & (( IExec (WH2,P,s)) . a1) = (s . a1) & (( IExec (WH2,P,s)) . a7) = 0 & (( IExec (WH2,P,s)) . a2) = (s . a2) & (( IExec (WH2,P,s)) . a4) = (s . a4) & (for i be Nat st i >= 8 holds (( IExec (WH2,P,s)) . ( intpos i)) = (s . ( intpos i))) & ex m5,mE3 be Nat st m5 = (( IExec (WH2,P,s)) . a5) & (( IExec (WH2,P,s)) . a3) = mE3 & (mE3 + m5) = m3 & m5 <= (s . a7) & (for i be Nat st (s . a3) <= i & i < mE3 holds (( IExec (WH2,P,s)) . ( intpos md)) > (( IExec (WH2,P,s)) . ( intpos i))) & (m5 = 0 or (( IExec (WH2,P,s)) . ( intpos md)) <= (( IExec (WH2,P,s)) . ( intpos mE3)))

 proof

 set a = GBP ;

 let s be 0 -started State of SCMPDS , m3,md be Nat;

 assume that

 A1: (s . a) = 0 and

 A2: (s . a7) > 0 and

 A3: ((s . a3) + (s . a7)) = m3 and

 A4: (s . a3) >= 8 and

 A5: (s . a2) = md and

 A6: md >= 8;

 (s . a7) >= (1 + 0 ) by A2, INT_1: 7;

 then

 reconsider m7 = ((s . a7) - 1) as Element of NAT by INT_1: 3, XREAL_1: 48;



 A7: (s . a7) = (m7 + 1);

 defpred P[ Nat] means for t be 0 -started State of SCMPDS , Q st (t . a) = 0 & (t . a7) = ($1 + 1) & ((t . a3) + (t . a7)) = m3 & (t . a3) >= 8 & (t . a2) = md holds (( IExec (WH2,Q,t)) . a) = 0 & (( IExec (WH2,Q,t)) . a1) = (t . a1) & (( IExec (WH2,Q,t)) . a7) = 0 & (( IExec (WH2,Q,t)) . a2) = (t . a2) & (( IExec (WH2,Q,t)) . a4) = (t . a4) & (for i be Nat st i >= 8 holds (( IExec (WH2,Q,t)) . ( intpos i)) = (t . ( intpos i))) & ex m5,mE3 be Nat st m5 = (( IExec (WH2,Q,t)) . a5) & (( IExec (WH2,Q,t)) . a3) = mE3 & (mE3 + m5) = m3 & m5 <= (t . a7) & (for i be Nat st (t . a3) <= i & i < mE3 holds (( IExec (WH2,Q,t)) . ( intpos md)) > (( IExec (WH2,Q,t)) . ( intpos i))) & (m5 = 0 or (( IExec (WH2,Q,t)) . ( intpos md)) <= (( IExec (WH2,Q,t)) . ( intpos mE3)));

 A8:

 now

 let k be Nat;

 assume

 A9: P[k];

 P[(k + 1)]

 proof

 let t be 0 -started State of SCMPDS ;

 let Q;

 set b = ( DataLoc ((t . a),7));

 assume that

 A10: (t . a) = 0 and

 A11: (t . a7) = ((k + 1) + 1) and

 A12: ((t . a3) + (t . a7)) = m3 and

 A13: (t . a3) >= 8 and

 A14: (t . a2) = md;

 reconsider me = (t . a3) as Element of NAT by A13, INT_1: 3;

 set It = ( IExec (WB2,Q,t)), IT = Q;



 A15: (t . ( intpos md)) > (t . ( intpos me)) implies (It . a7) = ((t . a7) - 1) & (It . a3) = ((t . a3) + 1) & (It . a5) = ((t . a7) - 1) by A6, A10, A13, A14, Lm13;



 A16: b = ( intpos ( 0 + 7)) by A10, SCMP_GCD: 1;

 A17:

 now

 let v be 0 -started State of SCMPDS ;

 let V;

 assume that

 A18: (v . a3) >= 8 and

 A19: (v . a2) = (t . a2) and

 A20: (v . a) = (t . a) and

 A21: (v . b) > 0 ;

 reconsider ME = (v . a3) as Element of NAT by A18, INT_1: 3;



 A22: ME = (v . a3);

 hence (( IExec (WB2,V,v)) . a) = (v . a) by A6, A10, A14, A18, A19, A20, Lm13;

 thus WB2 is_closed_on (v,V) & WB2 is_halting_on (v,V) by SCMPDS_6: 20, SCMPDS_6: 21;



 A23: (v . ( intpos md)) > (v . ( intpos ME)) implies (( IExec (WB2,V,v)) . a7) = ((v . a7) - 1) & (( IExec (WB2,V,v)) . a3) = ((v . a3) + 1) & (( IExec (WB2,V,v)) . a5) = ((v . a7) - 1) by A6, A10, A14, A18, A19, A20, Lm13;

 hereby

 per cases ;

 suppose

 A24: (v . ( intpos md)) > (v . ( intpos ME));

 hence (( IExec (WB2,V,v)) . b) < (v . b) by A16, A23, XREAL_1: 146;



 A25: ((v . a3) + 1) > (v . a3) by XREAL_1: 29;

 (( IExec (WB2,V,v)) . a3) = ((v . a3) + 1) by A6, A10, A14, A18, A19, A20, A24, Lm13;

 hence (( IExec (WB2,V,v)) . a3) >= 8 by A18, A25, XXREAL_0: 2;

 end;

 suppose

 A26: (v . ( intpos md)) <= (v . ( intpos ME));

 hence (( IExec (WB2,V,v)) . b) < (v . b) by A6, A10, A14, A16, A18, A19, A20, A21, Lm13;

 thus (( IExec (WB2,V,v)) . a3) >= 8 by A6, A10, A14, A18, A19, A20, A26, Lm13;

 end;

 end;

 thus (( IExec (WB2,V,v)) . a2) = (v . a2) by A6, A10, A14, A18, A19, A20, A22, Lm13;

 end;



 A27: (t . ( intpos md)) <= (t . ( intpos me)) implies (It . a7) = 0 & (It . a3) = (t . a3) & (It . a5) = (t . a7) by A6, A10, A13, A14, Lm13;



 A28: me = (t . a3);

 then

 A29: (It . a) = 0 by A6, A10, A13, A14, Lm13;

 then

 A30: ( DataLoc ((It . a),7)) = ( intpos ( 0 + 7)) by SCMP_GCD: 1;



 A31: (It . a1) = (t . a1) by A6, A10, A13, A14, A28, Lm13;



 A32: (It . a4) = (t . a4) by A6, A10, A13, A14, A28, Lm13;



 A33: (It . a2) = (t . a2) by A6, A10, A13, A14, A28, Lm13;



 A34: (( Initialize It) . a1) = (It . a1) by SCMPDS_5: 15;



 A35: (( Initialize It) . a4) = (It . a4) by SCMPDS_5: 15;



 A36: (( Initialize It) . a2) = (It . a2) by SCMPDS_5: 15;

 per cases ;

 suppose

 A37: (t . ( intpos md)) > (t . ( intpos me));

 then (t . a3) < (It . a3) by A15, XREAL_1: 29;

 then

 A38: (It . a3) >= 8 by A13, XXREAL_0: 2;



 A39: ((It . a3) + (It . a7)) = m3 by A12, A15, A37;



 A40: (( Initialize It) . a) = (It . a) by SCMPDS_5: 15;



 A41: (( Initialize It) . a3) = (It . a3) by SCMPDS_5: 15;



 A42: (( Initialize It) . a7) = (It . a7) by SCMPDS_5: 15;

 consider m5,mE3 be Nat such that

 A43: m5 = (( IExec (WH2,IT,( Initialize It))) . a5) and

 A44: (( IExec (WH2,IT,( Initialize It))) . a3) = mE3 and

 A45: (mE3 + m5) = m3 and

 A46: m5 <= (It . a7) and

 A47: for i be Nat st (It . a3) <= i & i < mE3 holds (( IExec (WH2,IT,( Initialize It))) . ( intpos md)) > (( IExec (WH2,IT,( Initialize It))) . ( intpos i)) and

 A48: m5 = 0 or (( IExec (WH2,IT,( Initialize It))) . ( intpos md)) <= (( IExec (WH2,IT,( Initialize It))) . ( intpos mE3)) by A9, A11, A12, A14, A29, A33, A36, A15, A27, A38, A39, A40, A41, A42;



 A49: ( IExec (WH2,Q,t)) = ( IExec (WH2,IT,( Initialize It))) by A11, A13, A16, A17, Th6;

 hence (( IExec (WH2,Q,t)) . a) = 0 by A9, A11, A12, A14, A29, A33, A36, A15, A27, A38, A39, A40, A41, A42;

 thus (( IExec (WH2,Q,t)) . a1) = (t . a1) by A9, A11, A12, A14, A29, A31, A34, A33, A36, A15, A27, A38, A39, A49, A40, A41, A42;

 thus (( IExec (WH2,Q,t)) . a7) = 0 by A9, A11, A12, A14, A29, A33, A36, A15, A27, A38, A39, A49, A40, A41, A42;

 thus (( IExec (WH2,Q,t)) . a2) = (t . a2) by A9, A11, A12, A14, A29, A33, A36, A15, A27, A38, A39, A49, A40, A41, A42;

 thus (( IExec (WH2,Q,t)) . a4) = (t . a4) by A9, A11, A12, A14, A29, A33, A36, A32, A35, A15, A27, A38, A39, A49, A40, A41, A42;

 A50:

 now

 let i be Nat;

 assume

 A51: i >= 8;

 (( Initialize It) . ( intpos i)) = (It . ( intpos i)) by SCMPDS_5: 15;



 hence (( IExec (WH2,Q,t)) . ( intpos i)) = (It . ( intpos i)) by A9, A11, A12, A14, A29, A33, A36, A15, A27, A38, A39, A49, A40, A41, A42, A51

 .= (t . ( intpos i)) by A6, A10, A13, A14, A28, A51, Lm13;

 end;

 hence for i be Nat st i >= 8 holds (( IExec (WH2,Q,t)) . ( intpos i)) = (t . ( intpos i));

 take m5;

 take mE3;

 thus m5 = (( IExec (WH2,Q,t)) . a5) by A11, A13, A16, A17, A43, Th6;

 thus (( IExec (WH2,Q,t)) . a3) = mE3 by A11, A13, A16, A17, A44, Th6;

 thus (mE3 + m5) = m3 by A45;

 thus m5 <= (t . a7) by A15, A37, A46, XREAL_1: 146, XXREAL_0: 2;



 A52: (It . a3) = ((t . a3) + 1) by A6, A10, A13, A14, A37, Lm13;

 hereby

 let i be Nat;

 assume that

 A53: (t . a3) <= i and

 A54: i < mE3;

 per cases ;

 suppose

 A55: i = (t . a3);

 (( IExec (WH2,Q,t)) . ( intpos me)) = (t . ( intpos me)) by A13, A50;

 hence (( IExec (WH2,Q,t)) . ( intpos md)) > (( IExec (WH2,Q,t)) . ( intpos i)) by A6, A37, A50, A55;

 end;

 suppose i <> (t . a3);

 then (t . a3) (( IExec (WH2,Q,t)) . ( intpos i)) by A49, A47, A54;

 end;

 end;

 thus m5 = 0 or (( IExec (WH2,Q,t)) . ( intpos md)) <= (( IExec (WH2,Q,t)) . ( intpos mE3)) by A49, A48;

 end;

 suppose

 A56: (t . ( intpos md)) <= (t . ( intpos me));

 A57:

 now

 let x;



 A58: (( Initialize It) . GBP ) = (It . GBP ) by SCMPDS_5: 15;



 A59: (( Initialize It) . ( DataLoc ((It . GBP ),7))) = (It . ( DataLoc ((It . GBP ),7))) by SCMPDS_5: 15;



 thus (( IExec (WH2,Q,t)) . x) = (( IExec (WH2,IT,( Initialize It))) . x) by A11, A13, A16, A17, Th6

 .= (( Initialize It) . x) by A27, A30, A56, A58, A59, SCMPDS_8: 23

 .= (It . x) by SCMPDS_5: 15;

 end;

 hence (( IExec (WH2,Q,t)) . a) = 0 by A29;

 thus (( IExec (WH2,Q,t)) . a1) = (t . a1) by A31, A57;

 thus (( IExec (WH2,Q,t)) . a7) = 0 by A27, A56, A57;

 thus (( IExec (WH2,Q,t)) . a2) = (t . a2) by A33, A57;

 thus (( IExec (WH2,Q,t)) . a4) = (t . a4) by A32, A57;

 A60:

 now

 let i be Nat;

 assume

 A61: i >= 8;



 thus (( IExec (WH2,Q,t)) . ( intpos i)) = (It . ( intpos i)) by A57

 .= (t . ( intpos i)) by A6, A10, A13, A14, A28, A61, Lm13;

 end;

 hence for i be Nat st i >= 8 holds (( IExec (WH2,Q,t)) . ( intpos i)) = (t . ( intpos i));

 reconsider m5 = ((k + 1) + 1) as Nat;

 take m5;

 take mE3 = me;

 thus (( IExec (WH2,Q,t)) . a5) = m5 by A11, A27, A56, A57;

 thus (( IExec (WH2,Q,t)) . a3) = mE3 by A27, A56, A57;

 thus (mE3 + m5) = m3 by A11, A12;

 thus m5 <= (t . a7) by A11;

 thus for i be Nat st (t . a3) <= i & i < mE3 holds (( IExec (WH2,Q,t)) . ( intpos md)) > (( IExec (WH2,Q,t)) . ( intpos i));

 (( IExec (WH2,Q,t)) . ( intpos me)) = (t . ( intpos me)) by A13, A60;

 hence m5 = 0 or (( IExec (WH2,Q,t)) . ( intpos md)) <= (( IExec (WH2,Q,t)) . ( intpos mE3)) by A6, A56, A60;

 end;

 end;

 hence P[(k + 1)];

 end;

 P[ 0 ]

 proof

 let t be 0 -started State of SCMPDS ;

 let Q;

 set b = ( DataLoc ((t . a),7));

 assume that

 A62: (t . a) = 0 and

 A63: (t . a7) = ( 0 + 1) and

 A64: ((t . a3) + (t . a7)) = m3 and

 A65: (t . a3) >= 8 and

 A66: (t . a2) = md;

 reconsider me = (t . a3) as Element of NAT by A65, INT_1: 3;

 set It = ( IExec (WB2,Q,t)), IT = Q;



 A67: (t . ( intpos md)) > (t . ( intpos me)) implies (It . a7) = ((t . a7) - 1) & (It . a3) = ((t . a3) + 1) & (It . a5) = ((t . a7) - 1) by A6, A62, A65, A66, Lm13;



 A68: b = ( intpos ( 0 + 7)) by A62, SCMP_GCD: 1;

 A69:

 now

 let v be 0 -started State of SCMPDS , V;

 assume that

 A70: (v . a3) >= 8 and

 A71: (v . a2) = (t . a2) and

 A72: (v . a) = (t . a) and

 A73: (v . b) > 0 ;

 reconsider ME = (v . a3) as Element of NAT by A70, INT_1: 3;



 A74: ME = (v . a3);



 A75: (v . ( intpos ME)) = (v . ( intpos ME));

 thus (( IExec (WB2,V,v)) . a) = (v . a) by A6, A62, A66, A70, A71, A72, Lm13, A75;

 thus WB2 is_closed_on (v,V) & WB2 is_halting_on (v,V) by SCMPDS_6: 20, SCMPDS_6: 21;



 A76: (v . ( intpos md)) > (v . ( intpos ME)) implies (( IExec (WB2,V,v)) . a7) = ((v . a7) - 1) & (( IExec (WB2,V,v)) . a3) = ((v . a3) + 1) & (( IExec (WB2,V,v)) . a5) = ((v . a7) - 1) by A6, A62, A66, A70, A71, A72, Lm13;

 hereby

 per cases ;

 suppose

 A77: (v . ( intpos md)) > (v . ( intpos ME));

 hence (( IExec (WB2,V,v)) . b) < (v . b) by A68, A76, XREAL_1: 146;



 A78: ((v . a3) + 1) > (v . a3) by XREAL_1: 29;

 (( IExec (WB2,V,v)) . a3) = ((v . a3) + 1) by A6, A62, A66, A70, A71, A72, A77, Lm13;

 hence (( IExec (WB2,V,v)) . a3) >= 8 by A70, A78, XXREAL_0: 2;

 end;

 suppose

 A79: (v . ( intpos md)) <= (v . ( intpos ME));

 hence (( IExec (WB2,V,v)) . b) < (v . b) by A6, A62, A66, A68, A70, A71, A72, A73, Lm13;

 thus (( IExec (WB2,V,v)) . a3) >= 8 by A6, A62, A66, A70, A71, A72, A79, Lm13;

 end;

 end;

 thus (( IExec (WB2,V,v)) . a2) = (v . a2) by A6, A62, A66, A70, A71, A72, A74, Lm13;

 end;



 A80: (t . ( intpos md)) <= (t . ( intpos me)) implies (It . a7) = 0 & (It . a3) = (t . a3) & (It . a5) = (t . a7) by A6, A62, A65, A66, Lm13;



 A81: me = (t . a3);

 then

 A82: (It . a) = 0 by A6, A62, A65, A66, Lm13;

 then

 A83: (It . ( DataLoc ((It . a),7))) = 0 by A63, A67, A80, SCMP_GCD: 1;

 A84:

 now

 let x;



 A85: (( Initialize It) . GBP ) = (It . GBP ) by SCMPDS_5: 15;



 A86: (( Initialize It) . ( DataLoc ((It . GBP ),7))) = (It . ( DataLoc ((It . GBP ),7))) by SCMPDS_5: 15;



 thus (( IExec (WH2,Q,t)) . x) = (( IExec (WH2,IT,( Initialize It))) . x) by A63, A65, A68, A69, Th6

 .= (( Initialize It) . x) by A83, A85, A86, SCMPDS_8: 23

 .= (It . x) by SCMPDS_5: 15;

 end;

 hence (( IExec (WH2,Q,t)) . a) = 0 by A82;

 (It . a1) = (t . a1) by A6, A62, A65, A66, A81, Lm13;

 hence (( IExec (WH2,Q,t)) . a1) = (t . a1) by A84;

 thus (( IExec (WH2,Q,t)) . a7) = 0 by A63, A67, A80, A84;

 (It . a2) = (t . a2) by A6, A62, A65, A66, A81, Lm13;

 hence (( IExec (WH2,Q,t)) . a2) = (t . a2) by A84;

 (It . a4) = (t . a4) by A6, A62, A65, A66, A81, Lm13;

 hence (( IExec (WH2,Q,t)) . a4) = (t . a4) by A84;

 A87:

 now

 let i be Nat;

 assume

 A88: i >= 8;



 thus (( IExec (WH2,Q,t)) . ( intpos i)) = (It . ( intpos i)) by A84

 .= (t . ( intpos i)) by A6, A62, A65, A66, A81, A88, Lm13;

 end;

 hence for i be Nat st i >= 8 holds (( IExec (WH2,Q,t)) . ( intpos i)) = (t . ( intpos i));



 A89: (( IExec (WH2,Q,t)) . ( intpos me)) = (t . ( intpos me)) by A65, A87;

 per cases ;

 suppose

 A90: (t . ( intpos md)) > (t . ( intpos me));

 reconsider m5 = 0 as Nat;

 take m5;

 take mE3 = m3;

 thus (( IExec (WH2,Q,t)) . a5) = m5 by A63, A67, A84, A90;

 thus (( IExec (WH2,Q,t)) . a3) = mE3 by A63, A64, A67, A84, A90;

 thus (mE3 + m5) = m3;

 thus m5 <= (t . a7) by A63;

 hereby

 let i be Nat;

 assume that

 A91: (t . a3) <= i and

 A92: i < mE3;

 i <= me by A63, A64, A92, NAT_1: 13;

 then i = (t . a3) by A91, XXREAL_0: 1;

 hence (( IExec (WH2,Q,t)) . ( intpos md)) > (( IExec (WH2,Q,t)) . ( intpos i)) by A6, A87, A89, A90;

 end;

 thus m5 = 0 or (( IExec (WH2,Q,t)) . ( intpos md)) <= (( IExec (WH2,Q,t)) . ( intpos mE3));

 end;

 suppose

 A93: (t . ( intpos md)) <= (t . ( intpos me));

 reconsider m5 = 1 as Nat;

 take m5;

 take mE3 = me;

 thus (( IExec (WH2,Q,t)) . a5) = m5 by A63, A80, A84, A93;

 thus (( IExec (WH2,Q,t)) . a3) = mE3 by A80, A84, A93;

 thus (mE3 + m5) = m3 by A63, A64;

 thus m5 <= (t . a7) by A63;

 thus for i be Nat st (t . a3) <= i & i < mE3 holds (( IExec (WH2,Q,t)) . ( intpos md)) > (( IExec (WH2,Q,t)) . ( intpos i));

 thus m5 = 0 or (( IExec (WH2,Q,t)) . ( intpos md)) <= (( IExec (WH2,Q,t)) . ( intpos mE3)) by A6, A87, A89, A93;

 end;

 end;

 then

 A94: P[ 0 ];

 for k be Nat holds P[k] from NAT_1:sch 2( A94, A8);

 hence thesis by A1, A3, A4, A5, A7;

 end;



 Lm15: for s be 0 -started State of SCMPDS , md be Nat st (s . GBP ) = 0 & (s . a3) >= 8 & (s . a2) = md & md >= 8 holds WH2 is_closed_on (s,P) & WH2 is_halting_on (s,P)

 proof

 set a = GBP ;

 let s be 0 -started State of SCMPDS , md be Nat;

 set b = ( DataLoc ((s . a),7));

 assume that

 A1: (s . a) = 0 and

 A2: (s . a3) >= 8 and

 A3: (s . a2) = md and

 A4: md >= 8;



 A5: b = ( intpos ( 0 + 7)) by A1, SCMP_GCD: 1;

 now

 let v be 0 -started State of SCMPDS , V;

 assume that

 A6: (v . a3) >= 8 and

 A7: (v . a2) = (s . a2) and

 A8: (v . a) = (s . a) and

 A9: (v . b) > 0 ;

 reconsider ME = (v . a3) as Element of NAT by A6, INT_1: 3;



 A10: ME = (v . a3);

 hence (( IExec (WB2,V,v)) . a) = (v . a) by A1, A3, A4, A6, A7, A8, Lm13;

 thus WB2 is_closed_on (v,V) & WB2 is_halting_on (v,V) by SCMPDS_6: 20, SCMPDS_6: 21;



 A11: (v . ( intpos md)) > (v . ( intpos ME)) implies (( IExec (WB2,V,v)) . a7) = ((v . a7) - 1) & (( IExec (WB2,V,v)) . a3) = ((v . a3) + 1) & (( IExec (WB2,V,v)) . a5) = ((v . a7) - 1) by A1, A3, A4, A6, A7, A8, Lm13;

 hereby

 per cases ;

 suppose

 A12: (v . ( intpos md)) > (v . ( intpos ME));

 hence (( IExec (WB2,V,v)) . b) < (v . b) by A5, A11, XREAL_1: 146;



 A13: ((v . a3) + 1) > (v . a3) by XREAL_1: 29;

 (( IExec (WB2,V,v)) . a3) = ((v . a3) + 1) by A1, A3, A4, A6, A7, A8, A12, Lm13;

 hence (( IExec (WB2,V,v)) . a3) >= 8 by A6, A13, XXREAL_0: 2;

 end;

 suppose

 A14: (v . ( intpos md)) <= (v . ( intpos ME));

 hence (( IExec (WB2,V,v)) . b) < (v . b) by A1, A3, A4, A5, A6, A7, A8, A9, Lm13;

 thus (( IExec (WB2,V,v)) . a3) >= 8 by A1, A3, A4, A6, A7, A8, A14, Lm13;

 end;

 end;

 thus (( IExec (WB2,V,v)) . a2) = (v . a2) by A1, A3, A4, A6, A7, A8, A10, Lm13;

 end;

 hence thesis by A2, Th6;

 end;



 Lm16: ( card WB3) = 29

 proof



 thus ( card WB3) = (( card (WH1 ';' WH2)) + ( card IF3)) by AFINSQ_1: 17

 .= ((11 + 11) + ( card IF3)) by Lm10, Lm12, AFINSQ_1: 17

 .= (22 + (( card (((((k5 ';' k6) ';' k7) ';' k8) ';' k9) ';' k0)) + 1)) by SCMPDS_6: 75

 .= (22 + ((( card ((((k5 ';' k6) ';' k7) ';' k8) ';' k9)) + 1) + 1)) by SCMP_GCD: 4

 .= (22 + (((( card (((k5 ';' k6) ';' k7) ';' k8)) + 1) + 1) + 1)) by SCMP_GCD: 4

 .= (22 + ((((( card ((k5 ';' k6) ';' k7)) + 1) + 1) + 1) + 1)) by SCMP_GCD: 4

 .= (22 + (((((( card (k5 ';' k6)) + 1) + 1) + 1) + 1) + 1)) by SCMP_GCD: 4

 .= (22 + (((((2 + 1) + 1) + 1) + 1) + 1)) by SCMP_GCD: 5

 .= 29;

 end;



 Lm17: ( card WH3) = 31

 proof



 thus ( card WH3) = (29 + 2) by Lm16, SCMPDS_8: 17

 .= 31;

 end;

 begin

 theorem :: SCPQSORT:9



 Th9: ( card Partition ) = 38

 proof



 thus ( card Partition ) = (( card (((K4 ';' WH3) ';' j8) ';' j9)) + 1) by SCMP_GCD: 4

 .= ((( card ((K4 ';' WH3) ';' j8)) + 1) + 1) by SCMP_GCD: 4

 .= (((( card (K4 ';' WH3)) + 1) + 1) + 1) by SCMP_GCD: 4

 .= ((((( card K4) + ( card WH3)) + 1) + 1) + 1) by AFINSQ_1: 17

 .= ((((4 + 31) + 1) + 1) + 1) by Lm17, Th4

 .= 38;

 end;



 Lm18: for s be 0 -started State of SCMPDS , m3,m4 be Nat st (s . GBP ) = 0 & (s . a5) > 0 & (s . a3) = m3 & (s . a4) = m4 & m3 > 6 & m4 > 6 holds (( IExec (IF3,P,( Initialize s))) . GBP ) = 0 & (( IExec (IF3,P,( Initialize s))) . a1) = (s . a1) & (( IExec (IF3,P,( Initialize s))) . a2) = (s . a2) & (( IExec (IF3,P,( Initialize s))) . ( intpos m3)) = (s . ( intpos m4)) & (( IExec (IF3,P,( Initialize s))) . ( intpos m4)) = (s . ( intpos m3)) & (( IExec (IF3,P,( Initialize s))) . a3) = ((s . a3) + 1) & (( IExec (IF3,P,( Initialize s))) . a4) = ((s . a4) - 1) & (( IExec (IF3,P,( Initialize s))) . a5) = ((s . a5) - 2) & for i be Nat st i >= 8 & i <> m3 & i <> m4 holds (( IExec (IF3,P,( Initialize s))) . ( intpos i)) = (s . ( intpos i))

 proof

 set a = GBP ;

 let s be 0 -started State of SCMPDS , m3,m4 be Nat;

 assume that

 A1: (s . a) = 0 and

 A2: (s . a5) > 0 and

 A3: (s . a3) = m3 and

 A4: (s . a4) = m4 and

 A5: m3 > 6 and

 A6: m4 > 6;



 A7: m4 > 3 by A6, XXREAL_0: 2;

 set x = ( intpos m3), y = ( intpos m4), t0 = ( Initialize s), t1 = ( IExec ((((((k5 ';' k6) ';' k7) ';' k8) ';' k9) ';' k0),P,( Initialize s))), t2 = ( IExec (((((k5 ';' k6) ';' k7) ';' k8) ';' k9),P,( Initialize s))), t3 = ( IExec ((((k5 ';' k6) ';' k7) ';' k8),P,( Initialize s))), t4 = ( IExec (((k5 ';' k6) ';' k7),P,( Initialize s))), t5 = ( IExec ((k5 ';' k6),P,( Initialize s))), t6 = ( Exec (k5,t0));



 A8: (t0 . a) = 0 by A1, SCMPDS_5: 15;

 then

 A9: ( DataLoc ((t0 . a),6)) = ( intpos ( 0 + 6)) by SCMP_GCD: 1;

 then

 A10: (t6 . a) = 0 by A8, AMI_3: 10, SCMPDS_2: 47;



 A11: (t0 . a4) = m4 by A4, SCMPDS_5: 15;

 then

 A12: (t6 . a4) = m4 by A9, AMI_3: 10, SCMPDS_2: 47;

 then

 A13: ( DataLoc ((t6 . a4), 0 )) = ( intpos (m4 + 0 )) by SCMP_GCD: 1;



 A14: (t5 . a) = (( Exec (k6,t6)) . a) by SCMPDS_5: 42

 .= 0 by A6, A10, A13, AMI_3: 10, SCMPDS_2: 47;



 A15: (t6 . a6) = (t0 . ( DataLoc ((t0 . a4), 0 ))) by A9, SCMPDS_2: 47

 .= (t0 . ( intpos (m4 + 0 ))) by A11, SCMP_GCD: 1

 .= (s . y) by SCMPDS_5: 15;

 (t5 . a6) = (( Exec (k6,t6)) . a6) by SCMPDS_5: 42

 .= (s . y) by A6, A15, A13, AMI_3: 10, SCMPDS_2: 47;

 then

 A16: (t5 . ( DataLoc ((t5 . a),6))) = (s . y) by A14, SCMP_GCD: 1;

 (t0 . a3) = m3 by A3, SCMPDS_5: 15;

 then

 A17: (t6 . a3) = m3 by A9, AMI_3: 10, SCMPDS_2: 47;



 A18: (t5 . a3) = (( Exec (k6,t6)) . a3) by SCMPDS_5: 42

 .= m3 by A7, A17, A13, AMI_3: 10, SCMPDS_2: 47;

 then

 A19: ( DataLoc ((t5 . a3), 0 )) = ( intpos (m3 + 0 )) by SCMP_GCD: 1;



 A20: (t0 . x) = (s . x) by SCMPDS_5: 15;



 A21: (t5 . y) = (( Exec (k6,t6)) . y) by SCMPDS_5: 42

 .= (t6 . ( DataLoc ((t6 . a3), 0 ))) by A13, SCMPDS_2: 47

 .= (t6 . ( intpos (m3 + 0 ))) by A17, SCMP_GCD: 1

 .= (s . x) by A5, A20, A9, AMI_3: 10, SCMPDS_2: 47;

 A22:

 now

 per cases ;

 suppose

 A23: y <> ( DataLoc ((t5 . a3), 0 ));



 thus (t4 . y) = (( Exec (k7,t5)) . y) by SCMPDS_5: 41

 .= (s . x) by A21, A23, SCMPDS_2: 47;

 end;

 suppose

 A24: y = ( DataLoc ((t5 . a3), 0 ));



 thus (t4 . y) = (( Exec (k7,t5)) . y) by SCMPDS_5: 41

 .= (s . x) by A19, A16, A24, SCMPDS_2: 47;

 end;

 end;

 (t0 . a2) = (s . a2) by SCMPDS_5: 15;

 then

 A25: (t6 . a2) = (s . a2) by A9, AMI_3: 10, SCMPDS_2: 47;



 A26: m4 > 2 by A6, XXREAL_0: 2;



 A27: m3 > 5 by A5, XXREAL_0: 2;



 A28: (t4 . a) = (( Exec (k7,t5)) . a) by SCMPDS_5: 41

 .= 0 by A5, A14, A19, AMI_3: 10, SCMPDS_2: 47;

 then

 A29: ( DataLoc ((t4 . a),5)) = ( intpos ( 0 + 5)) by SCMP_GCD: 1;



 A30: (t3 . a) = (( Exec (k8,t4)) . a) by SCMPDS_5: 41

 .= 0 by A28, A29, AMI_3: 10, SCMPDS_2: 48;

 then

 A31: a <> ( DataLoc ((t3 . a),3)) by AMI_3: 10, SCMP_GCD: 1;



 A32: m3 > 3 by A5, XXREAL_0: 2;

 then

 A33: x <> ( DataLoc ((t3 . a),3)) by A30, AMI_3: 10, SCMP_GCD: 1;



 A34: (t4 . a3) = (( Exec (k7,t5)) . a3) by SCMPDS_5: 41

 .= m3 by A32, A18, A19, AMI_3: 10, SCMPDS_2: 47;



 A35: (t3 . a3) = (( Exec (k8,t4)) . a3) by SCMPDS_5: 41

 .= m3 by A34, A29, AMI_3: 10, SCMPDS_2: 48;



 A36: ( DataLoc ((t3 . a),3)) = ( intpos ( 0 + 3)) by A30, SCMP_GCD: 1;



 A37: (t2 . a3) = (( Exec (k9,t3)) . a3) by SCMPDS_5: 41

 .= (m3 + 1) by A35, A36, SCMPDS_2: 48;



 A38: m3 > 2 by A5, XXREAL_0: 2;



 A39: m4 > 5 by A6, XXREAL_0: 2;



 A40: (t5 . a2) = (( Exec (k6,t6)) . a2) by SCMPDS_5: 42

 .= (s . a2) by A25, A13, A26, AMI_3: 10, SCMPDS_2: 47;



 A41: (t4 . x) = (( Exec (k7,t5)) . x) by SCMPDS_5: 41

 .= (s . y) by A19, A16, SCMPDS_2: 47;



 A42: (t3 . x) = (( Exec (k8,t4)) . x) by SCMPDS_5: 41

 .= (s . y) by A27, A41, A29, AMI_3: 10, SCMPDS_2: 48;



 A43: (t2 . x) = (( Exec (k9,t3)) . x) by SCMPDS_5: 41

 .= (s . y) by A42, A33, SCMPDS_2: 48;



 A44: (t3 . y) = (( Exec (k8,t4)) . y) by SCMPDS_5: 41

 .= (s . x) by A39, A22, A29, AMI_3: 10, SCMPDS_2: 48;



 A45: y <> ( DataLoc ((t3 . a),3)) by A7, A30, AMI_3: 10, SCMP_GCD: 1;



 A46: (t2 . y) = (( Exec (k9,t3)) . y) by SCMPDS_5: 41

 .= (s . x) by A44, A45, SCMPDS_2: 48;



 A47: (t4 . a2) = (( Exec (k7,t5)) . a2) by SCMPDS_5: 41

 .= (s . a2) by A40, A19, A38, AMI_3: 10, SCMPDS_2: 47;



 A48: (t3 . a2) = (( Exec (k8,t4)) . a2) by SCMPDS_5: 41

 .= (s . a2) by A47, A29, AMI_3: 10, SCMPDS_2: 48;



 A49: a2 <> ( DataLoc ((t3 . a),3)) by A30, AMI_3: 10, SCMP_GCD: 1;



 A50: (t2 . a2) = (( Exec (k9,t3)) . a2) by SCMPDS_5: 41

 .= (s . a2) by A48, A49, SCMPDS_2: 48;



 A51: m4 > 1 by A6, XXREAL_0: 2;



 A52: ( DataLoc ((s . a),5)) = ( intpos ( 0 + 5)) by A1, SCMP_GCD: 1;

 (t0 . a5) = (s . a5) by SCMPDS_5: 15;

 then

 A53: (t6 . a5) = (s . a5) by A9, AMI_3: 10, SCMPDS_2: 47;



 A54: (t5 . a5) = (( Exec (k6,t6)) . a5) by SCMPDS_5: 42

 .= (s . a5) by A39, A53, A13, AMI_3: 10, SCMPDS_2: 47;



 A55: (t4 . a5) = (( Exec (k7,t5)) . a5) by SCMPDS_5: 41

 .= (s . a5) by A27, A54, A19, AMI_3: 10, SCMPDS_2: 47;



 A56: (t3 . a5) = (( Exec (k8,t4)) . a5) by SCMPDS_5: 41

 .= ((t4 . a5) + ( - 2)) by A29, SCMPDS_2: 48

 .= ((s . a5) - 2) by A55;



 A57: a5 <> ( DataLoc ((t3 . a),3)) by A30, AMI_3: 10, SCMP_GCD: 1;



 A58: (t2 . a5) = (( Exec (k9,t3)) . a5) by SCMPDS_5: 41

 .= ((s . a5) - 2) by A56, A57, SCMPDS_2: 48;

 (t0 . a1) = (s . a1) by SCMPDS_5: 15;

 then

 A59: (t6 . a1) = (s . a1) by A9, AMI_3: 10, SCMPDS_2: 47;



 A60: m3 > 1 by A5, XXREAL_0: 2;



 A61: (t2 . a) = (( Exec (k9,t3)) . a) by SCMPDS_5: 41

 .= 0 by A30, A31, SCMPDS_2: 48;

 then

 A62: a <> ( DataLoc ((t2 . a),4)) by AMI_3: 10, SCMP_GCD: 1;



 A63: m4 > 4 by A6, XXREAL_0: 2;

 then

 A64: y <> ( DataLoc ((t2 . a),4)) by A61, AMI_3: 10, SCMP_GCD: 1;



 A65: m3 > 4 by A5, XXREAL_0: 2;

 then

 A66: x <> ( DataLoc ((t2 . a),4)) by A61, AMI_3: 10, SCMP_GCD: 1;



 A67: a4 <> ( DataLoc ((t3 . a),3)) by A30, AMI_3: 10, SCMP_GCD: 1;



 A68: (t5 . a4) = (( Exec (k6,t6)) . a4) by SCMPDS_5: 42

 .= m4 by A63, A12, A13, AMI_3: 10, SCMPDS_2: 47;



 A69: (t4 . a4) = (( Exec (k7,t5)) . a4) by SCMPDS_5: 41

 .= m4 by A65, A68, A19, AMI_3: 10, SCMPDS_2: 47;



 A70: (t3 . a4) = (( Exec (k8,t4)) . a4) by SCMPDS_5: 41

 .= m4 by A69, A29, AMI_3: 10, SCMPDS_2: 48;



 A71: (t2 . a4) = (( Exec (k9,t3)) . a4) by SCMPDS_5: 41

 .= m4 by A70, A67, SCMPDS_2: 48;



 A72: a3 <> ( DataLoc ((t2 . a),4)) by A61, AMI_3: 10, SCMP_GCD: 1;



 A73: (t5 . a1) = (( Exec (k6,t6)) . a1) by SCMPDS_5: 42

 .= (s . a1) by A59, A13, A51, AMI_3: 10, SCMPDS_2: 47;



 A74: (t4 . a1) = (( Exec (k7,t5)) . a1) by SCMPDS_5: 41

 .= (s . a1) by A73, A19, A60, AMI_3: 10, SCMPDS_2: 47;



 A75: (t3 . a1) = (( Exec (k8,t4)) . a1) by SCMPDS_5: 41

 .= (s . a1) by A74, A29, AMI_3: 10, SCMPDS_2: 48;



 A76: a1 <> ( DataLoc ((t3 . a),3)) by A30, AMI_3: 10, SCMP_GCD: 1;



 A77: (t2 . a1) = (( Exec (k9,t3)) . a1) by SCMPDS_5: 41

 .= (s . a1) by A75, A76, SCMPDS_2: 48;



 A78: a1 <> ( DataLoc ((t2 . a),4)) by A61, AMI_3: 10, SCMP_GCD: 1;

 (t1 . a) = (( Exec (k0,t2)) . a) by SCMPDS_5: 41

 .= 0 by A61, A62, SCMPDS_2: 48;

 hence (( IExec (IF3,P,( Initialize s))) . a) = 0 by A2, A52, SCMPDS_6: 83;

 (t1 . a1) = (( Exec (k0,t2)) . a1) by SCMPDS_5: 41

 .= (s . a1) by A77, A78, SCMPDS_2: 48;

 hence (( IExec (IF3,P,( Initialize s))) . a1) = (s . a1) by A2, A52, SCMPDS_6: 83;



 A79: a2 <> ( DataLoc ((t2 . a),4)) by A61, AMI_3: 10, SCMP_GCD: 1;

 (t1 . a2) = (( Exec (k0,t2)) . a2) by SCMPDS_5: 41

 .= (s . a2) by A50, A79, SCMPDS_2: 48;

 hence (( IExec (IF3,P,( Initialize s))) . a2) = (s . a2) by A2, A52, SCMPDS_6: 83;

 (t1 . x) = (( Exec (k0,t2)) . x) by SCMPDS_5: 41

 .= (s . y) by A43, A66, SCMPDS_2: 48;

 hence (( IExec (IF3,P,( Initialize s))) . x) = (s . y) by A2, A52, SCMPDS_6: 83;

 (t1 . y) = (( Exec (k0,t2)) . y) by SCMPDS_5: 41

 .= (s . x) by A46, A64, SCMPDS_2: 48;

 hence (( IExec (IF3,P,( Initialize s))) . y) = (s . x) by A2, A52, SCMPDS_6: 83;



 A80: ( DataLoc ((t2 . a),4)) = ( intpos ( 0 + 4)) by A61, SCMP_GCD: 1;

 (t1 . a3) = (( Exec (k0,t2)) . a3) by SCMPDS_5: 41

 .= (m3 + 1) by A37, A72, SCMPDS_2: 48;

 hence (( IExec (IF3,P,( Initialize s))) . a3) = ((s . a3) + 1) by A2, A3, A52, SCMPDS_6: 83;

 (t1 . a4) = (( Exec (k0,t2)) . a4) by SCMPDS_5: 41

 .= ((t2 . a4) + ( - 1)) by A80, SCMPDS_2: 48

 .= ((s . a4) - 1) by A4, A71;

 hence (( IExec (IF3,P,( Initialize s))) . a4) = ((s . a4) - 1) by A2, A52, SCMPDS_6: 83;



 A81: a5 <> ( DataLoc ((t2 . a),4)) by A61, AMI_3: 10, SCMP_GCD: 1;

 (t1 . a5) = (( Exec (k0,t2)) . a5) by SCMPDS_5: 41

 .= ((s . a5) - 2) by A58, A81, SCMPDS_2: 48;

 hence (( IExec (IF3,P,( Initialize s))) . a5) = ((s . a5) - 2) by A2, A52, SCMPDS_6: 83;

 A82:

 now

 let i be Nat;

 assume that

 A83: i >= 8 and i <> m3 and i <> m4;

 i > 6 by A83, XXREAL_0: 2;



 hence (t6 . ( intpos i)) = (t0 . ( intpos i)) by A9, AMI_3: 10, SCMPDS_2: 47

 .= (s . ( intpos i)) by SCMPDS_5: 15;

 end;

 A84:

 now

 let i be Nat;

 assume that

 A85: i >= 8 and

 A86: i <> m3 and

 A87: i <> m4;



 thus (t5 . ( intpos i)) = (( Exec (k6,t6)) . ( intpos i)) by SCMPDS_5: 42

 .= (t6 . ( intpos i)) by A13, A87, AMI_3: 10, SCMPDS_2: 47

 .= (s . ( intpos i)) by A82, A85, A86, A87;

 end;

 A88:

 now

 let i be Nat;

 assume that

 A89: i >= 8 and

 A90: i <> m3 and

 A91: i <> m4;



 thus (t4 . ( intpos i)) = (( Exec (k7,t5)) . ( intpos i)) by SCMPDS_5: 41

 .= (t5 . ( intpos i)) by A19, A90, AMI_3: 10, SCMPDS_2: 47

 .= (s . ( intpos i)) by A84, A89, A90, A91;

 end;

 A92:

 now

 let i be Nat;

 assume that

 A93: i >= 8 and

 A94: i <> m3 and

 A95: i <> m4;



 A96: i > 5 by A93, XXREAL_0: 2;



 thus (t3 . ( intpos i)) = (( Exec (k8,t4)) . ( intpos i)) by SCMPDS_5: 41

 .= (t4 . ( intpos i)) by A29, A96, AMI_3: 10, SCMPDS_2: 48

 .= (s . ( intpos i)) by A88, A93, A94, A95;

 end;

 A97:

 now

 let i be Nat;

 assume that

 A98: i >= 8 and

 A99: i <> m3 and

 A100: i <> m4;

 i > 3 by A98, XXREAL_0: 2;

 then

 A101: ( intpos i) <> ( DataLoc ((t3 . a),3)) by A30, AMI_3: 10, SCMP_GCD: 1;



 thus (t2 . ( intpos i)) = (( Exec (k9,t3)) . ( intpos i)) by SCMPDS_5: 41

 .= (t3 . ( intpos i)) by A101, SCMPDS_2: 48

 .= (s . ( intpos i)) by A92, A98, A99, A100;

 end;

 A102:

 now

 let i be Nat;

 assume that

 A103: i >= 8 and

 A104: i <> m3 and

 A105: i <> m4;

 i > 4 by A103, XXREAL_0: 2;

 then

 A106: ( intpos i) <> ( DataLoc ((t2 . a),4)) by A61, AMI_3: 10, SCMP_GCD: 1;



 thus (t1 . ( intpos i)) = (( Exec (k0,t2)) . ( intpos i)) by SCMPDS_5: 41

 .= (t2 . ( intpos i)) by A106, SCMPDS_2: 48

 .= (s . ( intpos i)) by A97, A103, A104, A105;

 end;

 hereby

 let i be Nat;

 assume that

 A107: i >= 8 and

 A108: i <> m3 and

 A109: i <> m4;



 thus (( IExec (IF3,P,( Initialize s))) . ( intpos i)) = (t1 . ( intpos i)) by A2, A52, SCMPDS_6: 83

 .= (s . ( intpos i)) by A102, A107, A108, A109;

 end;

 end;



 Lm19: for s be 0 -started State of SCMPDS , md,m3 be Nat st (s . GBP ) = 0 & (s . a5) > 0 & (s . a4) = (m3 + (s . a5)) & m3 = ((s . a3) - 1) & (s . a2) = md & md >= 8 & md <= m3 holds WB3 is_closed_on (s,P) & WB3 is_halting_on (s,P) & (( IExec (WB3,P,s)) . GBP ) = 0 & (( IExec (WB3,P,s)) . a1) = (s . a1) & (( IExec (WB3,P,s)) . a2) = md & (( IExec (WB3,P,s)) . a3) >= (s . a3) & (( IExec (WB3,P,s)) . a4) <= (s . a4) & (( IExec (WB3,P,s)) . a4) >= m3 & (( IExec (WB3,P,s)) . a5) < (s . a5) & (( IExec (WB3,P,s)) . a5) >= ( - 1) & (( IExec (WB3,P,s)) . a4) = (((( IExec (WB3,P,s)) . a3) - 1) + (( IExec (WB3,P,s)) . a5)) & ex n1,n2 be Nat st n1 = ((( IExec (WB3,P,s)) . a3) - 1) & n2 = ((( IExec (WB3,P,s)) . a4) + 1) & (for i be Nat st i >= 8 & i <> n1 & i <> n2 holds (( IExec (WB3,P,s)) . ( intpos i)) = (s . ( intpos i))) & ((( IExec (WB3,P,s)) . ( intpos n1)) = (s . ( intpos n1)) & (( IExec (WB3,P,s)) . ( intpos n2)) = (s . ( intpos n2)) or n1 >= (s . a3) & n2 <= (s . a4) & (( IExec (WB3,P,s)) . ( intpos n1)) = (s . ( intpos n2)) & (( IExec (WB3,P,s)) . ( intpos n2)) = (s . ( intpos n1))) & (for i be Nat st (s . a3) <= i & i <= n1 holds (( IExec (WB3,P,s)) . ( intpos md)) >= (( IExec (WB3,P,s)) . ( intpos i))) & for i be Nat st n2 <= i & i <= (s . a4) holds (( IExec (WB3,P,s)) . ( intpos md)) <= (( IExec (WB3,P,s)) . ( intpos i))

 proof

 set a = GBP ;

 let s be 0 -started State of SCMPDS , md,m3 be Nat;



 A1: ( Initialize s) = s by MEMSTR_0: 44;

 assume that

 A2: (s . a) = 0 and

 A3: (s . a5) > 0 and

 A4: (s . a4) = (m3 + (s . a5)) and

 A5: m3 = ((s . a3) - 1) and

 A6: (s . a2) = md and

 A7: md >= 8 and

 A8: md <= m3;

 set s1 = ( IExec (WH1,P,s));



 A9: (s1 . a) = (( Initialize s1) . a) by SCMPDS_5: 15;



 A10: (s1 . a1) = (( Initialize s1) . a1) by SCMPDS_5: 15;



 A11: (s1 . a2) = (( Initialize s1) . a2) by SCMPDS_5: 15;



 A12: (s1 . a3) = (( Initialize s1) . a3) by SCMPDS_5: 15;



 A13: (s1 . a4) = (( Initialize s1) . a4) by SCMPDS_5: 15;



 A14: (s1 . a7) = (( Initialize s1) . a7) by SCMPDS_5: 15;



 A15: m3 >= 8 by A7, A8, XXREAL_0: 2;

 then

 consider mE be Nat such that

 A16: mE = (s1 . a7) and

 A17: (s1 . a4) = (m3 + mE) and

 A18: mE <= (s . a5) and

 A19: for i be Nat st (m3 + mE) = (s1 . ( intpos (m3 + mE))) by A2, A3, A4, A6, A7, Lm8;



 A21: (s1 . a) = 0 by A2, A3, A4, A6, A7, A15, Lm8;



 A22: (s1 . a1) = (s . a1) by A2, A3, A4, A6, A7, A15, Lm8;



 A23: WH1 is_halting_on (s,P) by A2, A4, A6, A7, A15, Lm9;

 set s2 = ( IExec (WH2,P,( Initialize s1))), sm = ((m3 + 1) + mE), s12 = ( IExec ((WH1 ';' WH2),P,s)), s3 = ( IExec (WB3,P,s)), P12 = P, P3 = P, m4 = (m3 + mE);



 A24: (s12 . a) = (( Initialize s12) . a) by SCMPDS_5: 15;



 A25: (s12 . a1) = (( Initialize s12) . a1) by SCMPDS_5: 15;



 A26: (s12 . a2) = (( Initialize s12) . a2) by SCMPDS_5: 15;



 A27: (s12 . a3) = (( Initialize s12) . a3) by SCMPDS_5: 15;



 A28: (s12 . a4) = (( Initialize s12) . a4) by SCMPDS_5: 15;



 A29: (s12 . a5) = (( Initialize s12) . a5) by SCMPDS_5: 15;



 A30: ((s1 . a3) + (s1 . a7)) = sm by A2, A3, A4, A5, A6, A7, A15, A16, Lm8;



 A31: m4 >= 8 by A15, NAT_1: 12;



 A32: (s1 . a2) = (s . a2) by A2, A3, A4, A6, A7, A15, Lm8;



 A33: (s1 . a3) = (s . a3) by A2, A3, A4, A6, A7, A15, Lm8;

 (s1 . a3) = (m3 + 1) by A2, A3, A4, A5, A6, A7, A15, Lm8;

 then m3 < (s1 . a3) by XREAL_1: 29;

 then

 A34: (s1 . a3) >= 8 by A15, XXREAL_0: 2;



 A35: (s1 . a5) = 0 by A2, A3, A4, A6, A7, A15, Lm8;



 A36: (m3 + mE) <= (m3 + (s . a5)) by A18, XREAL_1: 6;



 A37: WH1 is_closed_on (s,P) by A2, A4, A6, A7, A15, Lm9;

 per cases ;

 suppose

 A38: mE <= 0 ;



 A39: (( Initialize s1) . GBP ) = (s1 . GBP ) by SCMPDS_5: 15;



 A40: (( Initialize s1) . ( DataLoc ((s1 . GBP ),7))) = (s1 . ( DataLoc ((s1 . GBP ),7))) by SCMPDS_5: 15;



 A41: (( Initialize s1) . a5) = (s1 . a5) by SCMPDS_5: 15;



 A42: (( Initialize s1) . a1) = (s1 . a1) by SCMPDS_5: 15;



 A43: (( Initialize s1) . a2) = (s1 . a2) by SCMPDS_5: 15;



 A44: (( Initialize s1) . a3) = (s1 . a3) by SCMPDS_5: 15;



 A45: (( Initialize s1) . a4) = (s1 . a4) by SCMPDS_5: 15;



 A46: ( DataLoc ((s1 . a),7)) = ( intpos ( 0 + 7)) by A21, SCMP_GCD: 1;

 then

 A47: WH2 is_halting_on (s1,P) by A16, A38, SCMPDS_8: 20;



 A48: WH2 is_closed_on (s1,P) by A16, A38, A46, SCMPDS_8: 20;



 then

 A49: (s12 . a5) = (s2 . a5) by A37, A23, A47, SCMPDS_7: 30

 .= 0 by A35, A41, A16, A38, A46, A39, A40, SCMPDS_8: 23;



 A50: (s12 . a) = (s2 . a) by A37, A23, A48, A47, SCMPDS_7: 30

 .= 0 by A21, A39, A16, A38, A46, A40, SCMPDS_8: 23;

 then

 A51: ( DataLoc ((s12 . a),5)) = ( intpos ( 0 + 5)) by SCMP_GCD: 1;



 A52: (WH1 ';' WH2) is_halting_on (s,P) by A37, A23, A48, A47, A1, SCMPDS_7: 24;



 A53: (WH1 ';' WH2) is_closed_on (s,P) by A37, A23, A48, A47, A1, SCMPDS_7: 24;

 hence WB3 is_closed_on (s,P) & WB3 is_halting_on (s,P) by A52, SCPISORT: 8;



 thus (( IExec (WB3,P,s)) . a) = (( IExec (IF3,P12,( Initialize s12))) . a) by A53, A52, SCPISORT: 6

 .= 0 by A50, A49, A51, SCMPDS_6: 84;



 thus (( IExec (WB3,P,s)) . a1) = (( IExec (IF3,P12,( Initialize s12))) . a1) by A53, A52, SCPISORT: 6

 .= (s12 . a1) by A49, A51, SCMPDS_6: 84

 .= (s2 . a1) by A37, A23, A48, A47, SCMPDS_7: 30

 .= (s . a1) by A22, A16, A38, A46, A39, A40, A42, SCMPDS_8: 23;



 thus (s3 . a2) = (( IExec (IF3,P12,( Initialize s12))) . a2) by A53, A52, SCPISORT: 6

 .= (s12 . a2) by A49, A51, SCMPDS_6: 84

 .= (s2 . a2) by A37, A23, A48, A47, SCMPDS_7: 30

 .= md by A6, A32, A16, A38, A46, A39, A40, A43, SCMPDS_8: 23;



 A54: (s3 . a3) = (( IExec (IF3,P12,( Initialize s12))) . a3) by A53, A52, SCPISORT: 6

 .= (s12 . a3) by A49, A51, SCMPDS_6: 84

 .= (s2 . a3) by A37, A23, A48, A47, SCMPDS_7: 30

 .= (s . a3) by A33, A16, A38, A46, A39, A40, A44, SCMPDS_8: 23;

 hence (s3 . a3) >= (s . a3);



 A55: (s3 . a4) = (( IExec (IF3,P12,( Initialize s12))) . a4) by A53, A52, SCPISORT: 6

 .= (s12 . a4) by A49, A51, SCMPDS_6: 84

 .= (s2 . a4) by A37, A23, A48, A47, SCMPDS_7: 30

 .= m4 by A16, A17, A38, A46, A39, A40, A45, SCMPDS_8: 23;

 hence (s3 . a4) <= (s . a4) by A4, A18, XREAL_1: 6;

 thus (s3 . a4) >= m3 by A55, NAT_1: 12;



 A56: (s3 . a5) = (( IExec (IF3,P12,( Initialize s12))) . a5) by A53, A52, SCPISORT: 6

 .= 0 by A49, A51, SCMPDS_6: 84;

 hence (s3 . a5) < (s . a5) by A3;

 thus (s3 . a5) >= ( - 1) by A56;

 thus (s3 . a4) = (((s3 . a3) - 1) + (s3 . a5)) by A5, A38, A54, A55, A56;

 take n1 = m4;

 take n2 = (m4 + 1);



 A57: mE = 0 by A38;

 hence n1 = ((s3 . a3) - 1) by A5, A54;

 thus n2 = ((s3 . a4) + 1) by A55;

 A58:

 now

 let x;



 thus (s3 . x) = (( IExec (IF3,P12,( Initialize s12))) . x) by A53, A52, SCPISORT: 6

 .= (s12 . x) by A49, A51, SCMPDS_6: 84

 .= (s2 . x) by A37, A23, A48, A47, SCMPDS_7: 30;

 end;

 A59:

 now

 let i be Nat;

 assume

 A60: i >= 8;



 A61: (( Initialize s1) . GBP ) = (s1 . GBP ) by SCMPDS_5: 15;



 A62: (( Initialize s1) . ( DataLoc ((s1 . GBP ),7))) = (s1 . ( DataLoc ((s1 . GBP ),7))) by SCMPDS_5: 15;



 thus (s3 . ( intpos i)) = (s2 . ( intpos i)) by A58

 .= (( Initialize s1) . ( intpos i)) by A16, A38, A46, A61, A62, SCMPDS_8: 23

 .= (s1 . ( intpos i)) by SCMPDS_5: 15

 .= (s . ( intpos i)) by A2, A3, A4, A6, A7, A15, A60, Lm8;

 end;

 hence for i be Nat st i >= 8 & i <> n1 & i <> n2 holds (s3 . ( intpos i)) = (s . ( intpos i));



 A63: m4 < n2 by XREAL_1: 29;

 hence (s3 . ( intpos n1)) = (s . ( intpos n1)) & (s3 . ( intpos n2)) = (s . ( intpos n2)) or n1 >= (s . a3) & n2 <= (s . a4) & (s3 . ( intpos n1)) = (s . ( intpos n2)) & (s3 . ( intpos n2)) = (s . ( intpos n1)) by A31, A59, XXREAL_0: 2;

 thus for i be Nat st (s . a3) <= i & i <= n1 holds (s3 . ( intpos md)) >= (s3 . ( intpos i)) by A5, A57, XREAL_1: 146, XXREAL_0: 2;

 hereby



 A64: (( Initialize s1) . GBP ) = (s1 . GBP ) by SCMPDS_5: 15;



 A65: (( Initialize s1) . ( DataLoc ((s1 . GBP ),7))) = (s1 . ( DataLoc ((s1 . GBP ),7))) by SCMPDS_5: 15;



 A66: (s3 . ( intpos md)) = (s2 . ( intpos md)) by A58

 .= (( Initialize s1) . ( intpos md)) by A16, A38, A46, A64, A65, SCMPDS_8: 23

 .= (s1 . ( intpos md)) by SCMPDS_5: 15;

 let i be Nat;

 assume that

 A67: n2 <= i and

 A68: i <= (s . a4);



 A69: (s3 . ( intpos i)) = (s2 . ( intpos i)) by A58

 .= (( Initialize s1) . ( intpos i)) by A16, A38, A46, A64, A65, SCMPDS_8: 23

 .= (s1 . ( intpos i)) by SCMPDS_5: 15;

 m4 0 ;

 then

 consider m5,mE3 be Nat such that

 A71: m5 = (s2 . a5) and

 A72: (s2 . a3) = mE3 and

 A73: (mE3 + m5) = sm and

 A74: m5 <= (( Initialize s1) . a7) and

 A75: (for i be Nat st (( Initialize s1) . a3) <= i & i < mE3 holds (s2 . ( intpos md)) > (s2 . ( intpos i))) and

 A76: (m5 = 0 or (s2 . ( intpos md)) <= (s2 . ( intpos mE3))) by A5, A6, A7, A16, A21, A32, A33, A34, Lm14, A9, A11, A14, A12;



 A77: (mE3 + m5) <= (mE3 + (s1 . a7)) by A74, A14, XREAL_1: 6;

 then

 A78: (s1 . a3) <= mE3 by A5, A33, A16, A73, XREAL_1: 6;



 A79: ((s1 . a3) + (s1 . a7)) <= (mE3 + (s1 . a7)) by A5, A33, A16, A73, A74, A14, XREAL_1: 6;

 then

 A80: (s . a3) <= mE3 by A33, XREAL_1: 6;

 then mE3 >= 8 by A33, A34, XXREAL_0: 2;

 then

 A81: mE3 > 6 by XXREAL_0: 2;



 A82: (( Initialize s1) . GBP ) = (s1 . GBP ) by SCMPDS_5: 15;



 A83: (( Initialize s1) . a2) = (s1 . a2) by SCMPDS_5: 15;



 A84: (( Initialize s1) . a3) = (s1 . a3) by SCMPDS_5: 15;



 A85: WH2 is_halting_on (( Initialize s1),P) by A6, A7, A21, A32, A34, Lm15, A82, A83, A84;



 A86: WH2 is_halting_on (s1,P)

 proof

 (P +* ( stop WH2)) halts_on ( Initialize ( Initialize s1)) by A85, SCMPDS_6:def 3;

 hence thesis by SCMPDS_6:def 3;

 end;



 A87: WH2 is_closed_on (( Initialize s1),P) by A6, A7, A21, A32, A34, Lm15, A82, A83, A84;



 A88: WH2 is_closed_on (s1,P)

 proof

 for k be Nat holds ( IC ( Comput ((P +* ( stop WH2)),( Initialize ( Initialize s1)),k))) in ( dom ( stop WH2)) by A87, SCMPDS_6:def 2;

 hence thesis by SCMPDS_6:def 2;

 end;

 then

 A89: (WH1 ';' WH2) is_closed_on (s,P) by A37, A23, A86, A1, SCMPDS_7: 24;

 (s2 . a1) = (s1 . a1) by A5, A6, A7, A21, A32, A33, A34, A30, A70, Lm14, A9, A10, A11, A12, A14;

 then

 A90: (s12 . a1) = (s . a1) by A22, A37, A23, A88, A86, SCMPDS_7: 30;

 (s2 . a4) = (s1 . a4) by A5, A6, A7, A21, A32, A33, A34, A30, A70, Lm14, A9, A11, A12, A13, A14;

 then

 A91: (s12 . a4) = m4 by A17, A37, A23, A88, A86, SCMPDS_7: 30;

 (s2 . a2) = (s1 . a2) by A5, A6, A7, A21, A32, A33, A34, A30, A70, Lm14, A9, A11, A12, A14;

 then

 A92: (s12 . a2) = md by A6, A32, A37, A23, A88, A86, SCMPDS_7: 30;

 (s2 . a) = 0 by A5, A6, A7, A21, A32, A33, A34, A30, A70, Lm14, A9, A11, A12, A14;

 then

 A93: (s12 . a) = 0 by A37, A23, A88, A86, SCMPDS_7: 30;



 A94: (s12 . a5) = m5 by A37, A23, A71, A88, A86, SCMPDS_7: 30;



 A95: (s12 . a3) = mE3 by A37, A23, A72, A88, A86, SCMPDS_7: 30;



 A96: (WH1 ';' WH2) is_halting_on (s,P) by A37, A23, A88, A86, A1, SCMPDS_7: 24;

 hence WB3 is_closed_on (s,P) & WB3 is_halting_on (s,P) by A89, SCPISORT: 8;



 A97: m4 > 6 by A31, XXREAL_0: 2;



 A98: m4 > (m3 + 0 ) by A70, XREAL_1: 6;

 then

 A99: m4 >= (m3 + 1) by INT_1: 7;



 A100: ( IExec (IF3,P12,( Initialize s12))) = ( IExec (IF3,P12,( Initialize ( Initialize s12))));

 hereby

 per cases ;

 suppose

 A101: (s12 . a5) > 0 ;

 then (( IExec (IF3,P12,( Initialize s12))) . a) = 0 by A93, A95, A81, A91, A97, Lm18, A24, A27, A28, A29, A100;

 hence (( IExec (WB3,P,s)) . a) = 0 by A89, A96, SCPISORT: 6;

 (( IExec (IF3,P12,( Initialize s12))) . a1) = (s12 . a1) by A93, A95, A81, A91, A97, A101, Lm18, A24, A25, A27, A28, A29, A100;

 hence (( IExec (WB3,P,s)) . a1) = (s . a1) by A89, A96, A90, SCPISORT: 6;

 (( IExec (IF3,P12,( Initialize s12))) . a2) = (s12 . a2) by A93, A95, A81, A91, A97, A101, Lm18, A24, A26, A27, A28, A29, A100;

 hence (s3 . a2) = md by A89, A96, A92, SCPISORT: 6;

 (( IExec (IF3,P12,( Initialize s12))) . a3) = ((s12 . a3) + 1) by A93, A95, A81, A91, A97, A101, Lm18, A24, A27, A28, A29, A100;

 then

 A102: (s3 . a3) = (mE3 + 1) by A89, A96, A95, SCPISORT: 6;

 then mE3 < (s3 . a3) by XREAL_1: 29;

 hence (s3 . a3) >= (s . a3) by A33, A78, XXREAL_0: 2;



 A103: (( IExec (IF3,P12,( Initialize s12))) . a4) = ((s12 . a4) - 1) by A93, A95, A81, A91, A97, A101, Lm18, A24, A27, A28, A29, A100;

 then

 A104: (s3 . a4) = (m4 - 1) by A89, A96, A91, SCPISORT: 6;

 hence (s3 . a4) <= (s . a4) by A4, A36, XREAL_1: 146, XXREAL_0: 2;



 A105: (( IExec (IF3,P12,( Initialize s12))) . a5) = ((s12 . a5) - 2) by A93, A95, A81, A91, A97, A101, Lm18, A24, A27, A28, A29, A100;

 then

 A106: (s3 . a5) = ((s12 . a5) - 2) by A89, A96, SCPISORT: 6;

 (s12 . a5) >= ( 0 + 1) by A101, INT_1: 7;

 then

 A107: ((s12 . a5) - 2) >= (1 - 2) by XREAL_1: 9;

 then

 A108: ((s3 . a5) + 1) >= (( - 1) + 1) by A106, XREAL_1: 6;

 thus (s3 . a4) >= m3 by A99, A104, XREAL_1: 19;

 m5 <= (s . a5) by A16, A18, A74, A14, XXREAL_0: 2;

 then (m5 - 2) < ((s . a5) - 0 ) by XREAL_1: 15;

 hence (s3 . a5) < (s . a5) by A37, A23, A71, A88, A86, A106, SCMPDS_7: 30;

 thus (s3 . a5) >= ( - 1) by A89, A96, A105, A107, SCPISORT: 6;

 thus (s3 . a4) = (((s3 . a3) - 1) + (s3 . a5)) by A73, A89, A96, A91, A94, A103, A102, A106, SCPISORT: 6;

 take n1 = mE3;

 take n2 = m4;

 n2 = (n1 + ((s3 . a5) + 1)) by A73, A94, A106;

 then

 A109: n2 >= (n1 + 0 ) by A108, XREAL_1: 6;

 thus n1 = ((s3 . a3) - 1) by A102;



 thus n2 = ((m4 + ( - 1)) + 1)

 .= ((s3 . a4) + 1) by A89, A96, A91, A103, SCPISORT: 6;

 A110:

 now

 let i be Nat;



 A111: (s1 . ( intpos i)) = (( Initialize s1) . ( intpos i)) by SCMPDS_5: 15;

 assume

 A112: i >= 8;



 thus (s12 . ( intpos i)) = (s2 . ( intpos i)) by A37, A23, A88, A86, SCMPDS_7: 30

 .= (s1 . ( intpos i)) by A5, A6, A7, A21, A32, A33, A34, A30, A70, A112, Lm14, A9, A11, A12, A14, A111

 .= (s . ( intpos i)) by A2, A3, A4, A6, A7, A15, A112, Lm8;

 end;

 A113:

 now

 let i be Nat;

 assume that

 A114: i >= 8 and

 A115: i <> n1 and

 A116: i <> n2;



 A117: (s12 . ( intpos i)) = (( Initialize s12) . ( intpos i)) by SCMPDS_5: 15;



 thus (s3 . ( intpos i)) = (( IExec (IF3,P12,( Initialize s12))) . ( intpos i)) by A89, A96, SCPISORT: 6

 .= (s12 . ( intpos i)) by A93, A95, A81, A91, A97, A101, A114, A115, A116, Lm18, A24, A27, A28, A29, A100, A117

 .= (s . ( intpos i)) by A110, A114;

 end;

 hence for i be Nat st i >= 8 & i <> n1 & i <> n2 holds (s3 . ( intpos i)) = (s . ( intpos i));



 A118: (s12 . ( intpos m4)) = (( Initialize s12) . ( intpos m4)) by SCMPDS_5: 15;

 (( IExec (IF3,P12,( Initialize s12))) . ( intpos mE3)) = (s12 . ( intpos m4)) by A93, A95, A81, A91, A97, A101, Lm18, A24, A27, A28, A29, A100, A118;



 then

 A119: (s3 . ( intpos n1)) = (s12 . ( intpos n2)) by A89, A96, SCPISORT: 6

 .= (s . ( intpos n2)) by A15, A110, NAT_1: 12;

 m3 < (m3 + 1) by XREAL_1: 29;

 then m3 < n1 by A5, A33, A78, XXREAL_0: 2;

 then

 A120: md < n1 by A8, XXREAL_0: 2;

 then md < n2 by A109, XXREAL_0: 2;

 then

 A121: n2 >= 8 by A7, XXREAL_0: 2;



 A122: (s3 . ( intpos md)) = (s . ( intpos md)) by A7, A113, A120, A109;

 then

 A123: (s1 . ( intpos md)) = (s3 . ( intpos md)) by A2, A3, A4, A6, A7, A15, Lm8;



 A124: (s12 . ( intpos mE3)) = (( Initialize s12) . ( intpos mE3)) by SCMPDS_5: 15;

 (( IExec (IF3,P12,( Initialize s12))) . ( intpos m4)) = (s12 . ( intpos mE3)) by A93, A95, A81, A91, A97, A101, Lm18, A24, A27, A28, A29, A100, A124;



 then

 A125: (s3 . ( intpos n2)) = (s12 . ( intpos n1)) by A89, A96, SCPISORT: 6

 .= (s . ( intpos n1)) by A33, A34, A80, A110, XXREAL_0: 2;

 hence (s3 . ( intpos n1)) = (s . ( intpos n1)) & (s3 . ( intpos n2)) = (s . ( intpos n2)) or (s . a3) <= n1 & n2 <= (s . a4) & (s3 . ( intpos n1)) = (s . ( intpos n2)) & (s3 . ( intpos n2)) = (s . ( intpos n1)) by A4, A33, A18, A79, A119, XREAL_1: 6;



 A126: (s1 . ( intpos md)) = (( Initialize s1) . ( intpos md)) by SCMPDS_5: 15;



 A127: (s2 . ( intpos md)) = (s1 . ( intpos md)) by A5, A6, A7, A21, A32, A33, A34, A30, A70, Lm14, A9, A11, A12, A14, A126

 .= (s3 . ( intpos md)) by A2, A3, A4, A6, A7, A15, A122, Lm8;

 hereby

 let i be Nat;

 assume that

 A128: (s . a3) <= i and

 A129: i <= n1;



 A130: i >= 8 by A33, A34, A128, XXREAL_0: 2;



 A131: (s1 . a3) <= i by A2, A3, A4, A6, A7, A15, A128, Lm8;

 per cases ;

 suppose

 A132: i < n1;



 A133: (s1 . ( intpos i)) = (( Initialize s1) . ( intpos i)) by SCMPDS_5: 15;

 (s2 . ( intpos i)) = (s1 . ( intpos i)) by A5, A6, A7, A21, A32, A33, A34, A30, A70, A130, A9, A11, A12, A14, A133, Lm14

 .= (s . ( intpos i)) by A2, A3, A4, A6, A7, A15, A130, Lm8

 .= (s3 . ( intpos i)) by A33, A34, A113, A109, A128, A132, XXREAL_0: 2;

 hence (s3 . ( intpos md)) >= (s3 . ( intpos i)) by A75, A127, A131, A132, A12;

 end;

 suppose i >= n1;

 then i = n1 by A129, XXREAL_0: 1;

 hence (s3 . ( intpos md)) >= (s3 . ( intpos i)) by A2, A3, A4, A6, A7, A15, A20, A70, A119, A121, A123, Lm8;

 end;

 end;



 A134: n1 >= 8 by A7, A120, XXREAL_0: 2;

 hereby

 let i be Nat;

 assume that

 A135: n2 <= i and

 A136: i <= (s . a4);



 A137: i >= 8 by A121, A135, XXREAL_0: 2;



 A138: (s1 . ( intpos n1)) = (( Initialize s1) . ( intpos n1)) by SCMPDS_5: 15;

 per cases ;

 suppose

 A139: n2 < i;

 (s1 . ( intpos i)) = (s . ( intpos i)) by A2, A3, A4, A6, A7, A15, A137, Lm8

 .= (s3 . ( intpos i)) by A113, A109, A121, A139, XXREAL_0: 2;

 hence (s3 . ( intpos md)) <= (s3 . ( intpos i)) by A19, A123, A136, A139;

 end;

 suppose n2 >= i;

 then i = n2 by A135, XXREAL_0: 1;



 then (s3 . ( intpos i)) = (s1 . ( intpos n1)) by A2, A3, A4, A6, A7, A15, A125, A134, Lm8

 .= (s2 . ( intpos n1)) by A5, A6, A7, A21, A32, A33, A34, A30, A70, A134, Lm14, A9, A11, A12, A14, A138;

 hence (s3 . ( intpos md)) <= (s3 . ( intpos i)) by A37, A23, A71, A76, A88, A86, A101, A127, SCMPDS_7: 30;

 end;

 end;

 end;

 suppose

 A140: (s12 . a5) <= 0 ;



 A141: ( DataLoc ((s12 . a),5)) = ( intpos ( 0 + 5)) by A93, SCMP_GCD: 1;



 thus (( IExec (WB3,P,s)) . a) = (( IExec (IF3,P12,( Initialize s12))) . a) by A89, A96, SCPISORT: 6

 .= 0 by A93, A140, A141, SCMPDS_6: 84;



 thus (( IExec (WB3,P,s)) . a1) = (( IExec (IF3,P12,( Initialize s12))) . a1) by A89, A96, SCPISORT: 6

 .= (s . a1) by A90, A140, A141, SCMPDS_6: 84;



 thus (s3 . a2) = (( IExec (IF3,P12,( Initialize s12))) . a2) by A89, A96, SCPISORT: 6

 .= md by A92, A140, A141, SCMPDS_6: 84;



 A142: (s3 . a3) = (( IExec (IF3,P12,( Initialize s12))) . a3) by A89, A96, SCPISORT: 6

 .= mE3 by A95, A140, A141, SCMPDS_6: 84;

 hence (s3 . a3) >= (s . a3) by A5, A16, A73, A77, XREAL_1: 6;



 A143: (s3 . a4) = (( IExec (IF3,P12,( Initialize s12))) . a4) by A89, A96, SCPISORT: 6

 .= m4 by A91, A140, A141, SCMPDS_6: 84;

 hence (s3 . a4) <= (s . a4) by A4, A18, XREAL_1: 6;



 A144: (s12 . a5) = 0 by A37, A23, A71, A88, A86, A140, SCMPDS_7: 30;

 thus (s3 . a4) >= m3 by A98, A143;



 A145: (s3 . a5) = (( IExec (IF3,P12,( Initialize s12))) . a5) by A89, A96, SCPISORT: 6

 .= 0 by A141, A144, SCMPDS_6: 84;

 hence (s3 . a5) < (s . a5) by A3;

 thus (s3 . a5) >= ( - 1) by A145;



 A146: m4 = (((m3 + mE) + 1) - 1)

 .= ((mE3 + 0 ) - 1) by A37, A23, A71, A73, A88, A86, A144, SCMPDS_7: 30;

 hence (s3 . a4) = (((s3 . a3) - 1) + (s3 . a5)) by A142, A143, A145;

 take n1 = m4;

 reconsider n2 = (m4 + 1) as Nat;

 take n2;

 thus n1 = ((s3 . a3) - 1) by A142, A146;

 thus n2 = ((s3 . a4) + 1) by A143;

 A147:

 now

 let x;



 thus (s3 . x) = (( IExec (IF3,P12,( Initialize s12))) . x) by A89, A96, SCPISORT: 6

 .= (s12 . x) by A140, A141, SCMPDS_6: 84

 .= (s2 . x) by A37, A23, A88, A86, SCMPDS_7: 30;

 end;

 A148:

 now

 let i be Nat;

 assume

 A149: i >= 8;



 A150: (s1 . ( intpos i)) = (( Initialize s1) . ( intpos i)) by SCMPDS_5: 15;



 thus (s3 . ( intpos i)) = (s2 . ( intpos i)) by A147

 .= (s1 . ( intpos i)) by A5, A6, A7, A21, A32, A33, A34, A30, A70, A149, Lm14, A9, A11, A12, A14, A150

 .= (s . ( intpos i)) by A2, A3, A4, A6, A7, A15, A149, Lm8;

 end;

 hence for i be Nat st i >= 8 & i <> n1 & i <> n2 holds (s3 . ( intpos i)) = (s . ( intpos i));



 A151: m4 < n2 by XREAL_1: 29;

 hence (s3 . ( intpos n1)) = (s . ( intpos n1)) & (s3 . ( intpos n2)) = (s . ( intpos n2)) or (s . a3) <= n1 & n2 <= (s . a4) & (s3 . ( intpos n1)) = (s . ( intpos n2)) & (s3 . ( intpos n2)) = (s . ( intpos n1)) by A31, A148, XXREAL_0: 2;

 hereby

 let i be Nat;

 assume that

 A152: (s . a3) <= i and

 A153: i <= n1;

 i < mE3 by A146, A153, XREAL_1: 146, XXREAL_0: 2;

 then

 A154: (s2 . ( intpos md)) > (s2 . ( intpos i)) by A33, A75, A152, A12;

 (s3 . ( intpos md)) = (s2 . ( intpos md)) by A147;

 hence (s3 . ( intpos md)) >= (s3 . ( intpos i)) by A147, A154;

 end;

 hereby

 let i be Nat;

 assume that

 A155: n2 <= i and

 A156: i <= (s . a4);



 A157: m4 = 8 by A31, XXREAL_0: 2;



 A159: (s1 . ( intpos md)) = (( Initialize s1) . ( intpos md)) by SCMPDS_5: 15;



 A160: (s1 . ( intpos i)) = (( Initialize s1) . ( intpos i)) by SCMPDS_5: 15;



 A161: (s3 . ( intpos md)) = (s2 . ( intpos md)) by A147

 .= (s1 . ( intpos md)) by A5, A6, A7, A21, A32, A33, A34, A30, A70, Lm14, A9, A11, A12, A14, A159;

 (s3 . ( intpos i)) = (s2 . ( intpos i)) by A147

 .= (s1 . ( intpos i)) by A5, A6, A7, A21, A32, A33, A34, A30, A70, A158, Lm14, A9, A11, A12, A14, A160;

 hence (s3 . ( intpos md)) <= (s3 . ( intpos i)) by A19, A156, A157, A161;

 end;

 end;

 end;

 end;

 end;



 Lm20: for i be Integer st i >= ( - 1) & i <= 0 holds i = ( - 1) or i = 0

 proof

 let i be Integer;

 assume that

 A1: i >= ( - 1) and

 A2: i <= 0 ;

 per cases ;

 suppose i <= ( - 1);

 hence thesis by A1, XXREAL_0: 1;

 end;

 suppose i > ( - 1);

 then i >= (( - 1) + 1) by INT_1: 7;

 hence thesis by A2;

 end;

 end;



 Lm21: for i1,i2 be Integer, n1,n2,i be Nat st i1 >= ( - 1) & i1 <= 0 & n2 = (i2 + 1) & i2 = (n1 + i1) & i < n2 holds i <= n1

 proof

 let i1,i2 be Integer, n1,n2,i be Nat;

 assume that

 A1: i1 >= ( - 1) and

 A2: i1 <= 0 and

 A3: n2 = (i2 + 1) and

 A4: i2 = (n1 + i1) and

 A5: i < n2;

 per cases by A1, A2, Lm20;

 suppose i1 = 0 ;

 hence thesis by A3, A4, A5, NAT_1: 13;

 end;

 suppose i1 = ( - 1);

 hence thesis by A3, A4, A5;

 end;

 end;



 Lm22: for i1,i2 be Integer, n1,n2 be Nat st i1 >= ( - 1) & n2 = (i2 + 1) & i2 = (n1 + i1) holds n1 <= n2

 proof

 let i1,i2 be Integer, n1,n2 be Nat;

 assume that

 A1: i1 >= ( - 1) and

 A2: n2 = (i2 + 1) and

 A3: i2 = (n1 + i1);

 (n1 + i1) >= (n1 + ( - 1)) by A1, XREAL_1: 6;

 then (i2 + 1) >= ((n1 + ( - 1)) + 1) by A3, XREAL_1: 6;

 hence thesis by A2;

 end;



 Lm23: for s,s1 be State of SCMPDS , n0,n1,n2,n be Nat, f,f1 be FinSequence of INT st f is_FinSequence_on (s,n0) & f1 is_FinSequence_on (s1,n0) & ( len f) = n & ( len f1) = n & (for i be Nat st i >= (n0 + 1) & i <> n1 & i <> n2 holds (s1 . ( intpos i)) = (s . ( intpos i))) & ((s1 . ( intpos n1)) = (s . ( intpos n1)) & (s1 . ( intpos n2)) = (s . ( intpos n2)) or n1 >= (n0 + 1) & n2 >= (n0 + 1) & n1 <= (n0 + n) & n2 <= (n0 + n) & (s1 . ( intpos n1)) = (s . ( intpos n2)) & (s1 . ( intpos n2)) = (s . ( intpos n1))) holds (f,f1) are_fiberwise_equipotent

 proof

 let s,s1 be State of SCMPDS , n0,n1,n2,n be Nat, f,f1 be FinSequence of INT ;

 assume that

 A1: f is_FinSequence_on (s,n0) and

 A2: f1 is_FinSequence_on (s1,n0);

 assume that

 A3: ( len f) = n and

 A4: ( len f1) = n;

 assume

 A5: for i be Nat st i >= (n0 + 1) & i <> n1 & i <> n2 holds (s1 . ( intpos i)) = (s . ( intpos i));

 assume that

 A6: (s1 . ( intpos n1)) = (s . ( intpos n1)) & (s1 . ( intpos n2)) = (s . ( intpos n2)) or n1 >= (n0 + 1) & n2 >= (n0 + 1) & n1 <= (n0 + n) & n2 <= (n0 + n) & (s1 . ( intpos n1)) = (s . ( intpos n2)) & (s1 . ( intpos n2)) = (s . ( intpos n1));

 per cases by A6;

 suppose

 A7: (s1 . ( intpos n1)) = (s . ( intpos n1)) & (s1 . ( intpos n2)) = (s . ( intpos n2));



 A8: ( dom f1) = ( Seg n) by A4, FINSEQ_1:def 3;

 now

 let i be Nat;

 reconsider a = i as Nat;

 assume

 A9: i in ( dom f1);

 then

 A10: 1 <= i by A8, FINSEQ_1: 1;

 then

 A11: (n0 + 1) <= (n0 + i) by XREAL_1: 6;



 A12: i <= n by A8, A9, FINSEQ_1: 1;

 per cases ;

 suppose

 A13: (n0 + i) <> n1 & (n0 + i) <> n2;



 thus (f1 . i) = (s1 . ( intpos (n0 + a))) by A2, A4, A10, A12

 .= (s . ( intpos (n0 + a))) by A5, A11, A13

 .= (f . i) by A1, A3, A10, A12;

 end;

 suppose

 A14: not ((n0 + i) <> n1 & (n0 + i) <> n2);

 hereby

 per cases by A14;

 suppose (n0 + i) = n1;



 hence (f1 . i) = (s . ( intpos (n0 + a))) by A2, A4, A7, A10, A12

 .= (f . i) by A1, A3, A10, A12;

 end;

 suppose (n0 + i) = n2;



 hence (f1 . i) = (s . ( intpos (n0 + a))) by A2, A4, A7, A10, A12

 .= (f . i) by A1, A3, A10, A12;

 end;

 end;

 end;

 end;

 hence thesis by A3, A4, FINSEQ_2: 9;

 end;

 suppose

 A15: n1 >= (n0 + 1) & n2 >= (n0 + 1) & n1 <= (n0 + n) & n2 <= (n0 + n) & (s1 . ( intpos n1)) = (s . ( intpos n2)) & (s1 . ( intpos n2)) = (s . ( intpos n1));

 then

 A16: (n1 - n0) >= 1 by XREAL_1: 19;

 then

 reconsider m1 = (n1 - n0) as Element of NAT by INT_1: 3;



 A17: m1 <= ( len f) by A3, A15, XREAL_1: 20;



 A18: (n2 - n0) >= 1 by A15, XREAL_1: 19;

 then

 reconsider m2 = (n2 - n0) as Element of NAT by INT_1: 3;



 A19: m2 <= ( len f1) by A4, A15, XREAL_1: 20;



 A20: n2 = (m2 + n0);



 A21: n1 = (m1 + n0);



 then

 A22: (f . m1) = (s1 . ( intpos n2)) by A1, A15, A16, A17

 .= (f1 . m2) by A2, A18, A19, A20;

 A23:

 now

 let k be Nat;

 assume that

 A24: k <> m1 and

 A25: k <> m2 and

 A26: 1 <= k and

 A27: k <= ( len f);



 A28: (k + n0) <> (m1 + n0) by A24;



 A29: (n0 + 1) <= (n0 + k) by A26, XREAL_1: 6;



 A30: (k + n0) <> (m2 + n0) by A25;



 thus (f . k) = (s . ( intpos (k + n0))) by A1, A26, A27

 .= (s1 . ( intpos (k + n0))) by A5, A28, A30, A29

 .= (f1 . k) by A2, A3, A4, A26, A27;

 end;



 A31: m2 <= ( len f) by A3, A15, XREAL_1: 20;



 then (f . m2) = (s1 . ( intpos n1)) by A1, A15, A18, A20

 .= (f1 . m1) by A2, A3, A4, A16, A17, A21;

 hence thesis by A3, A4, A16, A18, A17, A31, A22, A23, SCPISORT: 3;

 end;

 end;



 Lm24: for s,s1 be State of SCMPDS , n0,n1,n2 be Nat, c1,c2 be Integer st (for i be Nat st i >= n0 & i <> n1 & i <> n2 holds (s1 . ( intpos i)) = (s . ( intpos i))) & n1 <= n2 & ((s1 . ( intpos n1)) = (s . ( intpos n1)) & (s1 . ( intpos n2)) = (s . ( intpos n2)) or c1 <= n1 & n2 <= c2 & (s1 . ( intpos n1)) = (s . ( intpos n2)) & (s1 . ( intpos n2)) = (s . ( intpos n1))) holds for i be Nat st i >= n0 & (i < c1 or i > c2) holds (s1 . ( intpos i)) = (s . ( intpos i))

 proof

 let s,s1 be State of SCMPDS , n0,n1,n2 be Nat, c1,c2 be Integer;

 assume

 A1: for i be Nat st i >= n0 & i <> n1 & i <> n2 holds (s1 . ( intpos i)) = (s . ( intpos i));

 assume

 A2: n1 <= n2;

 assume

 A3: (s1 . ( intpos n1)) = (s . ( intpos n1)) & (s1 . ( intpos n2)) = (s . ( intpos n2)) or c1 <= n1 & n2 <= c2 & (s1 . ( intpos n1)) = (s . ( intpos n2)) & (s1 . ( intpos n2)) = (s . ( intpos n1));

 per cases by A3;

 suppose

 A4: (s1 . ( intpos n1)) = (s . ( intpos n1)) & (s1 . ( intpos n2)) = (s . ( intpos n2));

 hereby

 let i be Nat;

 assume that

 A5: i >= n0 and i < c1 or i > c2;

 per cases ;

 suppose i <> n1 & i <> n2;

 hence (s1 . ( intpos i)) = (s . ( intpos i)) by A1, A5;

 end;

 suppose

 A6: not (i <> n1 & i <> n2);

 hereby

 per cases by A6;

 suppose i = n1;

 hence (s1 . ( intpos i)) = (s . ( intpos i)) by A4;

 end;

 suppose i = n2;

 hence (s1 . ( intpos i)) = (s . ( intpos i)) by A4;

 end;

 end;

 end;

 end;

 end;

 suppose

 A7: c1 <= n1 & n2 <= c2 & (s1 . ( intpos n1)) = (s . ( intpos n2)) & (s1 . ( intpos n2)) = (s . ( intpos n1));

 hereby

 let i be Nat;

 assume that

 A8: i >= n0 and

 A9: i < c1 or i > c2;

 per cases by A9;

 suppose i < c1;

 then i < n1 by A7, XXREAL_0: 2;

 hence (s1 . ( intpos i)) = (s . ( intpos i)) by A1, A2, A8;

 end;

 suppose i > c2;

 then i > n2 by A7, XXREAL_0: 2;

 hence (s1 . ( intpos i)) = (s . ( intpos i)) by A1, A2, A8;

 end;

 end;

 end;

 end;



 Lm25: for s be 0 -started State of SCMPDS , md,m3,n0 be Nat, f,f1 be FinSequence of INT st (s . GBP ) = 0 & (s . a5) > 0 & (s . a4) = (m3 + (s . a5)) & m3 = ((s . a3) - 1) & (s . a2) = md & md >= (n0 + 1) & md <= m3 & (n0 + 1) <= (s . a3) & (s . a4) <= (n0 + n) & f is_FinSequence_on (s,n0) & f1 is_FinSequence_on (( IExec (WH3,P,s)),n0) & n0 >= 7 & ( len f) = n & ( len f1) = n holds (( IExec (WH3,P,s)) . GBP ) = 0 & (( IExec (WH3,P,s)) . a1) = (s . a1) & (( IExec (WH3,P,s)) . a2) = md & (( IExec (WH3,P,s)) . a4) >= md & (( IExec (WH3,P,s)) . a4) <= (s . a4) & (f,f1) are_fiberwise_equipotent & (for i be Nat st (s . a3) <= i & i <= (( IExec (WH3,P,s)) . a4) holds (( IExec (WH3,P,s)) . ( intpos md)) >= (( IExec (WH3,P,s)) . ( intpos i))) & (for i be Nat st (( IExec (WH3,P,s)) . a4) = (n0 + 1) & (i < (s . a3) or i > (s . a4)) holds (( IExec (WH3,P,s)) . ( intpos i)) = (s . ( intpos i))

 proof

 set a = GBP ;

 let s be 0 -started State of SCMPDS , md,m3,n0 be Nat, f,f1 be FinSequence of INT ;

 assume that

 A1: (s . a) = 0 and

 A2: (s . a5) > 0 and

 A3: (s . a4) = (m3 + (s . a5)) and

 A4: m3 = ((s . a3) - 1) and

 A5: (s . a2) = md and

 A6: md >= (n0 + 1) and

 A7: md <= m3 and

 A8: (n0 + 1) <= (s . a3) and

 A9: (s . a4) <= (n0 + n);

 defpred P[ Nat] means for t be 0 -started State of SCMPDS , Q holds for n3 be Nat, g,g1 be FinSequence of INT st (t . a) = 0 & (t . a5) > 0 & (t . a5) <= ($1 + 1) & (t . a4) = (n3 + (t . a5)) & n3 = ((t . a3) - 1) & (t . a2) = md & md <= n3 & (n0 + 1) <= (t . a3) & (t . a4) <= (n0 + n) & g is_FinSequence_on (t,n0) & g1 is_FinSequence_on (( IExec (WH3,Q,t)),n0) & ( len g) = n & ( len g1) = n holds (( IExec (WH3,Q,t)) . a) = 0 & (( IExec (WH3,Q,t)) . a1) = (t . a1) & (( IExec (WH3,Q,t)) . a2) = md & (( IExec (WH3,Q,t)) . a4) >= md & (( IExec (WH3,Q,t)) . a4) <= (t . a4) & (g,g1) are_fiberwise_equipotent & (for i be Nat st (t . a3) <= i & i <= (( IExec (WH3,Q,t)) . a4) holds (( IExec (WH3,Q,t)) . ( intpos md)) >= (( IExec (WH3,Q,t)) . ( intpos i))) & (for i be Nat st (( IExec (WH3,Q,t)) . a4) = (n0 + 1) & (i < (t . a3) or i > (t . a4)) holds (( IExec (WH3,Q,t)) . ( intpos i)) = (t . ( intpos i)));

 assume

 A10: f is_FinSequence_on (s,n0);

 assume

 A11: f1 is_FinSequence_on (( IExec (WH3,P,s)),n0);

 assume that

 A12: n0 >= 7 and

 A13: ( len f) = n and

 A14: ( len f1) = n;

 (n0 + 1) >= (7 + 1) by A12, XREAL_1: 6;

 then

 A15: md >= 8 by A6, XXREAL_0: 2;

 A16:

 now

 let k be Nat;

 assume

 A17: P[k];

 now

 let t be 0 -started State of SCMPDS , n3 be Nat, g,g1 be FinSequence of INT ;

 let Q;

 assume that

 A18: (t . a) = 0 and

 A19: (t . a5) > 0 and

 A20: (t . a5) <= ((k + 1) + 1) and

 A21: (t . a4) = (n3 + (t . a5)) and

 A22: n3 = ((t . a3) - 1) and

 A23: (t . a2) = md and

 A24: md <= n3 and

 A25: (n0 + 1) <= (t . a3) and

 A26: (t . a4) <= (n0 + n);

 set t1 = ( IExec (WB3,Q,t)), Q1 = Q;



 A27: (t1 . a) = (( Initialize t1) . a) by SCMPDS_5: 15;



 A28: (t1 . a1) = (( Initialize t1) . a1) by SCMPDS_5: 15;



 A29: (t1 . a2) = (( Initialize t1) . a2) by SCMPDS_5: 15;



 A30: (t1 . a3) = (( Initialize t1) . a3) by SCMPDS_5: 15;



 A31: (t1 . a4) = (( Initialize t1) . a4) by SCMPDS_5: 15;



 A32: (t1 . a5) = (( Initialize t1) . a5) by SCMPDS_5: 15;



 A33: (t1 . a4) = (((t1 . a3) - 1) + (t1 . a5)) by A15, A18, A19, A21, A22, A23, A24, Lm19;

 A34:

 now



 A35: ( DataLoc ((t . a),5)) = ( intpos ( 0 + 5)) by A18, SCMP_GCD: 1;

 let v be 0 -started State of SCMPDS ;

 let V;

 assume that

 A36: (v . a4) = (((v . a3) - 1) + (v . a5)) and

 A37: md <= ((v . a3) - 1) and

 A38: (v . a2) = (t . a2) and

 A39: (v . a) = (t . a) and

 A40: (v . ( DataLoc ((t . a),5))) > 0 ;

 reconsider mm = ((v . a3) - 1) as Element of NAT by A37, INT_1: 3;

 set Iv = ( IExec (WB3,V,v));



 A41: (v . a4) = (mm + (v . a5)) by A36;

 hence (Iv . a) = (v . a) by A15, A18, A23, A37, A38, A39, A40, A35, Lm19;

 thus WB3 is_closed_on (v,V) & WB3 is_halting_on (v,V) by A15, A18, A23, A37, A38, A39, A40, A35, A41, Lm19;

 thus (Iv . ( DataLoc ((t . a),5))) < (v . ( DataLoc ((t . a),5))) by A15, A18, A23, A37, A38, A39, A40, A35, A41, Lm19;

 thus (Iv . a4) = (((Iv . a3) - 1) + (Iv . a5)) by A15, A18, A23, A37, A38, A39, A40, A35, A41, Lm19;

 (Iv . a3) >= (v . a3) by A15, A18, A23, A37, A38, A39, A40, A35, A41, Lm19;

 then ((Iv . a3) - 1) >= ((v . a3) - 1) by XREAL_1: 9;

 hence md <= ((Iv . a3) - 1) by A37, XXREAL_0: 2;

 thus (Iv . a2) = (v . a2) by A15, A18, A23, A37, A38, A39, A40, A35, A41, Lm19;

 end;

 (t1 . a5) < (t . a5) by A15, A18, A19, A21, A22, A23, A24, Lm19;

 then ((t1 . a5) + 1) <= (t . a5) by INT_1: 7;

 then ((t1 . a5) + 1) <= ((k + 1) + 1) by A20, XXREAL_0: 2;

 then

 A42: (t1 . a5) <= (k + 1) by XREAL_1: 6;

 set t2 = ( IExec (WH3,Q,t));

 assume

 A43: g is_FinSequence_on (t,n0);

 assume

 A44: g1 is_FinSequence_on (( IExec (WH3,Q,t)),n0);



 A45: (t1 . a) = 0 by A15, A18, A19, A21, A22, A23, A24, Lm19;

 then

 A46: ( DataLoc ((t1 . a),5)) = ( intpos ( 0 + 5)) by SCMP_GCD: 1;

 assume that

 A47: ( len g) = n and

 A48: ( len g1) = n;



 A49: (t1 . a2) = md by A15, A18, A19, A21, A22, A23, A24, Lm19;



 A50: (t1 . a1) = (t . a1) by A15, A18, A19, A21, A22, A23, A24, Lm19;



 A51: (t1 . a3) >= (t . a3) by A15, A18, A19, A21, A22, A23, A24, Lm19;



 A52: (t1 . a4) >= n3 by A15, A18, A19, A21, A22, A23, A24, Lm19;



 A53: ( DataLoc ((t . a),5)) = ( intpos ( 0 + 5)) by A18, SCMP_GCD: 1;



 A54: (t1 . a4) <= (t . a4) by A15, A18, A19, A21, A22, A23, A24, Lm19;

 consider n1,n2 be Nat such that

 A55: n1 = ((t1 . a3) - 1) and

 A56: n2 = ((t1 . a4) + 1) and

 A57: for i be Nat st i >= 8 & i <> n1 & i <> n2 holds (t1 . ( intpos i)) = (t . ( intpos i)) and

 A58: (t1 . ( intpos n1)) = (t . ( intpos n1)) & (t1 . ( intpos n2)) = (t . ( intpos n2)) or (t . a3) <= n1 & n2 <= (t . a4) & (t1 . ( intpos n1)) = (t . ( intpos n2)) & (t1 . ( intpos n2)) = (t . ( intpos n1)) and

 A59: for i be Nat st (t . a3) <= i & i <= n1 holds (t1 . ( intpos md)) >= (t1 . ( intpos i)) and

 A60: for i be Nat st n2 <= i & i <= (t . a4) holds (t1 . ( intpos md)) <= (t1 . ( intpos i)) by A15, A18, A19, A21, A22, A23, A24, Lm19;



 A61: (t1 . a5) >= ( - 1) by A15, A18, A19, A21, A22, A23, A24, Lm19;

 then

 A62: n1 <= n2 by A33, A55, A56, Lm22;

 per cases ;

 suppose

 A63: (t1 . a5) > 0 ;

 set t3 = ( IExec (WH3,Q1,( Initialize t1)));

 consider f3 be FinSequence of INT such that

 A64: ( len f3) = n and

 A65: for i be Nat st 1 <= i & i <= ( len f3) holds (f3 . i) = (t1 . ( intpos (n0 + i))) by SCPISORT: 1;



 A66: f3 is_FinSequence_on (t1,n0) by A65;



 A67: f3 is_FinSequence_on (( Initialize t1),n0)

 proof

 let i be Nat;

 assume 1 <= i & i <= ( len f3);

 then (f3 . i) = (t1 . ( intpos (n0 + i))) by A65;

 hence (f3 . i) = (( Initialize t1) . ( intpos (n0 + i))) by SCMPDS_5: 15;

 end;



 A68: (t1 . ( intpos n1)) = (t . ( intpos n1)) & (t1 . ( intpos n2)) = (t . ( intpos n2)) or n1 >= (n0 + 1) & n2 >= (n0 + 1) & n1 <= (n0 + n) & n2 <= (n0 + n) & (t1 . ( intpos n1)) = (t . ( intpos n2)) & (t1 . ( intpos n2)) = (t . ( intpos n1))

 proof

 per cases by A58;

 suppose (t1 . ( intpos n1)) = (t . ( intpos n1)) & (t1 . ( intpos n2)) = (t . ( intpos n2));

 hence thesis;

 end;

 suppose

 A69: (t . a3) <= n1 & n2 <= (t . a4) & (t1 . ( intpos n1)) = (t . ( intpos n2)) & (t1 . ( intpos n2)) = (t . ( intpos n1));

 then

 A70: n2 <= (n0 + n) by A26, XXREAL_0: 2;

 n1 >= (n0 + 1) by A25, A69, XXREAL_0: 2;

 hence thesis by A62, A69, A70, XXREAL_0: 2;

 end;

 end;



 A71: (n0 + 1) <= (t1 . a3) by A25, A51, XXREAL_0: 2;

 ((t1 . a3) - 1) >= ((t . a3) - 1) by A51, XREAL_1: 9;

 then

 A72: md <= n1 by A22, A24, A55, XXREAL_0: 2;

 now

 let i be Nat;

 assume that

 A73: i >= (n0 + 1) and

 A74: i <> n1 and

 A75: i <> n2;

 (n0 + 1) >= (7 + 1) by A12, XREAL_1: 6;

 hence (t1 . ( intpos i)) = (t . ( intpos i)) by A57, A73, A74, A75, XXREAL_0: 2;

 end;

 then

 A76: (g,f3) are_fiberwise_equipotent by A43, A47, A64, A66, A68, Lm23;



 A77: (t1 . a4) <= (n0 + n) by A26, A54, XXREAL_0: 2;



 A78: t2 = t3 by A19, A21, A22, A24, A53, A34, Th7;

 consider f4 be FinSequence of INT such that

 A79: ( len f4) = n and

 A80: for i be Nat st 1 <= i & i <= ( len f4) holds (f4 . i) = (t3 . ( intpos (n0 + i))) by SCPISORT: 1;



 A81: f4 is_FinSequence_on (t3,n0) by A80;

 thus (t2 . a) = 0 by A17, A33, A42, A45, A49, A55, A63, A64, A67, A78, A79, A81, A72, A71, A77, A27, A29, A30, A31, A32;



 A82: (md < (t1 . a3) or md > (t1 . a4)) implies (t2 . ( intpos md)) = (( Initialize t1) . ( intpos md)) by A6, A17, A33, A42, A45, A49, A55, A63, A64, A67, A71, A72, A77, A78, A79, A81, A27, A29, A30, A31, A32;



 A83: md < (t1 . a3) by A55, A72, XREAL_1: 146, XXREAL_0: 2;



 A84: ( dom f4) = ( Seg n) by A79, FINSEQ_1:def 3;

 now

 let i be Nat;

 reconsider a = i as Nat;

 assume

 A85: i in ( dom f4);

 then

 A86: i <= n by A84, FINSEQ_1: 1;



 A87: 1 <= i by A84, A85, FINSEQ_1: 1;



 hence (f4 . i) = (t2 . ( intpos (n0 + a))) by A79, A80, A78, A86

 .= (g1 . i) by A44, A48, A87, A86;

 end;

 then

 A88: f4 = g1 by A48, A79, FINSEQ_2: 9;

 thus (t2 . a1) = (t . a1) by A17, A45, A50, A49, A33, A55, A42, A63, A64, A67, A79, A81, A72, A71, A77, A78, A27, A28, A29, A30, A31, A32;

 thus (t2 . a2) = md by A17, A45, A49, A33, A55, A42, A63, A64, A67, A79, A81, A72, A71, A77, A78, A27, A29, A30, A31, A32;

 thus (t2 . a4) >= md by A17, A45, A49, A33, A55, A42, A63, A64, A67, A79, A81, A72, A71, A77, A78, A27, A29, A30, A31, A32;

 (t3 . a4) <= (t1 . a4) by A17, A45, A49, A33, A55, A42, A63, A64, A67, A79, A81, A72, A71, A77, A27, A29, A30, A31, A32;

 hence (t2 . a4) <= (t . a4) by A54, A78, XXREAL_0: 2;

 (f3,f4) are_fiberwise_equipotent by A17, A45, A49, A33, A55, A42, A63, A64, A67, A79, A81, A72, A71, A77, A27, A29, A30, A31, A32;

 hence (g,g1) are_fiberwise_equipotent by A76, A88, CLASSES1: 76;



 A89: (t1 . a3) = (n1 + 1) by A55;

 hereby

 let i be Nat;

 assume that

 A90: (t . a3) <= i and

 A91: i <= (t2 . a4);



 A92: (t1 . ( intpos i)) = (( Initialize t1) . ( intpos i)) by SCMPDS_5: 15;



 A93: (n0 + 1) <= i by A25, A90, XXREAL_0: 2;

 hereby

 per cases ;

 suppose

 A94: i < (t1 . a3);

 then

 A95: i <= n1 by A89, INT_1: 7;



 A96: (t2 . ( intpos i)) = (t1 . ( intpos i)) by A17, A45, A49, A33, A55, A42, A63, A64, A67, A79, A81, A72, A71, A77, A78, A93, A94, A92, A27, A29, A30, A31, A32;

 (t2 . ( intpos md)) = (t1 . ( intpos md)) by A83, A82, SCMPDS_5: 15;

 hence (t2 . ( intpos md)) >= (t2 . ( intpos i)) by A59, A90, A95, A96;

 end;

 suppose

 A97: i >= (t1 . a3);

 thus (t2 . ( intpos md)) >= (t2 . ( intpos i)) by A17, A45, A49, A33, A55, A42, A63, A64, A67, A79, A81, A72, A71, A77, A78, A91, A97, A27, A29, A30, A31, A32;

 end;

 end;

 end;

 hereby

 let i be Nat;

 assume that

 A98: (t2 . a4) < i and

 A99: i <= (t . a4);



 A100: (t1 . ( intpos i)) = (( Initialize t1) . ( intpos i)) by SCMPDS_5: 15;

 hereby

 per cases ;

 suppose

 A101: i > (t1 . a4);

 then

 A102: i >= n2 by A56, INT_1: 7;

 (t1 . a4) > (n1 + 0 ) by A33, A55, A63, XREAL_1: 6;

 then n2 > (n1 + 1) by A56, XREAL_1: 6;

 then n2 >= (n0 + 1) by A55, A71, XXREAL_0: 2;

 then i >= (n0 + 1) by A102, XXREAL_0: 2;

 then

 A103: (t2 . ( intpos i)) = (t1 . ( intpos i)) by A17, A45, A49, A33, A55, A42, A63, A64, A67, A79, A81, A72, A71, A77, A78, A101, A27, A29, A30, A31, A32, A100;

 (t2 . ( intpos md)) = (t1 . ( intpos md)) by A83, A82, SCMPDS_5: 15;

 hence (t2 . ( intpos md)) <= (t2 . ( intpos i)) by A60, A99, A102, A103;

 end;

 suppose

 A104: i <= (t1 . a4);

 thus (t2 . ( intpos md)) <= (t2 . ( intpos i)) by A17, A45, A49, A33, A55, A42, A63, A64, A67, A79, A81, A72, A71, A77, A78, A98, A104, A27, A29, A30, A31, A32;

 end;

 end;

 end;

 hereby

 let i be Nat;



 A105: (t1 . ( intpos i)) = (( Initialize t1) . ( intpos i)) by SCMPDS_5: 15;

 assume that

 A106: i >= (n0 + 1) and

 A107: i < (t . a3) or i > (t . a4);

 (n0 + 1) >= (7 + 1) by A12, XREAL_1: 6;

 then

 A108: i >= 8 by A106, XXREAL_0: 2;

 then

 A109: (t1 . ( intpos i)) = (t . ( intpos i)) by A57, A58, A62, A107, Lm24;

 per cases by A107;

 suppose i < (t . a3);

 then i < (t1 . a3) by A51, XXREAL_0: 2;



 hence (t2 . ( intpos i)) = (t1 . ( intpos i)) by A17, A45, A49, A33, A55, A42, A63, A64, A67, A79, A81, A72, A71, A77, A78, A106, A27, A29, A30, A31, A32, A105

 .= (t . ( intpos i)) by A57, A58, A62, A107, Lm24, A108;

 end;

 suppose i > (t . a4);

 then i > (t1 . a4) by A54, XXREAL_0: 2;

 hence (t2 . ( intpos i)) = (t . ( intpos i)) by A109, A17, A45, A49, A33, A55, A42, A63, A64, A67, A79, A81, A72, A71, A77, A78, A106, A27, A29, A30, A31, A32, A105;

 end;

 end;

 end;

 suppose

 A110: (t1 . a5) <= 0 ;

 A111:

 now

 let x;



 A112: (( Initialize t1) . GBP ) = (t1 . GBP ) by SCMPDS_5: 15;



 A113: (( Initialize t1) . ( DataLoc ((t1 . GBP ),5))) = (t1 . ( DataLoc ((t1 . GBP ),5))) by SCMPDS_5: 15;



 thus (t2 . x) = (( IExec (WH3,Q1,( Initialize t1))) . x) by A19, A21, A22, A24, A53, A34, Th7

 .= (( Initialize t1) . x) by A46, A110, A112, A113, SCMPDS_8: 23

 .= (t1 . x) by SCMPDS_5: 15;

 end;

 hence (t2 . a) = 0 by A45;

 thus (t2 . a1) = (t . a1) by A50, A111;

 thus (t2 . a2) = md by A49, A111;

 (t2 . a4) >= n3 by A52, A111;

 hence (t2 . a4) >= md by A24, XXREAL_0: 2;

 thus (t2 . a4) <= (t . a4) by A54, A111;

 A114:

 now



 A115: (n0 + 1) >= (7 + 1) by A12, XREAL_1: 6;

 let i be Nat;

 assume that

 A116: i >= (n0 + 1) and

 A117: i <> n1 and

 A118: i <> n2;



 thus (t2 . ( intpos i)) = (t1 . ( intpos i)) by A111

 .= (t . ( intpos i)) by A57, A116, A117, A118, A115, XXREAL_0: 2;

 end;



 A119: (t2 . ( intpos n1)) = (t . ( intpos n1)) & (t2 . ( intpos n2)) = (t . ( intpos n2)) or (t . a3) <= n1 & n2 <= (t . a4) & (t2 . ( intpos n1)) = (t . ( intpos n2)) & (t2 . ( intpos n2)) = (t . ( intpos n1))

 proof

 per cases by A58;

 suppose (t1 . ( intpos n1)) = (t . ( intpos n1)) & (t1 . ( intpos n2)) = (t . ( intpos n2));

 hence thesis by A111;

 end;

 suppose (t . a3) <= n1 & n2 <= (t . a4) & (t1 . ( intpos n1)) = (t . ( intpos n2)) & (t1 . ( intpos n2)) = (t . ( intpos n1));

 hence thesis by A111;

 end;

 end;

 (t2 . ( intpos n1)) = (t . ( intpos n1)) & (t2 . ( intpos n2)) = (t . ( intpos n2)) or n1 >= (n0 + 1) & n2 >= (n0 + 1) & n1 <= (n0 + n) & n2 <= (n0 + n) & (t2 . ( intpos n1)) = (t . ( intpos n2)) & (t2 . ( intpos n2)) = (t . ( intpos n1))

 proof

 per cases by A119;

 suppose (t2 . ( intpos n1)) = (t . ( intpos n1)) & (t2 . ( intpos n2)) = (t . ( intpos n2));

 hence thesis;

 end;

 suppose

 A120: (t . a3) <= n1 & n2 <= (t . a4) & (t2 . ( intpos n1)) = (t . ( intpos n2)) & (t2 . ( intpos n2)) = (t . ( intpos n1));

 then

 A121: n2 <= (n0 + n) by A26, XXREAL_0: 2;

 n1 >= (n0 + 1) by A25, A120, XXREAL_0: 2;

 hence thesis by A62, A120, A121, XXREAL_0: 2;

 end;

 end;

 hence (g,g1) are_fiberwise_equipotent by A43, A44, A47, A48, A114, Lm23;



 A122: (t2 . a4) = (n2 - 1) by A56, A111;

 hereby

 let i be Nat;

 assume that

 A123: (t . a3) <= i and

 A124: i <= (t2 . a4);

 i < n2 by A122, A124, XREAL_1: 146, XXREAL_0: 2;

 then i <= n1 by A61, A33, A55, A56, A110, Lm21;

 then (t1 . ( intpos md)) >= (t1 . ( intpos i)) by A59, A123;

 then (t2 . ( intpos md)) >= (t1 . ( intpos i)) by A111;

 hence (t2 . ( intpos md)) >= (t2 . ( intpos i)) by A111;

 end;

 hereby

 let i be Nat;

 assume that

 A125: (t2 . a4) < i and

 A126: i <= (t . a4);

 ((t2 . a4) + 1) = n2 by A56, A111;

 then n2 <= i by A125, INT_1: 7;

 then (t1 . ( intpos md)) <= (t1 . ( intpos i)) by A60, A126;

 then (t2 . ( intpos md)) <= (t1 . ( intpos i)) by A111;

 hence (t2 . ( intpos md)) <= (t2 . ( intpos i)) by A111;

 end;

 thus for i be Nat st i >= (n0 + 1) & (i < (t . a3) or i > (t . a4)) holds (t2 . ( intpos i)) = (t . ( intpos i)) by A62, A114, A119, Lm24;

 end;

 end;

 hence P[(k + 1)];

 end;

 (s . a5) >= (1 + 0 ) by A2, INT_1: 7;

 then

 reconsider m5 = ((s . a5) - 1) as Element of NAT by INT_1: 3, XREAL_1: 48;



 A127: (s . a5) = (m5 + 1);



 A128: P[ 0 qua Nat]

 proof

 let t be 0 -started State of SCMPDS , Q;

 let n3 be Nat, g,g1 be FinSequence of INT ;

 assume that

 A129: (t . a) = 0 and

 A130: (t . a5) > 0 and

 A131: (t . a5) <= ( 0 + 1) and

 A132: (t . a4) = (n3 + (t . a5)) and

 A133: n3 = ((t . a3) - 1) and

 A134: (t . a2) = md and

 A135: md <= n3 and

 A136: (n0 + 1) <= (t . a3) and

 A137: (t . a4) <= (n0 + n);

 set t2 = ( IExec (WH3,Q,t)), Q2 = Q;

 assume

 A138: g is_FinSequence_on (t,n0);

 set t1 = ( IExec (WB3,Q,t)), Q1 = Q;

 assume

 A139: g1 is_FinSequence_on (( IExec (WH3,Q,t)),n0);



 A140: (t1 . a4) = (((t1 . a3) - 1) + (t1 . a5)) by A15, A129, A130, A132, A133, A134, A135, Lm19;

 consider n1,n2 be Nat such that

 A141: n1 = ((t1 . a3) - 1) and

 A142: n2 = ((t1 . a4) + 1) and

 A143: for i be Nat st i >= 8 & i <> n1 & i <> n2 holds (t1 . ( intpos i)) = (t . ( intpos i)) and

 A144: (t1 . ( intpos n1)) = (t . ( intpos n1)) & (t1 . ( intpos n2)) = (t . ( intpos n2)) or (t . a3) <= n1 & n2 <= (t . a4) & (t1 . ( intpos n1)) = (t . ( intpos n2)) & (t1 . ( intpos n2)) = (t . ( intpos n1)) and

 A145: for i be Nat st (t . a3) <= i & i <= n1 holds (t1 . ( intpos md)) >= (t1 . ( intpos i)) and

 A146: for i be Nat st n2 <= i & i <= (t . a4) holds (t1 . ( intpos md)) <= (t1 . ( intpos i)) by A15, A129, A130, A132, A133, A134, A135, Lm19;



 A147: (t1 . a5) >= ( - 1) by A15, A129, A130, A132, A133, A134, A135, Lm19;

 then

 A148: n1 <= n2 by A140, A141, A142, Lm22;

 (t1 . a5) < (t . a5) by A15, A129, A130, A132, A133, A134, A135, Lm19;

 then ((t1 . a5) + 1) <= (t . a5) by INT_1: 7;

 then ((t1 . a5) + 1) <= ( 0 + 1) by A131, XXREAL_0: 2;

 then

 A149: (t1 . a5) <= 0 by XREAL_1: 6;

 A150:

 now



 A151: ( DataLoc ((t . a),5)) = ( intpos ( 0 + 5)) by A129, SCMP_GCD: 1;

 let v be 0 -started State of SCMPDS ;

 let V;

 assume that

 A152: (v . a4) = (((v . a3) - 1) + (v . a5)) and

 A153: md <= ((v . a3) - 1) and

 A154: (v . a2) = (t . a2) and

 A155: (v . a) = (t . a) and

 A156: (v . ( DataLoc ((t . a),5))) > 0 ;

 reconsider mm = ((v . a3) - 1) as Element of NAT by A153, INT_1: 3;

 set Iv = ( IExec (WB3,V,v));



 A157: (v . a4) = (mm + (v . a5)) by A152;

 hence (Iv . a) = (v . a) by A15, A129, A134, A153, A154, A155, A156, A151, Lm19;

 thus WB3 is_closed_on (v,V) & WB3 is_halting_on (v,V) by A15, A129, A134, A153, A154, A155, A156, A151, A157, Lm19;

 thus (Iv . ( DataLoc ((t . a),5))) < (v . ( DataLoc ((t . a),5))) by A15, A129, A134, A153, A154, A155, A156, A151, A157, Lm19;

 thus (Iv . a4) = (((Iv . a3) - 1) + (Iv . a5)) by A15, A129, A134, A153, A154, A155, A156, A151, A157, Lm19;

 (Iv . a3) >= (v . a3) by A15, A129, A134, A153, A154, A155, A156, A151, A157, Lm19;

 then ((Iv . a3) - 1) >= ((v . a3) - 1) by XREAL_1: 9;

 hence md <= ((Iv . a3) - 1) by A153, XXREAL_0: 2;

 thus (Iv . a2) = (v . a2) by A15, A129, A134, A153, A154, A155, A156, A151, A157, Lm19;

 end;



 A158: ( DataLoc ((t . a),5)) = ( intpos ( 0 + 5)) by A129, SCMP_GCD: 1;

 assume that

 A159: ( len g) = n and

 A160: ( len g1) = n;



 A161: (t1 . a) = 0 by A15, A129, A130, A132, A133, A134, A135, Lm19;

 then

 A162: ( DataLoc ((t1 . a),5)) = ( intpos ( 0 + 5)) by SCMP_GCD: 1;

 A163:

 now



 A164: (( Initialize t1) . GBP ) = (t1 . GBP ) by SCMPDS_5: 15;



 A165: (( Initialize t1) . ( DataLoc ((t1 . GBP ),5))) = (t1 . ( DataLoc ((t1 . GBP ),5))) by SCMPDS_5: 15;

 let x;



 thus (t2 . x) = (( IExec (WH3,Q1,( Initialize t1))) . x) by A130, A132, A133, A135, A158, A150, Th7

 .= (( Initialize t1) . x) by A162, A149, A164, A165, SCMPDS_8: 23

 .= (t1 . x) by SCMPDS_5: 15;

 end;

 hence (t2 . a) = 0 by A161;

 (t1 . a1) = (t . a1) by A15, A129, A130, A132, A133, A134, A135, Lm19;

 hence (t2 . a1) = (t . a1) by A163;

 (t1 . a2) = md by A15, A129, A130, A132, A133, A134, A135, Lm19;

 hence (t2 . a2) = md by A163;

 (t1 . a4) >= n3 by A15, A129, A130, A132, A133, A134, A135, Lm19;

 then (t2 . a4) >= n3 by A163;

 hence (t2 . a4) >= md by A135, XXREAL_0: 2;

 (t1 . a4) <= (t . a4) by A15, A129, A130, A132, A133, A134, A135, Lm19;

 hence (t2 . a4) <= (t . a4) by A163;

 A166:

 now



 A167: (n0 + 1) >= (7 + 1) by A12, XREAL_1: 6;

 let i be Nat;

 assume that

 A168: i >= (n0 + 1) and

 A169: i <> n1 and

 A170: i <> n2;



 thus (t2 . ( intpos i)) = (t1 . ( intpos i)) by A163

 .= (t . ( intpos i)) by A143, A168, A169, A170, A167, XXREAL_0: 2;

 end;



 A171: (t2 . ( intpos n1)) = (t . ( intpos n1)) & (t2 . ( intpos n2)) = (t . ( intpos n2)) or (t . a3) <= n1 & n2 <= (t . a4) & (t2 . ( intpos n1)) = (t . ( intpos n2)) & (t2 . ( intpos n2)) = (t . ( intpos n1))

 proof

 per cases by A144;

 suppose (t1 . ( intpos n1)) = (t . ( intpos n1)) & (t1 . ( intpos n2)) = (t . ( intpos n2));

 hence thesis by A163;

 end;

 suppose (t . a3) <= n1 & n2 <= (t . a4) & (t1 . ( intpos n1)) = (t . ( intpos n2)) & (t1 . ( intpos n2)) = (t . ( intpos n1));

 hence thesis by A163;

 end;

 end;

 (t2 . ( intpos n1)) = (t . ( intpos n1)) & (t2 . ( intpos n2)) = (t . ( intpos n2)) or n1 >= (n0 + 1) & n2 >= (n0 + 1) & n1 <= (n0 + n) & n2 <= (n0 + n) & (t2 . ( intpos n1)) = (t . ( intpos n2)) & (t2 . ( intpos n2)) = (t . ( intpos n1))

 proof

 per cases by A171;

 suppose (t2 . ( intpos n1)) = (t . ( intpos n1)) & (t2 . ( intpos n2)) = (t . ( intpos n2));

 hence thesis;

 end;

 suppose

 A172: (t . a3) <= n1 & n2 <= (t . a4) & (t2 . ( intpos n1)) = (t . ( intpos n2)) & (t2 . ( intpos n2)) = (t . ( intpos n1));

 then

 A173: n2 <= (n0 + n) by A137, XXREAL_0: 2;

 n1 >= (n0 + 1) by A136, A172, XXREAL_0: 2;

 hence thesis by A148, A172, A173, XXREAL_0: 2;

 end;

 end;

 hence (g,g1) are_fiberwise_equipotent by A138, A139, A159, A160, A166, Lm23;



 A174: (t2 . a4) = (n2 - 1) by A142, A163;

 hereby

 let i be Nat;

 assume that

 A175: (t . a3) <= i and

 A176: i <= (t2 . a4);

 i < n2 by A174, A176, XREAL_1: 146, XXREAL_0: 2;

 then i <= n1 by A147, A140, A141, A142, A149, Lm21;

 then (t1 . ( intpos md)) >= (t1 . ( intpos i)) by A145, A175;

 then (t2 . ( intpos md)) >= (t1 . ( intpos i)) by A163;

 hence (t2 . ( intpos md)) >= (t2 . ( intpos i)) by A163;

 end;

 hereby

 let i be Nat;

 assume that

 A177: (t2 . a4) < i and

 A178: i <= (t . a4);

 ((t2 . a4) + 1) = n2 by A142, A163;

 then n2 <= i by A177, INT_1: 7;

 then (t1 . ( intpos md)) <= (t1 . ( intpos i)) by A146, A178;

 then (t2 . ( intpos md)) <= (t1 . ( intpos i)) by A163;

 hence (t2 . ( intpos md)) <= (t2 . ( intpos i)) by A163;

 end;

 thus thesis by A166, A148, A171, Lm24;

 end;

 for k be Nat holds P[k] from NAT_1:sch 2( A128, A16);

 hence thesis by A1, A3, A4, A5, A7, A8, A9, A10, A11, A13, A14, A127;

 end;



 Lm26: for s be 0 -started State of SCMPDS , md,m3,n0 be Nat st (s . GBP ) = 0 & (s . a4) = (m3 + (s . a5)) & m3 = ((s . a3) - 1) & (s . a2) = md & md >= (n0 + 1) & md <= m3 & n0 >= 7 holds WH3 is_closed_on (s,P) & WH3 is_halting_on (s,P)

 proof

 set a = GBP ;

 let s be 0 -started State of SCMPDS , md,m3,n0 be Nat;

 assume that

 A1: (s . a) = 0 and

 A2: (s . a4) = (m3 + (s . a5)) and

 A3: m3 = ((s . a3) - 1) and

 A4: (s . a2) = md and

 A5: md >= (n0 + 1) and

 A6: md <= m3 and

 A7: n0 >= 7;

 (n0 + 1) >= (7 + 1) by A7, XREAL_1: 6;

 then

 A8: md >= 8 by A5, XXREAL_0: 2;



 A9: ( DataLoc ((s . a),5)) = ( intpos ( 0 + 5)) by A1, SCMP_GCD: 1;

 now

 let v be 0 -started State of SCMPDS ;

 let V;

 assume that

 A10: (v . a4) = (((v . a3) - 1) + (v . a5)) and

 A11: md <= ((v . a3) - 1) and

 A12: (v . a2) = (s . a2) and

 A13: (v . a) = (s . a) and

 A14: (v . ( DataLoc ((s . a),5))) > 0 ;

 set Iv = ( IExec (WB3,V,v));

 reconsider mm = ((v . a3) - 1) as Element of NAT by A11, INT_1: 3;



 A15: (v . a4) = (mm + (v . a5)) by A10;

 hence (Iv . a) = (v . a) by A1, A4, A8, A9, A11, A12, A13, A14, Lm19;

 thus WB3 is_closed_on (v,V) & WB3 is_halting_on (v,V) by A1, A4, A8, A9, A11, A12, A13, A14, A15, Lm19;

 thus (Iv . ( DataLoc ((s . a),5))) < (v . ( DataLoc ((s . a),5))) by A1, A4, A8, A9, A11, A12, A13, A14, A15, Lm19;

 thus (Iv . a4) = (((Iv . a3) - 1) + (Iv . a5)) by A1, A4, A8, A9, A11, A12, A13, A14, A15, Lm19;

 (Iv . a3) >= (v . a3) by A1, A4, A8, A9, A11, A12, A13, A14, A15, Lm19;

 then ((Iv . a3) - 1) >= ((v . a3) - 1) by XREAL_1: 9;

 hence md <= ((Iv . a3) - 1) by A11, XXREAL_0: 2;

 thus (Iv . a2) = (v . a2) by A1, A4, A8, A9, A11, A12, A13, A14, A15, Lm19;

 end;

 hence thesis by A2, A3, A6, Th7;

 end;



 Lm27: for s be 0 -started State of SCMPDS st (s . GBP ) = 0 holds (( IExec (K4,P,( Initialize s))) . GBP ) = 0 & (( IExec (K4,P,( Initialize s))) . a1) = (s . a1) & (( IExec (K4,P,( Initialize s))) . a2) = (s . a2) & (( IExec (K4,P,( Initialize s))) . a3) = ((s . a2) + 1) & (( IExec (K4,P,( Initialize s))) . a4) = (s . a4) & (( IExec (K4,P,( Initialize s))) . a5) = ((s . a4) - (s . a2)) & for i be Nat st i >= 8 holds (( IExec (K4,P,( Initialize s))) . ( intpos i)) = (s . ( intpos i))

 proof

 set a = GBP ;

 let s be 0 -started State of SCMPDS ;

 set t0 = ( Initialize s), t1 = ( IExec ((((k1 ';' k2) ';' k3) ';' k4),P,( Initialize s))), t2 = ( IExec (((k1 ';' k2) ';' k3),P,( Initialize s))), t3 = ( IExec ((k1 ';' k2),P,( Initialize s))), t4 = ( Exec (k1,t0));

 assume (s . a) = 0 ;

 then

 A1: (t0 . a) = 0 by SCMPDS_5: 15;

 then

 A2: ( DataLoc ((t0 . a),5)) = ( intpos ( 0 + 5)) by SCMP_GCD: 1;



 then

 A3: (t4 . a5) = (t0 . ( DataLoc ((t0 . a),4))) by SCMPDS_2: 47

 .= (t0 . ( intpos ( 0 + 4))) by A1, SCMP_GCD: 1

 .= (s . a4) by SCMPDS_5: 15;

 (t0 . a4) = (s . a4) by SCMPDS_5: 15;

 then

 A4: (t4 . a4) = (s . a4) by A2, AMI_3: 10, SCMPDS_2: 47;

 (t0 . a1) = (s . a1) by SCMPDS_5: 15;

 then

 A5: (t4 . a1) = (s . a1) by A2, AMI_3: 10, SCMPDS_2: 47;



 A6: (t4 . a) = 0 by A1, A2, AMI_3: 10, SCMPDS_2: 47;

 then

 A7: ( DataLoc ((t4 . a),5)) = ( intpos ( 0 + 5)) by SCMP_GCD: 1;



 A8: (t3 . a) = (( Exec (k2,t4)) . a) by SCMPDS_5: 42

 .= 0 by A6, A7, AMI_3: 10, SCMPDS_2: 50;

 then

 A9: ( DataLoc ((t3 . a),3)) = ( intpos ( 0 + 3)) by SCMP_GCD: 1;



 A10: (t3 . a4) = (( Exec (k2,t4)) . a4) by SCMPDS_5: 42

 .= (s . a4) by A4, A7, AMI_3: 10, SCMPDS_2: 50;



 A11: (t2 . a4) = (( Exec (k3,t3)) . a4) by SCMPDS_5: 41

 .= (s . a4) by A10, A9, AMI_3: 10, SCMPDS_2: 47;



 A12: (t0 . a2) = (s . a2) by SCMPDS_5: 15;

 then

 A13: (t4 . a2) = (s . a2) by A2, AMI_3: 10, SCMPDS_2: 47;



 A14: (t3 . a5) = (( Exec (k2,t4)) . a5) by SCMPDS_5: 42

 .= ((t4 . a5) - (t4 . ( DataLoc ((t4 . a),2)))) by A7, SCMPDS_2: 50

 .= ((t4 . a5) - (t4 . ( intpos ( 0 + 2)))) by A6, SCMP_GCD: 1

 .= ((s . a4) - (s . a2)) by A12, A2, A3, AMI_3: 10, SCMPDS_2: 47;



 A15: (t2 . a5) = (( Exec (k3,t3)) . a5) by SCMPDS_5: 41

 .= ((s . a4) - (s . a2)) by A14, A9, AMI_3: 10, SCMPDS_2: 47;



 A16: (t3 . a2) = (( Exec (k2,t4)) . a2) by SCMPDS_5: 42

 .= (s . a2) by A13, A7, AMI_3: 10, SCMPDS_2: 50;



 A17: (t3 . a1) = (( Exec (k2,t4)) . a1) by SCMPDS_5: 42

 .= (s . a1) by A5, A7, AMI_3: 10, SCMPDS_2: 50;



 A18: (t2 . a1) = (( Exec (k3,t3)) . a1) by SCMPDS_5: 41

 .= (s . a1) by A17, A9, AMI_3: 10, SCMPDS_2: 47;



 A19: (t2 . a) = (( Exec (k3,t3)) . a) by SCMPDS_5: 41

 .= 0 by A8, A9, AMI_3: 10, SCMPDS_2: 47;

 then

 A20: ( DataLoc ((t2 . a),3)) = ( intpos ( 0 + 3)) by SCMP_GCD: 1;



 A21: (t2 . a3) = (( Exec (k3,t3)) . a3) by SCMPDS_5: 41

 .= (t3 . ( DataLoc ((t3 . a),2))) by A9, SCMPDS_2: 47

 .= (s . a2) by A8, A16, SCMP_GCD: 1;



 A22: (t2 . a2) = (( Exec (k3,t3)) . a2) by SCMPDS_5: 41

 .= (s . a2) by A16, A9, AMI_3: 10, SCMPDS_2: 47;



 thus (t1 . a) = (( Exec (k4,t2)) . a) by SCMPDS_5: 41

 .= 0 by A19, A20, AMI_3: 10, SCMPDS_2: 48;



 thus (t1 . a1) = (( Exec (k4,t2)) . a1) by SCMPDS_5: 41

 .= (s . a1) by A18, A20, AMI_3: 10, SCMPDS_2: 48;



 thus (t1 . a2) = (( Exec (k4,t2)) . a2) by SCMPDS_5: 41

 .= (s . a2) by A22, A20, AMI_3: 10, SCMPDS_2: 48;



 thus (t1 . a3) = (( Exec (k4,t2)) . a3) by SCMPDS_5: 41

 .= ((s . a2) + 1) by A21, A20, SCMPDS_2: 48;



 thus (t1 . a4) = (( Exec (k4,t2)) . a4) by SCMPDS_5: 41

 .= (s . a4) by A11, A20, AMI_3: 10, SCMPDS_2: 48;



 thus (t1 . a5) = (( Exec (k4,t2)) . a5) by SCMPDS_5: 41

 .= ((s . a4) - (s . a2)) by A15, A20, AMI_3: 10, SCMPDS_2: 48;

 A23:

 now

 let i be Nat;

 assume i >= 8;

 then i > 5 by XXREAL_0: 2;



 hence (t4 . ( intpos i)) = (t0 . ( intpos i)) by A2, AMI_3: 10, SCMPDS_2: 47

 .= (s . ( intpos i)) by SCMPDS_5: 15;

 end;

 A24:

 now

 let i be Nat;

 assume

 A25: i >= 8;

 then

 A26: i > 5 by XXREAL_0: 2;



 thus (t3 . ( intpos i)) = (( Exec (k2,t4)) . ( intpos i)) by SCMPDS_5: 42

 .= (t4 . ( intpos i)) by A7, A26, AMI_3: 10, SCMPDS_2: 50

 .= (s . ( intpos i)) by A23, A25;

 end;

 A27:

 now

 let i be Nat;

 assume

 A28: i >= 8;

 then

 A29: i > 3 by XXREAL_0: 2;



 thus (t2 . ( intpos i)) = (( Exec (k3,t3)) . ( intpos i)) by SCMPDS_5: 41

 .= (t3 . ( intpos i)) by A9, A29, AMI_3: 10, SCMPDS_2: 47

 .= (s . ( intpos i)) by A24, A28;

 end;

 hereby

 let i be Nat;

 assume

 A30: i >= 8;

 then

 A31: i > 3 by XXREAL_0: 2;



 thus (t1 . ( intpos i)) = (( Exec (k4,t2)) . ( intpos i)) by SCMPDS_5: 41

 .= (t2 . ( intpos i)) by A20, A31, AMI_3: 10, SCMPDS_2: 48

 .= (s . ( intpos i)) by A27, A30;

 end;

 end;

 theorem :: SCPQSORT:10



 Th10: for s be 0 -started State of SCMPDS , md,p0 be Nat st (s . GBP ) = 0 & (s . ( intpos 2)) = md & md >= (p0 + 1) & p0 >= 7 holds Partition is_closed_on (s,P) & Partition is_halting_on (s,P)

 proof

 set KW = (K4 ';' WH3);

 let s be 0 -started State of SCMPDS , md,n0 be Nat;

 set s2 = ( IExec (K4,P,( Initialize s))), a = GBP , P2 = P;

 assume that

 A1: (s . a) = 0 and

 A2: (s . a2) = md and

 A3: md >= (n0 + 1) and

 A4: n0 >= 7;



 A5: (s2 . a2) = (s . a2) by A1, Lm27;

 set m3 = md;

 (s2 . a5) = ((s . a4) - (s . a2)) by A1, Lm27;

 then

 A6: (s2 . a4) = (m3 + (s2 . a5)) by A1, A2, Lm27;

 (s2 . a3) = ((s . a2) + 1) by A1, Lm27;

 then

 A7: (s . a2) = ((s2 . a3) - 1);



 A8: (s2 . a) = 0 by A1, Lm27;



 A9: (( Initialize s2) . GBP ) = (s2 . GBP ) by SCMPDS_5: 15;



 A10: (( Initialize s2) . a2) = (s2 . a2) by SCMPDS_5: 15;



 A11: (( Initialize s2) . a3) = (s2 . a3) by SCMPDS_5: 15;



 A12: (( Initialize s2) . a4) = (s2 . a4) by SCMPDS_5: 15;



 A13: (( Initialize s2) . a5) = (s2 . a5) by SCMPDS_5: 15;



 A14: WH3 is_halting_on (( Initialize s2),P2) by A2, A3, A4, A5, A7, A6, Lm26, A8, A9, A10, A11, A12, A13;



 A15: WH3 is_halting_on (s2,P2)

 proof

 (P +* ( stop WH3)) halts_on ( Initialize ( Initialize s2)) by A14, SCMPDS_6:def 3;

 hence thesis by SCMPDS_6:def 3;

 end;



 A16: WH3 is_closed_on (( Initialize s2),P2) by A2, A3, A4, A8, A5, A7, A6, Lm26, A9, A10, A11, A12, A13;



 A17: WH3 is_closed_on (s2,P2)

 proof

 for k be Nat holds ( IC ( Comput ((P +* ( stop WH3)),( Initialize ( Initialize s2)),k))) in ( dom ( stop WH3)) by A16, SCMPDS_6:def 2;

 hence thesis by SCMPDS_6:def 2;

 end;

 then

 A18: KW is_closed_on (s,P) by A15, SCPISORT: 9;



 A19: KW is_halting_on (s,P) by A17, A15, SCPISORT: 9;

 then

 A20: (KW ';' j8) is_closed_on (s,P) by A18, SCPISORT: 10;



 A21: (KW ';' j8) is_halting_on (s,P) by A18, A19, SCPISORT: 10;

 then

 A22: ((KW ';' j8) ';' j9) is_halting_on (s,P) by A20, SCPISORT: 10;

 ((KW ';' j8) ';' j9) is_closed_on (s,P) by A20, A21, SCPISORT: 10;

 hence thesis by A22, SCPISORT: 10;

 end;

 theorem :: SCPQSORT:11



 Th11: for s be 0 -started State of SCMPDS , md,p0,n be Nat, f,f1 be FinSequence of INT st (s . GBP ) = 0 & ((s . ( intpos 4)) - (s . ( intpos 2))) > 0 & (s . ( intpos 2)) = md & md >= (p0 + 1) & (s . ( intpos 4)) <= (p0 + n) & p0 >= 7 & f is_FinSequence_on (s,p0) & ( len f) = n & f1 is_FinSequence_on (( IExec ( Partition ,P,s)),p0) & ( len f1) = n holds (( IExec ( Partition ,P,s)) . GBP ) = 0 & (( IExec ( Partition ,P,s)) . ( intpos 1)) = (s . ( intpos 1)) & (f,f1) are_fiberwise_equipotent & ex m4 be Nat st m4 = (( IExec ( Partition ,P,s)) . ( intpos 4)) & md <= m4 & m4 <= (s . ( intpos 4)) & (for i be Nat st md <= i & i < m4 holds (( IExec ( Partition ,P,s)) . ( intpos m4)) >= (( IExec ( Partition ,P,s)) . ( intpos i))) & (for i be Nat st m4 = (p0 + 1) & (i < (s . ( intpos 2)) or i > (s . ( intpos 4))) holds (( IExec ( Partition ,P,s)) . ( intpos i)) = (s . ( intpos i))

 proof

 set KW = (K4 ';' WH3);

 let s be 0 -started State of SCMPDS , md,n0,n be Nat, f,f1 be FinSequence of INT ;



 A1: ( Initialize s) = s by MEMSTR_0: 44;

 set s1 = ( IExec ( Partition ,P,s)), s2 = ( IExec (K4,P,s)), P2 = P, s3 = ( IExec ((K4 ';' WH3),P,s)), a = GBP ;



 A2: (s2 . a) = (( Initialize s2) . a) by SCMPDS_5: 15;



 A3: (s2 . a1) = (( Initialize s2) . a1) by SCMPDS_5: 15;



 A4: (s2 . a2) = (( Initialize s2) . a2) by SCMPDS_5: 15;



 A5: (s2 . a3) = (( Initialize s2) . a3) by SCMPDS_5: 15;



 A6: (s2 . a4) = (( Initialize s2) . a4) by SCMPDS_5: 15;



 A7: (s2 . a5) = (( Initialize s2) . a5) by SCMPDS_5: 15;

 assume that

 A8: (s . a) = 0 and

 A9: ((s . a4) - (s . a2)) > 0 and

 A10: (s . a2) = md and

 A11: md >= (n0 + 1) and

 A12: (s . a4) <= (n0 + n) and

 A13: n0 >= 7;



 A14: (s2 . GBP ) = 0 by A8, Lm27, A1;



 A15: (s2 . a1) = (s . a1) by A8, Lm27, A1;



 A16: (n0 + 1) >= (7 + 1) by A13, XREAL_1: 6;

 then

 A17: md >= 8 by A11, XXREAL_0: 2;

 then

 A18: md > 1 by XXREAL_0: 2;



 A19: md > 4 by A17, XXREAL_0: 2;



 A20: (md - n0) >= 1 by A11, XREAL_1: 19;

 then

 reconsider n1 = (md - n0) as Element of NAT by INT_1: 3;



 A21: md = (n0 + n1);



 A22: (s2 . a4) = (s . a4) by A8, Lm27, A1;

 set m3 = md;

 (s2 . a5) = ((s . a4) - (s . a2)) by A8, Lm27, A1;

 then

 A23: (s2 . a4) = (m3 + (s2 . a5)) by A8, A10, Lm27, A1;

 set s4 = ( IExec (WH3,P2,( Initialize s2)));

 consider f3 be FinSequence of INT such that

 A24: ( len f3) = n and

 A25: for i be Nat st 1 <= i & i <= ( len f3) holds (f3 . i) = (s4 . ( intpos (n0 + i))) by SCPISORT: 1;



 A26: f3 is_FinSequence_on (s4,n0) by A25;

 md < (s . a4) by A9, A10, XREAL_1: 47;

 then md < (n0 + n) by A12, XXREAL_0: 2;

 then

 A27: n1 <= ( len f3) by A24, XREAL_1: 20;



 A28: (s2 . a2) = (s . a2) by A8, Lm27, A1;



 A29: (s2 . a3) = ((s . a2) + 1) by A8, Lm27, A1;

 then

 A30: (s . a2) = ((s2 . a3) - 1);



 A31: (( Initialize s2) . GBP ) = (s2 . GBP ) by SCMPDS_5: 15;



 A32: (( Initialize s2) . a2) = (s2 . a2) by SCMPDS_5: 15;



 A33: (( Initialize s2) . a3) = (s2 . a3) by SCMPDS_5: 15;



 A34: (( Initialize s2) . a4) = (s2 . a4) by SCMPDS_5: 15;



 A35: (( Initialize s2) . a5) = (s2 . a5) by SCMPDS_5: 15;



 A36: WH3 is_halting_on (( Initialize s2),P2) by A10, A11, A13, A14, A28, A23, Lm26, A30, A31, A32, A33, A34, A35;



 A37: WH3 is_halting_on (s2,P2)

 proof

 (P +* ( stop WH3)) halts_on ( Initialize ( Initialize s2)) by A36, SCMPDS_6:def 3;

 hence thesis by SCMPDS_6:def 3;

 end;

 (s . a2) < (s2 . a3) by A29, XREAL_1: 29;

 then

 A38: (n0 + 1) <= (s2 . a3) by A10, A11, XXREAL_0: 2;



 A39: WH3 is_closed_on (( Initialize s2),P2) by A10, A11, A13, A14, A28, A30, A23, Lm26, A31, A32, A33, A34, A35;



 A40: WH3 is_closed_on (s2,P2)

 proof

 for k be Nat holds ( IC ( Comput ((P +* ( stop WH3)),( Initialize ( Initialize s2)),k))) in ( dom ( stop WH3)) by A39, SCMPDS_6:def 2;

 hence thesis by SCMPDS_6:def 2;

 end;

 then

 A41: KW is_halting_on (s,P) by A37, A1, SCPISORT: 9;

 consider f2 be FinSequence of INT such that

 A42: ( len f2) = n and

 A43: for i be Nat st 1 <= i & i <= ( len f2) holds (f2 . i) = (s2 . ( intpos (n0 + i))) by SCPISORT: 1;



 A44: f2 is_FinSequence_on (( Initialize s2),n0)

 proof

 let i be Nat;

 assume 1 <= i & i <= ( len f2);

 then (f2 . i) = (s2 . ( intpos (n0 + i))) by A43;

 hence thesis by SCMPDS_5: 15;

 end;

 then

 A45: (f2,f3) are_fiberwise_equipotent by A9, A10, A11, A12, A13, A14, A28, A22, A30, A23, A38, A42, A24, A26, Lm25, A2, A4, A5, A6, A7;

 (s4 . a4) > 0 by A9, A10, A11, A12, A13, A14, A28, A22, A30, A23, A38, A42, A44, A24, A26, Lm25, A2, A4, A5, A6, A7;

 then

 reconsider m4 = (s4 . a4) as Element of NAT by INT_1: 3;



 A46: (s4 . a4) >= md by A9, A10, A11, A12, A13, A14, A28, A22, A30, A23, A38, A42, A44, A24, A26, Lm25, A2, A4, A5, A6, A7;

 then

 A47: (m4 - n0) >= (md - n0) by XREAL_1: 9;

 then

 reconsider n2 = (m4 - n0) as Element of NAT by A20, INT_1: 3;



 A48: (m4 - n0) >= 1 by A20, A47, XXREAL_0: 2;

 (s4 . a2) = md by A9, A10, A11, A12, A13, A14, A28, A22, A30, A23, A38, A42, A44, A24, A26, Lm25, A2, A4, A5, A6, A7;

 then

 A49: (s3 . a2) = md by A40, A37, SCPISORT: 7;



 A50: (s3 . a4) = m4 by A40, A37, SCPISORT: 7;

 (s4 . a) = 0 by A9, A10, A11, A12, A13, A14, A28, A22, A30, A23, A38, A42, A44, A24, A26, Lm25, A2, A4, A5, A6, A7;

 then

 A51: (s3 . a) = 0 by A40, A37, SCPISORT: 7;

 then

 A52: ( DataLoc ((s3 . a),6)) = ( intpos ( 0 + 6)) by SCMP_GCD: 1;

 set t2 = ( IExec (((KW ';' j8) ';' j9),P,s)), t3 = ( IExec ((KW ';' j8),P,s));

 assume that

 A53: f is_FinSequence_on (s,n0) and

 A54: ( len f) = n;

 assume that

 A55: f1 is_FinSequence_on (s1,n0) and

 A56: ( len f1) = n;



 A57: ( dom f2) = ( Seg n) by A42, FINSEQ_1:def 3;

 A58:

 now

 let i be Nat;

 reconsider a = i as Nat;

 assume

 A59: i in ( dom f2);

 then

 A60: 1 <= i by A57, FINSEQ_1: 1;

 then (i + n0) >= (n0 + 1) by XREAL_1: 6;

 then

 A61: (n0 + i) >= 8 by A16, XXREAL_0: 2;



 A62: i <= n by A57, A59, FINSEQ_1: 1;



 hence (f2 . i) = (s2 . ( intpos (n0 + a))) by A42, A43, A60

 .= (s . ( intpos (n0 + a))) by A8, A61, Lm27, A1

 .= (f . i) by A53, A54, A60, A62;

 end;



 A63: KW is_closed_on (s,P) by A40, A37, A1, SCPISORT: 9;

 then

 A64: (KW ';' j8) is_closed_on (s,P) by A41, SCPISORT: 10;



 A65: (s4 . a4) <= (s2 . a4) by A9, A10, A11, A12, A13, A14, A28, A22, A30, A23, A38, A42, A44, A24, A26, Lm25, A2, A4, A5, A6, A7;

 then m4 <= (n0 + n) by A12, A22, XXREAL_0: 2;

 then

 A66: n2 <= ( len f3) by A24, XREAL_1: 20;



 A67: (t3 . a6) = (( Exec (j8,s3)) . a6) by A63, A41, SCMPDS_7: 31

 .= (s3 . ( DataLoc ((s3 . a4), 0 ))) by A52, SCMPDS_2: 47

 .= (s3 . ( intpos (m4 + 0 ))) by A50, SCMP_GCD: 1

 .= (s4 . ( intpos m4)) by A40, A37, SCPISORT: 7;



 A68: (t3 . a4) = (( Exec (j8,s3)) . a4) by A63, A41, SCMPDS_7: 31

 .= m4 by A50, A52, AMI_3: 10, SCMPDS_2: 47;

 then

 A69: ( DataLoc ((t3 . a4), 0 )) = ( intpos (m4 + 0 )) by SCMP_GCD: 1;



 A70: (s4 . a1) = (s2 . a1) by A9, A10, A11, A12, A13, A14, A28, A22, A30, A23, A38, A42, A44, A24, A26, Lm25, A2, A3, A4, A5, A6, A7;



 A71: (t3 . a1) = (( Exec (j8,s3)) . a1) by A63, A41, SCMPDS_7: 31

 .= (s3 . a1) by A52, AMI_3: 10, SCMPDS_2: 47

 .= (s . a1) by A15, A70, A40, A37, SCPISORT: 7;



 A72: (s4 . a4) >= 8 by A46, A17, XXREAL_0: 2;

 then

 A73: m4 > 2 by XXREAL_0: 2;



 A74: (t3 . a) = (( Exec (j8,s3)) . a) by A63, A41, SCMPDS_7: 31

 .= 0 by A51, A52, AMI_3: 10, SCMPDS_2: 47;



 A75: (t3 . a2) = (( Exec (j8,s3)) . a2) by A63, A41, SCMPDS_7: 31

 .= md by A49, A52, AMI_3: 10, SCMPDS_2: 47;



 A76: (KW ';' j8) is_halting_on (s,P) by A63, A41, SCPISORT: 10;

 then

 A77: ((KW ';' j8) ';' j9) is_halting_on (s,P) by A64, SCPISORT: 10;

 (t2 . a2) = (( Exec (j9,t3)) . a2) by A64, A76, SCMPDS_7: 31

 .= md by A75, A69, A73, AMI_3: 10, SCMPDS_2: 47;

 then

 A78: ( DataLoc ((t2 . a2), 0 )) = ( intpos (md + 0 )) by SCMP_GCD: 1;



 A79: m4 > 6 by A72, XXREAL_0: 2;



 A80: m4 > 1 by A72, XXREAL_0: 2;



 A81: (t2 . a6) = (( Exec (j9,t3)) . a6) by A64, A76, SCMPDS_7: 31

 .= (s4 . ( intpos m4)) by A67, A69, A79, AMI_3: 10, SCMPDS_2: 47;



 A82: (t2 . a) = (( Exec (j9,t3)) . a) by A64, A76, SCMPDS_7: 31

 .= 0 by A11, A46, A74, A69, AMI_3: 10, SCMPDS_2: 47;



 A83: ((KW ';' j8) ';' j9) is_closed_on (s,P) by A64, A76, SCPISORT: 10;



 then

 A84: (s1 . ( intpos md)) = (( Exec (j0,t2)) . ( intpos md)) by A77, SCMPDS_7: 31

 .= (t2 . ( DataLoc ((t2 . a),6))) by A78, SCMPDS_2: 47

 .= (s4 . ( intpos m4)) by A82, A81, SCMP_GCD: 1;

 A85:

 now

 let i be Nat;

 assume i >= 8;

 then

 A86: i > 6 by XXREAL_0: 2;



 thus (t3 . ( intpos i)) = (( Exec (j8,s3)) . ( intpos i)) by A63, A41, SCMPDS_7: 31

 .= (s3 . ( intpos i)) by A52, A86, AMI_3: 10, SCMPDS_2: 47

 .= (s4 . ( intpos i)) by A40, A37, SCPISORT: 7;

 end;

 A87:

 now

 let i be Nat;

 assume that

 A88: i >= 8 and

 A89: i <> m4;



 thus (t2 . ( intpos i)) = (( Exec (j9,t3)) . ( intpos i)) by A64, A76, SCMPDS_7: 31

 .= (t3 . ( intpos i)) by A69, A89, AMI_3: 10, SCMPDS_2: 47

 .= (s4 . ( intpos i)) by A85, A88;

 end;

 A90:

 now

 let i be Nat;

 assume that

 A91: i >= 8 and

 A92: i <> m4 and

 A93: i <> md;



 thus (s1 . ( intpos i)) = (( Exec (j0,t2)) . ( intpos i)) by A83, A77, SCMPDS_7: 31

 .= (t2 . ( intpos i)) by A78, A93, AMI_3: 10, SCMPDS_2: 47

 .= (s4 . ( intpos i)) by A87, A91, A92;

 end;

 A94:

 now

 let k be Nat;

 assume that

 A95: k <> n1 and

 A96: k <> n2 and

 A97: 1 <= k and

 A98: k <= ( len f3);



 A99: (k + n0) <> md by A95;



 A100: (k + n0) >= (n0 + 1) by A97, XREAL_1: 6;



 A101: (k + n0) <> m4 by A96;



 thus (f3 . k) = (s4 . ( intpos (n0 + k))) by A25, A97, A98

 .= (s1 . ( intpos (n0 + k))) by A16, A90, A99, A101, A100, XXREAL_0: 2

 .= (f1 . k) by A55, A56, A24, A97, A98;

 end;



 A102: m4 = (n0 + n2);



 then

 A103: (f3 . n2) = (s1 . ( intpos md)) by A25, A84, A48, A66

 .= (f1 . n1) by A55, A56, A24, A20, A27, A21;



 A104: (t2 . a1) = (( Exec (j9,t3)) . a1) by A64, A76, SCMPDS_7: 31

 .= (s . a1) by A71, A69, A80, AMI_3: 10, SCMPDS_2: 47;



 thus (s1 . a) = (( Exec (j0,t2)) . a) by A83, A77, SCMPDS_7: 31

 .= 0 by A11, A82, A78, AMI_3: 10, SCMPDS_2: 47;



 thus (s1 . a1) = (( Exec (j0,t2)) . a1) by A83, A77, SCMPDS_7: 31

 .= (s . a1) by A104, A78, A18, AMI_3: 10, SCMPDS_2: 47;



 A105: m4 > 4 by A72, XXREAL_0: 2;



 A106: (t2 . a4) = (( Exec (j9,t3)) . a4) by A64, A76, SCMPDS_7: 31

 .= m4 by A68, A69, A105, AMI_3: 10, SCMPDS_2: 47;



 A107: (t2 . ( intpos m4)) = (( Exec (j9,t3)) . ( intpos m4)) by A64, A76, SCMPDS_7: 31

 .= (t3 . ( DataLoc ((t3 . a2), 0 ))) by A69, SCMPDS_2: 47

 .= (t3 . ( intpos (md + 0 ))) by A75, SCMP_GCD: 1

 .= (s4 . ( intpos md)) by A11, A16, A85, XXREAL_0: 2;

 A108:

 now

 per cases ;

 suppose m4 = md;

 hence (s1 . ( intpos m4)) = (s4 . ( intpos md)) by A84;

 end;

 suppose

 A109: m4 <> md;



 thus (s1 . ( intpos m4)) = (( Exec (j0,t2)) . ( intpos m4)) by A83, A77, SCMPDS_7: 31

 .= (s4 . ( intpos md)) by A107, A78, A109, AMI_3: 10, SCMPDS_2: 47;

 end;

 end;



 then (f3 . n1) = (s1 . ( intpos m4)) by A25, A20, A27, A21

 .= (f1 . n2) by A55, A56, A24, A48, A66, A102;

 then (f3,f1) are_fiberwise_equipotent by A56, A24, A20, A27, A48, A66, A103, A94, SCPISORT: 3;

 then (f2,f1) are_fiberwise_equipotent by A45, CLASSES1: 76;

 hence (f,f1) are_fiberwise_equipotent by A54, A42, A58, FINSEQ_2: 9;

 take m4;



 thus (s1 . a4) = (( Exec (j0,t2)) . a4) by A83, A77, SCMPDS_7: 31

 .= m4 by A106, A78, A19, AMI_3: 10, SCMPDS_2: 47;



 A110: m3 = ((s2 . a3) - 1) by A10, A29;

 hence md <= m4 by A9, A10, A11, A12, A13, A14, A28, A22, A23, A38, A42, A44, A24, A26, Lm25, A2, A4, A5, A6, A7;

 thus m4 <= (s . a4) by A9, A10, A11, A12, A13, A14, A28, A22, A30, A23, A38, A42, A44, A24, A26, Lm25, A2, A4, A5, A6, A7;



 A111: (s2 . a3) = (md + 1) by A8, A10, Lm27, A1;

 hereby

 let i be Nat;

 assume that

 A112: md <= i and

 A113: i < m4;

 per cases ;

 suppose

 A114: i = md;

 then (s2 . a3) <= m4 by A111, A113, INT_1: 7;

 hence (s1 . ( intpos m4)) >= (s1 . ( intpos i)) by A114, A84, A9, A10, A11, A12, A13, A14, A28, A22, A110, A23, A38, A42, A44, A24, A26, A108, Lm25, A2, A4, A5, A6, A7;

 end;

 suppose

 A115: i <> md;

 then md = (s4 . ( intpos i)) by A9, A10, A11, A12, A13, A14, A28, A22, A110, A23, A38, A42, A44, A24, A26, A113, Lm25, A2, A4, A5, A6, A7;

 hence (s1 . ( intpos m4)) >= (s1 . ( intpos i)) by A17, A108, A90, A112, A115, XXREAL_0: 2;

 end;

 end;

 hereby

 let i be Nat;

 assume that

 A116: m4 < i and

 A117: i <= (s . a4);



 A118: md < i by A46, A116, XXREAL_0: 2;

 (s4 . ( intpos md)) <= (s4 . ( intpos i)) by A9, A10, A11, A12, A13, A14, A28, A22, A30, A23, A38, A42, A44, A24, A26, A116, A117, Lm25, A2, A4, A5, A6, A7;

 hence (s1 . ( intpos m4)) <= (s1 . ( intpos i)) by A17, A108, A90, A118, XXREAL_0: 2;

 end;

 hereby

 let i be Nat;

 assume that

 A119: i >= (n0 + 1) and

 A120: i < (s . a2) or i > (s . a4);



 A121: i >= 8 by A16, A119, XXREAL_0: 2;

 A122:

 now

 per cases by A120;

 case

 A123: i < (s . a2);

 (s . a2) < (s2 . a3) by A29, XREAL_1: 29;

 hence i < (s2 . a3) by A123, XXREAL_0: 2;

 end;

 case i > (s . a4);

 hence i > (s2 . a4) by A8, Lm27, A1;

 end;

 end;



 A124: (s2 . ( intpos i)) = (( Initialize s2) . ( intpos i)) by SCMPDS_5: 15;



 A125: for i be Nat st i >= (n0 + 1) & (i < (( Initialize s2) . a3) or i > (( Initialize s2) . a4)) holds (( IExec (WH3,P2,( Initialize s2))) . ( intpos i)) = (( Initialize s2) . ( intpos i)) by A9, A10, A11, A12, A13, A14, A28, A22, A110, A23, A38, A42, A44, A24, A26, Lm25, A2, A4, A5, A6, A7;

 i <> md & i <> m4

 proof

 per cases by A120;

 suppose i < (s . a2);

 hence thesis by A9, A10, A11, A12, A13, A14, A28, A22, A30, A23, A38, A42, A44, A24, A26, Lm25, A2, A4, A5, A6, A7;

 end;

 suppose i > (s . a4);

 hence thesis by A22, A46, A65, XXREAL_0: 2;

 end;

 end;



 hence (s1 . ( intpos i)) = (s4 . ( intpos i)) by A16, A90, A119, XXREAL_0: 2

 .= (s2 . ( intpos i)) by A119, A122, A125, A5, A6, A124

 .= (s . ( intpos i)) by A8, A121, Lm27, A1;

 end;

 end;

 theorem :: SCPQSORT:12

 Partition is halt-free shiftable;



 Lm28: for s be 0 -started State of SCMPDS , p0,n be Nat holds ( card ( QuickSort (n,p0))) = 57 & (p0 >= 7 implies ( QuickSort (n,p0)) is_halting_on (s,P) & ex f,g be FinSequence of INT st ( len f) = n & f is_FinSequence_on (s,p0) & ( len g) = n & g is_FinSequence_on (( IExec (( QuickSort (n,p0)),P,s)),p0) & (f,g) are_fiberwise_equipotent & g is_non_decreasing_on (1,n))

 proof

 set b1 = ( DataLoc ( 0 ,1));

 reconsider Pt = Partition as halt-free shiftable Program of SCMPDS ;

 let s be 0 -started State of SCMPDS , p0,n be Nat;



 A1: ( Initialize s) = s by MEMSTR_0: 44;

 set pn = (p0 + n);

 set l1 = (( GBP ,2) := ( SBP ,(pn + 1))), l2 = ( SubFrom ( GBP ,2, SBP ,pn)), l3 = (( GBP ,2) := ( SBP ,pn)), l4 = (( GBP ,4) := ( SBP ,(pn + 1))), l5 = (( SBP ,(pn + 3)) := ( SBP ,(pn + 1))), l6 = (( SBP ,(pn + 1)) := ( GBP ,4)), l7 = (( SBP ,(pn + 2)) := ( GBP ,4)), l8 = ( AddTo ( SBP ,(pn + 1),( - 1))), l9 = ( AddTo ( SBP ,(pn + 2),1)), l0 = ( AddTo ( GBP ,1,2)), lb = ( Load ( AddTo ( GBP ,1,( - 2)))), L5 = (((((l5 ';' l6) ';' l7) ';' l8) ';' l9) ';' l0), TR = (((l3 ';' l4) ';' Pt) ';' L5), IF4 = ( if>0 ( GBP ,2,TR,lb)), WB4 = ((l1 ';' l2) ';' IF4), WH4 = ( while>0 ( GBP ,1,WB4));

 set m1 = ( GBP := 0 ), m2 = ( SBP := 1), m3 = (( SBP ,pn) := (p0 + 1)), m4 = (( SBP ,(pn + 1)) := pn), M4 = (((m1 ';' m2) ';' m3) ';' m4);

 set s1 = ( IExec (M4,P,s)), a = GBP , P1 = P;



 A2: (7 + n) >= 7 by NAT_1: 11;



 A3: ( card L5) = (( card ((((l5 ';' l6) ';' l7) ';' l8) ';' l9)) + 1) by SCMP_GCD: 4

 .= ((( card (((l5 ';' l6) ';' l7) ';' l8)) + 1) + 1) by SCMP_GCD: 4

 .= ((4 + 1) + 1) by Th4;



 A4: ( card WB4) = (( card (l1 ';' l2)) + ( card IF4)) by AFINSQ_1: 17

 .= (2 + ( card IF4)) by SCMP_GCD: 5

 .= (2 + ((( card TR) + ( card lb)) + 2)) by SCMPDS_6: 65

 .= (2 + ((( card TR) + 1) + 2)) by COMPOS_1: 54

 .= (2 + (((( card ((l3 ';' l4) ';' Pt)) + ( card L5)) + 1) + 2)) by AFINSQ_1: 17

 .= (2 + ((((( card (l3 ';' l4)) + ( card Pt)) + ( card L5)) + 1) + 2)) by AFINSQ_1: 17

 .= (2 + ((((2 + 38) + ( card L5)) + 1) + 2)) by Th9, SCMP_GCD: 5

 .= 51 by A3;



 thus ( card ( QuickSort (n,p0))) = (( card M4) + ( card WH4)) by AFINSQ_1: 17

 .= (4 + ( card WH4)) by Th4

 .= (4 + (51 + 2)) by A4, SCMPDS_8: 17

 .= 57;

 assume

 A5: p0 >= 7;

 then pn >= (7 + n) by XREAL_1: 6;

 then

 A6: pn >= 7 by A2, XXREAL_0: 2;



 A7: for t be State of SCMPDS , Q holds for m1 be Nat st (t . a) = 0 & (t . SBP ) = m1 holds (( IExec ((l3 ';' l4),Q,( Initialize t))) . a2) = (t . ( intpos (m1 + pn))) & (( IExec ((l3 ';' l4),Q,( Initialize t))) . a4) = (t . ( intpos ((m1 + pn) + 1))) & for i be Nat st i <> 2 & i <> 4 holds (( IExec ((l3 ';' l4),Q,( Initialize t))) . ( intpos i)) = (t . ( intpos i))

 proof

 let t be State of SCMPDS , Q;

 let m1 be Nat;

 set t0 = ( Initialize t), t2 = ( IExec ((l3 ';' l4),Q,( Initialize t))), t3 = ( Exec (l3,t0)), mp = (m1 + pn), Q0 = Q;

 assume that

 A8: (t . a) = 0 and

 A9: (t . SBP ) = m1;



 A10: (t0 . a) = 0 by A8, SCMPDS_5: 15;

 then

 A11: ( DataLoc ((t0 . a),2)) = ( intpos ( 0 + 2)) by SCMP_GCD: 1;

 then (t3 . a) = 0 by A10, AMI_3: 10, SCMPDS_2: 47;

 then

 A12: ( DataLoc ((t3 . a),4)) = ( intpos ( 0 + 4)) by SCMP_GCD: 1;



 A13: (t0 . SBP ) = m1 by A9, SCMPDS_5: 15;

 then

 A14: (t3 . SBP ) = m1 by A11, AMI_3: 10, SCMPDS_2: 47;

 mp >= ( 0 + 7) by A6, XREAL_1: 7;

 then (mp + 1) >= (7 + 1) by XREAL_1: 6;

 then (mp + 1) > 2 by XXREAL_0: 2;



 then

 A15: (t3 . ( intpos (mp + 1))) = (t0 . ( intpos (mp + 1))) by A11, AMI_3: 10, SCMPDS_2: 47

 .= (t . ( intpos (mp + 1))) by SCMPDS_5: 15;



 A16: (t3 . a2) = (t0 . ( DataLoc ((t0 . SBP ),pn))) by A11, SCMPDS_2: 47

 .= (t0 . ( intpos mp)) by A13, SCMP_GCD: 1

 .= (t . ( intpos mp)) by SCMPDS_5: 15;



 thus (t2 . a2) = (( Exec (l4,t3)) . a2) by SCMPDS_5: 42

 .= (t . ( intpos mp)) by A16, A12, AMI_3: 10, SCMPDS_2: 47;



 thus (t2 . a4) = (( Exec (l4,t3)) . a4) by SCMPDS_5: 42

 .= (t3 . ( DataLoc ((t3 . SBP ),(pn + 1)))) by A12, SCMPDS_2: 47

 .= (t . ( intpos (mp + 1))) by A14, A15, SCMP_GCD: 1;

 hereby

 let i be Nat;

 assume that

 A17: i <> 2 and

 A18: i <> 4;



 thus (t2 . ( intpos i)) = (( Exec (l4,t3)) . ( intpos i)) by SCMPDS_5: 42

 .= (t3 . ( intpos i)) by A12, A18, AMI_3: 10, SCMPDS_2: 47

 .= (t0 . ( intpos i)) by A11, A17, AMI_3: 10, SCMPDS_2: 47

 .= (t . ( intpos i)) by SCMPDS_5: 15;

 end;

 end;



 A19: for t be State of SCMPDS , Q holds for m,m1 be Nat st (t . a) = 0 & (t . SBP ) = m1 & m1 = (m + 1) holds (( IExec (L5,Q,( Initialize t))) . a) = 0 & (( IExec (L5,Q,( Initialize t))) . SBP ) = (m1 + 2) & (( IExec (L5,Q,( Initialize t))) . ( intpos (m1 + pn))) = (t . ( intpos (m1 + pn))) & (( IExec (L5,Q,( Initialize t))) . ( intpos ((m1 + pn) + 1))) = ((t . a4) - 1) & (( IExec (L5,Q,( Initialize t))) . ( intpos ((m1 + pn) + 2))) = ((t . a4) + 1) & (( IExec (L5,Q,( Initialize t))) . ( intpos ((m1 + pn) + 3))) = (t . ( intpos ((m1 + pn) + 1))) & for i be Nat st i >= 8 & i < (m1 + pn) holds (( IExec (L5,Q,( Initialize t))) . ( intpos i)) = (t . ( intpos i))

 proof

 let t be State of SCMPDS , Q;

 let m,m1 be Nat;

 assume that

 A20: (t . a) = 0 and

 A21: (t . SBP ) = m1 and

 A22: m1 = (m + 1);

 set t0 = ( Initialize t), t1 = ( IExec (L5,Q,( Initialize t))), Q0 = Q, t2 = ( IExec (((((l5 ';' l6) ';' l7) ';' l8) ';' l9),Q,( Initialize t))), t3 = ( IExec ((((l5 ';' l6) ';' l7) ';' l8),Q,( Initialize t))), t4 = ( IExec (((l5 ';' l6) ';' l7),Q,( Initialize t))), t5 = ( IExec ((l5 ';' l6),Q,( Initialize t))), t6 = ( Exec (l5,t0)), mp = (m1 + pn);



 A23: (mp + 3) <> (mp + 1);

 m1 >= 1 by A22, NAT_1: 11;

 then

 A24: mp >= (1 + 7) by A6, XREAL_1: 7;

 then

 A25: (mp + 1) >= (8 + 1) by XREAL_1: 6;

 then

 A26: (mp + 1) > 4 by XXREAL_0: 2;



 A27: (mp + 1) > 1 by A25, XXREAL_0: 2;



 A28: (mp + 0 ) <> (mp + 2);



 A29: mp > 1 by A24, XXREAL_0: 2;



 A30: (mp + 1) <> (mp + 2);



 A31: (t0 . SBP ) = m1 by A21, SCMPDS_5: 15;

 then

 A32: ( DataLoc ((t0 . SBP ),(pn + 3))) = ( intpos (mp + 3)) by SCMP_GCD: 1;

 A33:

 now

 let i be Nat;

 assume i <> (mp + 3);



 hence (t6 . ( intpos i)) = (t0 . ( intpos i)) by A32, AMI_3: 10, SCMPDS_2: 47

 .= (t . ( intpos i)) by SCMPDS_5: 15;

 end;

 (mp + 0 ) <> (mp + 3);

 then

 A34: (t6 . ( intpos mp)) = (t . ( intpos mp)) by A33;



 A35: (mp + 1) <> (mp + 2);



 A36: (mp + 3) <> (mp + 2);



 A37: (mp + 3) >= (8 + 3) by A24, XREAL_1: 6;

 then

 A38: (mp + 3) > 1 by XXREAL_0: 2;

 (mp + 3) > 1 by A37, XXREAL_0: 2;

 then

 A39: (t6 . SBP ) = m1 by A21, A33;

 then

 A40: ( DataLoc ((t6 . SBP ),(pn + 1))) = ( intpos (mp + 1)) by SCMP_GCD: 1;

 A41:

 now

 let i be Nat;

 assume

 A42: i <> (mp + 1);



 thus (t5 . ( intpos i)) = (( Exec (l6,t6)) . ( intpos i)) by SCMPDS_5: 42

 .= (t6 . ( intpos i)) by A40, A42, AMI_3: 10, SCMPDS_2: 47;

 end;

 (mp + 1) > 1 by A25, XXREAL_0: 2;

 then

 A43: (t5 . SBP ) = m1 by A39, A41;

 then

 A44: ( DataLoc ((t5 . SBP ),(pn + 2))) = ( intpos (mp + 2)) by SCMP_GCD: 1;

 A45:

 now

 let i be Nat;

 assume

 A46: i <> (mp + 2);



 thus (t4 . ( intpos i)) = (( Exec (l7,t5)) . ( intpos i)) by SCMPDS_5: 41

 .= (t5 . ( intpos i)) by A44, A46, AMI_3: 10, SCMPDS_2: 47;

 end;

 (t6 . ( intpos (mp + 3))) = (t0 . ( DataLoc ((t0 . SBP ),(pn + 1)))) by A32, SCMPDS_2: 47

 .= (t0 . ( intpos (m1 + (pn + 1)))) by A31, SCMP_GCD: 1

 .= (t . ( intpos (mp + 1))) by SCMPDS_5: 15;

 then (t5 . ( intpos (mp + 3))) = (t . ( intpos (mp + 1))) by A41, A23;

 then

 A47: (t4 . ( intpos (mp + 3))) = (t . ( intpos (mp + 1))) by A45, A36;



 A48: (mp + 3) > 4 by A37, XXREAL_0: 2;

 then

 A49: (t6 . a4) = (t . a4) by A33;



 A50: (mp + 2) >= (8 + 2) by A24, XREAL_1: 6;

 then

 A51: (mp + 2) > 1 by XXREAL_0: 2;

 (mp + 2) > 1 by A50, XXREAL_0: 2;

 then

 A52: (t4 . SBP ) = m1 by A43, A45;

 then

 A53: ( DataLoc ((t4 . SBP ),(pn + 1))) = ( intpos (mp + 1)) by SCMP_GCD: 1;

 A54:

 now

 let i be Nat;

 assume

 A55: i <> (mp + 1);



 thus (t3 . ( intpos i)) = (( Exec (l8,t4)) . ( intpos i)) by SCMPDS_5: 41

 .= (t4 . ( intpos i)) by A53, A55, AMI_3: 10, SCMPDS_2: 48;

 end;

 ((m1 + pn) + 1) > 1 by A25, XXREAL_0: 2;

 then

 A56: (t3 . SBP ) = m1 by A52, A54;

 then

 A57: ( DataLoc ((t3 . SBP ),(pn + 2))) = ( intpos (mp + 2)) by SCMP_GCD: 1;

 A58:

 now

 let i be Nat;

 assume

 A59: i <> (mp + 2);



 thus (t2 . ( intpos i)) = (( Exec (l9,t3)) . ( intpos i)) by SCMPDS_5: 41

 .= (t3 . ( intpos i)) by A57, A59, AMI_3: 10, SCMPDS_2: 48;

 end;



 A60: (t6 . a) = 0 by A20, A33;

 then

 A61: (t5 . a) = 0 by A41;

 then (t4 . a) = 0 by A45;

 then (t3 . a) = 0 by A54;

 then

 A62: (t2 . a) = 0 by A58;

 then

 A63: ( DataLoc ((t2 . a),1)) = ( intpos ( 0 + 1)) by SCMP_GCD: 1;

 (mp + 2) > 1 by A50, XXREAL_0: 2;

 then

 A64: (t2 . SBP ) = m1 by A56, A58;

 A65:

 now

 let i be Nat;

 assume i <> 1;

 then

 A66: ( intpos i) <> ( DataLoc ((t2 . a),1)) by A62, AMI_3: 10, SCMP_GCD: 1;



 thus (t1 . ( intpos i)) = (( Exec (l0,t2)) . ( intpos i)) by SCMPDS_5: 41

 .= (t2 . ( intpos i)) by A66, SCMPDS_2: 48;

 end;

 hence (t1 . a) = 0 by A62;



 A67: (mp + 0 ) <> (mp + 2);

 (mp + 0 ) <> (mp + 1);

 then (t5 . ( intpos mp)) = (t . ( intpos mp)) by A34, A41;

 then

 A68: (t4 . ( intpos mp)) = (t . ( intpos mp)) by A45, A67;



 thus (t1 . SBP ) = (( Exec (l0,t2)) . SBP ) by SCMPDS_5: 41

 .= (m1 + 2) by A64, A63, SCMPDS_2: 48;

 (mp + 0 ) <> (mp + 1);

 then (t3 . ( intpos mp)) = (t . ( intpos mp)) by A68, A54;

 then (t2 . ( intpos mp)) = (t . ( intpos mp)) by A58, A28;

 hence (t1 . ( intpos mp)) = (t . ( intpos mp)) by A65, A29;



 A69: (mp + 1) <> (mp + 2);

 (t4 . ( intpos (mp + 2))) = (( Exec (l7,t5)) . ( intpos (mp + 2))) by SCMPDS_5: 41

 .= (t5 . ( DataLoc ((t5 . GBP ),4))) by A44, SCMPDS_2: 47

 .= (t5 . ( intpos ( 0 + 4))) by A61, SCMP_GCD: 1

 .= (t . a4) by A49, A41, A26;

 then

 A70: (t3 . ( intpos (mp + 2))) = (t . a4) by A54, A30;

 (t5 . ( intpos (mp + 1))) = (( Exec (l6,t6)) . ( intpos (mp + 1))) by SCMPDS_5: 42

 .= (t6 . ( DataLoc ((t6 . GBP ),4))) by A40, SCMPDS_2: 47

 .= (t6 . ( intpos ( 0 + 4))) by A60, SCMP_GCD: 1

 .= (t . a4) by A33, A48;

 then

 A71: (t4 . ( intpos (mp + 1))) = (t . a4) by A45, A69;

 (t3 . ( intpos (mp + 1))) = (( Exec (l8,t4)) . ( intpos (mp + 1))) by SCMPDS_5: 41

 .= ((t . a4) + ( - 1)) by A71, A53, SCMPDS_2: 48

 .= ((t . a4) - 1);

 then (t2 . ( intpos (mp + 1))) = ((t . a4) - 1) by A58, A35;

 hence (t1 . ( intpos (mp + 1))) = ((t . a4) - 1) by A65, A27;

 (t2 . ( intpos (mp + 2))) = (( Exec (l9,t3)) . ( intpos (mp + 2))) by SCMPDS_5: 41

 .= ((t . a4) + 1) by A70, A57, SCMPDS_2: 48;

 hence (t1 . ( intpos (mp + 2))) = ((t . a4) + 1) by A65, A51;



 A72: (mp + 3) <> (mp + 2);

 (mp + 3) <> (mp + 1);

 then (t3 . ( intpos (mp + 3))) = (t . ( intpos (mp + 1))) by A47, A54;

 then (t2 . ( intpos (mp + 3))) = (t . ( intpos (mp + 1))) by A58, A72;

 hence (t1 . ( intpos (mp + 3))) = (t . ( intpos (mp + 1))) by A65, A38;

 hereby



 A73: mp <= (mp + 3) by NAT_1: 11;



 A74: mp <= (mp + 2) by NAT_1: 11;



 A75: mp <= (mp + 1) by NAT_1: 11;

 let i be Nat;

 assume that

 A76: i >= 8 and

 A77: i < mp;

 i > 1 by A76, XXREAL_0: 2;



 hence (t1 . ( intpos i)) = (t2 . ( intpos i)) by A65

 .= (t3 . ( intpos i)) by A58, A77, A74

 .= (t4 . ( intpos i)) by A54, A77, A75

 .= (t5 . ( intpos i)) by A45, A77, A74

 .= (t6 . ( intpos i)) by A41, A77, A75

 .= (t . ( intpos i)) by A33, A77, A73;

 end;

 end;



 A78: for t be 0 -started State of SCMPDS , Q holds for m,m1,md be Nat, n4 be Integer, f1,f2 be FinSequence of INT st (t . a) = 0 & (t . SBP ) = m1 & m1 = (m + 1) & (t . ( intpos (m1 + pn))) = md & md >= (p0 + 1) & n4 = (t . ( intpos ((m1 + pn) + 1))) & (n4 - md) > 0 & n4 <= pn & f1 is_FinSequence_on (t,p0) & ( len f1) = n & f2 is_FinSequence_on (( IExec (TR,Q,t)),p0) & ( len f2) = n holds (( IExec (TR,Q,t)) . a) = 0 & (( IExec (TR,Q,t)) . a1) = (m1 + 2) & md = (( IExec (TR,Q,t)) . ( intpos (m1 + pn))) & n4 = (( IExec (TR,Q,t)) . ( intpos ((m1 + pn) + 3))) & (for j be Nat st 1 <= j & j < m1 holds (( IExec (TR,Q,t)) . ( intpos (pn + j))) = (t . ( intpos (pn + j)))) & (f1,f2) are_fiberwise_equipotent & ex m4 be Nat st md <= m4 & m4 <= n4 & (m4 - 1) = (( IExec (TR,Q,t)) . ( intpos ((m1 + pn) + 1))) & (m4 + 1) = (( IExec (TR,Q,t)) . ( intpos ((m1 + pn) + 2))) & (for i be Nat st md <= i & i < m4 holds (( IExec (TR,Q,t)) . ( intpos m4)) >= (( IExec (TR,Q,t)) . ( intpos i))) & (for i be Nat st m4 < i & i <= n4 holds (( IExec (TR,Q,t)) . ( intpos m4)) <= (( IExec (TR,Q,t)) . ( intpos i))) & for j be Nat st (p0 + 1) <= j & j < md or n4 < j & j <= (p0 + n) holds (( IExec (TR,Q,t)) . ( intpos j)) = (t . ( intpos j))

 proof

 let t be 0 -started State of SCMPDS , Q;

 let m,m1,md be Nat, n4 be Integer, f1,f2 be FinSequence of INT ;



 A79: ( Initialize t) = t by MEMSTR_0: 44;

 set t1 = ( IExec (TR,Q,t)), lPt = ((l3 ';' l4) ';' Pt), t2 = ( IExec (lPt,Q,t)), Q2 = Q, t4 = ( IExec ((l3 ';' l4),Q,t)), mp = (m1 + pn), Q4 = Q;



 A80: (t4 . a) = (( Initialize t4) . a) by SCMPDS_5: 15;



 A81: (t4 . a2) = (( Initialize t4) . a2) by SCMPDS_5: 15;



 A82: (t4 . a4) = (( Initialize t4) . a4) by SCMPDS_5: 15;



 A83: (t4 . a1) = (( Initialize t4) . a1) by SCMPDS_5: 15;

 set tp = ( IExec (Pt,Q4,( Initialize t4)));

 assume that

 A84: (t . a) = 0 and

 A85: (t . SBP ) = m1 and

 A86: m1 = (m + 1) and

 A87: (t . ( intpos mp)) = md and

 A88: md >= (p0 + 1) and

 A89: n4 = (t . ( intpos (mp + 1))) and

 A90: (n4 - md) > 0 and

 A91: n4 <= pn;



 A92: (t4 . a) = 0 by A7, A84, A85, A79;

 assume that

 A93: f1 is_FinSequence_on (t,p0) and

 A94: ( len f1) = n;



 A95: (p0 + 1) >= (7 + 1) by A5, XREAL_1: 6;

 A96:

 now

 let i be Nat;

 assume that

 A97: 1 <= i and

 A98: i <= ( len f1);

 (p0 + i) >= (p0 + 1) by A97, XREAL_1: 6;

 then

 A99: (p0 + i) >= 8 by A95, XXREAL_0: 2;

 then

 A100: (p0 + i) > 2 by XXREAL_0: 2;



 A101: (p0 + i) > 4 by A99, XXREAL_0: 2;



 thus (f1 . i) = (t . ( intpos (p0 + i))) by A93, A97, A98

 .= (t4 . ( intpos (p0 + i))) by A7, A84, A85, A100, A101, A79;

 end;



 A102: (t4 . a4) = n4 by A7, A84, A85, A89, A79;



 A103: (t4 . a4) = (t . ( intpos (mp + 1))) by A7, A84, A85, A79;



 A104: (t4 . a2) = (t . ( intpos mp)) by A7, A84, A85, A79;

 then Pt is_closed_on (( Initialize t4),Q4) by A5, A87, A88, A92, Th10, A80, A81;

 then

 A105: Pt is_closed_on (t4,Q4) by SCMPDS_6: 125;

 Pt is_halting_on (( Initialize t4),Q4) by A5, A87, A88, A104, A92, Th10, A80, A81;

 then

 A106: Pt is_halting_on (t4,Q4) by SCMPDS_6: 126;

 then

 A107: lPt is_closed_on (t,Q) by A105, A79, SCPISORT: 9;

 assume that

 A108: f2 is_FinSequence_on (( IExec (TR,Q,t)),p0) and

 A109: ( len f2) = n;

 consider f4 be FinSequence of INT such that

 A110: ( len f4) = n and

 A111: for i be Nat st 1 <= i & i <= ( len f4) holds (f4 . i) = (tp . ( intpos (p0 + i))) by SCPISORT: 1;



 A112: f1 is_FinSequence_on (( Initialize t4),p0)

 proof

 let i be Nat;

 assume 1 <= i & i <= ( len f1);

 then (f1 . i) = (t4 . ( intpos (p0 + i))) by A96;

 hence thesis by SCMPDS_5: 15;

 end;



 A113: f4 is_FinSequence_on (tp,p0) by A111;

 then

 consider m4 be Nat such that

 A114: m4 = (tp . a4) and

 A115: md <= m4 and

 A116: m4 <= (t4 . a4) and

 A117: for i be Nat st md <= i & i < m4 holds (tp . ( intpos m4)) >= (tp . ( intpos i)) and

 A118: for i be Nat st m4 = (p0 + 1) & (i < (t4 . a2) or i > (t4 . a4)) holds (tp . ( intpos i)) = (( Initialize t4) . ( intpos i)) by A5, A87, A88, A89, A90, A91, A94, A104, A103, A92, A112, A110, Th11, A80, A81, A82;

 (tp . a) = 0 by A5, A87, A88, A89, A90, A91, A94, A104, A103, A92, A112, A110, A113, Th11, A80, A81, A82;

 then

 A120: (t2 . a) = 0 by A105, A106, SCPISORT: 7;



 A121: lPt is_halting_on (t,Q) by A105, A106, A79, SCPISORT: 9;

 (tp . a1) = (t4 . a1) by A5, A87, A88, A89, A90, A91, A94, A104, A103, A92, A112, A110, A113, Th11, A80, A81, A82, A83;



 then

 A122: (t2 . SBP ) = (t4 . a1) by A105, A106, SCPISORT: 7

 .= m1 by A7, A84, A85, A79;

 then

 A123: (( IExec (L5,Q2,( Initialize t2))) . ( intpos mp)) = (t2 . ( intpos mp)) by A19, A86, A120;

 (( IExec (L5,Q2,( Initialize t2))) . a) = 0 by A19, A86, A120, A122;

 hence (t1 . a) = 0 by A107, A121, SCPISORT: 6;

 (( IExec (L5,Q2,( Initialize t2))) . SBP ) = (m1 + 2) by A19, A86, A120, A122;

 hence (t1 . a1) = (m1 + 2) by A107, A121, SCPISORT: 6;



 A124: (( IExec (L5,Q2,( Initialize t2))) . ( intpos (mp + 3))) = (t2 . ( intpos (mp + 1))) by A19, A86, A120, A122;



 A125: (( IExec (L5,Q2,( Initialize t2))) . ( intpos (mp + 2))) = ((t2 . a4) + 1) by A19, A86, A120, A122;



 A126: (( IExec (L5,Q2,( Initialize t2))) . ( intpos (mp + 1))) = ((t2 . a4) - 1) by A19, A86, A120, A122;



 A127: (1 + pn) > pn by XREAL_1: 29;



 A128: m1 >= 1 by A86, NAT_1: 11;

 then mp >= (1 + pn) by XREAL_1: 6;

 then

 A129: mp > pn by A127, XXREAL_0: 2;

 then

 A130: mp > (t4 . a4) by A91, A102, XXREAL_0: 2;

 then

 A131: m4 < mp by A116, XXREAL_0: 2;



 A132: ( dom f2) = ( Seg n) by A109, FINSEQ_1:def 3;

 A133:

 now

 let i be Nat;

 reconsider a = i as Nat;

 assume

 A134: i in ( dom f2);

 then

 A135: 1 <= i by A132, FINSEQ_1: 1;

 then (p0 + i) >= (p0 + 1) by XREAL_1: 6;

 then

 A136: (p0 + i) >= 8 by A95, XXREAL_0: 2;



 A137: i <= n by A132, A134, FINSEQ_1: 1;

 then (p0 + i) <= pn by XREAL_1: 6;

 then

 A138: (p0 + i) < mp by A129, XXREAL_0: 2;



 thus (f2 . i) = (t1 . ( intpos (p0 + a))) by A108, A109, A135, A137

 .= (( IExec (L5,Q2,( Initialize t2))) . ( intpos (p0 + a))) by A107, A121, SCPISORT: 6

 .= (t2 . ( intpos (p0 + a))) by A19, A86, A120, A122, A136, A138

 .= (tp . ( intpos (p0 + a))) by A105, A106, SCPISORT: 7

 .= (f4 . i) by A110, A111, A135, A137;

 end;



 A139: mp >= (1 + 7) by A6, A128, XREAL_1: 7;

 then

 A140: mp > 4 by XXREAL_0: 2;



 A141: (mp + 1) > mp by XREAL_1: 29;

 then

 A142: (mp + 1) > (t4 . a4) by A130, XXREAL_0: 2;



 A143: pn >= p0 by NAT_1: 11;

 then

 A144: mp >= (p0 + 1) by A128, XREAL_1: 7;

 then

 A145: (mp + 1) >= (p0 + 1) by A141, XXREAL_0: 2;



 A146: (mp + 1) >= (8 + 1) by A139, XREAL_1: 6;

 then

 A147: (mp + 1) > 4 by XXREAL_0: 2;



 A148: (t4 . ( intpos mp)) = (( Initialize t4) . ( intpos mp)) by SCMPDS_5: 15;



 A149: (t4 . ( intpos (mp + 1))) = (( Initialize t4) . ( intpos (mp + 1))) by SCMPDS_5: 15;

 mp > 2 by A139, XXREAL_0: 2;



 hence md = (t4 . ( intpos mp)) by A7, A84, A85, A87, A140, A79

 .= (tp . ( intpos mp)) by A119, A144, A130, A148

 .= (( IExec (L5,Q2,( Initialize t2))) . ( intpos mp)) by A105, A106, A123, SCPISORT: 7

 .= (t1 . ( intpos mp)) by A107, A121, SCPISORT: 6;

 (mp + 1) > 2 by A146, XXREAL_0: 2;



 hence n4 = (t4 . ( intpos (mp + 1))) by A7, A84, A85, A89, A147, A79

 .= (tp . ( intpos (mp + 1))) by A119, A145, A142, A149

 .= (( IExec (L5,Q2,( Initialize t2))) . ( intpos (mp + 3))) by A105, A106, A124, SCPISORT: 7

 .= (t1 . ( intpos (mp + 3))) by A107, A121, SCPISORT: 6;

 hereby

 let j be Nat;

 assume that

 A150: 1 <= j and

 A151: j < m1;



 A152: (pn + j) < mp by A151, XREAL_1: 6;

 (pn + j) >= (pn + 1) by A150, XREAL_1: 6;

 then (pn + j) > pn by A127, XXREAL_0: 2;

 then

 A153: (pn + j) > (t4 . a4) by A91, A102, XXREAL_0: 2;



 A154: (pn + j) >= (p0 + 1) by A143, A150, XREAL_1: 7;



 A155: (pn + j) >= (1 + 7) by A6, A150, XREAL_1: 7;

 then

 A156: (pn + j) > 2 by XXREAL_0: 2;



 A157: (pn + j) > 4 by A155, XXREAL_0: 2;



 A158: (t4 . ( intpos (pn + j))) = (( Initialize t4) . ( intpos (pn + j))) by SCMPDS_5: 15;



 thus (t1 . ( intpos (pn + j))) = (( IExec (L5,Q2,( Initialize t2))) . ( intpos (pn + j))) by A107, A121, SCPISORT: 6

 .= (t2 . ( intpos (pn + j))) by A19, A86, A120, A122, A155, A152

 .= (tp . ( intpos (pn + j))) by A105, A106, SCPISORT: 7

 .= (t4 . ( intpos (pn + j))) by A119, A154, A153, A158

 .= (t . ( intpos (pn + j))) by A7, A84, A85, A156, A157, A79;

 end;

 (f1,f4) are_fiberwise_equipotent by A5, A87, A88, A89, A90, A91, A94, A104, A103, A92, A112, A110, A113, Th11, A80, A81, A82;

 hence (f1,f2) are_fiberwise_equipotent by A109, A110, A133, FINSEQ_2: 9;

 take m4;

 thus md <= m4 & m4 <= n4 by A7, A84, A85, A89, A115, A116, A79;



 thus (m4 - 1) = ((t2 . a4) - 1) by A114, A105, A106, SCPISORT: 7

 .= (t1 . ( intpos (mp + 1))) by A107, A121, A126, SCPISORT: 6;



 thus (m4 + 1) = ((t2 . a4) + 1) by A114, A105, A106, SCPISORT: 7

 .= (t1 . ( intpos (mp + 2))) by A107, A121, A125, SCPISORT: 6;



 A159: md >= 8 by A88, A95, XXREAL_0: 2;

 then

 A160: m4 >= 8 by A115, XXREAL_0: 2;

 hereby



 A161: (tp . ( intpos m4)) = (t2 . ( intpos m4)) by A105, A106, SCPISORT: 7

 .= (( IExec (L5,Q2,( Initialize t2))) . ( intpos m4)) by A19, A86, A120, A122, A131, A160

 .= (t1 . ( intpos m4)) by A107, A121, SCPISORT: 6;

 let i be Nat;

 assume that

 A162: md <= i and

 A163: i < m4;



 A164: i < mp by A131, A163, XXREAL_0: 2;



 A165: i >= 8 by A159, A162, XXREAL_0: 2;

 (tp . ( intpos i)) = (t2 . ( intpos i)) by A105, A106, SCPISORT: 7

 .= (( IExec (L5,Q2,( Initialize t2))) . ( intpos i)) by A19, A86, A120, A122, A164, A165

 .= (t1 . ( intpos i)) by A107, A121, SCPISORT: 6;

 hence (t1 . ( intpos m4)) >= (t1 . ( intpos i)) by A117, A162, A163, A161;

 end;

 hereby



 A166: (tp . ( intpos m4)) = (t2 . ( intpos m4)) by A105, A106, SCPISORT: 7

 .= (( IExec (L5,Q2,( Initialize t2))) . ( intpos m4)) by A19, A86, A120, A122, A131, A160

 .= (t1 . ( intpos m4)) by A107, A121, SCPISORT: 6;

 let i be Nat;

 assume that

 A167: m4 = md by A115, A167, XXREAL_0: 2;

 then

 A170: i >= 8 by A159, XXREAL_0: 2;

 (tp . ( intpos i)) = (t2 . ( intpos i)) by A105, A106, SCPISORT: 7

 .= (( IExec (L5,Q2,( Initialize t2))) . ( intpos i)) by A19, A86, A120, A122, A169, A170

 .= (t1 . ( intpos i)) by A107, A121, SCPISORT: 6;

 hence (t1 . ( intpos m4)) <= (t1 . ( intpos i)) by A102, A118, A167, A168, A166;

 end;

 hereby

 let j be Nat;

 assume that

 A171: (p0 + 1) <= j & j < md or n4 < j & j <= pn;

 A172:

 now

 per cases by A171;

 suppose

 A173: (p0 + 1) <= j & j < md;

 hence j >= (p0 + 1) & (j < (t4 . a2) or j > (t4 . a4)) by A7, A84, A85, A87, A79;

 md < mp by A115, A131, XXREAL_0: 2;

 hence j < mp by A173, XXREAL_0: 2;

 end;

 suppose

 A174: n4 < j & j <= pn;

 n4 >= md by A102, A115, A116, XXREAL_0: 2;

 then j >= md by A174, XXREAL_0: 2;

 hence j >= (p0 + 1) by A88, XXREAL_0: 2;

 thus j < (t4 . a2) or j > (t4 . a4) by A7, A84, A85, A89, A174, A79;

 thus j < mp by A129, A174, XXREAL_0: 2;

 end;

 end;

 then

 A175: j >= 8 by A95, XXREAL_0: 2;

 then

 A176: j > 2 by XXREAL_0: 2;



 A177: j > 4 by A175, XXREAL_0: 2;



 A178: (t4 . ( intpos j)) = (( Initialize t4) . ( intpos j)) by SCMPDS_5: 15;



 thus (t1 . ( intpos j)) = (( IExec (L5,Q2,( Initialize t2))) . ( intpos j)) by A107, A121, SCPISORT: 6

 .= (t2 . ( intpos j)) by A19, A86, A120, A122, A172, A175

 .= (tp . ( intpos j)) by A105, A106, SCPISORT: 7

 .= (t4 . ( intpos j)) by A119, A172, A178

 .= (t . ( intpos j)) by A7, A84, A85, A176, A177, A79;

 end;

 end;



 A179: for t be State of SCMPDS , Q holds for m be Nat st (t . a) = 0 & (t . SBP ) = m holds (( IExec ((l1 ';' l2),Q,( Initialize t))) . a2) = ((t . ( intpos ((m + pn) + 1))) - (t . ( intpos (m + pn)))) & for i be Nat st i <> 2 holds (( IExec ((l1 ';' l2),Q,( Initialize t))) . ( intpos i)) = (t . ( intpos i))

 proof

 let t be State of SCMPDS , Q;

 let m be Nat;

 assume that

 A180: (t . a) = 0 and

 A181: (t . SBP ) = m;

 set t0 = ( Initialize t), t1 = ( IExec ((l1 ';' l2),Q,( Initialize t))), t2 = ( Exec (l1,t0)), mp = (m + pn);



 A182: (t0 . a) = 0 by A180, SCMPDS_5: 15;

 then

 A183: ( DataLoc ((t0 . a),2)) = ( intpos ( 0 + 2)) by SCMP_GCD: 1;

 then (t2 . a) = 0 by A182, AMI_3: 10, SCMPDS_2: 47;

 then

 A184: ( DataLoc ((t2 . a),2)) = ( intpos ( 0 + 2)) by SCMP_GCD: 1;



 A185: (t0 . SBP ) = m by A181, SCMPDS_5: 15;

 then

 A186: (t2 . SBP ) = m by A183, AMI_3: 10, SCMPDS_2: 47;

 mp >= ( 0 + 7) by A6, XREAL_1: 7;

 then mp > 2 by XXREAL_0: 2;



 then

 A187: (t2 . ( intpos mp)) = (t0 . ( intpos mp)) by A183, AMI_3: 10, SCMPDS_2: 47

 .= (t . ( intpos mp)) by SCMPDS_5: 15;



 A188: (t2 . a2) = (t0 . ( DataLoc ((t0 . SBP ),(pn + 1)))) by A183, SCMPDS_2: 47

 .= (t0 . ( intpos (m + (pn + 1)))) by A185, SCMP_GCD: 1

 .= (t . ( intpos (mp + 1))) by SCMPDS_5: 15;



 thus (t1 . a2) = (( Exec (l2,t2)) . a2) by SCMPDS_5: 42

 .= ((t2 . a2) - (t2 . ( DataLoc ((t2 . SBP ),pn)))) by A184, SCMPDS_2: 50

 .= ((t . ( intpos (mp + 1))) - (t . ( intpos mp))) by A186, A188, A187, SCMP_GCD: 1;

 hereby

 let i be Nat;

 assume

 A189: i <> 2;



 thus (t1 . ( intpos i)) = (( Exec (l2,t2)) . ( intpos i)) by SCMPDS_5: 42

 .= (t2 . ( intpos i)) by A184, A189, AMI_3: 10, SCMPDS_2: 50

 .= (t0 . ( intpos i)) by A183, A189, AMI_3: 10, SCMPDS_2: 47

 .= (t . ( intpos i)) by SCMPDS_5: 15;

 end;

 end;



 A190: (pn + 1) >= (7 + 1) by A6, XREAL_1: 6;



 A191: for t be 0 -started State of SCMPDS , Q holds for m be Nat st (t . a) = 0 & (t . SBP ) = m & (t . ( intpos ((m + pn) + 1))) <= (t . ( intpos (m + pn))) holds (( IExec (WB4,Q,t)) . a) = 0 & (( IExec (WB4,Q,t)) . a1) = (m - 2) & (for j be Nat st 1 <= j & j < m holds (( IExec (WB4,Q,t)) . ( intpos (pn + j))) = (t . ( intpos (pn + j)))) & for j be Nat st 1 <= j & j <= n holds (( IExec (WB4,Q,t)) . ( intpos (p0 + j))) = (t . ( intpos (p0 + j)))

 proof

 let t be 0 -started State of SCMPDS , Q;

 let m be Nat;

 set mp = (m + pn), t1 = ( IExec ((l1 ';' l2),Q,t)), Q1 = Q;



 A192: ( Initialize t) = t by MEMSTR_0: 44;

 assume that

 A193: (t . a) = 0 and

 A194: (t . SBP ) = m and

 A195: (t . ( intpos (mp + 1))) <= (t . ( intpos mp));



 A196: (t1 . a1) = m by A179, A193, A194, A192;



 A197: (t1 . a2) = ((t . ( intpos (mp + 1))) - (t . ( intpos mp))) by A179, A193, A194, A192;

 then

 A198: (t1 . a2) <= 0 by A195, XREAL_1: 47;

 set li = ( AddTo ( GBP ,1,( - 2))), t0 = ( Initialize t1);



 A199: lb is_halting_on (t1,Q1) by SCMPDS_6: 21;



 A200: (t1 . a) = 0 by A179, A193, A194, A192;

 then

 A201: ( DataLoc ((t1 . a),2)) = ( intpos ( 0 + 2)) by SCMP_GCD: 1;



 A202: (t0 . a) = 0 by A200, SCMPDS_5: 15;

 then

 A203: ( DataLoc ((t0 . a),1)) = ( intpos ( 0 + 1)) by SCMP_GCD: 1;



 A204: lb is_closed_on (t1,Q1) by SCMPDS_6: 20;

 then

 A205: IF4 is_halting_on (t1,Q1) by A201, A198, A199, SCMPDS_6: 69;



 A206: IF4 is_closed_on (t1,Q1) by A201, A198, A204, A199, SCMPDS_6: 69;

 A207:

 now

 let x;



 A208: (( Initialize t1) . GBP ) = (t1 . GBP ) by SCMPDS_5: 15;



 A209: (( Initialize t1) . ( DataLoc ((t1 . GBP ),2))) = (t1 . ( DataLoc ((t1 . GBP ),2))) by SCMPDS_5: 15;



 thus (( IExec (WB4,Q,t)) . x) = (( IExec (IF4,Q1,( Initialize t1))) . x) by A206, A205, SCPISORT: 7

 .= (( IExec (lb,Q1,( Initialize t1))) . x) by A195, A197, A201, A208, A209, SCMPDS_6: 74, XREAL_1: 47

 .= (( Exec (li,t0)) . x) by SCMPDS_5: 40;

 end;

 A210:

 now

 let i be Nat;

 assume i <> 1;

 then

 A211: ( intpos i) <> ( DataLoc ((t0 . a),1)) by A202, AMI_3: 10, SCMP_GCD: 1;



 thus (( IExec (WB4,Q,t)) . ( intpos i)) = (( Exec (li,t0)) . ( intpos i)) by A207

 .= (t0 . ( intpos i)) by A211, SCMPDS_2: 48

 .= (t1 . ( intpos i)) by SCMPDS_5: 15;

 end;

 hence (( IExec (WB4,Q,t)) . a) = 0 by A200;



 thus (( IExec (WB4,Q,t)) . a1) = (( Exec (li,t0)) . a1) by A207

 .= ((t0 . a1) + ( - 2)) by A203, SCMPDS_2: 48

 .= ((t0 . a1) - 2)

 .= (m - 2) by A196, SCMPDS_5: 15;

 hereby

 let j be Nat;

 assume that

 A212: 1 <= j and j < m;

 (pn + j) >= (pn + 1) by A212, XREAL_1: 6;

 then

 A213: (pn + j) >= 8 by A190, XXREAL_0: 2;

 then

 A214: (pn + j) > 2 by XXREAL_0: 2;

 (pn + j) > 1 by A213, XXREAL_0: 2;



 hence (( IExec (WB4,Q,t)) . ( intpos (pn + j))) = (t1 . ( intpos (pn + j))) by A210

 .= (t . ( intpos (pn + j))) by A179, A193, A194, A214, A192;

 end;

 hereby

 let j be Nat;

 assume that

 A215: 1 <= j and j <= n;



 A216: (p0 + 1) >= (7 + 1) by A5, XREAL_1: 6;

 (p0 + j) >= (p0 + 1) by A215, XREAL_1: 6;

 then

 A217: (p0 + j) >= 8 by A216, XXREAL_0: 2;

 then

 A218: (p0 + j) > 2 by XXREAL_0: 2;

 (p0 + j) > 1 by A217, XXREAL_0: 2;



 hence (( IExec (WB4,Q,t)) . ( intpos (p0 + j))) = (t1 . ( intpos (p0 + j))) by A210

 .= (t . ( intpos (p0 + j))) by A179, A193, A194, A218, A192;

 end;

 end;



 A219: for t be State of SCMPDS , Q holds for m1,md be Nat st (t . a) = 0 & (t . SBP ) = m1 & (t . ( intpos (m1 + pn))) = md & md >= (p0 + 1) & ((t . ( intpos ((m1 + pn) + 1))) - md) > 0 holds TR is_closed_on (t,Q) & TR is_halting_on (t,Q)

 proof

 let t be State of SCMPDS , Q;

 let m1,md be Nat;

 set lPt = ((l3 ';' l4) ';' Pt), t2 = ( IExec ((l3 ';' l4),Q,( Initialize t))), mp = (m1 + pn), Q2 = Q;



 A220: (t2 . a) = (( Initialize t2) . a) by SCMPDS_5: 15;



 A221: (t2 . a2) = (( Initialize t2) . a2) by SCMPDS_5: 15;

 assume that

 A222: (t . a) = 0 and

 A223: (t . SBP ) = m1 and

 A224: (t . ( intpos mp)) = md and

 A225: md >= (p0 + 1) and ((t . ( intpos (mp + 1))) - md) > 0 ;



 A226: (t2 . a) = 0 by A7, A222, A223;



 A227: (t2 . a2) = (t . ( intpos mp)) by A7, A222, A223;

 then Pt is_halting_on (( Initialize t2),Q2) by A5, A224, A225, A226, Th10, A220, A221;

 then

 A228: Pt is_halting_on (t2,Q2) by SCMPDS_6: 126;

 Pt is_closed_on (( Initialize t2),Q2) by A5, A224, A225, A227, A226, Th10, A220, A221;

 then

 A229: Pt is_closed_on (t2,Q2) by SCMPDS_6: 125;

 then

 A230: lPt is_halting_on (t,Q) by A228, SCPISORT: 9;

 lPt is_closed_on (t,Q) by A229, A228, SCPISORT: 9;

 hence thesis by A230, SCPISORT: 8;

 end;



 A231: for t be 0 -started State of SCMPDS , Q holds for m,m1,md be Nat, n4 be Integer, f1,f2 be FinSequence of INT st (t . a) = 0 & (t . SBP ) = m1 & m1 = (m + 1) & (t . ( intpos (m1 + pn))) = md & md >= (p0 + 1) & n4 = (t . ( intpos ((m1 + pn) + 1))) & (n4 - md) > 0 & n4 <= pn & f1 is_FinSequence_on (t,p0) & ( len f1) = n & f2 is_FinSequence_on (( IExec (WB4,Q,t)),p0) & ( len f2) = n holds (( IExec (WB4,Q,t)) . a) = 0 & (( IExec (WB4,Q,t)) . a1) = (m1 + 2) & md = (( IExec (WB4,Q,t)) . ( intpos (m1 + pn))) & n4 = (( IExec (WB4,Q,t)) . ( intpos ((m1 + pn) + 3))) & (for j be Nat st (1 <= j & j < m1) holds (( IExec (WB4,Q,t)) . ( intpos (pn + j))) = (t . ( intpos (pn + j)))) & (f1,f2) are_fiberwise_equipotent & ex m4 be Nat st md <= m4 & m4 <= n4 & (m4 - 1) = (( IExec (WB4,Q,t)) . ( intpos ((m1 + pn) + 1))) & (m4 + 1) = (( IExec (WB4,Q,t)) . ( intpos ((m1 + pn) + 2))) & (for i be Nat st md <= i & i < m4 holds (( IExec (WB4,Q,t)) . ( intpos m4)) >= (( IExec (WB4,Q,t)) . ( intpos i))) & (for i be Nat st m4 < i & i <= n4 holds (( IExec (WB4,Q,t)) . ( intpos m4)) <= (( IExec (WB4,Q,t)) . ( intpos i))) & for j be Nat st (p0 + 1) <= j & j < md or n4 < j & j <= (p0 + n) holds (( IExec (WB4,Q,t)) . ( intpos j)) = (t . ( intpos j))

 proof

 let t be 0 -started State of SCMPDS , Q;

 let m,m1,md be Nat, n4 be Integer, f1,f2 be FinSequence of INT ;



 A232: ( Initialize t) = t by MEMSTR_0: 44;

 set mp = (m1 + pn), t1 = ( IExec ((l1 ';' l2),Q,t)), Q1 = Q;

 assume that

 A233: (t . a) = 0 and

 A234: (t . SBP ) = m1 and

 A235: m1 = (m + 1) and

 A236: (t . ( intpos mp)) = md and

 A237: md >= (p0 + 1) and

 A238: n4 = (t . ( intpos (mp + 1))) and

 A239: (n4 - md) > 0 and

 A240: n4 <= pn;



 A241: (t1 . a2) > 0 by A179, A233, A234, A236, A238, A239, A232;



 A242: mp >= ( 0 + 7) by A6, XREAL_1: 7;

 then mp > 2 by XXREAL_0: 2;

 then

 A243: (t1 . ( intpos mp)) = md by A179, A233, A234, A236, A232;

 assume that

 A244: f1 is_FinSequence_on (t,p0) and

 A245: ( len f1) = n;



 A246: (p0 + 1) >= (7 + 1) by A5, XREAL_1: 6;

 now

 let i be Nat;

 assume that

 A247: 1 <= i and

 A248: i <= ( len f1);

 (p0 + i) >= (p0 + 1) by A247, XREAL_1: 6;

 then (p0 + i) >= 8 by A246, XXREAL_0: 2;

 then

 A249: (p0 + i) > 2 by XXREAL_0: 2;



 thus (f1 . i) = (t . ( intpos (p0 + i))) by A244, A247, A248

 .= (t1 . ( intpos (p0 + i))) by A179, A233, A234, A249, A232;

 end;

 then

 A250: f1 is_FinSequence_on (t1,p0);

 (mp + 1) >= (7 + 1) by A242, XREAL_1: 6;

 then (mp + 1) > 2 by XXREAL_0: 2;

 then

 A251: n4 = (t1 . ( intpos (mp + 1))) by A179, A233, A234, A238, A232;



 A252: (t1 . a) = 0 by A179, A233, A234, A232;

 then

 A253: ( DataLoc ((t1 . a),2)) = ( intpos ( 0 + 2)) by SCMP_GCD: 1;



 A254: (t1 . SBP ) = m1 by A179, A233, A234, A232;

 then

 A255: TR is_halting_on (t1,Q1) by A219, A237, A239, A243, A251, A252;



 A256: TR is_halting_on (( Initialize t1),Q1)

 proof

 (Q1 +* ( stop TR)) halts_on ( Initialize ( Initialize t1)) by A255, SCMPDS_6:def 3;

 hence thesis by SCMPDS_6:def 3;

 end;



 A257: (( Initialize t1) . GBP ) = (t1 . GBP ) by SCMPDS_5: 15;



 A258: (( Initialize t1) . ( DataLoc ((t1 . GBP ),2))) = (t1 . ( DataLoc ((t1 . GBP ),2))) by SCMPDS_5: 15;



 A259: TR is_closed_on (t1,Q1) by A219, A237, A239, A254, A243, A251, A252;



 A260: TR is_closed_on (( Initialize t1),Q1)

 proof

 for k be Nat holds ( IC ( Comput ((Q1 +* ( stop TR)),( Initialize ( Initialize t1)),k))) in ( dom ( stop TR)) by A259, SCMPDS_6:def 2;

 hence thesis by SCMPDS_6:def 2;

 end;

 then

 A261: IF4 is_halting_on (( Initialize t1),Q1) by A253, A241, A257, A258, A256, SCMPDS_6: 68;



 A262: IF4 is_halting_on (t1,Q1)

 proof

 (Q1 +* ( stop IF4)) halts_on ( Initialize ( Initialize t1)) by A261, SCMPDS_6:def 3;

 hence thesis by SCMPDS_6:def 3;

 end;

 assume that

 A263: f2 is_FinSequence_on (( IExec (WB4,Q,t)),p0) and

 A264: ( len f2) = n;



 A265: IF4 is_closed_on (( Initialize t1),Q1) by A253, A241, A257, A258, A256, A260, SCMPDS_6: 68;



 A266: IF4 is_closed_on (t1,Q1)

 proof

 for k be Nat holds ( IC ( Comput ((Q1 +* ( stop IF4)),( Initialize ( Initialize t1)),k))) in ( dom ( stop IF4)) by A265, SCMPDS_6:def 2;

 hence thesis by SCMPDS_6:def 2;

 end;

 A267:

 now

 let x;



 thus (( IExec (WB4,Q,t)) . x) = (( IExec (IF4,Q1,( Initialize t1))) . x) by A266, A262, SCPISORT: 7

 .= (( IExec (TR,Q1,( Initialize t1))) . x) by A253, A241, Th1, A257, A258, A260, A256;

 end;

 now

 let i be Nat;

 assume that

 A268: 1 <= i and

 A269: i <= ( len f2);



 thus (f2 . i) = (( IExec (WB4,Q,t)) . ( intpos (p0 + i))) by A263, A268, A269

 .= (( IExec (TR,Q1,( Initialize t1))) . ( intpos (p0 + i))) by A267;

 end;

 then

 A270: f2 is_FinSequence_on (( IExec (TR,Q1,( Initialize t1))),p0);



 A271: (( Initialize t1) . SBP ) = (t1 . SBP ) by SCMPDS_5: 15;



 A272: (( Initialize t1) . a) = (t1 . a) by SCMPDS_5: 15;



 A273: (( Initialize t1) . ( intpos (m1 + pn))) = (t1 . ( intpos (m1 + pn))) by SCMPDS_5: 15;



 A274: (( Initialize t1) . ( intpos ((m1 + pn) + 1))) = (t1 . ( intpos ((m1 + pn) + 1))) by SCMPDS_5: 15;



 A275: f1 is_FinSequence_on (( Initialize t1),p0)

 proof

 let i be Nat;

 assume 1 <= i & i <= ( len f1);

 then (f1 . i) = (t1 . ( intpos (p0 + i))) by A250;

 hence thesis by SCMPDS_5: 15;

 end;

 (( IExec (TR,Q1,( Initialize t1))) . a) = 0 by A78, A235, A237, A239, A240, A245, A264, A254, A243, A251, A252, A270, A271, A272, A273, A274, A275;

 hence (( IExec (WB4,Q,t)) . a) = 0 by A267;

 consider m4 be Nat such that

 A276: md <= m4 and

 A277: m4 <= n4 and

 A278: (m4 - 1) = (( IExec (TR,Q1,( Initialize t1))) . ( intpos (mp + 1))) and

 A279: (m4 + 1) = (( IExec (TR,Q1,( Initialize t1))) . ( intpos (mp + 2))) and

 A280: for i be Nat st md <= i & i < m4 holds (( IExec (TR,Q1,( Initialize t1))) . ( intpos m4)) >= (( IExec (TR,Q1,( Initialize t1))) . ( intpos i)) and

 A281: for i be Nat st m4 < i & i <= n4 holds (( IExec (TR,Q1,( Initialize t1))) . ( intpos m4)) <= (( IExec (TR,Q1,( Initialize t1))) . ( intpos i)) and

 A282: for j be Nat st (p0 + 1) <= j & j < md or n4 < j & j <= (p0 + n) holds (( IExec (TR,Q1,( Initialize t1))) . ( intpos j)) = (( Initialize t1) . ( intpos j)) by A78, A235, A237, A239, A240, A245, A264, A254, A243, A251, A252, A270, A271, A272, A273, A274, A275;



 A283: for j be Nat st (p0 + 1) <= j & j < md or n4 < j & j <= (p0 + n) holds (( IExec (TR,Q1,( Initialize t1))) . ( intpos j)) = (t1 . ( intpos j))

 proof

 let j be Nat;

 assume (p0 + 1) <= j & j < md or n4 < j & j <= (p0 + n);

 then (( IExec (TR,Q1,( Initialize t1))) . ( intpos j)) = (( Initialize t1) . ( intpos j)) by A282;

 hence thesis by SCMPDS_5: 15;

 end;

 (( IExec (TR,Q1,( Initialize t1))) . a1) = (m1 + 2) by A78, A235, A237, A239, A240, A245, A264, A254, A243, A251, A252, A270, A271, A272, A273, A274, A275;

 hence (( IExec (WB4,Q,t)) . a1) = (m1 + 2) by A267;

 md = (( IExec (TR,Q1,( Initialize t1))) . ( intpos mp)) by A78, A235, A237, A239, A240, A245, A264, A254, A243, A251, A252, A270, A271, A272, A273, A274, A275;

 hence md = (( IExec (WB4,Q,t)) . ( intpos mp)) by A267;

 n4 = (( IExec (TR,Q1,( Initialize t1))) . ( intpos (mp + 3))) by A78, A235, A237, A239, A240, A245, A264, A254, A243, A251, A252, A270, A271, A272, A273, A274, A275;

 hence n4 = (( IExec (WB4,Q,t)) . ( intpos (mp + 3))) by A267;

 hereby

 let j be Nat;

 assume that

 A284: 1 <= j and

 A285: j < m1;

 (pn + j) >= (pn + 1) by A284, XREAL_1: 6;

 then (pn + j) >= 8 by A190, XXREAL_0: 2;

 then

 A286: (pn + j) > 2 by XXREAL_0: 2;



 thus (( IExec (WB4,Q,t)) . ( intpos (pn + j))) = (( IExec (TR,Q1,( Initialize t1))) . ( intpos (pn + j))) by A267

 .= (( Initialize t1) . ( intpos (pn + j))) by A78, A235, A237, A239, A240, A245, A264, A254, A243, A251, A252, A270, A284, A285, A271, A272, A273, A274, A275

 .= (t1 . ( intpos (pn + j))) by SCMPDS_5: 15

 .= (t . ( intpos (pn + j))) by A179, A233, A234, A286, A232;

 end;

 thus (f1,f2) are_fiberwise_equipotent by A78, A235, A237, A239, A240, A245, A264, A254, A243, A251, A252, A270, A271, A272, A273, A274, A275;

 take m4;

 thus md <= m4 & m4 <= n4 by A276, A277;

 thus (m4 - 1) = (( IExec (WB4,Q,t)) . ( intpos (mp + 1))) by A267, A278;

 thus (m4 + 1) = (( IExec (WB4,Q,t)) . ( intpos (mp + 2))) by A267, A279;

 hereby

 let i be Nat;

 assume that

 A287: md <= i and

 A288: i < m4;

 (( IExec (TR,Q1,( Initialize t1))) . ( intpos m4)) >= (( IExec (TR,Q1,( Initialize t1))) . ( intpos i)) by A280, A287, A288;

 then (( IExec (WB4,Q,t)) . ( intpos m4)) >= (( IExec (TR,Q1,( Initialize t1))) . ( intpos i)) by A267;

 hence (( IExec (WB4,Q,t)) . ( intpos m4)) >= (( IExec (WB4,Q,t)) . ( intpos i)) by A267;

 end;

 hereby

 let i be Nat;

 assume that

 A289: m4 < i and

 A290: i <= n4;

 (( IExec (TR,Q1,( Initialize t1))) . ( intpos m4)) <= (( IExec (TR,Q1,( Initialize t1))) . ( intpos i)) by A281, A289, A290;

 then (( IExec (WB4,Q,t)) . ( intpos m4)) <= (( IExec (TR,Q1,( Initialize t1))) . ( intpos i)) by A267;

 hence (( IExec (WB4,Q,t)) . ( intpos m4)) <= (( IExec (WB4,Q,t)) . ( intpos i)) by A267;

 end;

 hereby

 let j be Nat;

 assume that

 A291: (p0 + 1) <= j & j < md or n4 < j & j <= (p0 + n);

 now

 per cases by A291;

 suppose (p0 + 1) <= j & j < md;

 hence (p0 + 1) <= j;

 end;

 suppose n4 < j & j <= (p0 + n);

 then j >= m4 by A277, XXREAL_0: 2;

 then j >= md by A276, XXREAL_0: 2;

 hence j >= (p0 + 1) by A237, XXREAL_0: 2;

 end;

 end;

 then j >= 8 by A246, XXREAL_0: 2;

 then

 A292: j > 2 by XXREAL_0: 2;



 thus (( IExec (WB4,Q,t)) . ( intpos j)) = (( IExec (TR,Q1,( Initialize t1))) . ( intpos j)) by A267

 .= (t1 . ( intpos j)) by A283, A291

 .= (t . ( intpos j)) by A179, A233, A234, A292, A232;

 end;

 end;

 consider f be FinSequence of INT such that

 A293: ( len f) = n and

 A294: f is_FinSequence_on (s,p0) by SCPISORT: 2;



 A295: (pn + 2) >= (7 + 2) by A6, XREAL_1: 6;



 A296: (s1 . a) = 0 & (s1 . a1) = 1 & (s1 . ( intpos (pn + 1))) = (p0 + 1) & (s1 . ( intpos (pn + 2))) = pn & for i be Nat st i >= 8 & i <= pn holds (s1 . ( intpos i)) = (s . ( intpos i))

 proof

 set t2 = ( IExec (((m1 ';' m2) ';' m3),P,s)), t3 = ( IExec ((m1 ';' m2),P,s)), t4 = ( Exec (m1,s));



 A297: (t4 . a) = 0 by SCMPDS_2: 45;



 A298: (t3 . a) = (( Exec (m2,t4)) . a) by SCMPDS_5: 42

 .= 0 by A297, SCMPDS_2: 45, SCMP_GCD: 3;



 A299: (t3 . SBP ) = (( Exec (m2,t4)) . SBP ) by SCMPDS_5: 42

 .= 1 by SCMPDS_2: 45;

 then

 A300: ( DataLoc ((t3 . SBP ),pn)) = ( intpos (pn + 1)) by SCMP_GCD: 1;



 A301: (t2 . a) = (( Exec (m3,t3)) . a) by SCMPDS_5: 41

 .= 0 by A298, A300, AMI_3: 10, SCMPDS_2: 46;



 A302: (t2 . ( intpos (pn + 1))) = (( Exec (m3,t3)) . ( intpos (pn + 1))) by SCMPDS_5: 41

 .= (p0 + 1) by A300, SCMPDS_2: 46;



 A303: (pn + 2) > 1 by A295, XXREAL_0: 2;



 A304: (pn + 1) > 1 by A190, XXREAL_0: 2;



 A305: (t2 . SBP ) = (( Exec (m3,t3)) . SBP ) by SCMPDS_5: 41

 .= 1 by A299, A300, A304, AMI_3: 10, SCMPDS_2: 46;

 then

 A306: ( DataLoc ((t2 . SBP ),(pn + 1))) = ( intpos (pn + (1 + 1))) by SCMP_GCD: 1;



 thus (s1 . a) = (( Exec (m4,t2)) . a) by SCMPDS_5: 41

 .= 0 by A301, A306, AMI_3: 10, SCMPDS_2: 46;



 thus (s1 . a1) = (( Exec (m4,t2)) . SBP ) by SCMPDS_5: 41

 .= 1 by A305, A306, A303, AMI_3: 10, SCMPDS_2: 46;



 A307: (pn + 2) > (pn + 1) by XREAL_1: 6;



 thus (s1 . ( intpos (pn + 1))) = (( Exec (m4,t2)) . ( intpos (pn + 1))) by SCMPDS_5: 41

 .= (p0 + 1) by A302, A306, A307, AMI_3: 10, SCMPDS_2: 46;



 thus (s1 . ( intpos (pn + 2))) = (( Exec (m4,t2)) . ( intpos (pn + 2))) by SCMPDS_5: 41

 .= pn by A306, SCMPDS_2: 46;



 A308: for i be Nat st i >= 8 holds (t4 . ( intpos i)) = (s . ( intpos i)) by AMI_3: 10, SCMPDS_2: 45;

 A309:

 now

 let i be Nat;

 assume

 A310: i >= 8;

 then

 A311: i > 1 by XXREAL_0: 2;



 thus (t3 . ( intpos i)) = (( Exec (m2,t4)) . ( intpos i)) by SCMPDS_5: 42

 .= (t4 . ( intpos i)) by A311, AMI_3: 10, SCMPDS_2: 45

 .= (s . ( intpos i)) by A308, A310;

 end;

 A312:

 now

 let i be Nat;

 assume that

 A313: i >= 8 and

 A314: i <= pn;



 A315: (pn + 1) > pn by XREAL_1: 29;



 thus (t2 . ( intpos i)) = (( Exec (m3,t3)) . ( intpos i)) by SCMPDS_5: 41

 .= (t3 . ( intpos i)) by A300, A314, A315, AMI_3: 10, SCMPDS_2: 46

 .= (s . ( intpos i)) by A309, A313;

 end;



 A316: (pn + 2) > (pn + 0 ) by XREAL_1: 6;

 let i be Nat;

 assume that

 A317: i >= 8 and

 A318: i <= pn;



 thus (s1 . ( intpos i)) = (( Exec (m4,t2)) . ( intpos i)) by SCMPDS_5: 41

 .= (t2 . ( intpos i)) by A306, A318, A316, AMI_3: 10, SCMPDS_2: 46

 .= (s . ( intpos i)) by A312, A317, A318;

 end;

 now



 A319: (p0 + 1) >= (7 + 1) by A5, XREAL_1: 6;

 let i be Nat;

 assume that

 A320: 1 <= i and

 A321: i <= ( len f);



 A322: (p0 + i) <= pn by A293, A321, XREAL_1: 6;

 (p0 + i) >= (p0 + 1) by A320, XREAL_1: 6;

 then

 A323: (p0 + i) >= 8 by A319, XXREAL_0: 2;



 thus (f . i) = (s . ( intpos (p0 + i))) by A294, A320, A321

 .= (s1 . ( intpos (p0 + i))) by A296, A323, A322;

 end;

 then

 A324: f is_FinSequence_on (s1,p0);

 consider g be FinSequence of INT such that

 A325: ( len g) = n and

 A326: g is_FinSequence_on (( IExec (( QuickSort (n,p0)),P,s)),p0) by SCPISORT: 2;



 A327: b1 = ( intpos ( 0 + 1)) by SCMP_GCD: 1;



 A328: for t be State of SCMPDS , Q holds for m,md be Nat st (t . a) = 0 & (t . SBP ) = m & (t . ( intpos (m + pn))) = md & (md >= (p0 + 1) or ((t . ( intpos ((m + pn) + 1))) - md) <= 0 ) holds WB4 is_closed_on (t,Q) & WB4 is_halting_on (t,Q)

 proof

 let t be State of SCMPDS , Q;

 let m,md be Nat;

 set mp = (m + pn), t1 = ( IExec ((l1 ';' l2),Q,( Initialize t))), Q1 = Q;

 assume that

 A329: (t . a) = 0 and

 A330: (t . SBP ) = m and

 A331: (t . ( intpos mp)) = md and

 A332: md >= (p0 + 1) or ((t . ( intpos (mp + 1))) - md) <= 0 ;



 A333: (t1 . a2) = ((t . ( intpos (mp + 1))) - (t . ( intpos mp))) by A179, A329, A330;



 A334: (t1 . a) = 0 by A179, A329, A330;

 then

 A335: ( DataLoc ((t1 . a),2)) = ( intpos ( 0 + 2)) by SCMP_GCD: 1;

 per cases ;

 suppose

 A336: (t1 . ( DataLoc ((t1 . a),2))) > 0 ;



 A337: mp >= ( 0 + 7) by A6, XREAL_1: 7;

 then (mp + 1) >= (7 + 1) by XREAL_1: 6;

 then (mp + 1) > 2 by XXREAL_0: 2;

 then

 A338: ((t1 . ( intpos (mp + 1))) - md) > 0 by A179, A329, A330, A331, A333, A335, A336;



 A339: (t1 . SBP ) = m by A179, A329, A330;

 mp > 2 by A337, XXREAL_0: 2;

 then

 A340: (t1 . ( intpos mp)) = md by A179, A329, A330, A331;

 then

 A341: TR is_halting_on (t1,Q1) by A219, A179, A329, A330, A331, A332, A334, A335, A336, A339, A338;



 A342: (( Initialize t1) . ( DataLoc ((t1 . a),2))) = (t1 . ( DataLoc ((t1 . a),2))) by SCMPDS_5: 15;



 A343: (( Initialize t1) . a) = (t1 . a) by SCMPDS_5: 15;



 A344: TR is_closed_on (t1,Q1) by A219, A179, A329, A330, A331, A332, A334, A335, A336, A339, A340, A338;



 A345: TR is_halting_on (( Initialize t1),Q1)

 proof

 (Q1 +* ( stop TR)) halts_on ( Initialize ( Initialize t1)) by A341, SCMPDS_6:def 3;

 hence thesis by SCMPDS_6:def 3;

 end;



 A346: TR is_closed_on (( Initialize t1),Q1)

 proof

 for k be Nat holds ( IC ( Comput ((Q1 +* ( stop TR)),( Initialize ( Initialize t1)),k))) in ( dom ( stop TR)) by A344, SCMPDS_6:def 2;

 hence thesis by SCMPDS_6:def 2;

 end;

 then

 A347: IF4 is_halting_on (( Initialize t1),Q1) by A336, A345, A342, A343, SCMPDS_6: 68;



 A348: IF4 is_halting_on (t1,Q1)

 proof

 (Q1 +* ( stop IF4)) halts_on ( Initialize ( Initialize t1)) by A347, SCMPDS_6:def 3;

 hence thesis by SCMPDS_6:def 3;

 end;



 A349: IF4 is_closed_on (( Initialize t1),Q1) by A336, A346, A345, A342, A343, SCMPDS_6: 68;

 IF4 is_closed_on (t1,Q1)

 proof

 for k be Nat holds ( IC ( Comput ((Q1 +* ( stop IF4)),( Initialize ( Initialize t1)),k))) in ( dom ( stop IF4)) by A349, SCMPDS_6:def 2;

 hence thesis by SCMPDS_6:def 2;

 end;

 hence thesis by A348, SCPISORT: 9;

 end;

 suppose

 A350: (t1 . ( DataLoc ((t1 . a),2))) <= 0 ;



 A351: lb is_halting_on (t1,Q1) by SCMPDS_6: 21;



 A352: lb is_closed_on (t1,Q1) by SCMPDS_6: 20;

 then

 A353: IF4 is_halting_on (t1,Q1) by A350, A351, SCMPDS_6: 69;

 IF4 is_closed_on (t1,Q1) by A350, A352, A351, SCMPDS_6: 69;

 hence thesis by A353, SCPISORT: 9;

 end;

 end;

 A354:

 now

 let t be 0 -started State of SCMPDS , Q;

 let f1,f2 be FinSequence of INT , k1,k2,y1,yn be Nat;

 set mm = ((2 * k1) + 1), md = (p0 + y1), n4 = (p0 + yn);

 assume that

 A355: (t . a) = 0 and

 A356: mm = (t . b1) and

 A357: k2 = ((pn + (2 * k1)) + 1) and

 A358: md = (t . ( intpos k2)) and

 A359: n4 = (t . ( intpos (k2 + 1))) and

 A360: 1 <= y1 & yn <= n or y1 >= yn;

 set mp = (mm + pn);



 A361: md = (t . ( intpos mp)) by A357, A358;

 now

 per cases by A360;

 case 1 <= y1 & yn <= n;

 hence (p0 + 1) <= md by XREAL_1: 6;

 end;

 case y1 >= yn;

 then md >= (t . ( intpos (mp + 1))) by A357, A359, XREAL_1: 6;

 hence ((t . ( intpos (mp + 1))) - md) <= 0 by XREAL_1: 47;

 end;

 end;

 hence WB4 is_closed_on (t,Q) & WB4 is_halting_on (t,Q) by A328, A327, A355, A356, A361;

 consider f3 be FinSequence of INT such that

 A362: ( len f3) = n and

 A363: for i be Nat st 1 <= i & i <= ( len f3) holds (f3 . i) = (t . ( intpos (p0 + i))) by SCPISORT: 1;

 consider f4 be FinSequence of INT such that

 A364: ( len f4) = n and

 A365: for i be Nat st 1 <= i & i <= ( len f4) holds (f4 . i) = (( IExec (WB4,Q,t)) . ( intpos (p0 + i))) by SCPISORT: 1;



 A366: f4 is_FinSequence_on (( IExec (WB4,Q,t)),p0) by A365;



 A367: f3 is_FinSequence_on (t,p0) by A363;

 hereby

 per cases ;

 suppose (t . ( intpos (mp + 1))) <= (t . ( intpos mp));

 hence (( IExec (WB4,Q,t)) . a) = (t . a) & for j be Nat st 1 <= j & j < mm holds (( IExec (WB4,Q,t)) . ( intpos (pn + j))) = (t . ( intpos (pn + j))) by A191, A327, A355, A356;

 end;

 suppose

 A368: (t . ( intpos (mp + 1))) > (t . ( intpos mp));

 then

 A369: (n4 - md) > 0 by A357, A358, A359, XREAL_1: 50;



 A370: n4 <= pn by A357, A358, A359, A360, A368, XREAL_1: 6;

 (p0 + 1) <= md by A357, A358, A359, A360, A368, XREAL_1: 6;

 hence (( IExec (WB4,Q,t)) . a) = (t . a) & for j be Nat st 1 <= j & j < mm holds (( IExec (WB4,Q,t)) . ( intpos (pn + j))) = (t . ( intpos (pn + j))) by A231, A327, A355, A356, A357, A359, A361, A362, A367, A364, A366, A369, A370;

 end;

 end;

 hereby

 assume y1 >= yn;

 then

 A371: (t . ( intpos (mp + 1))) <= (t . ( intpos mp)) by A357, A358, A359, XREAL_1: 6;



 hence (( IExec (WB4,Q,t)) . b1) = (((2 * k1) + 1) - 2) by A191, A327, A355, A356

 .= ((2 * k1) - 1);

 thus for j be Nat st 1 <= j & j <= n holds (( IExec (WB4,Q,t)) . ( intpos (p0 + j))) = (t . ( intpos (p0 + j))) by A191, A327, A355, A356, A371;

 end;



 A372: k2 = mp by A357;

 set s = ( Initialize s1), a = GBP , c = 0 , m = p0, i = 1, I = WB4, P = P1, pn = (p0 + n), m1 = (pn + 1);

 thus y1 < yn implies (( IExec (I,Q,t)) . ( DataLoc (c,i))) = ((2 * k1) + 3) & (for j be Nat st 1 <= j & j < y1 or yn < j & j <= n holds (( IExec (I,Q,t)) . ( intpos (m + j))) = (t . ( intpos (m + j)))) & (ex ym be Nat st y1 <= ym & ym <= yn & (m + y1) = (( IExec (I,Q,t)) . ( intpos k2)) & ((m + ym) - 1) = (( IExec (I,Q,t)) . ( intpos (k2 + 1))) & ((m + ym) + 1) = (( IExec (I,Q,t)) . ( intpos (k2 + 2))) & (m + yn) = (( IExec (I,Q,t)) . ( intpos (k2 + 3))) & (for j be Nat st y1 <= j & j < ym holds (( IExec (I,Q,t)) . ( intpos (m + j))) <= (( IExec (I,Q,t)) . ( intpos (m + ym)))) & (for j be Nat st ym < j & j <= yn holds (( IExec (I,Q,t)) . ( intpos (m + j))) >= (( IExec (I,Q,t)) . ( intpos (m + ym)))))

 proof

 assume

 A373: y1 < yn;

 then

 A374: (p0 + 1) <= md by A360, XREAL_1: 6;

 n4 > md by A373, XREAL_1: 6;

 then

 A375: (n4 - md) > 0 by XREAL_1: 50;



 A376: n4 <= pn by A360, A373, XREAL_1: 6;



 hence (( IExec (WB4,Q,t)) . b1) = (((2 * k1) + 1) + 2) by A231, A327, A355, A356, A357, A359, A361, A362, A367, A364, A366, A375, A374

 .= ((2 * k1) + 3);

 consider m4 be Nat such that

 A377: md <= m4 and

 A378: m4 <= n4 and

 A379: (m4 - 1) = (( IExec (WB4,Q,t)) . ( intpos (mp + 1))) and

 A380: (m4 + 1) = (( IExec (WB4,Q,t)) . ( intpos (mp + 2))) and

 A381: for i be Nat st md <= i & i < m4 holds (( IExec (WB4,Q,t)) . ( intpos m4)) >= (( IExec (WB4,Q,t)) . ( intpos i)) and

 A382: for i be Nat st m4 < i & i <= n4 holds (( IExec (WB4,Q,t)) . ( intpos m4)) <= (( IExec (WB4,Q,t)) . ( intpos i)) and

 A383: for j be Nat st (p0 + 1) <= j & j < md or n4 < j & j <= (p0 + n) holds (( IExec (WB4,Q,t)) . ( intpos j)) = (t . ( intpos j)) by A231, A327, A355, A356, A357, A358, A359, A362, A367, A364, A366, A375, A374, A376;

 hereby

 let j be Nat;

 assume 1 <= j & j < y1 or yn < j & j <= n;

 then (p0 + 1) <= (p0 + j) & (p0 + j) < md or n4 < (p0 + j) & (p0 + j) <= (p0 + n) by XREAL_1: 6;

 hence (( IExec (WB4,Q,t)) . ( intpos (p0 + j))) = (t . ( intpos (p0 + j))) by A383;

 end;

 (m4 - p0) >= y1 by A377, XREAL_1: 19;

 then

 reconsider ym = (m4 - p0) as Element of NAT by INT_1: 3;

 take ym;

 m4 = (p0 + ym);

 hence y1 <= ym & ym <= yn by A377, A378, XREAL_1: 6;

 thus md = (( IExec (WB4,Q,t)) . ( intpos k2)) by A231, A327, A355, A356, A358, A359, A372, A362, A367, A364, A366, A375, A374, A376;

 thus ((p0 + ym) - 1) = (( IExec (WB4,Q,t)) . ( intpos (k2 + 1))) by A357, A379;

 thus ((p0 + ym) + 1) = (( IExec (WB4,Q,t)) . ( intpos (k2 + 2))) by A357, A380;

 thus n4 = (( IExec (WB4,Q,t)) . ( intpos (k2 + 3))) by A231, A327, A355, A356, A358, A359, A372, A362, A367, A364, A366, A375, A374, A376;

 hereby

 let j be Nat;

 assume that

 A384: y1 <= j and

 A385: j < ym;



 A386: (p0 + j) < (p0 + ym) by A385, XREAL_1: 6;

 (p0 + y1) <= (p0 + j) by A384, XREAL_1: 6;

 hence (( IExec (WB4,Q,t)) . ( intpos (p0 + j))) <= (( IExec (WB4,Q,t)) . ( intpos (p0 + ym))) by A381, A386;

 end;

 hereby

 let j be Nat;

 assume that

 A387: ym < j and

 A388: j <= yn;



 A389: (p0 + j) <= (p0 + yn) by A388, XREAL_1: 6;

 (p0 + ym) < (p0 + j) by A387, XREAL_1: 6;

 hence (( IExec (WB4,Q,t)) . ( intpos (p0 + j))) >= (( IExec (WB4,Q,t)) . ( intpos (p0 + ym))) by A382, A389;

 end;

 end;

 hereby

 assume that

 A390: f1 is_FinSequence_on (t,p0) and

 A391: f2 is_FinSequence_on (( IExec (WB4,Q,t)),p0) and

 A392: ( len f1) = n and

 A393: ( len f2) = n;

 per cases ;

 suppose

 A394: (t . ( intpos (mp + 1))) <= (t . ( intpos mp));



 A395: ( dom f1) = ( Seg n) by A392, FINSEQ_1:def 3;

 now

 let i be Nat;

 reconsider a = i as Nat;

 assume

 A396: i in ( dom f1);

 then

 A397: 1 <= i by A395, FINSEQ_1: 1;



 A398: i <= n by A395, A396, FINSEQ_1: 1;



 hence (f1 . i) = (t . ( intpos (p0 + a))) by A390, A392, A397

 .= (( IExec (WB4,Q,t)) . ( intpos (p0 + a))) by A191, A327, A355, A356, A394, A397, A398

 .= (f2 . i) by A391, A393, A397, A398;

 end;

 hence (f1,f2) are_fiberwise_equipotent by A392, A393, FINSEQ_2: 9;

 end;

 suppose

 A399: (t . ( intpos (mp + 1))) > (t . ( intpos mp));

 then

 A400: (n4 - md) > 0 by A357, A358, A359, XREAL_1: 50;



 A401: n4 <= pn by A357, A358, A359, A360, A399, XREAL_1: 6;

 (p0 + 1) <= md by A357, A358, A359, A360, A399, XREAL_1: 6;

 hence (f1,f2) are_fiberwise_equipotent by A231, A327, A355, A356, A357, A359, A361, A390, A391, A392, A393, A400, A401;

 end;

 end;

 end;



 A402: (( Initialize s1) . GBP ) = (s1 . GBP ) by SCMPDS_5: 15;



 A403: (( Initialize s1) . ( intpos (pn + 1))) = (s1 . ( intpos (pn + 1))) by SCMPDS_5: 15;



 A404: (( Initialize s1) . ( intpos ((pn + 1) + 1))) = (s1 . ( intpos ((pn + 1) + 1))) by SCMPDS_5: 15;



 A405: (( Initialize s1) . ( DataLoc ( 0 ,1))) = (s1 . ( DataLoc ( 0 ,1))) by SCMPDS_5: 15;



 A406: WH4 is_closed_on (( Initialize s1),P) by A327, A354, A296, Lm5, A402, A403, A404, A405;



 A407: WH4 is_closed_on (s1,P)

 proof

 for k be Nat holds ( IC ( Comput ((P +* ( stop WH4)),( Initialize ( Initialize s1)),k))) in ( dom ( stop WH4)) by A406, SCMPDS_6:def 2;

 hence thesis by SCMPDS_6:def 2;

 end;



 A408: WH4 is_halting_on (( Initialize s1),P) by A327, A354, A296, Lm5, A402, A403, A404, A405;



 A409: WH4 is_halting_on (s1,P)

 proof

 (P +* ( stop WH4)) halts_on ( Initialize ( Initialize s1)) by A408, SCMPDS_6:def 3;

 hence thesis by SCMPDS_6:def 3;

 end;

 hence ( QuickSort (n,p0)) is_halting_on (s,P) by A407, A1, SCPISORT: 9;

 take f, g;

 thus ( len f) = n & f is_FinSequence_on (s,p0) & ( len g) = n & g is_FinSequence_on (( IExec (( QuickSort (n,p0)),P,s)),p0) by A293, A294, A325, A326;

 now

 let i be Nat;

 assume that

 A410: 1 <= i and

 A411: i <= ( len g);



 thus (g . i) = (( IExec ((M4 ';' WH4),P,s)) . ( intpos (p0 + i))) by A326, A410, A411

 .= (( IExec (WH4,P1,( Initialize s1))) . ( intpos (p0 + i))) by A409, A407, SCPISORT: 7;

 end;

 then

 A412: g is_FinSequence_on (( IExec (WH4,P1,( Initialize s1))),p0);



 A413: (( Initialize s1) . GBP ) = (s1 . GBP ) by SCMPDS_5: 15;



 A414: (( Initialize s1) . ( DataLoc ( 0 ,1))) = (s1 . ( DataLoc ( 0 ,1))) by SCMPDS_5: 15;



 A415: (( Initialize s1) . ( intpos (pn + 1))) = (s1 . ( intpos (pn + 1))) by SCMPDS_5: 15;



 A416: (( Initialize s1) . ( intpos ((pn + 1) + 1))) = (s1 . ( intpos ((pn + 1) + 1))) by SCMPDS_5: 15;

 f is_FinSequence_on (( Initialize s1),p0)

 proof

 let i be Nat;

 assume 1 <= i & i <= ( len f);

 then (f . i) = (s1 . ( intpos (p0 + i))) by A324;

 hence thesis by SCMPDS_5: 15;

 end;

 hence thesis by Lm3, A325, A293, A412, A414, A327, A296, A413, A415, A416, A354;

 end;

 begin

 theorem :: SCPQSORT:13

 ( card ( QuickSort (n,p0))) = 57 by Lm28;

 theorem :: SCPQSORT:14

 for p0,n be Nat st p0 >= 7 holds ( QuickSort (n,p0)) is parahalting

 proof

 let p0,n be Nat;

 assume

 A1: p0 >= 7;

 for s be State of SCMPDS , P holds ( QuickSort (n,p0)) is_halting_on (s,P)

 proof

 let s be State of SCMPDS , P;

 ( QuickSort (n,p0)) is_halting_on (( Initialize s),P) by A1, Lm28;

 hence thesis by SCMPDS_6: 126;

 end;

 hence thesis by SCMPDS_6: 21;

 end;

 theorem :: SCPQSORT:15

 for s be 0 -started State of SCMPDS , p0,n be Nat st p0 >= 7 holds ex f,g be FinSequence of INT st ( len f) = n & f is_FinSequence_on (s,p0) & ( len g) = n & g is_FinSequence_on (( IExec (( QuickSort (n,p0)),P,s)),p0) & (f,g) are_fiberwise_equipotent & g is_non_decreasing_on (1,n) by Lm28;