:: A Mathematical Model of CPU
::  by Yatsuka Nakamura and Andrzej Trybulec
::
:: Received October 14, 1992
:: Copyright (c) 1992 Association of Mizar Users

environ

 vocabularies BOOLE, FUNCT_2, FUNCT_1, RELAT_1, FUNCOP_1, CAT_1, FUNCT_4,
      CARD_3, TARSKI, FRAENKEL, PARTFUN1, FINSET_1, AMI_1, NAT_1, NEWTON,
      AMI_5, MCART_1, AMISTD_2, FUNCT_3, FINSEQ_1, FINSEQ_4, GRAPH_2, TREES_2,
      ORDINAL1;
 notations TARSKI, XBOOLE_0, ZFMISC_1, MCART_1, SUBSET_1, SETFAM_1, ORDINAL1,
      NUMBERS, CARD_3, XCMPLX_0, RELAT_1, FUNCT_1, PARTFUN1, FUNCT_2, BINOP_1,
      FINSET_1, NAT_1, FRAENKEL, FUNCOP_1, FUNCT_4, FINSEQ_1, FINSEQ_4,
      GRAPH_2, DOMAIN_1, TREES_1, TREES_2, STRUCT_0, XXREAL_0;
 constructors BINOP_1, SETFAM_1, PARTFUN1, DOMAIN_1, FUNCT_4, FRAENKEL,
      XXREAL_0, NAT_1, INT_1, CARD_3, CQC_LANG, STRUCT_0, FINSEQ_4, GRAPH_2,
      TREES_2;
 registrations XBOOLE_0, SUBSET_1, SETFAM_1, RELAT_1, FUNCT_1, ORDINAL1,
      FUNCT_2, FUNCOP_1, FUNCT_4, ARYTM_3, FINSET_1, FRAENKEL, XREAL_0,
      FINSEQ_1, CARD_3, STRUCT_0, ALTCAT_1, AFINSQ_1, TREES_2, TREES_1;
 requirements NUMERALS, BOOLE, SUBSET, ARITHM;


begin :: General concepts

definition let N be set;
 struct (1-sorted) AMI-Struct over N
  (# carrier -> set,
    Instruction-Counter -> Element of the carrier,
    Instruction-Locations -> Subset of the carrier,
    Instructions -> non empty set,
    Object-Kind ->
     Function of the carrier,
                 N \/ { the Instructions, the Instruction-Locations },
    Execution ->
     Function of the Instructions,
       Funcs(product the Object-Kind, product the Object-Kind)
  #);
end;

definition let N be set;
 canceled;

 func Trivial-AMI N -> strict AMI-Struct over N means
:: AMI_1:def 2

  the carrier of it = {0,1} &
  the Instruction-Counter of it = 0 &
  the Instruction-Locations of it = {1} &
  the Instructions of it = {[0,{}]} &
  the Object-Kind of it = (0,1) --> ({1},{[0,{}]}) &
  the Execution of it = [0,{}] .--> id product (0,1) --> ({1},{[0,{}]});
end;

definition let N be set; let S be AMI-Struct over N;
 attr S is void means
:: AMI_1:def 3
 the Instruction-Locations of S is empty;
end;

registration let N be set;
 cluster Trivial-AMI N -> non empty non void;
end;

registration let N be set;
 cluster non empty non void AMI-Struct over N;
end;

registration let N be set; let S be non void AMI-Struct over N;
 cluster the Instruction-Locations of S -> non empty;
end;

definition let N be set; let S be non empty AMI-Struct over N;
 mode Object of S is Element of S;
end;

definition let N be set; let S be non empty non void AMI-Struct over N;
 mode Instruction-Location of S -> Element of the Instruction-Locations of S
      means
:: AMI_1:def 4
  not contradiction;
end;

definition let N be set; let S be AMI-Struct over N;
 mode Instruction of S is Element of the Instructions of S;
end;

definition let N be set; let S be non empty AMI-Struct over N;
 func IC S -> Object of S equals
:: AMI_1:def 5
  the Instruction-Counter of S;
end;

definition let N be set; let S be non empty AMI-Struct over N;
 let o be Object of S;
 func ObjectKind o ->
  Element of N \/ { the Instructions of S, the Instruction-Locations of S }
   equals
:: AMI_1:def 6
  (the Object-Kind of S).o;
end;

definition let N be set; let S be AMI-Struct over N;
  mode State of S is Element of product the Object-Kind of S;
end;

definition let N be with_non-empty_elements set;
 let S be non void AMI-Struct over N;
 let I be Instruction of S, s be State of S;
 func Exec(I,s) -> State of S equals
:: AMI_1:def 7
    ((the Execution of S).I).s;
end;

reserve N for with_non-empty_elements set;

definition let N; let S be non void AMI-Struct over N;
           let I be Instruction of S;
 attr I is halting means
:: AMI_1:def 8
  for s being State of S holds Exec(I,s) = s;
end;

definition let N; let S be non void AMI-Struct over N;
 attr S is halting means
:: AMI_1:def 9
  ex I being Instruction of S st I is halting;
end;

reserve E for set;

canceled 5;

theorem :: AMI_1:6
 Trivial-AMI N is halting;

registration let N;
 cluster Trivial-AMI N -> halting;
end;

registration let N;
 cluster halting (non void AMI-Struct over N);
end;

registration let N; let S be halting (non void AMI-Struct over N);
 cluster halting Instruction of S;
end;

definition let N; let S be halting (non void AMI-Struct over N);
 func halt S -> Instruction of S equals
:: AMI_1:def 10
  choose { I where I is Instruction of S: I is halting };
end;

registration let N; let S be halting (non void AMI-Struct over N);
 cluster halt S -> halting;
end;

definition let N be set; let IT be non empty AMI-Struct over N;
 attr IT is IC-Ins-separated means
:: AMI_1:def 11
  ObjectKind IC IT = the Instruction-Locations of IT;
end;

definition let N be with_non-empty_elements set;
 let IT be non empty non void AMI-Struct over N;
canceled;
 attr IT is steady-programmed means
:: AMI_1:def 13
  for s being State of IT, i being Instruction of IT,
      l being Instruction-Location of IT
   holds Exec(i,s).l = s.l;
end;

definition let N be set; let IT be non empty non void AMI-Struct over N;
 attr IT is definite means
:: AMI_1:def 14
  for l being Instruction-Location of IT holds
   ObjectKind l = the Instructions of IT;
end;

theorem :: AMI_1:7
 Trivial-AMI E is IC-Ins-separated;

canceled;

theorem :: AMI_1:9
 for s1, s2 being State of Trivial-AMI E holds s1=s2;

theorem :: AMI_1:10
 Trivial-AMI N is steady-programmed;

theorem :: AMI_1:11
 Trivial-AMI E is definite;

registration let E be set;
  cluster Trivial-AMI E -> IC-Ins-separated definite;
end;

registration let N be with_non-empty_elements set;
  cluster Trivial-AMI N -> steady-programmed;
end;

registration let E be set;
 cluster strict AMI-Struct over E;
end;

registration let M be set;
 cluster IC-Ins-separated definite strict
              (non empty non void AMI-Struct over M);
end;

registration let N;
 cluster IC-Ins-separated halting steady-programmed definite
         strict (non empty non void AMI-Struct over N);
end;

definition let N be with_non-empty_elements set;
  let S be IC-Ins-separated (non empty non void AMI-Struct over N);
 let s be State of S;
 func IC s -> Instruction-Location of S equals
:: AMI_1:def 15
  s.IC S;
end;

begin :: Preliminaries

reserve x,y,z,A,B for set,
        f,g,h for Function,
        i,j,k for Element of NAT;

canceled 2;

theorem :: AMI_1:14
 f tolerates g & [x,y] in f & [x,z] in g implies y = z;

theorem :: AMI_1:15
 (for x st x in A holds x is Function) &
 (for f,g being Function st f in A & g in A holds f tolerates g)
  implies union A is Function;

definition
 canceled;
end;

canceled 9;

theorem :: AMI_1:25
 g in sproduct f implies dom g c= dom f &
   for x st x in dom g holds g.x in f.x;

theorem :: AMI_1:26
 {} in sproduct f;

theorem :: AMI_1:27
 product f c= sproduct f;

theorem :: AMI_1:28
 x in sproduct f implies x is PartFunc of dom f, union rng f;

theorem :: AMI_1:29
 g in product f & h in sproduct f implies g +* h in product f;

theorem :: AMI_1:30
 product f <> {} implies
 (g in sproduct f iff ex h st h in product f & g <= h);

theorem :: AMI_1:31
 sproduct f c= PFuncs(dom f,union rng f);

theorem :: AMI_1:32
 f c= g implies sproduct f c= sproduct g;

theorem :: AMI_1:33
  sproduct {} = {{}};

theorem :: AMI_1:34
     PFuncs(A,B) = sproduct (A --> B);

theorem :: AMI_1:35
    for A, B being non empty set
 for f being Function of A,B
  holds sproduct f = sproduct(f|{x where x is Element of A: f.x <> {} });

theorem :: AMI_1:36
 x in dom f & y in f.x implies x .--> y in sproduct f;

theorem :: AMI_1:37
    sproduct f = {{}} iff for x st x in dom f holds f.x = {};

theorem :: AMI_1:38
 A c= sproduct f &
 (for h1,h2 being Function st h1 in A & h2 in A holds h1 tolerates h2)
 implies union A in sproduct f;

theorem :: AMI_1:39
    g tolerates h & g in sproduct f & h in sproduct f
  implies g \/ h in sproduct f;

theorem :: AMI_1:40
 g c= h & h in sproduct f implies g in sproduct f;

theorem :: AMI_1:41
 g in sproduct f implies g|A in sproduct f;

theorem :: AMI_1:42
 g in sproduct f implies g|A in sproduct f|A;

theorem :: AMI_1:43
    h in sproduct(f+*g) implies
  ex f',g' being Function st f' in sproduct f & g' in sproduct g & h = f'+*g';

theorem :: AMI_1:44
 for f',g' being Function st dom g misses dom f' \ dom g' &
  f' in sproduct f & g' in sproduct g
 holds f'+*g' in sproduct(f+*g);

theorem :: AMI_1:45
    for f',g' being Function st dom f' misses dom g \ dom g' &
  f' in sproduct f & g' in sproduct g
 holds f'+*g' in sproduct(f+*g);

theorem :: AMI_1:46
 g in sproduct f & h in sproduct f
  implies g +* h in sproduct f;

theorem :: AMI_1:47
    for x1,x2,y1,y2 being set holds
  x1 in dom f & y1 in f.x1 & x2 in dom f & y2 in f.x2
   implies (x1,x2)-->(y1,y2) in sproduct f;

begin :: General theory

definition let N;
 let S be IC-Ins-separated definite (non empty non void AMI-Struct over N);
 let s be State of S;
 func CurInstr s -> Instruction of S equals
:: AMI_1:def 17
 s.IC s;
end;

definition let N;
 let S be IC-Ins-separated definite (non empty non void AMI-Struct over N);
 let s be State of S;
 func Following s -> State of S equals
:: AMI_1:def 18
 Exec(CurInstr s,s);
end;

definition let N;
 let S be IC-Ins-separated definite (non empty non void AMI-Struct over N);
 let s be State of S;
 func Computation s -> Function of NAT, product the Object-Kind of S
  means
:: AMI_1:def 19

  it.0 = s &
  for i holds it.(i+1) = Following(it.i);
end;

definition let N; let S be non void AMI-Struct over N;
 let f be Function of NAT, product the Object-Kind of S;
 let k;
 redefine func f.k -> State of S;
end;

definition let N;
 let S be halting IC-Ins-separated definite
       (non empty non void AMI-Struct over N);
 let IT be State of S;
 attr IT is halting means
:: AMI_1:def 20
 ex k st CurInstr((Computation IT).k) = halt S;
end;

definition let N be set; let IT be AMI-Struct over N;
 attr IT is realistic means
:: AMI_1:def 21

  not the Instruction-Counter of IT in the Instruction-Locations of IT;
end;

theorem :: AMI_1:48
 for S being IC-Ins-separated definite (non empty non void AMI-Struct over E)
  st S is realistic holds
  not ex l being Instruction-Location of S st IC S = l;

 reserve
   S for IC-Ins-separated definite (non empty non void AMI-Struct over N),
   s for State of S;

canceled 2;

theorem :: AMI_1:51
 for k holds
 (Computation s).(i+k) = (Computation (Computation s).i).k;

theorem :: AMI_1:52
 i <= j implies
  for N for S being
   halting IC-Ins-separated definite (non empty non void AMI-Struct over N)
  for s being State of S st CurInstr((Computation s).i) = halt S
   holds (Computation s).j = (Computation s).i;

definition let N;
 let S be halting IC-Ins-separated definite
     (non empty non void AMI-Struct over N);
 let s be State of S such that
 s is halting;
 func Result s -> State of S means
:: AMI_1:def 22
  ex k st it = (Computation s).k & CurInstr(it) = halt S;
end;

theorem :: AMI_1:53
    for S being steady-programmed
  IC-Ins-separated definite (non empty non void AMI-Struct over N)
 for s being State of S, i be Instruction-Location of S
  holds s.i = (Following s).i;

definition let N; let S be definite (non empty non void AMI-Struct over N);
 let s be State of S, l be Instruction-Location of S;
 redefine func s.l -> Instruction of S;
end;

theorem :: AMI_1:54
 for S being steady-programmed IC-Ins-separated definite
   (non empty non void AMI-Struct over N)
 for s being State of S, i be Instruction-Location of S, k
  holds s.i = (Computation s).k.i;

theorem :: AMI_1:55
    for S being steady-programmed IC-Ins-separated definite
    (non empty non void AMI-Struct over N)
 for s being State of S
  holds (Computation s).(k+1)
    = Exec(s.(IC (Computation s).k),(Computation s).k);

theorem :: AMI_1:56
 for S being steady-programmed IC-Ins-separated
 halting definite (non empty non void AMI-Struct over N)
 for s being State of S, k st s.IC (Computation s).k = halt S
  holds Result s = (Computation s).k;

theorem :: AMI_1:57
    for S being steady-programmed IC-Ins-separated
 halting definite (non empty non void AMI-Struct over N)
 for s being State of S st
   ex k st s.IC (Computation s).k = halt S
 for i holds Result s = Result (Computation s).i;

registration let N;
  let S be non empty non void AMI-Struct over N, o be Object of S;
  cluster ObjectKind o -> non empty;
end;

begin :: Finite substates

definition let N be set; let S be AMI-Struct over N;
 func FinPartSt S -> Subset of sproduct the Object-Kind of S equals
:: AMI_1:def 23
     { p where p is Element of sproduct the Object-Kind of S: p is finite };
end;

registration let N be set; let S be AMI-Struct over N;
 cluster finite Element of sproduct the Object-Kind of S;
end;

registration let N be set; let S be AMI-Struct over N;
 cluster FinPartSt S -> non empty functional;
end;

definition let N be set; let S be AMI-Struct over N;
 mode FinPartState of S is Element of FinPartSt S;
end;

registration let N be set; let S be AMI-Struct over N;
 cluster -> finite FinPartState of S;
end;

definition let N;
 canceled;
 let S be IC-Ins-separated definite (non empty non void AMI-Struct over N);
 let IT be FinPartState of S;
 attr IT is autonomic means
:: AMI_1:def 25
  for s1,s2 being State of S st IT c= s1 & IT c= s2
   for i holds (Computation s1).i|dom IT = (Computation s2).i|dom IT;
end;

definition let N;
 let S be halting IC-Ins-separated definite
     (non empty non void AMI-Struct over N);
 let IT be FinPartState of S;
 attr IT is halting means
:: AMI_1:def 26
  for s being State of S st IT c= s holds s is halting;
end;

definition let N;
 let IT be IC-Ins-separated definite (non empty non void AMI-Struct over N);
 attr IT is programmable means
:: AMI_1:def 27
 ex s being FinPartState of IT st s is non empty autonomic;
end;

theorem :: AMI_1:58
 for S being IC-Ins-separated definite (non empty non void AMI-Struct over N)
 for A,B being set, la,lb being Object of S st
  ObjectKind la = A & ObjectKind lb = B
 for a being Element of A, b being Element of B holds
  (la,lb) --> (a,b) is FinPartState of S;

theorem :: AMI_1:59
 for S being IC-Ins-separated definite (non empty non void AMI-Struct over N)
 for A being set, la being Object of S st ObjectKind la = A
 for a being Element of A holds la .--> a is FinPartState of S;

definition let N;
 let S be IC-Ins-separated definite (non empty non void AMI-Struct over N);
 let la be Object of S;
 let a be Element of ObjectKind la;
 redefine func la .--> a -> FinPartState of S;
end;

definition let N;
 let S be IC-Ins-separated definite (non empty non void AMI-Struct over N);
 let la,lb be Object of S;
 let a be Element of ObjectKind la,
     b be Element of ObjectKind lb;
 redefine func (la,lb) --> (a,b) -> FinPartState of S;
end;

theorem :: AMI_1:60
 Trivial-AMI E is realistic;

theorem :: AMI_1:61
 Trivial-AMI N is programmable;

registration let E;
  cluster Trivial-AMI E -> realistic;
end;

registration let N;
  cluster Trivial-AMI N -> programmable;
end;

registration let E;
 cluster realistic strict AMI-Struct over E;
end;

registration let M be set;
 cluster realistic strict IC-Ins-separated
 definite (non empty non void AMI-Struct over M);
end;

registration let N;
 cluster halting steady-programmed realistic programmable
         strict (IC-Ins-separated definite
         (non empty non void AMI-Struct over N));
end;

theorem :: AMI_1:62
 for S being non void AMI-Struct over N,
     s being State of S, p being FinPartState of S
  holds s|dom p is FinPartState of S;

theorem :: AMI_1:63
for N being set for S being AMI-Struct over N holds {} is FinPartState of S;

registration let N;
 let S be programmable
  (IC-Ins-separated definite (non empty non void AMI-Struct over N));
 cluster non empty autonomic FinPartState of S;
end;

definition let N be set;
 let S be AMI-Struct over N;
 let f,g be FinPartState of S;
 redefine func f +* g -> FinPartState of S;
end;

begin :: Preprograms

theorem :: AMI_1:64
for S being halting realistic IC-Ins-separated definite
  (non empty non void AMI-Struct over N)
for loc being Instruction-Location of S
for l being Element of ObjectKind IC S st l = loc
for h being Element of ObjectKind loc st h = halt S
for s being State of S st (IC S,loc) --> (l, h) c= s
 holds CurInstr s = halt S;

theorem :: AMI_1:65
for S being halting realistic IC-Ins-separated definite
   (non empty non void AMI-Struct over N)
for loc being Instruction-Location of S
for l being Element of ObjectKind IC S st l = loc
for h being Element of ObjectKind loc st h = halt S
 holds (IC S,loc) --> (l, h) is halting;

theorem :: AMI_1:66
for S being realistic halting IC-Ins-separated definite
 (non empty non void AMI-Struct over N)
for loc being Instruction-Location of S
for l being Element of ObjectKind IC S st l = loc
for h being Element of ObjectKind loc st h = halt S
for s being State of S st (IC S,loc) --> (l, h) c= s
for i holds (Computation s).i = s;

theorem :: AMI_1:67
for S be realistic halting IC-Ins-separated definite
   (non empty non void AMI-Struct over N)
for loc being Instruction-Location of S
for l being Element of ObjectKind IC S st l = loc
for h being Element of ObjectKind loc st h = halt S
 holds (IC S,loc) --> (l, h) is autonomic;

registration let N;
 let S be realistic halting IC-Ins-separated definite
  (non empty non void AMI-Struct over N);
 cluster autonomic halting FinPartState of S;
end;

definition let N;
 let S be realistic halting IC-Ins-separated definite
  (non empty non void AMI-Struct over N);
 mode pre-program of S is autonomic halting FinPartState of S;
end;

definition let N;
 let S be realistic halting IC-Ins-separated definite
  (non empty non void AMI-Struct over N);
 let s be FinPartState of S;
 assume
 s is pre-program of S;
 func Result(s) -> FinPartState of S means
:: AMI_1:def 28
    for s' being State of S st s c= s' holds it = (Result s')|dom s;
end;

begin :: Computability

definition let N;
 let S be realistic halting IC-Ins-separated definite
  (non empty non void AMI-Struct over N);
 let p be FinPartState of S, F be Function;
  pred p computes F means
:: AMI_1:def 29

  for x being set st x in dom F ex s being FinPartState of S st x = s &
   p +* s is pre-program of S & F.s c= Result(p +* s);
 end;

notation let N;
 let S be realistic halting IC-Ins-separated definite
  (non empty non void AMI-Struct over N);
 let p be FinPartState of S, F be Function;
  antonym p does_not_compute F for p computes F;
end;

theorem :: AMI_1:68
 for S being realistic halting IC-Ins-separated definite
    (non empty non void AMI-Struct over N)
 for p being FinPartState of S
  holds p computes {};

theorem :: AMI_1:69
 for S being realistic halting IC-Ins-separated definite
  (non empty non void AMI-Struct over N)
 for p being FinPartState of S
  holds p is pre-program of S iff p computes {} .--> Result(p);

theorem :: AMI_1:70
 for S being realistic halting IC-Ins-separated definite
  (non empty non void AMI-Struct over N)
 for p being FinPartState of S
  holds p is pre-program of S iff p computes {} .--> {};

definition let N;
 let S be realistic halting IC-Ins-separated definite
    (non empty non void AMI-Struct over N);
 let IT be PartFunc of FinPartSt S, FinPartSt S;
 attr IT is computable means
:: AMI_1:def 30
 ex p being FinPartState of S st p computes IT;
end;

theorem :: AMI_1:71
 for S being realistic halting IC-Ins-separated definite
  (non empty non void AMI-Struct over N)
 for F being PartFunc of FinPartSt S, FinPartSt S st F = {}
  holds F is computable;

theorem :: AMI_1:72
 for S being realistic halting IC-Ins-separated definite
  (non empty non void AMI-Struct over N)
 for F being PartFunc of FinPartSt S, FinPartSt S st F = {} .--> {}
  holds F is computable;

theorem :: AMI_1:73
 for S being realistic halting IC-Ins-separated definite
  (non empty non void AMI-Struct over N)
 for p being pre-program of S
 for F being PartFunc of FinPartSt S, FinPartSt S st F = {} .--> Result(p)
  holds F is computable;

definition let N;
 let S be realistic halting IC-Ins-separated definite
  (non empty non void AMI-Struct over N);
 let F be PartFunc of FinPartSt S, FinPartSt S such that
 F is computable;
  mode Program of F -> FinPartState of S means
:: AMI_1:def 31
it computes F;
end;

theorem :: AMI_1:74
    for S being realistic halting IC-Ins-separated definite
  (non empty non void AMI-Struct over N)
 for F being PartFunc of FinPartSt S, FinPartSt S st F = {}
 for p being FinPartState of S
  holds p is Program of F;

theorem :: AMI_1:75
    for S being realistic halting IC-Ins-separated definite
  (non empty non void AMI-Struct over N)
 for F being PartFunc of FinPartSt S, FinPartSt S st F = {} .--> {}
 for p being pre-program of S holds p is Program of F;

theorem :: AMI_1:76
    for S being realistic halting IC-Ins-separated definite
  (non empty non void AMI-Struct over N)
 for p being pre-program of S
 for F being PartFunc of FinPartSt S, FinPartSt S st F = {} .--> Result p
  holds p is Program of F;

begin :: InsType & InsCode

notation let x;
 synonym InsCode x for x`1;
 synonym AddressPart x for x`2;
end;

definition
 let N be set, S be AMI-Struct over N;
 attr S is standard-ins means
:: AMI_1:def 32
 the Instructions of S c= [: NAT, ((union N) \/ the carrier of S)* :];
end;

registration let N be set;
 cluster Trivial-AMI N -> standard-ins;
end;

registration let N be set;
 cluster standard-ins non empty non void AMI-Struct over N;
end;

registration
 let N be set, S be standard-ins AMI-Struct over N;
 cluster the Instructions of S -> Relation-like;
end;

definition
 let N be set, S be standard-ins AMI-Struct over N;
 func InsCodes S equals
:: AMI_1:def 33
  dom the Instructions of S;
end;

definition
 let N be set, S be standard-ins AMI-Struct over N;
 mode InsType of S is Element of InsCodes S;
end;

definition
 let N be set, S be standard-ins AMI-Struct over N;
 let I be Element of the Instructions of S;
 redefine func InsCode I -> InsType of S;
end;

theorem :: AMI_1:77
 for N being set, S being AMI-Struct over N
 for x being finite Element of sproduct the Object-Kind of S
         holds x in FinPartSt S;

begin :: On the instruction locations

definition let N be set, S be AMI-Struct over N;
 mode IL-FinSequence of S -> FinSequence of the Instruction-Locations of S
  means
:: AMI_1:def 34
 not contradiction;
end;

reserve N for set, S for non empty non void AMI-Struct over N;

definition let N,S;
 let f be IL-FinSequence of S; let x be set;
 func f/.x -> Instruction-Location of S equals
:: AMI_1:def 35
   f/.x;
end;

definition let N,S; let l1 be Instruction-Location of S;
 redefine func <*l1*> -> IL-FinSequence of S;
 let l2 be Instruction-Location of S;
 redefine func <*l1,l2*> -> IL-FinSequence of S;
end;

registration let N,S;
 cluster non empty IL-FinSequence of S;
end;

definition let N,S;
 let f1,f2 be IL-FinSequence of S;
 redefine func f1^'f2 -> IL-FinSequence of S;
end;

definition let D be set;
 let N, S;
 mode IL-Function of D,S -> Function of D, the Instruction-Locations of S means
:: AMI_1:def 36
  not contradiction;
end;

definition let D be non empty set;
 let N,S;
 let f be IL-Function of D,S, d be Element of D;
 redefine func f.d -> Instruction-Location of S;
end;

definition let N,S;
 mode IL-DecoratedTree of S -> DecoratedTree of the Instruction-Locations of S
      means
:: AMI_1:def 37
  not contradiction;
end;

definition let N,S; let T be IL-DecoratedTree of S;
 let x be set such that
 x in dom T;
 func T.x -> Instruction-Location of S equals
:: AMI_1:def 38
  T.x;
end;

scheme :: AMI_1:sch 1
 ILFraenkelFin {N() -> set,
      S() -> non empty non void AMI-Struct over N(),
      X() -> set, F(set) -> set }:
 { F(w) where w is Instruction-Location of S(): w in X() } is finite
provided
  X() is finite;

scheme :: AMI_1:sch 2
 {N,D()-> set, S()-> non empty non void AMI-Struct over N(),
        F(set) -> set,
        T() -> Instruction-Location of S(),
        P[set,set]}:
  { F(i) where i is Element of D():
    ex l being Instruction-Location of S() st l = T() & P[i,l] }
    = { F(j) where j is Element of D(): P[j,T()] };