let s be State of SCM+FSA; :: thesis: for P being the Instructions of SCM+FSA -valued ManySortedSet of NAT
for I being paraclosed Program of SCM+FSA st P +* I halts_on s +* I & Directed I c= s & Directed I c= P & Start-At (0,SCM+FSA) c= s holds
IC (Comput (P,s,((LifeSpan ((P +* I),(s +* I))) + 1))) = card I
let P be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: for I being paraclosed Program of SCM+FSA st P +* I halts_on s +* I & Directed I c= s & Directed I c= P & Start-At (0,SCM+FSA) c= s holds
IC (Comput (P,s,((LifeSpan ((P +* I),(s +* I))) + 1))) = card I
set A = NAT ;
let I be paraclosed Program of SCM+FSA; :: thesis: ( P +* I halts_on s +* I & Directed I c= s & Directed I c= P & Start-At (0,SCM+FSA) c= s implies IC (Comput (P,s,((LifeSpan ((P +* I),(s +* I))) + 1))) = card I )
assume that
A1: P +* I halts_on s +* I and
A2: Directed I c= s and
A3: Directed I c= P and
A4: Start-At (0,SCM+FSA) c= s ; :: thesis: IC (Comput (P,s,((LifeSpan ((P +* I),(s +* I))) + 1))) = card I
A5: I c= P +* I by FUNCT_4:26;
set sISA0 = s +* (Initialize I);
A6: s +* (Initialize I) = Initialize (s +* I) by FUNCT_4:15
.= (Initialize s) +* I by COMPOS_1:83
.= s +* I by A4, FUNCT_4:79 ;
set s2 = (s +* (Initialize I)) +* (Directed I);
set IAt = Initialize I;
A7: dom (Directed I) = dom I by FUNCT_4:105;
set m = LifeSpan ((P +* I),(s +* (Initialize I)));
set l1 = IC (Comput ((P +* I),(s +* (Initialize I)),(LifeSpan ((P +* I),(s +* (Initialize I))))));
A8: Initialize I c= s +* (Initialize I) by FUNCT_4:26;
A9: I c= P +* I by FUNCT_4:26;
A10: IC (Comput ((P +* I),(s +* (Initialize I)),(LifeSpan ((P +* I),(s +* (Initialize I)))))) in dom I by Def2, A8, A9;
set s1 = (s +* (Initialize I)) +* (I ';' I);
A11: P +* (I ';' I) = P +* (I +* (I ';' I)) by SCMFSA6A:57
.= (P +* I) +* (I ';' I) by FUNCT_4:15 ;
A12: now
let k be Element of NAT ; :: thesis: ( k <= LifeSpan ((P +* I),(s +* (Initialize I))) implies Comput ((P +* I),(s +* (Initialize I)),k), Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),k) equal_outside NAT )
defpred S1[ Nat] means ( $1 <= k implies Comput ((P +* (I ';' I)),((s +* (Initialize I)) +* (I ';' I)),$1), Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),$1) equal_outside NAT );
assume A13: k <= LifeSpan ((P +* I),(s +* (Initialize I))) ; :: thesis: Comput ((P +* I),(s +* (Initialize I)),k), Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),k) equal_outside NAT
A14: for n being Element of NAT st S1[n] holds
S1[n + 1]
proof
let n be Element of NAT ; :: thesis: ( S1[n] implies S1[n + 1] )
assume A15: ( n <= k implies Comput ((P +* (I ';' I)),((s +* (Initialize I)) +* (I ';' I)),n), Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),n) equal_outside NAT ) ; :: thesis: S1[n + 1]
A16: Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),(n + 1)) = Following ((P +* (Directed I)),(Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),n))) by EXTPRO_1:4
.= Exec ((CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),n)))),(Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),n))) ;
A17: Comput ((P +* (I ';' I)),((s +* (Initialize I)) +* (I ';' I)),(n + 1)) = Following ((P +* (I ';' I)),(Comput ((P +* (I ';' I)),((s +* (Initialize I)) +* (I ';' I)),n))) by EXTPRO_1:4
.= Exec ((CurInstr ((P +* (I ';' I)),(Comput ((P +* (I ';' I)),((s +* (Initialize I)) +* (I ';' I)),n)))),(Comput ((P +* (I ';' I)),((s +* (Initialize I)) +* (I ';' I)),n))) ;
A18: n <= n + 1 by NAT_1:12;
assume A19: n + 1 <= k ; :: thesis: Comput ((P +* (I ';' I)),((s +* (Initialize I)) +* (I ';' I)),(n + 1)), Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),(n + 1)) equal_outside NAT
then A20: IC (Comput ((P +* (I ';' I)),((s +* (Initialize I)) +* (I ';' I)),n)) = IC (Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),n)) by A15, A18, COMPOS_1:24, XXREAL_0:2;
n <= k by A19, A18, XXREAL_0:2;
then n <= LifeSpan ((P +* I),(s +* (Initialize I))) by A13, XXREAL_0:2;
then IC (Comput ((P +* I),(s +* (Initialize I)),n)) = IC (Comput ((P +* (I ';' I)),((s +* (Initialize I)) +* (I ';' I)),n)) by A1, A8, A6, Th36, COMPOS_1:24, A9, A11;
then IC (Comput ((P +* (I ';' I)),((s +* (Initialize I)) +* (I ';' I)),n)) in dom I by Def2, A5, A8;
then A21: IC (Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),n)) in dom (Directed I) by A20, FUNCT_4:105;
dom (P +* (Directed I)) = NAT by PARTFUN1:def 4;
then A22: (P +* (Directed I)) /. (IC (Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),n))) = (P +* (Directed I)) . (IC (Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),n))) by PARTFUN1:def 8;
A23: dom (P +* (I ';' I)) = NAT by PARTFUN1:def 4;
Directed I c= P +* (Directed I) by FUNCT_4:26;
then A24: CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),n))) = (Directed I) . (IC (Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),n))) by A21, GRFUNC_1:8, A22;
A25: ( dom I c= dom (I ';' I) & CurInstr ((P +* (I ';' I)),(Comput ((P +* (I ';' I)),((s +* (Initialize I)) +* (I ';' I)),n))) = (P +* (I ';' I)) . (IC (Comput ((P +* (I ';' I)),((s +* (Initialize I)) +* (I ';' I)),n))) ) by PARTFUN1:def 8, SCMFSA6A:56, A23;
A26: Directed I c= I ';' I by SCMFSA6A:55;
I ';' I c= P +* (I ';' I) by FUNCT_4:26;
then A27: Directed I c= P +* (I ';' I) by A26, XBOOLE_1:1;
CurInstr ((P +* (I ';' I)),(Comput ((P +* (I ';' I)),((s +* (Initialize I)) +* (I ';' I)),n))) = (Directed I) . (IC (Comput ((P +* (I ';' I)),((s +* (Initialize I)) +* (I ';' I)),n))) by A20, A21, GRFUNC_1:8, A27, A25;
hence Comput ((P +* (I ';' I)),((s +* (Initialize I)) +* (I ';' I)),(n + 1)), Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),(n + 1)) equal_outside NAT by A15, A19, A18, A20, A24, A17, A16, AMISTD_2:def 20, XXREAL_0:2; :: thesis: verum
end;
( Comput ((P +* (I ';' I)),((s +* (Initialize I)) +* (I ';' I)),0) = (s +* (Initialize I)) +* (I ';' I) & Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),0) = (s +* (Initialize I)) +* (Directed I) ) by EXTPRO_1:3;
then Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),0), Comput ((P +* (I ';' I)),((s +* (Initialize I)) +* (I ';' I)),0) equal_outside NAT by FUNCT_7:107, FUNCT_7:133;
then A28: S1[ 0 ] by FUNCT_7:28;
for n being Element of NAT holds S1[n] from NAT_1:sch 1(A28, A14);
then A29: Comput ((P +* (I ';' I)),((s +* (Initialize I)) +* (I ';' I)),k), Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),k) equal_outside NAT ;
Comput ((P +* I),(s +* (Initialize I)),k), Comput ((P +* (I ';' I)),((s +* (Initialize I)) +* (I ';' I)),k) equal_outside NAT by A1, A6, A13, Th36, A11, A5, A8;
hence Comput ((P +* I),(s +* (Initialize I)),k), Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),k) equal_outside NAT by A29, FUNCT_7:29; :: thesis: verum
end;
then A30: IC (Comput ((P +* I),(s +* (Initialize I)),(LifeSpan ((P +* I),(s +* (Initialize I)))))) = IC (Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),(LifeSpan ((P +* I),(s +* (Initialize I)))))) by COMPOS_1:24;
A31: dom (P +* I) = NAT by PARTFUN1:def 4;
I c= P +* I by FUNCT_4:26;
then A32: I . (IC (Comput ((P +* I),(s +* (Initialize I)),(LifeSpan ((P +* I),(s +* (Initialize I))))))) = (P +* I) . (IC (Comput ((P +* I),(s +* (Initialize I)),(LifeSpan ((P +* I),(s +* (Initialize I))))))) by A10, GRFUNC_1:8
.= CurInstr ((P +* I),(Comput ((P +* I),(s +* (Initialize I)),(LifeSpan ((P +* I),(s +* (Initialize I))))))) by A31, PARTFUN1:def 8
.= halt SCM+FSA by A1, A6, EXTPRO_1:def 14 ;
IC (Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),(LifeSpan ((P +* I),(s +* (Initialize I)))))) in dom I by A12, A10, COMPOS_1:24;
then IC (Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),(LifeSpan ((P +* I),(s +* (Initialize I)))))) in dom (Directed I) by FUNCT_4:105;
then A33: (P +* (Directed I)) . (IC (Comput ((P +* I),(s +* (Initialize I)),(LifeSpan ((P +* I),(s +* (Initialize I))))))) = (Directed I) . (IC (Comput ((P +* I),(s +* (Initialize I)),(LifeSpan ((P +* I),(s +* (Initialize I))))))) by A30, FUNCT_4:14
.= goto (card I) by A10, A32, FUNCT_4:112 ;
A34: (s +* (Initialize I)) +* (Directed I) = (Initialize (s +* I)) +* (Directed I) by FUNCT_4:15
.= ((Initialize s) +* I) +* (Directed I) by COMPOS_1:83
.= (s +* I) +* (Directed I) by A4, FUNCT_4:79
.= s +* (I +* (Directed I)) by FUNCT_4:15
.= s +* (Directed I) by A7, FUNCT_4:20
.= s by A2, FUNCT_4:79 ;
A35: P +* (Directed I) = P by A3, FUNCT_4:79;
dom (P +* (Directed I)) = NAT by PARTFUN1:def 4;
then A36: CurInstr ((P +* (Directed I)),(Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),(LifeSpan ((P +* I),(s +* (Initialize I))))))) = goto (card I) by A30, A33, PARTFUN1:def 8;
Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),((LifeSpan ((P +* I),(s +* (Initialize I)))) + 1)) = Following ((P +* (Directed I)),(Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),(LifeSpan ((P +* I),(s +* (Initialize I))))))) by EXTPRO_1:4
.= Exec ((goto (card I)),(Comput ((P +* (Directed I)),((s +* (Initialize I)) +* (Directed I)),(LifeSpan ((P +* I),(s +* (Initialize I))))))) by A36 ;
hence IC (Comput (P,s,((LifeSpan ((P +* I),(s +* I))) + 1))) = card I by A6, A34, A35, SCMFSA_2:95; :: thesis: verum