let k be Nat; :: thesis: for R being non trivial Ring
for s1, s2 being State of (SCM R)
for q being NAT -defined the InstructionsF of (SCM b1) -valued finite non halt-free Function
for p being non empty b4 -autonomic FinPartState of (SCM R) st p c= s1 & IncIC (p,k) c= s2 holds
for P1, P2 being Instruction-Sequence of (SCM R) st q c= P1 & Reloc (q,k) c= P2 holds
for i being Nat holds
( (IC (Comput (P1,s1,i))) + k = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),k) = CurInstr (P2,(Comput (P2,s2,i))) & (Comput (P1,s1,i)) | (dom (DataPart p)) = (Comput (P2,s2,i)) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),i)) = DataPart (Comput (P2,s2,i)) )
let R be non trivial Ring; :: thesis: for s1, s2 being State of (SCM R)
for q being NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function
for p being non empty b3 -autonomic FinPartState of (SCM R) st p c= s1 & IncIC (p,k) c= s2 holds
for P1, P2 being Instruction-Sequence of (SCM R) st q c= P1 & Reloc (q,k) c= P2 holds
for i being Nat holds
( (IC (Comput (P1,s1,i))) + k = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),k) = CurInstr (P2,(Comput (P2,s2,i))) & (Comput (P1,s1,i)) | (dom (DataPart p)) = (Comput (P2,s2,i)) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),i)) = DataPart (Comput (P2,s2,i)) )
let s1, s2 be State of (SCM R); :: thesis: for q being NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function
for p being non empty b1 -autonomic FinPartState of (SCM R) st p c= s1 & IncIC (p,k) c= s2 holds
for P1, P2 being Instruction-Sequence of (SCM R) st q c= P1 & Reloc (q,k) c= P2 holds
for i being Nat holds
( (IC (Comput (P1,s1,i))) + k = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),k) = CurInstr (P2,(Comput (P2,s2,i))) & (Comput (P1,s1,i)) | (dom (DataPart p)) = (Comput (P2,s2,i)) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),i)) = DataPart (Comput (P2,s2,i)) )
let q be NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function; :: thesis: for p being non empty q -autonomic FinPartState of (SCM R) st p c= s1 & IncIC (p,k) c= s2 holds
for P1, P2 being Instruction-Sequence of (SCM R) st q c= P1 & Reloc (q,k) c= P2 holds
for i being Nat holds
( (IC (Comput (P1,s1,i))) + k = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),k) = CurInstr (P2,(Comput (P2,s2,i))) & (Comput (P1,s1,i)) | (dom (DataPart p)) = (Comput (P2,s2,i)) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),i)) = DataPart (Comput (P2,s2,i)) )
let p be non empty q -autonomic FinPartState of (SCM R); :: thesis: ( p c= s1 & IncIC (p,k) c= s2 implies for P1, P2 being Instruction-Sequence of (SCM R) st q c= P1 & Reloc (q,k) c= P2 holds
for i being Nat holds
( (IC (Comput (P1,s1,i))) + k = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),k) = CurInstr (P2,(Comput (P2,s2,i))) & (Comput (P1,s1,i)) | (dom (DataPart p)) = (Comput (P2,s2,i)) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),i)) = DataPart (Comput (P2,s2,i)) ) )
assume that
A1: p c= s1 and
A2: IncIC (p,k) c= s2 ; :: thesis: for P1, P2 being Instruction-Sequence of (SCM R) st q c= P1 & Reloc (q,k) c= P2 holds
for i being Nat holds
( (IC (Comput (P1,s1,i))) + k = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),k) = CurInstr (P2,(Comput (P2,s2,i))) & (Comput (P1,s1,i)) | (dom (DataPart p)) = (Comput (P2,s2,i)) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),i)) = DataPart (Comput (P2,s2,i)) )
A3: IC in dom p by AMISTD_5:6;
let P1, P2 be Instruction-Sequence of (SCM R); :: thesis: ( q c= P1 & Reloc (q,k) c= P2 implies for i being Nat holds
( (IC (Comput (P1,s1,i))) + k = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),k) = CurInstr (P2,(Comput (P2,s2,i))) & (Comput (P1,s1,i)) | (dom (DataPart p)) = (Comput (P2,s2,i)) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),i)) = DataPart (Comput (P2,s2,i)) ) )
assume A4: ( q c= P1 & Reloc (q,k) c= P2 ) ; :: thesis: for i being Nat holds
( (IC (Comput (P1,s1,i))) + k = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),k) = CurInstr (P2,(Comput (P2,s2,i))) & (Comput (P1,s1,i)) | (dom (DataPart p)) = (Comput (P2,s2,i)) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),i)) = DataPart (Comput (P2,s2,i)) )
set s = s1 +* (DataPart s2);
defpred S1[ Nat] means ( (IC (Comput (P1,s1,$1))) + k = IC (Comput (P2,s2,$1)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,$1)))),k) = CurInstr (P2,(Comput (P2,s2,$1))) & (Comput (P1,s1,$1)) | (dom (DataPart p)) = (Comput (P2,s2,$1)) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),$1)) = DataPart (Comput (P2,s2,$1)) );
A5: IC p = IC s1 by A1, A3, GRFUNC_1:2;
then IC p = IC (Comput (P1,s1,0)) ;
then A6: IC p in dom q by A1, A4, AMISTD_5:def 4;
A7: p c= s1 +* (DataPart s2) by A1, A2, MEMSTR_0:61;
A8: for i being Nat st S1[i] holds
S1[i + 1]
proof
set DPp = DataPart p;
let i be Nat; :: thesis: ( S1[i] implies S1[i + 1] )
assume that
A9: (IC (Comput (P1,s1,i))) + k = IC (Comput (P2,s2,i)) and
A10: IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),k) = CurInstr (P2,(Comput (P2,s2,i))) and
A11: (Comput (P1,s1,i)) | (dom (DataPart p)) = (Comput (P2,s2,i)) | (dom (DataPart p)) and
A12: DataPart (Comput (P1,(s1 +* (DataPart s2)),i)) = DataPart (Comput (P2,s2,i)) ; :: thesis: S1[i + 1]
set Cs2i1 = Comput (P2,s2,(i + 1));
set Cs3i = Comput (P1,(s1 +* (DataPart s2)),i);
set Cs2i = Comput (P2,s2,i);
A13: Comput (P2,s2,(i + 1)) = Following (P2,(Comput (P2,s2,i))) by EXTPRO_1:3
.= Exec ((CurInstr (P2,(Comput (P2,s2,i)))),(Comput (P2,s2,i))) ;
A14: now :: thesis: for s being State of (SCM R)
for d being Data-Location of R holds d in dom (DataPart s)
let s be State of (SCM R); :: thesis: for d being Data-Location of R holds d in dom (DataPart s)
let d be Data-Location of R; :: thesis: d in dom (DataPart s)
d in Data-Locations by SCMRING2:23;
hence d in dom (DataPart s) by MEMSTR_0:9; :: thesis: verum
end;
A15: now :: thesis: for d being Data-Location of R holds (Comput (P1,(s1 +* (DataPart s2)),i)) . d = (Comput (P2,s2,i)) . d
let d be Data-Location of R; :: thesis: (Comput (P1,(s1 +* (DataPart s2)),i)) . d = (Comput (P2,s2,i)) . d
A16: d in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),i))) by A14;
hence (Comput (P1,(s1 +* (DataPart s2)),i)) . d = (DataPart (Comput (P1,(s1 +* (DataPart s2)),i))) . d by FUNCT_1:47
.= (Comput (P2,s2,i)) . d by A12, A16, FUNCT_1:47 ;
:: thesis: verum
end;
set Cs1i1 = Comput (P1,s1,(i + 1));
set Cs1i = Comput (P1,s1,i);
dom (Comput (P1,s1,(i + 1))) = the carrier of (SCM R) by PARTFUN1:def 2;
then A17: dom (Comput (P1,s1,(i + 1))) = {(IC )} \/ (Data-Locations ) by XBOOLE_1:45;
A18: ((IC (Comput (P1,s1,i))) + k) + 1 = ((IC (Comput (P1,s1,i))) + 1) + k ;
A19: now :: thesis: ( (IC (Comput (P1,s1,(i + 1)))) + k = IC (Comput (P2,s2,(i + 1))) implies IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1)))) )
reconsider loc = IC (Comput (P1,s1,(i + 1))) as Element of NAT ;
assume A20: (IC (Comput (P1,s1,(i + 1)))) + k = IC (Comput (P2,s2,(i + 1))) ; :: thesis: IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1))))
A21: loc in dom q by A1, A4, AMISTD_5:def 4;
loc + k in dom (Reloc (q,k)) by A21, COMPOS_1:46;
then A22: (Reloc (q,k)) . (loc + k) = P2 . (loc + k) by A4, GRFUNC_1:2;
A23: P2 /. (IC (Comput (P2,s2,(i + 1)))) = P2 . (IC (Comput (P2,s2,(i + 1)))) by PBOOLE:143;
CurInstr (P1,(Comput (P1,s1,(i + 1)))) = P1 . loc by PBOOLE:143
.= q . loc by A21, A4, GRFUNC_1:2 ;
hence IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1)))) by A20, A21, A22, A23, COMPOS_1:35; :: thesis: verum
end;
dom (Comput (P2,s2,i)) = the carrier of (SCM R) by PARTFUN1:def 2;
then A24: dom (Comput (P2,s2,i)) = {(IC )} \/ (Data-Locations ) by XBOOLE_1:45;
dom (DataPart p) = (dom p) /\ (Data-Locations ) by RELAT_1:61;
then A25: dom (DataPart p) c= {(IC )} \/ (Data-Locations ) by XBOOLE_1:10, XBOOLE_1:17;
set Cs3i1 = Comput (P1,(s1 +* (DataPart s2)),(i + 1));
A26: dom (DataPart (Comput (P2,s2,i))) = Data-Locations by MEMSTR_0:9;
A27: dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) = Data-Locations by MEMSTR_0:9;
then A28: dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) c= dom (DataPart (Comput (P2,s2,(i + 1)))) by MEMSTR_0:9;
A29: dom (DataPart (Comput (P2,s2,(i + 1)))) = Data-Locations by MEMSTR_0:9;
A30: now :: thesis: for x being set st x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) & (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . x = (Comput (P2,s2,(i + 1))) . x holds
(DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x
let x be set ; :: thesis: ( x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) & (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . x = (Comput (P2,s2,(i + 1))) . x implies (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x )
assume that
A31: x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) and
A32: (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . x = (Comput (P2,s2,(i + 1))) . x ; :: thesis: (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x
thus (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (Comput (P2,s2,(i + 1))) . x by A31, A32, FUNCT_1:47
.= (DataPart (Comput (P2,s2,(i + 1)))) . x by A27, A29, A31, FUNCT_1:47 ; :: thesis: verum
end;
A33: dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),i))) = Data-Locations by MEMSTR_0:9;
A34: now :: thesis: for x being set st x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) & (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . x = (Comput (P1,(s1 +* (DataPart s2)),i)) . x & (Comput (P2,s2,(i + 1))) . x = (Comput (P2,s2,i)) . x holds
(DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x
let x be set ; :: thesis: ( x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) & (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . x = (Comput (P1,(s1 +* (DataPart s2)),i)) . x & (Comput (P2,s2,(i + 1))) . x = (Comput (P2,s2,i)) . x implies (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x )
assume that
A35: x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) and
A36: ( (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . x = (Comput (P1,(s1 +* (DataPart s2)),i)) . x & (Comput (P2,s2,(i + 1))) . x = (Comput (P2,s2,i)) . x ) ; :: thesis: (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x
(DataPart (Comput (P1,(s1 +* (DataPart s2)),i))) . x = (Comput (P1,(s1 +* (DataPart s2)),i)) . x by A33, A27, A35, FUNCT_1:47;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A12, A26, A27, A30, A35, A36, FUNCT_1:47; :: thesis: verum
end;
dom (Comput (P1,s1,i)) = the carrier of (SCM R) by PARTFUN1:def 2;
then A37: dom (Comput (P1,s1,i)) = {(IC )} \/ (Data-Locations ) by XBOOLE_1:45;
dom (Comput (P2,s2,(i + 1))) = the carrier of (SCM R) by PARTFUN1:def 2;
then A38: dom (Comput (P2,s2,(i + 1))) = {(IC )} \/ (Data-Locations ) by XBOOLE_1:45;
set I = CurInstr (P1,(Comput (P1,s1,i)));
A39: Comput (P1,s1,(i + 1)) = Following (P1,(Comput (P1,s1,i))) by EXTPRO_1:3
.= Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P1,s1,i))) ;
A40: dom ((Comput (P1,s1,i)) | (dom (DataPart p))) = (dom (Comput (P1,s1,i))) /\ (dom (DataPart p)) by RELAT_1:61
.= dom (DataPart p) by A37, A25, XBOOLE_1:28 ;
A41: dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) = (dom (Comput (P1,s1,(i + 1)))) /\ (dom (DataPart p)) by RELAT_1:61
.= dom (DataPart p) by A17, A25, XBOOLE_1:28 ;
A42: dom ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) = (dom (Comput (P2,s2,(i + 1)))) /\ (dom (DataPart p)) by RELAT_1:61
.= dom (DataPart p) by A38, A25, XBOOLE_1:28 ;
then A43: dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) c= dom ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) by A41;
A44: dom ((Comput (P2,s2,i)) | (dom (DataPart p))) = (dom (Comput (P2,s2,i))) /\ (dom (DataPart p)) by RELAT_1:61
.= dom (DataPart p) by A24, A25, XBOOLE_1:28 ;
A45: now :: thesis: for x being set
for d being Data-Location of R st d = x & d in dom (DataPart p) & (Comput (P1,s1,(i + 1))) . d = (Comput (P1,s1,i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d holds
((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x
let x be set ; :: thesis: for d being Data-Location of R st d = x & d in dom (DataPart p) & (Comput (P1,s1,(i + 1))) . d = (Comput (P1,s1,i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d holds
((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x
let d be Data-Location of R; :: thesis: ( d = x & d in dom (DataPart p) & (Comput (P1,s1,(i + 1))) . d = (Comput (P1,s1,i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d implies ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x )
assume that
A46: d = x and
A47: d in dom (DataPart p) and
A48: ( (Comput (P1,s1,(i + 1))) . d = (Comput (P1,s1,i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) ; :: thesis: ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x
A49: ( ((Comput (P1,s1,i)) | (dom (DataPart p))) . d = (Comput (P1,s1,i)) . d & ((Comput (P2,s2,i)) | (dom (DataPart p))) . d = (Comput (P2,s2,i)) . d ) by A40, A44, A47, FUNCT_1:47;
thus ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = (Comput (P1,s1,(i + 1))) . d by A41, A46, A47, FUNCT_1:47
.= ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A11, A42, A46, A47, A48, A49, FUNCT_1:47 ; :: thesis: verum
end;
A50: now :: thesis: for x being set
for d being Data-Location of R st d = x & d in dom (DataPart p) & (Comput (P1,s1,(i + 1))) . d = (Comput (P2,s2,(i + 1))) . d holds
((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x
let x be set ; :: thesis: for d being Data-Location of R st d = x & d in dom (DataPart p) & (Comput (P1,s1,(i + 1))) . d = (Comput (P2,s2,(i + 1))) . d holds
((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x
let d be Data-Location of R; :: thesis: ( d = x & d in dom (DataPart p) & (Comput (P1,s1,(i + 1))) . d = (Comput (P2,s2,(i + 1))) . d implies ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x )
assume that
A51: ( d = x & d in dom (DataPart p) ) and
A52: (Comput (P1,s1,(i + 1))) . d = (Comput (P2,s2,(i + 1))) . d ; :: thesis: ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x
thus ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = (Comput (P2,s2,(i + 1))) . d by A41, A51, A52, FUNCT_1:47
.= ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A42, A51, FUNCT_1:47 ; :: thesis: verum
end;
A53: Comput (P1,(s1 +* (DataPart s2)),(i + 1)) = Following (P1,(Comput (P1,(s1 +* (DataPart s2)),i))) by EXTPRO_1:3
.= Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P1,(s1 +* (DataPart s2)),i))) by A1, A7, A4, AMISTD_5:7 ;
not not InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 0 & ... & not InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 7 by SCMRING3:39;
per cases then ( InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 0 or InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 1 or InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 2 or InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 3 or InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 4 or InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 5 or InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 6 or InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 7 ) ;
suppose InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 0 ; :: thesis: S1[i + 1]
then A54: CurInstr (P1,(Comput (P1,s1,i))) = halt (SCM R) by SCMRING3:12;
hence (IC (Comput (P1,s1,(i + 1)))) + k = (IC (Comput (P1,s1,i))) + k by A39, EXTPRO_1:def 3
.= IC (Comput (P2,s2,(i + 1))) by A9, A10, A13, A54, EXTPRO_1:def 3 ;
:: thesis: ( IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1)))) & (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) = (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1))) )
hence IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1)))) by A19; :: thesis: ( (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) = (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1))) )
A55: Comput (P2,s2,(i + 1)) = Comput (P2,s2,i) by A10, A13, A54, EXTPRO_1:def 3;
hence (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) = (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) by A11, A39, A54, EXTPRO_1:def 3; :: thesis: DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1)))
thus DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1))) by A12, A53, A54, A55, EXTPRO_1:def 3; :: thesis: verum
end;
suppose InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 1 ; :: thesis: S1[i + 1]
then consider da, db being Data-Location of R such that
A56: CurInstr (P1,(Comput (P1,s1,i))) = da := db by SCMRING3:13;
A57: IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),k) = da := db by A56, COMPOS_0:4;
A58: (Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P1,s1,i)))) . (IC ) = (IC (Comput (P1,s1,i))) + 1 by A56, SCMRING2:11;
hence (IC (Comput (P1,s1,(i + 1)))) + k = IC (Comput (P2,s2,(i + 1))) by A9, A10, A39, A13, A18, A57, SCMRING2:11; :: thesis: ( IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1)))) & (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) = (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1))) )
thus IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1)))) by A9, A10, A19, A39, A13, A18, A57, A58, SCMRING2:11; :: thesis: ( (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) = (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1))) )
A59: (Comput (P1,(s1 +* (DataPart s2)),i)) . db = (Comput (P2,s2,i)) . db by A15;
now :: thesis: for x being object st x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) holds
((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x
DataPart p c= p by RELAT_1:59;
then A60: dom (DataPart p) c= dom p by GRFUNC_1:2;
let x be object ; :: thesis: ( x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) implies ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . b1 = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . b1 )
assume A61: x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) ; :: thesis: ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . b1 = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . b1
dom (DataPart p) c= Data-Locations by RELAT_1:58;
then reconsider d = x as Data-Location of R by A41, A61, SCMRING2:23;
per cases ( da = d or da <> d ) ;
suppose A62: da = d ; :: thesis: ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . b1 = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . b1
then ( (Comput (P1,s1,(i + 1))) . d = (Comput (P1,s1,i)) . db & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . db ) by A10, A39, A13, A56, A57, SCMRING2:11;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A1, A7, A41, A50, A56, A59, A61, A60, A62, Th7, A4; :: thesis: verum
end;
suppose da <> d ; :: thesis: ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . b1 = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . b1
then ( (Comput (P1,s1,(i + 1))) . d = (Comput (P1,s1,i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A10, A39, A13, A56, A57, SCMRING2:11;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A41, A45, A61; :: thesis: verum
end;
end;
end;
then (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) c= (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) by A43, GRFUNC_1:2;
hence (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) = (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) by A41, A42, GRFUNC_1:3; :: thesis: DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1)))
now :: thesis: for x being object st x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) holds
(DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x
let x be object ; :: thesis: ( x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) implies (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . b1 = (DataPart (Comput (P2,s2,(i + 1)))) . b1 )
assume A63: x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) ; :: thesis: (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . b1 = (DataPart (Comput (P2,s2,(i + 1)))) . b1
then reconsider d = x as Data-Location of R by A27, SCMRING2:23;
per cases ( da = d or da <> d ) ;
suppose da = d ; :: thesis: (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . b1 = (DataPart (Comput (P2,s2,(i + 1)))) . b1
then ( (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . db & (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = (Comput (P1,(s1 +* (DataPart s2)),i)) . db ) by A10, A13, A53, A56, A57, SCMRING2:11;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A15, A30, A63; :: thesis: verum
end;
suppose da <> d ; :: thesis: (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . b1 = (DataPart (Comput (P2,s2,(i + 1)))) . b1
then ( (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = (Comput (P1,(s1 +* (DataPart s2)),i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A10, A13, A53, A56, A57, SCMRING2:11;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A34, A63; :: thesis: verum
end;
end;
end;
then DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) c= DataPart (Comput (P2,s2,(i + 1))) by A28, GRFUNC_1:2;
hence DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1))) by A27, A29, GRFUNC_1:3; :: thesis: verum
end;
suppose InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 2 ; :: thesis: S1[i + 1]
then consider da, db being Data-Location of R such that
A64: CurInstr (P1,(Comput (P1,s1,i))) = AddTo (da,db) by SCMRING3:14;
A65: IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),k) = AddTo (da,db) by A64, COMPOS_0:4;
A66: (Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P1,s1,i)))) . (IC ) = (IC (Comput (P1,s1,i))) + 1 by A64, SCMRING2:12;
hence (IC (Comput (P1,s1,(i + 1)))) + k = IC (Comput (P2,s2,(i + 1))) by A9, A10, A39, A13, A18, A65, SCMRING2:12; :: thesis: ( IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1)))) & (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) = (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1))) )
thus IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1)))) by A9, A10, A19, A39, A13, A18, A65, A66, SCMRING2:12; :: thesis: ( (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) = (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1))) )
A67: ( (Comput (P1,(s1 +* (DataPart s2)),i)) . da = (Comput (P2,s2,i)) . da & (Comput (P1,(s1 +* (DataPart s2)),i)) . db = (Comput (P2,s2,i)) . db ) by A15;
now :: thesis: for x being object st x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) holds
((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x
DataPart p c= p by RELAT_1:59;
then A68: dom (DataPart p) c= dom p by GRFUNC_1:2;
let x be object ; :: thesis: ( x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) implies ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . b1 = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . b1 )
assume A69: x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) ; :: thesis: ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . b1 = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . b1
dom (DataPart p) c= Data-Locations by RELAT_1:58;
then reconsider d = x as Data-Location of R by A41, A69, SCMRING2:23;
per cases ( da = d or da <> d ) ;
suppose A70: da = d ; :: thesis: ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . b1 = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . b1
then ( (Comput (P1,s1,(i + 1))) . d = ((Comput (P1,s1,i)) . da) + ((Comput (P1,s1,i)) . db) & (Comput (P2,s2,(i + 1))) . d = ((Comput (P2,s2,i)) . da) + ((Comput (P2,s2,i)) . db) ) by A10, A39, A13, A64, A65, SCMRING2:12;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A1, A7, A41, A50, A64, A67, A69, A68, A70, Th8, A4; :: thesis: verum
end;
suppose da <> d ; :: thesis: ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . b1 = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . b1
then ( (Comput (P1,s1,(i + 1))) . d = (Comput (P1,s1,i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A10, A39, A13, A64, A65, SCMRING2:12;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A41, A45, A69; :: thesis: verum
end;
end;
end;
then (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) c= (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) by A43, GRFUNC_1:2;
hence (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) = (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) by A41, A42, GRFUNC_1:3; :: thesis: DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1)))
now :: thesis: for x being object st x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) holds
(DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x
let x be object ; :: thesis: ( x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) implies (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . b1 = (DataPart (Comput (P2,s2,(i + 1)))) . b1 )
assume A71: x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) ; :: thesis: (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . b1 = (DataPart (Comput (P2,s2,(i + 1)))) . b1
then reconsider d = x as Data-Location of R by A27, SCMRING2:23;
per cases ( da = d or da <> d ) ;
suppose da = d ; :: thesis: (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . b1 = (DataPart (Comput (P2,s2,(i + 1)))) . b1
then ( (Comput (P2,s2,(i + 1))) . d = ((Comput (P2,s2,i)) . da) + ((Comput (P2,s2,i)) . db) & (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = ((Comput (P1,(s1 +* (DataPart s2)),i)) . da) + ((Comput (P1,(s1 +* (DataPart s2)),i)) . db) ) by A10, A13, A53, A64, A65, SCMRING2:12;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A30, A67, A71; :: thesis: verum
end;
suppose da <> d ; :: thesis: (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . b1 = (DataPart (Comput (P2,s2,(i + 1)))) . b1
then ( (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = (Comput (P1,(s1 +* (DataPart s2)),i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A10, A13, A53, A64, A65, SCMRING2:12;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A34, A71; :: thesis: verum
end;
end;
end;
then DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) c= DataPart (Comput (P2,s2,(i + 1))) by A28, GRFUNC_1:2;
hence DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1))) by A27, A29, GRFUNC_1:3; :: thesis: verum
end;
suppose InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 3 ; :: thesis: S1[i + 1]
then consider da, db being Data-Location of R such that
A72: CurInstr (P1,(Comput (P1,s1,i))) = SubFrom (da,db) by SCMRING3:15;
A73: IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),k) = SubFrom (da,db) by A72, COMPOS_0:4;
A74: (Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P1,s1,i)))) . (IC ) = (IC (Comput (P1,s1,i))) + 1 by A72, SCMRING2:13;
hence (IC (Comput (P1,s1,(i + 1)))) + k = IC (Comput (P2,s2,(i + 1))) by A9, A10, A39, A13, A18, A73, SCMRING2:13; :: thesis: ( IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1)))) & (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) = (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1))) )
thus IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1)))) by A9, A10, A19, A39, A13, A18, A73, A74, SCMRING2:13; :: thesis: ( (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) = (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1))) )
A75: ( (Comput (P1,(s1 +* (DataPart s2)),i)) . da = (Comput (P2,s2,i)) . da & (Comput (P1,(s1 +* (DataPart s2)),i)) . db = (Comput (P2,s2,i)) . db ) by A15;
now :: thesis: for x being object st x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) holds
((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x
DataPart p c= p by RELAT_1:59;
then A76: dom (DataPart p) c= dom p by GRFUNC_1:2;
let x be object ; :: thesis: ( x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) implies ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . b1 = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . b1 )
assume A77: x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) ; :: thesis: ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . b1 = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . b1
dom (DataPart p) c= Data-Locations by RELAT_1:58;
then reconsider d = x as Data-Location of R by A41, A77, SCMRING2:23;
per cases ( da = d or da <> d ) ;
suppose A78: da = d ; :: thesis: ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . b1 = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . b1
then ( (Comput (P1,s1,(i + 1))) . d = ((Comput (P1,s1,i)) . da) - ((Comput (P1,s1,i)) . db) & (Comput (P2,s2,(i + 1))) . d = ((Comput (P2,s2,i)) . da) - ((Comput (P2,s2,i)) . db) ) by A10, A39, A13, A72, A73, SCMRING2:13;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A1, A7, A41, A50, A72, A75, A77, A76, A78, Th9, A4; :: thesis: verum
end;
suppose da <> d ; :: thesis: ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . b1 = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . b1
then ( (Comput (P1,s1,(i + 1))) . d = (Comput (P1,s1,i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A10, A39, A13, A72, A73, SCMRING2:13;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A41, A45, A77; :: thesis: verum
end;
end;
end;
then (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) c= (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) by A43, GRFUNC_1:2;
hence (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) = (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) by A41, A42, GRFUNC_1:3; :: thesis: DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1)))
now :: thesis: for x being object st x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) holds
(DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x
let x be object ; :: thesis: ( x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) implies (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . b1 = (DataPart (Comput (P2,s2,(i + 1)))) . b1 )
assume A79: x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) ; :: thesis: (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . b1 = (DataPart (Comput (P2,s2,(i + 1)))) . b1
then reconsider d = x as Data-Location of R by A27, SCMRING2:23;
per cases ( da = d or da <> d ) ;
suppose da = d ; :: thesis: (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . b1 = (DataPart (Comput (P2,s2,(i + 1)))) . b1
then ( (Comput (P2,s2,(i + 1))) . d = ((Comput (P2,s2,i)) . da) - ((Comput (P2,s2,i)) . db) & (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = ((Comput (P1,(s1 +* (DataPart s2)),i)) . da) - ((Comput (P1,(s1 +* (DataPart s2)),i)) . db) ) by A10, A13, A53, A72, A73, SCMRING2:13;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A30, A75, A79; :: thesis: verum
end;
suppose da <> d ; :: thesis: (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . b1 = (DataPart (Comput (P2,s2,(i + 1)))) . b1
then ( (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = (Comput (P1,(s1 +* (DataPart s2)),i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A10, A13, A53, A72, A73, SCMRING2:13;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A34, A79; :: thesis: verum
end;
end;
end;
then DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) c= DataPart (Comput (P2,s2,(i + 1))) by A28, GRFUNC_1:2;
hence DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1))) by A27, A29, GRFUNC_1:3; :: thesis: verum
end;
suppose InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 4 ; :: thesis: S1[i + 1]
then consider da, db being Data-Location of R such that
A80: CurInstr (P1,(Comput (P1,s1,i))) = MultBy (da,db) by SCMRING3:16;
A81: IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),k) = MultBy (da,db) by A80, COMPOS_0:4;
A82: (Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P1,s1,i)))) . (IC ) = (IC (Comput (P1,s1,i))) + 1 by A80, SCMRING2:14;
hence (IC (Comput (P1,s1,(i + 1)))) + k = IC (Comput (P2,s2,(i + 1))) by A9, A10, A39, A13, A18, A81, SCMRING2:14; :: thesis: ( IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1)))) & (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) = (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1))) )
thus IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1)))) by A9, A10, A19, A39, A13, A18, A81, A82, SCMRING2:14; :: thesis: ( (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) = (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1))) )
A83: ( (Comput (P1,(s1 +* (DataPart s2)),i)) . da = (Comput (P2,s2,i)) . da & (Comput (P1,(s1 +* (DataPart s2)),i)) . db = (Comput (P2,s2,i)) . db ) by A15;
now :: thesis: for x being object st x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) holds
((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x
DataPart p c= p by RELAT_1:59;
then A84: dom (DataPart p) c= dom p by GRFUNC_1:2;
let x be object ; :: thesis: ( x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) implies ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . b1 = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . b1 )
assume A85: x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) ; :: thesis: ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . b1 = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . b1
dom (DataPart p) c= Data-Locations by RELAT_1:58;
then reconsider d = x as Data-Location of R by A41, A85, SCMRING2:23;
per cases ( da = d or da <> d ) ;
suppose A86: da = d ; :: thesis: ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . b1 = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . b1
then ( (Comput (P1,s1,(i + 1))) . d = ((Comput (P1,s1,i)) . da) * ((Comput (P1,s1,i)) . db) & (Comput (P2,s2,(i + 1))) . d = ((Comput (P2,s2,i)) . da) * ((Comput (P2,s2,i)) . db) ) by A10, A39, A13, A80, A81, SCMRING2:14;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A1, A7, A41, A50, A80, A83, A85, A84, A86, Th10, A4; :: thesis: verum
end;
suppose da <> d ; :: thesis: ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . b1 = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . b1
then ( (Comput (P1,s1,(i + 1))) . d = (Comput (P1,s1,i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A10, A39, A13, A80, A81, SCMRING2:14;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A41, A45, A85; :: thesis: verum
end;
end;
end;
then (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) c= (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) by A43, GRFUNC_1:2;
hence (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) = (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) by A41, A42, GRFUNC_1:3; :: thesis: DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1)))
now :: thesis: for x being object st x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) holds
(DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x
let x be object ; :: thesis: ( x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) implies (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . b1 = (DataPart (Comput (P2,s2,(i + 1)))) . b1 )
assume A87: x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) ; :: thesis: (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . b1 = (DataPart (Comput (P2,s2,(i + 1)))) . b1
then reconsider d = x as Data-Location of R by A27, SCMRING2:23;
per cases ( da = d or da <> d ) ;
suppose da = d ; :: thesis: (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . b1 = (DataPart (Comput (P2,s2,(i + 1)))) . b1
then ( (Comput (P2,s2,(i + 1))) . d = ((Comput (P2,s2,i)) . da) * ((Comput (P2,s2,i)) . db) & (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = ((Comput (P1,(s1 +* (DataPart s2)),i)) . da) * ((Comput (P1,(s1 +* (DataPart s2)),i)) . db) ) by A10, A13, A53, A80, A81, SCMRING2:14;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A30, A83, A87; :: thesis: verum
end;
suppose da <> d ; :: thesis: (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . b1 = (DataPart (Comput (P2,s2,(i + 1)))) . b1
then ( (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = (Comput (P1,(s1 +* (DataPart s2)),i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A10, A13, A53, A80, A81, SCMRING2:14;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A34, A87; :: thesis: verum
end;
end;
end;
then DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) c= DataPart (Comput (P2,s2,(i + 1))) by A28, GRFUNC_1:2;
hence DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1))) by A27, A29, GRFUNC_1:3; :: thesis: verum
end;
suppose InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 5 ; :: thesis: S1[i + 1]
then consider da being Data-Location of R, r being Element of R such that
A88: CurInstr (P1,(Comput (P1,s1,i))) = da := r by SCMRING3:17;
A89: IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),k) = da := r by A88, COMPOS_0:4;
A90: (Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P1,s1,i)))) . (IC ) = (IC (Comput (P1,s1,i))) + 1 by A88, SCMRING2:17;
hence (IC (Comput (P1,s1,(i + 1)))) + k = IC (Comput (P2,s2,(i + 1))) by A9, A10, A39, A13, A18, A89, SCMRING2:17; :: thesis: ( IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1)))) & (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) = (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1))) )
thus IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1)))) by A9, A10, A19, A39, A13, A18, A89, A90, SCMRING2:17; :: thesis: ( (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) = (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1))) )
now :: thesis: for x being object st x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) holds
((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x
let x be object ; :: thesis: ( x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) implies ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . b1 = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . b1 )
assume A91: x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) ; :: thesis: ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . b1 = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . b1
dom (DataPart p) c= Data-Locations by RELAT_1:58;
then reconsider d = x as Data-Location of R by A41, A91, SCMRING2:23;
per cases ( da = d or da <> d ) ;
suppose A92: da = d ; :: thesis: ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . b1 = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . b1
thus ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = (Comput (P1,s1,(i + 1))) . d by A41, A91, FUNCT_1:49
.= r by A39, A88, A92, SCMRING2:17
.= (Comput (P2,s2,(i + 1))) . d by A10, A13, A89, A92, SCMRING2:17
.= ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A41, A91, FUNCT_1:49 ; :: thesis: verum
end;
suppose da <> d ; :: thesis: ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . b1 = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . b1
then ( (Comput (P1,s1,(i + 1))) . d = (Comput (P1,s1,i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A10, A39, A13, A88, A89, SCMRING2:17;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A41, A45, A91; :: thesis: verum
end;
end;
end;
then (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) c= (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) by A43, GRFUNC_1:2;
hence (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) = (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) by A41, A42, GRFUNC_1:3; :: thesis: DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1)))
now :: thesis: for x being object st x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) holds
(DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x
let x be object ; :: thesis: ( x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) implies (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . b1 = (DataPart (Comput (P2,s2,(i + 1)))) . b1 )
assume A93: x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) ; :: thesis: (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . b1 = (DataPart (Comput (P2,s2,(i + 1)))) . b1
then reconsider d = x as Data-Location of R by A27, SCMRING2:23;
per cases ( da = d or da <> d ) ;
suppose da = d ; :: thesis: (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . b1 = (DataPart (Comput (P2,s2,(i + 1)))) . b1
then ( (Comput (P2,s2,(i + 1))) . d = r & (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = r ) by A10, A13, A53, A88, A89, SCMRING2:17;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A30, A93; :: thesis: verum
end;
suppose da <> d ; :: thesis: (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . b1 = (DataPart (Comput (P2,s2,(i + 1)))) . b1
then ( (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = (Comput (P1,(s1 +* (DataPart s2)),i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A10, A13, A53, A88, A89, SCMRING2:17;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A34, A93; :: thesis: verum
end;
end;
end;
then DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) c= DataPart (Comput (P2,s2,(i + 1))) by A28, GRFUNC_1:2;
hence DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1))) by A27, A29, GRFUNC_1:3; :: thesis: verum
end;
suppose InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 6 ; :: thesis: S1[i + 1]
then consider loc being Nat such that
A94: CurInstr (P1,(Comput (P1,s1,i))) = goto (loc,R) by SCMRING3:18;
A95: CurInstr (P2,(Comput (P2,s2,i))) = goto ((loc + k),R) by A10, A94, SCMRING3:37;
thus (IC (Comput (P1,s1,(i + 1)))) + k = loc + k by A39, A94, SCMRING2:15
.= IC (Comput (P2,s2,(i + 1))) by A13, A95, SCMRING2:15 ; :: thesis: ( IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1)))) & (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) = (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1))) )
hence IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1)))) by A19; :: thesis: ( (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) = (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1))) )
now :: thesis: for x being object st x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) holds
((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x
let x be object ; :: thesis: ( x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) implies ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x )
assume A96: x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) ; :: thesis: ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x
dom (DataPart p) c= Data-Locations by RELAT_1:58;
then reconsider d = x as Data-Location of R by A41, A96, SCMRING2:23;
( (Comput (P1,s1,(i + 1))) . d = (Comput (P1,s1,i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A39, A13, A94, A95, SCMRING2:15;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A41, A45, A96; :: thesis: verum
end;
then (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) c= (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) by A43, GRFUNC_1:2;
hence (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) = (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) by A41, A42, GRFUNC_1:3; :: thesis: DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1)))
now :: thesis: for x being object st x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) holds
(DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x
let x be object ; :: thesis: ( x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) implies (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x )
assume A97: x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) ; :: thesis: (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x
then reconsider d = x as Data-Location of R by A27, SCMRING2:23;
( (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = (Comput (P1,(s1 +* (DataPart s2)),i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A13, A53, A94, A95, SCMRING2:15;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A34, A97; :: thesis: verum
end;
then DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) c= DataPart (Comput (P2,s2,(i + 1))) by A28, GRFUNC_1:2;
hence DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1))) by A27, A29, GRFUNC_1:3; :: thesis: verum
end;
suppose InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 7 ; :: thesis: S1[i + 1]
then consider da being Data-Location of R, loc being Nat such that
A98: CurInstr (P1,(Comput (P1,s1,i))) = da =0_goto loc by SCMRING3:19;
A99: now :: thesis: ( ( (Comput (P1,s1,i)) . da = 0. R & (IC (Comput (P1,s1,(i + 1)))) + k = loc + k ) or ( (Comput (P1,s1,i)) . da <> 0. R & (IC (Comput (P1,s1,(i + 1)))) + k = (IC (Comput (P2,s2,i))) + 1 ) )
per cases ( (Comput (P1,s1,i)) . da = 0. R or (Comput (P1,s1,i)) . da <> 0. R ) ;
case (Comput (P1,s1,i)) . da = 0. R ; :: thesis: (IC (Comput (P1,s1,(i + 1)))) + k = loc + k
hence (IC (Comput (P1,s1,(i + 1)))) + k = loc + k by A39, A98, SCMRING2:16; :: thesis: verum
end;
case (Comput (P1,s1,i)) . da <> 0. R ; :: thesis: (IC (Comput (P1,s1,(i + 1)))) + k = (IC (Comput (P2,s2,i))) + 1
hence (IC (Comput (P1,s1,(i + 1)))) + k = (IC (Comput (P2,s2,i))) + 1 by A9, A39, A18, A98, SCMRING2:16; :: thesis: verum
end;
end;
end;
A100: CurInstr (P2,(Comput (P2,s2,i))) = da =0_goto (loc + k) by A10, A98, SCMRING3:38;
A101: now :: thesis: ( ( (Comput (P2,s2,i)) . da = 0. R & IC (Comput (P2,s2,(i + 1))) = loc + k ) or ( (Comput (P2,s2,i)) . da <> 0. R & IC (Comput (P2,s2,(i + 1))) = (IC (Comput (P2,s2,i))) + 1 ) )
per cases ( (Comput (P2,s2,i)) . da = 0. R or (Comput (P2,s2,i)) . da <> 0. R ) ;
case (Comput (P2,s2,i)) . da = 0. R ; :: thesis: IC (Comput (P2,s2,(i + 1))) = loc + k
hence IC (Comput (P2,s2,(i + 1))) = loc + k by A13, A100, SCMRING2:16; :: thesis: verum
end;
case (Comput (P2,s2,i)) . da <> 0. R ; :: thesis: IC (Comput (P2,s2,(i + 1))) = (IC (Comput (P2,s2,i))) + 1
hence IC (Comput (P2,s2,(i + 1))) = (IC (Comput (P2,s2,i))) + 1 by A13, A100, SCMRING2:16; :: thesis: verum
end;
end;
end;
A102: (Comput (P1,(s1 +* (DataPart s2)),i)) . da = (Comput (P2,s2,i)) . da by A15;
now :: thesis: (IC (Comput (P1,s1,(i + 1)))) + k = IC (Comput (P2,s2,(i + 1)))
per cases ( loc <> (IC (Comput (P1,s1,i))) + 1 or loc = (IC (Comput (P1,s1,i))) + 1 ) ;
suppose loc <> (IC (Comput (P1,s1,i))) + 1 ; :: thesis: (IC (Comput (P1,s1,(i + 1)))) + k = IC (Comput (P2,s2,(i + 1)))
hence (IC (Comput (P1,s1,(i + 1)))) + k = IC (Comput (P2,s2,(i + 1))) by A1, A7, A98, A102, A99, A101, Th11, A4; :: thesis: verum
end;
suppose loc = (IC (Comput (P1,s1,i))) + 1 ; :: thesis: (IC (Comput (P1,s1,(i + 1)))) + k = IC (Comput (P2,s2,(i + 1)))
hence (IC (Comput (P1,s1,(i + 1)))) + k = IC (Comput (P2,s2,(i + 1))) by A9, A99, A101; :: thesis: verum
end;
end;
end;
hence ( (IC (Comput (P1,s1,(i + 1)))) + k = IC (Comput (P2,s2,(i + 1))) & IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1)))) ) by A19; :: thesis: ( (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) = (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1))) )
now :: thesis: for x being object st x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) holds
((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x
let x be object ; :: thesis: ( x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) implies ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x )
assume A103: x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) ; :: thesis: ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x
dom (DataPart p) c= Data-Locations by RELAT_1:58;
then reconsider d = x as Data-Location of R by A41, A103, SCMRING2:23;
( (Comput (P1,s1,(i + 1))) . d = (Comput (P1,s1,i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A39, A13, A98, A100, SCMRING2:16;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A41, A45, A103; :: thesis: verum
end;
then (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) c= (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) by A43, GRFUNC_1:2;
hence (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) = (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) by A41, A42, GRFUNC_1:3; :: thesis: DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1)))
now :: thesis: for x being object st x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) holds
(DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x
let x be object ; :: thesis: ( x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) implies (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x )
assume A104: x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) ; :: thesis: (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x
then reconsider d = x as Data-Location of R by A27, SCMRING2:23;
( (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = (Comput (P1,(s1 +* (DataPart s2)),i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A13, A53, A98, A100, SCMRING2:16;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A34, A104; :: thesis: verum
end;
then DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) c= DataPart (Comput (P2,s2,(i + 1))) by A28, GRFUNC_1:2;
hence DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1))) by A27, A29, GRFUNC_1:3; :: thesis: verum
end;
end;
end;
A105: DataPart p c= p by RELAT_1:59;
A106: DataPart (IncIC (p,k)) = DataPart p by MEMSTR_0:51;
A107: DataPart p c= IncIC (p,k) by A106, MEMSTR_0:12;
A108: (Comput (P1,s1,0)) | (dom (DataPart p)) = s1 | (dom (DataPart p))
.= DataPart p by A1, A105, GRFUNC_1:23, XBOOLE_1:1
.= s2 | (dom (DataPart p)) by A107, A2, GRFUNC_1:23, XBOOLE_1:1
.= (Comput (P2,s2,0)) | (dom (DataPart p)) ;
A109: DataPart (Comput (P1,(s1 +* (DataPart s2)),0)) = DataPart (s1 +* (DataPart s2))
.= DataPart s2 by PBOOLE:142
.= DataPart (Comput (P2,s2,0)) ;
A110: IC in dom (IncIC (p,k)) by MEMSTR_0:52;
A111: (IC (Comput (P1,s1,0))) + k = (IC s1) + k
.= (IC p) + k by A1, A3, GRFUNC_1:2
.= (IC p) + k
.= IC (IncIC (p,k)) by MEMSTR_0:53
.= IC s2 by A2, A110, GRFUNC_1:2
.= IC (Comput (P2,s2,0)) ;
A112: IC in dom (IncIC (p,k)) by MEMSTR_0:52;
A113: (IC p) + k in dom (Reloc (q,k)) by A6, COMPOS_1:46;
A114: P2 /. (IC s2) = P2 . (IC s2) by PBOOLE:143;
A115: CurInstr (P2,(Comput (P2,s2,0))) = P2 . (IC s2) by A114
.= P2 . (IC (IncIC (p,k))) by A2, A112, GRFUNC_1:2
.= P2 . ((IC p) + k) by MEMSTR_0:53
.= P2 . ((IC p) + k)
.= (Reloc (q,k)) . ((IC p) + k) by A113, A4, GRFUNC_1:2 ;
A116: q . (IC p) = P1 . (IC s1) by A5, A6, A4, GRFUNC_1:2;
A117: CurInstr (P1,s1) = q . (IC p) by A116, PBOOLE:143;
A118: IncAddr ((CurInstr (P1,(Comput (P1,s1,0)))),k) = IncAddr ((CurInstr (P1,s1)),k)
.= CurInstr (P2,(Comput (P2,s2,0))) by A115, A6, A117, COMPOS_1:35 ;
A119: S1[ 0 ] by A111, A118, A108, A109;
for n being Nat holds S1[n] from NAT_1:sch 2(A119, A8);
 hence for i being Nat holds
( (IC (Comput (P1,s1,i))) + k = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),k) = CurInstr (P2,(Comput (P2,s2,i))) & (Comput (P1,s1,i)) | (dom (DataPart p)) = (Comput (P2,s2,i)) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),i)) = DataPart (Comput (P2,s2,i)) ) ; :: thesis: verum