:: On the Composition of Macro Instructions of Standard Computers
:: by Artur Korni{\l}owicz
::
:: Received April 14, 2000
:: Copyright (c) 2000-2021 Association of Mizar Users

theorem :: AMISTD_2:1

for N being with_zero set
for I being Instruction of (STC N) holds JumpPart I = 0 ;

definition

let N be with_zero set ;
let S be non empty with_non-empty_values IC-Ins-separated AMI-Struct over N;
let I be Instruction of S;

attr I is with_explicit_jumps means :Def1: :: AMISTD_2:def 1
JUMP I = rng (JumpPart I);

end;

:: deftheorem Def1 defines with_explicit_jumps AMISTD_2:def 1 :
for N being with_zero set
for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N
for I being Instruction of S holds
( I is with_explicit_jumps iff JUMP I = rng (JumpPart I) );

definition

let N be with_zero set ;
let S be non empty with_non-empty_values IC-Ins-separated AMI-Struct over N;

attr S is with_explicit_jumps means :Def2: :: AMISTD_2:def 2
for I being Instruction of S holds I is with_explicit_jumps ;

end;

:: deftheorem Def2 defines with_explicit_jumps AMISTD_2:def 2 :
for N being with_zero set
for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N holds
( S is with_explicit_jumps iff for I being Instruction of S holds I is with_explicit_jumps );

registration

let N be with_zero set ;

cluster non empty with_non-empty_values IC-Ins-separated standard for AMI-Struct over N;

existence
ex b₁ being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N st b₁ is standard

proof end;

end;

theorem Th2: :: AMISTD_2:2

for N being with_zero set
for S being non empty with_non-empty_values IC-Ins-separated standard AMI-Struct over N
for I being Instruction of S st ( for f being Element of NAT holds NIC (I,f) = {(f + 1)} ) holds
JUMP I is empty

proof end;

registration

let N be with_zero set ;
let I be Instruction of (STC N);

cluster JUMP I -> empty ;

coherence
JUMP I is empty

proof end;

end;

theorem :: AMISTD_2:3

for N being with_zero set
for T being InsType of the InstructionsF of (STC N) holds JumpParts T = {0}

proof end;

Lm1: for N being with_zero set
for I being Instruction of (Trivial-AMI N) holds JumpPart I = 0

proof end;

Lm2: for N being with_zero set
for T being InsType of the InstructionsF of (Trivial-AMI N) holds JumpParts T = {0}

proof end;

registration

let N be with_zero set ;

cluster STC N -> with_explicit_jumps ;

coherence
STC N is with_explicit_jumps

proof end;

end;

registration

let N be with_zero set ;

cluster non empty with_non-empty_values IC-Ins-separated halting standard with_explicit_jumps for AMI-Struct over N;

existence
ex b₁ being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N st
( b₁ is standard & b₁ is halting & b₁ is with_explicit_jumps )

proof end;

end;

registration

let N be with_zero set ;
let I be Instruction of (Trivial-AMI N);

cluster JUMP I -> empty ;

coherence
JUMP I is empty

proof end;

end;

registration

let N be with_zero set ;

cluster Trivial-AMI N -> with_explicit_jumps ;

coherence
Trivial-AMI N is with_explicit_jumps

proof end;

end;

registration

let N be with_zero set ;

cluster non empty with_non-empty_values IC-Ins-separated halting with_explicit_jumps for AMI-Struct over N;

existence
ex b₁ being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N st
( b₁ is with_explicit_jumps & b₁ is halting )

proof end;

end;

registration

let N be with_zero set ;
let S be non empty with_non-empty_values IC-Ins-separated with_explicit_jumps AMI-Struct over N;

cluster -> with_explicit_jumps for Element of the InstructionsF of S;

coherence
for b₁ being Instruction of S holds b₁ is with_explicit_jumps by Def2;

end;

theorem Th4: :: AMISTD_2:4

for N being with_zero set
for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N
for I being Instruction of S st I is halting holds
JUMP I is empty

proof end;

registration

let N be with_zero set ;
let S be non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N;
let I be halting Instruction of S;

cluster JUMP I -> empty ;

coherence
JUMP I is empty by Th4;

end;

theorem :: AMISTD_2:5

for N being with_zero set
for S being non empty with_non-empty_values IC-Ins-separated halting with_explicit_jumps AMI-Struct over N
for I being Instruction of S st I is ins-loc-free holds
JUMP I is empty

proof end;

registration

let N be with_zero set ;
let S be non empty with_non-empty_values IC-Ins-separated with_explicit_jumps AMI-Struct over N;

cluster halting -> ins-loc-free for Element of the InstructionsF of S;

coherence
for b₁ being Instruction of S st b₁ is halting holds
b₁ is ins-loc-free

proof end;

end;

registration

let N be with_zero set ;
let S be non empty with_non-empty_values IC-Ins-separated with_explicit_jumps AMI-Struct over N;

cluster sequential -> ins-loc-free for Element of the InstructionsF of S;

coherence
for b₁ being Instruction of S st b₁ is sequential holds
b₁ is ins-loc-free

proof end;

end;

registration

let N be with_zero set ;
let S be non empty with_non-empty_values IC-Ins-separated halting with_explicit_jumps AMI-Struct over N;
let I be halting Instruction of S;
let k be Nat;

cluster IncAddr (I,k) -> halting ;

coherence
IncAddr (I,k) is halting by COMPOS_0:4;

end;

theorem :: AMISTD_2:6

for k being Nat
for N being with_zero set
for S being non empty with_non-empty_values IC-Ins-separated halting standard with_explicit_jumps AMI-Struct over N
for I being Instruction of S st I is sequential holds
IncAddr (I,k) is sequential by COMPOS_0:4;

definition

let N be with_zero set ;
let S be non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N;
let I be Instruction of S;

attr I is IC-relocable means :Def3: :: AMISTD_2:def 3
for j, k being Nat
for s being State of S holds (IC (Exec ((IncAddr (I,j)),s))) + k = IC (Exec ((IncAddr (I,(j + k))),(IncIC (s,k))));

end;

:: deftheorem Def3 defines IC-relocable AMISTD_2:def 3 :
for N being with_zero set
for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N
for I being Instruction of S holds
( I is IC-relocable iff for j, k being Nat
for s being State of S holds (IC (Exec ((IncAddr (I,j)),s))) + k = IC (Exec ((IncAddr (I,(j + k))),(IncIC (s,k)))) );

definition

let N be with_zero set ;
let S be non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N;

attr S is IC-relocable means :Def4: :: AMISTD_2:def 4
for I being Instruction of S holds I is IC-relocable ;

end;

:: deftheorem Def4 defines IC-relocable AMISTD_2:def 4 :
for N being with_zero set
for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N holds
( S is IC-relocable iff for I being Instruction of S holds I is IC-relocable );

registration

let N be with_zero set ;
let S be non empty with_non-empty_values IC-Ins-separated halting with_explicit_jumps AMI-Struct over N;

cluster sequential -> IC-relocable for Element of the InstructionsF of S;

coherence
for b₁ being Instruction of S st b₁ is sequential holds
b₁ is IC-relocable

proof end;

end;

registration

let N be with_zero set ;
let S be non empty with_non-empty_values IC-Ins-separated halting with_explicit_jumps AMI-Struct over N;

cluster halting -> IC-relocable for Element of the InstructionsF of S;

coherence
for b₁ being Instruction of S st b₁ is halting holds
b₁ is IC-relocable

proof end;

end;

registration

let N be with_zero set ;

cluster STC N -> IC-relocable ;

coherence
STC N is IC-relocable

proof end;

end;

registration

let N be with_zero set ;

cluster non empty with_non-empty_values IC-Ins-separated halting standard with_explicit_jumps for AMI-Struct over N;

existence
ex b₁ being non empty with_non-empty_values IC-Ins-separated standard AMI-Struct over N st
( b₁ is halting & b₁ is with_explicit_jumps )

proof end;

end;

registration

let N be with_zero set ;

cluster non empty with_non-empty_values IC-Ins-separated halting standard with_explicit_jumps IC-relocable for AMI-Struct over N;

existence
ex b₁ being non empty with_non-empty_values IC-Ins-separated halting standard with_explicit_jumps AMI-Struct over N st b₁ is IC-relocable

proof end;

end;

registration

let N be with_zero set ;
let S be non empty with_non-empty_values IC-Ins-separated halting IC-relocable AMI-Struct over N;

cluster -> IC-relocable for Element of the InstructionsF of S;

coherence
for b₁ being Instruction of S holds b₁ is IC-relocable by Def4;

end;

registration

let N be with_zero set ;
let S be non empty with_non-empty_values IC-Ins-separated halting with_explicit_jumps AMI-Struct over N;

cluster with_explicit_jumps IC-relocable for Element of the InstructionsF of S;

existence
ex b₁ being Instruction of S st b₁ is IC-relocable

proof end;

end;

theorem Th7: :: AMISTD_2:7

for N being with_zero set
for S being non empty with_non-empty_values IC-Ins-separated halting with_explicit_jumps AMI-Struct over N
for I being IC-relocable Instruction of S
for k being Nat
for s being State of S holds (IC (Exec (I,s))) + k = IC (Exec ((IncAddr (I,k)),(IncIC (s,k))))

proof end;

registration

let N be with_zero set ;
let S be non empty with_non-empty_values IC-Ins-separated halting standard with_explicit_jumps IC-relocable AMI-Struct over N;
let F, G be really-closed Program of ;

cluster F ';' G -> really-closed ;

coherence
F ';' G is really-closed

proof end;

end;

theorem :: AMISTD_2:8

for N being with_zero set
for I being Instruction of (Trivial-AMI N) holds JumpPart I = 0 by Lm1;

theorem :: AMISTD_2:9

for N being with_zero set
for T being InsType of the InstructionsF of (Trivial-AMI N) holds JumpParts T = {0} by Lm2;

theorem :: AMISTD_2:10

for N being with_zero set
for n being Nat
for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N
for s being State of S
for I being Program of
for P1, P2 being Instruction-Sequence of S st I c= P1 & I c= P2 & ( for m being Nat st m < n holds
IC (Comput (P2,s,m)) in dom I ) holds
for m being Nat st m <= n holds
Comput (P1,s,m) = Comput (P2,s,m)

proof end;

theorem :: AMISTD_2:11

for N being with_zero set
for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N
for P being Instruction-Sequence of S
for s being State of S st s = Following (P,s) holds
for n being Nat holds Comput (P,s,n) = s

proof end;