let INS be Instruction of SCM; :: according to AMISTD_5:def 2 :: thesis: INS is relocable
let j, k be Nat; :: according to AMISTD_5:def 1 :: thesis: for b₁ being set holds Exec ((IncAddr (INS,(j + k))),(IncIC (b₁,k))) = IncIC ((Exec ((IncAddr (INS,j)),b₁)),k)
reconsider k = k as Element of NAT by ORDINAL1:def 12;
let s be State of SCM; :: thesis: Exec ((IncAddr (INS,(j + k))),(IncIC (s,k))) = IncIC ((Exec ((IncAddr (INS,j)),s)),k)
A1: IC (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) = (IC (Exec ((IncAddr (INS,j)),s))) + k by MEMSTR_0:53
.= IC (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) by AMISTD_2:def 3 ;
not not InsCode INS = 0 & ... & not InsCode INS = 8 by AMI_5:5;

per cases then ( InsCode INS = 0 or InsCode INS = 1 or InsCode INS = 2 or InsCode INS = 3 or InsCode INS = 4 or InsCode INS = 5 or InsCode INS = 6 or InsCode INS = 7 or InsCode INS = 8 ) ;

suppose InsCode INS = 0 ; :: thesis: Exec ((IncAddr (INS,(j + k))),(IncIC (s,k))) = IncIC ((Exec ((IncAddr (INS,j)),s)),k)

then A2: INS = halt SCM by AMI_5:7;
Exec ((IncAddr (INS,(j + k))),(IncIC (s,k))) = Exec (INS,(IncIC (s,k))) by A2, COMPOS_0:4
.= IncIC (s,k) by A2, EXTPRO_1:def 3
.= IncIC ((Exec ((IncAddr (INS,j)),s)),k) by A2, EXTPRO_1:def 3 ;
hence Exec ((IncAddr (INS,(j + k))),(IncIC (s,k))) = IncIC ((Exec ((IncAddr (INS,j)),s)),k) ; :: thesis: verum

end;

suppose InsCode INS = 1 ; :: thesis: Exec ((IncAddr (INS,(j + k))),(IncIC (s,k))) = IncIC ((Exec ((IncAddr (INS,j)),s)),k)

then consider da, db being Data-Location such that
A3: INS = da := db by AMI_5:8;

now :: thesis: for d being Data-Location holds (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d

let d be Data-Location; :: thesis: (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . b₁ = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . b₁

per cases ( da = d or da <> d ) ;

suppose A4: da = d ; :: thesis: (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . b₁ = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . b₁

thus (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (Exec (INS,(IncIC (s,k)))) . d by A3, COMPOS_0:4
.= (IncIC (s,k)) . db by A3, A4, AMI_3:2
.= s . db by AMI_5:16
.= (Exec (INS,s)) . d by A3, A4, AMI_3:2
.= (Exec ((IncAddr (INS,j)),s)) . d by A3, COMPOS_0:4
.= (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d by AMI_5:16 ; :: thesis: verum

end;

suppose A5: da <> d ; :: thesis: (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . b₁ = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . b₁

thus (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (Exec (INS,(IncIC (s,k)))) . d by A3, COMPOS_0:4
.= (IncIC (s,k)) . d by A3, A5, AMI_3:2
.= s . d by AMI_5:16
.= (Exec (INS,s)) . d by A3, A5, AMI_3:2
.= (Exec ((IncAddr (INS,j)),s)) . d by A3, COMPOS_0:4
.= (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d by AMI_5:16 ; :: thesis: verum

end;

end;

end;

hence Exec ((IncAddr (INS,(j + k))),(IncIC (s,k))) = IncIC ((Exec ((IncAddr (INS,j)),s)),k) by A1, AMI_5:25; :: thesis: verum

end;

suppose InsCode INS = 2 ; :: thesis: Exec ((IncAddr (INS,(j + k))),(IncIC (s,k))) = IncIC ((Exec ((IncAddr (INS,j)),s)),k)

then consider da, db being Data-Location such that
A6: INS = AddTo (da,db) by AMI_5:9;

now :: thesis: for d being Data-Location holds (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d

let d be Data-Location; :: thesis: (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . b₁ = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . b₁

per cases ( da = d or da <> d ) ;

suppose A7: da = d ; :: thesis: (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . b₁ = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . b₁

thus (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (Exec (INS,(IncIC (s,k)))) . d by A6, COMPOS_0:4
.= ((IncIC (s,k)) . da) + ((IncIC (s,k)) . db) by A7, A6, AMI_3:3
.= (s . da) + ((IncIC (s,k)) . db) by AMI_5:16
.= (s . da) + (s . db) by AMI_5:16
.= (Exec (INS,s)) . d by A6, A7, AMI_3:3
.= (Exec ((IncAddr (INS,j)),s)) . d by A6, COMPOS_0:4
.= (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d by AMI_5:16 ; :: thesis: verum

end;

suppose A8: da <> d ; :: thesis: (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . b₁ = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . b₁

thus (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (Exec (INS,(IncIC (s,k)))) . d by A6, COMPOS_0:4
.= (IncIC (s,k)) . d by A6, A8, AMI_3:3
.= s . d by AMI_5:16
.= (Exec (INS,s)) . d by A6, A8, AMI_3:3
.= (Exec ((IncAddr (INS,j)),s)) . d by A6, COMPOS_0:4
.= (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d by AMI_5:16 ; :: thesis: verum

end;

end;

end;

hence Exec ((IncAddr (INS,(j + k))),(IncIC (s,k))) = IncIC ((Exec ((IncAddr (INS,j)),s)),k) by A1, AMI_5:25; :: thesis: verum

end;

suppose InsCode INS = 3 ; :: thesis: Exec ((IncAddr (INS,(j + k))),(IncIC (s,k))) = IncIC ((Exec ((IncAddr (INS,j)),s)),k)

then consider da, db being Data-Location such that
A9: INS = SubFrom (da,db) by AMI_5:10;

now :: thesis: for d being Data-Location holds (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d

let d be Data-Location; :: thesis: (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . b₁ = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . b₁

per cases ( da = d or da <> d ) ;

suppose A10: da = d ; :: thesis: (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . b₁ = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . b₁

thus (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (Exec (INS,(IncIC (s,k)))) . d by A9, COMPOS_0:4
.= ((IncIC (s,k)) . da) - ((IncIC (s,k)) . db) by A10, A9, AMI_3:4
.= (s . da) - ((IncIC (s,k)) . db) by AMI_5:16
.= (s . da) - (s . db) by AMI_5:16
.= (Exec (INS,s)) . d by A9, A10, AMI_3:4
.= (Exec ((IncAddr (INS,j)),s)) . d by A9, COMPOS_0:4
.= (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d by AMI_5:16 ; :: thesis: verum

end;

suppose A11: da <> d ; :: thesis: (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . b₁ = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . b₁

thus (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (Exec (INS,(IncIC (s,k)))) . d by A9, COMPOS_0:4
.= (IncIC (s,k)) . d by A9, A11, AMI_3:4
.= s . d by AMI_5:16
.= (Exec (INS,s)) . d by A9, A11, AMI_3:4
.= (Exec ((IncAddr (INS,j)),s)) . d by A9, COMPOS_0:4
.= (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d by AMI_5:16 ; :: thesis: verum

end;

end;

end;

hence Exec ((IncAddr (INS,(j + k))),(IncIC (s,k))) = IncIC ((Exec ((IncAddr (INS,j)),s)),k) by A1, AMI_5:25; :: thesis: verum

end;

suppose InsCode INS = 4 ; :: thesis: Exec ((IncAddr (INS,(j + k))),(IncIC (s,k))) = IncIC ((Exec ((IncAddr (INS,j)),s)),k)

then consider da, db being Data-Location such that
A12: INS = MultBy (da,db) by AMI_5:11;

now :: thesis: for d being Data-Location holds (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d

let d be Data-Location; :: thesis: (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . b₁ = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . b₁

per cases ( da = d or da <> d ) ;

suppose A13: da = d ; :: thesis: (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . b₁ = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . b₁

thus (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (Exec (INS,(IncIC (s,k)))) . d by A12, COMPOS_0:4
.= ((IncIC (s,k)) . da) * ((IncIC (s,k)) . db) by A13, A12, AMI_3:5
.= (s . da) * ((IncIC (s,k)) . db) by AMI_5:16
.= (s . da) * (s . db) by AMI_5:16
.= (Exec (INS,s)) . d by A12, A13, AMI_3:5
.= (Exec ((IncAddr (INS,j)),s)) . d by A12, COMPOS_0:4
.= (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d by AMI_5:16 ; :: thesis: verum

end;

suppose A14: da <> d ; :: thesis: (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . b₁ = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . b₁

thus (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (Exec (INS,(IncIC (s,k)))) . d by A12, COMPOS_0:4
.= (IncIC (s,k)) . d by A12, A14, AMI_3:5
.= s . d by AMI_5:16
.= (Exec (INS,s)) . d by A12, A14, AMI_3:5
.= (Exec ((IncAddr (INS,j)),s)) . d by A12, COMPOS_0:4
.= (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d by AMI_5:16 ; :: thesis: verum

end;

end;

end;

hence Exec ((IncAddr (INS,(j + k))),(IncIC (s,k))) = IncIC ((Exec ((IncAddr (INS,j)),s)),k) by A1, AMI_5:25; :: thesis: verum

end;

suppose InsCode INS = 5 ; :: thesis: Exec ((IncAddr (INS,(j + k))),(IncIC (s,k))) = IncIC ((Exec ((IncAddr (INS,j)),s)),k)

then consider da, db being Data-Location such that
A15: INS = Divide (da,db) by AMI_5:12;

now :: thesis: for d being Data-Location holds (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d

let d be Data-Location; :: thesis: (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . b₁ = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . b₁

per cases ( da <> db or da = db ) ;

suppose A16: da <> db ; :: thesis: (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . b₁ = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . b₁

hereby :: thesis: verum

per cases ( da = d or db = d or ( da <> d & db <> d ) ) ;

suppose A17: da = d ; :: thesis: (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d

thus (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (Exec (INS,(IncIC (s,k)))) . d by A15, COMPOS_0:4
.= ((IncIC (s,k)) . da) div ((IncIC (s,k)) . db) by A16, A17, A15, AMI_3:6
.= (s . da) div ((IncIC (s,k)) . db) by AMI_5:16
.= (s . da) div (s . db) by AMI_5:16
.= (Exec (INS,s)) . d by A15, A16, A17, AMI_3:6
.= (Exec ((IncAddr (INS,j)),s)) . d by A15, COMPOS_0:4
.= (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d by AMI_5:16 ; :: thesis: verum

end;

suppose A18: db = d ; :: thesis: (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d

thus (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (Exec (INS,(IncIC (s,k)))) . d by A15, COMPOS_0:4
.= ((IncIC (s,k)) . da) mod ((IncIC (s,k)) . db) by A18, A15, AMI_3:6
.= (s . da) mod ((IncIC (s,k)) . db) by AMI_5:16
.= (s . da) mod (s . db) by AMI_5:16
.= (Exec (INS,s)) . d by A15, A18, AMI_3:6
.= (Exec ((IncAddr (INS,j)),s)) . d by A15, COMPOS_0:4
.= (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d by AMI_5:16 ; :: thesis: verum

end;

suppose A19: ( da <> d & db <> d ) ; :: thesis: (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d

thus (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (Exec (INS,(IncIC (s,k)))) . d by A15, COMPOS_0:4
.= (IncIC (s,k)) . d by A15, A19, AMI_3:6
.= s . d by AMI_5:16
.= (Exec (INS,s)) . d by A15, A19, AMI_3:6
.= (Exec ((IncAddr (INS,j)),s)) . d by A15, COMPOS_0:4
.= (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d by AMI_5:16 ; :: thesis: verum

end;

end;

end;

end;

suppose A20: da = db ; :: thesis: (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . b₁ = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . b₁

per cases ( da = d or da <> d ) ;

suppose A21: da = d ; :: thesis: (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . b₁ = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . b₁

thus (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (Exec (INS,(IncIC (s,k)))) . d by A15, COMPOS_0:4
.= ((IncIC (s,k)) . da) mod ((IncIC (s,k)) . db) by A20, A21, A15, AMI_3:6
.= (s . da) mod ((IncIC (s,k)) . db) by AMI_5:16
.= (s . da) mod (s . db) by AMI_5:16
.= (Exec (INS,s)) . d by A15, A20, A21, AMI_3:6
.= (Exec ((IncAddr (INS,j)),s)) . d by A15, COMPOS_0:4
.= (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d by AMI_5:16 ; :: thesis: verum

end;

suppose A22: da <> d ; :: thesis: (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . b₁ = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . b₁

thus (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (Exec (INS,(IncIC (s,k)))) . d by A15, COMPOS_0:4
.= (IncIC (s,k)) . d by A15, A20, A22, AMI_3:6
.= s . d by AMI_5:16
.= (Exec (INS,s)) . d by A15, A20, A22, AMI_3:6
.= (Exec ((IncAddr (INS,j)),s)) . d by A15, COMPOS_0:4
.= (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d by AMI_5:16 ; :: thesis: verum

end;

end;

end;

end;

end;

hence Exec ((IncAddr (INS,(j + k))),(IncIC (s,k))) = IncIC ((Exec ((IncAddr (INS,j)),s)),k) by A1, AMI_5:25; :: thesis: verum

end;

suppose InsCode INS = 6 ; :: thesis: Exec ((IncAddr (INS,(j + k))),(IncIC (s,k))) = IncIC ((Exec ((IncAddr (INS,j)),s)),k)

then consider loc being Nat such that
A23: INS = SCM-goto loc by AMI_5:13;
A24: IncAddr (INS,(j + k)) = SCM-goto (loc + (j + k)) by A23, Th1;
A25: IncAddr (INS,j) = SCM-goto (loc + j) by A23, Th1;

now :: thesis: for d being Data-Location holds (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d

let d be Data-Location; :: thesis: (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d
thus (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (IncIC (s,k)) . d by A24, AMI_3:7
.= s . d by AMI_5:16
.= (Exec ((IncAddr (INS,j)),s)) . d by A25, AMI_3:7
.= (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d by AMI_5:16 ; :: thesis: verum

end;

hence Exec ((IncAddr (INS,(j + k))),(IncIC (s,k))) = IncIC ((Exec ((IncAddr (INS,j)),s)),k) by A1, AMI_5:25; :: thesis: verum

end;

suppose InsCode INS = 7 ; :: thesis: Exec ((IncAddr (INS,(j + k))),(IncIC (s,k))) = IncIC ((Exec ((IncAddr (INS,j)),s)),k)

then consider loc being Nat, da being Data-Location such that
A26: INS = da =0_goto loc by AMI_5:14;
A27: IncAddr (INS,(j + k)) = da =0_goto (loc + (j + k)) by A26, Th2;
A28: IncAddr (INS,j) = da =0_goto (loc + j) by A26, Th2;

now :: thesis: for d being Data-Location holds (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d

let d be Data-Location; :: thesis: (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d
thus (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (IncIC (s,k)) . d by A27, AMI_3:8
.= s . d by AMI_5:16
.= (Exec ((IncAddr (INS,j)),s)) . d by A28, AMI_3:8
.= (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d by AMI_5:16 ; :: thesis: verum

end;

hence Exec ((IncAddr (INS,(j + k))),(IncIC (s,k))) = IncIC ((Exec ((IncAddr (INS,j)),s)),k) by A1, AMI_5:25; :: thesis: verum

end;

suppose InsCode INS = 8 ; :: thesis: Exec ((IncAddr (INS,(j + k))),(IncIC (s,k))) = IncIC ((Exec ((IncAddr (INS,j)),s)),k)

then consider loc being Nat, da being Data-Location such that
A29: INS = da >0_goto loc by AMI_5:15;
A30: IncAddr (INS,(j + k)) = da >0_goto (loc + (j + k)) by A29, Th3;
A31: IncAddr (INS,j) = da >0_goto (loc + j) by A29, Th3;

now :: thesis: for d being Data-Location holds (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d

let d be Data-Location; :: thesis: (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d
thus (Exec ((IncAddr (INS,(j + k))),(IncIC (s,k)))) . d = (IncIC (s,k)) . d by A30, AMI_3:9
.= s . d by AMI_5:16
.= (Exec ((IncAddr (INS,j)),s)) . d by A31, AMI_3:9
.= (IncIC ((Exec ((IncAddr (INS,j)),s)),k)) . d by AMI_5:16 ; :: thesis: verum

end;

hence Exec ((IncAddr (INS,(j + k))),(IncIC (s,k))) = IncIC ((Exec ((IncAddr (INS,j)),s)),k) by A1, AMI_5:25; :: thesis: verum

end;

end;