:: 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;
 definitions TARSKI, XBOOLE_0, STRUCT_0, FUNCOP_1;
 theorems ZFMISC_1, FUNCT_2, TARSKI, NAT_1, CQC_LANG, CARD_3, CARD_5, FINSEQ_1,
      FUNCT_4, FUNCOP_1, FRAENKEL, FINSET_1, PARTFUN1, FUNCT_1, GRFUNC_1,
      RELAT_1, RELSET_1, XBOOLE_0, XBOOLE_1, ORDINAL1, MCART_1, TREES_1,
      XREAL_1, XXREAL_0;
 schemes NAT_1, RECDEF_1, FRAENKEL, XBOOLE_0, FUNCT_1;

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
:Def2:
  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,{}]});
 existence
  proof
    reconsider y = 0 as Element of {0,1}by TARSKI:def 2;
       0 in NAT & {} in (union N \/ {0,1})* by FINSEQ_1:66;
    then [0,{}] in [: NAT, (union N \/ {0,1})* :] by ZFMISC_1:106;
    then reconsider I = {[0,{}]} as non empty Subset of
        [: NAT, (union(N) \/ {0,1})* :] by ZFMISC_1:37;
    reconsider S = {1} as non empty Subset of {0,1} by ZFMISC_1:12;
    set f = (0,1) --> (S,I);
       rng f c= { I,S} & { I,S } c= N \/ {I, S} by FUNCT_4:65,XBOOLE_1:7;
    then dom f = {0,1} & rng f c= N \/ {I, S} by FUNCT_4:65,XBOOLE_1:1;
    then reconsider f as Function of {0,1}, N \/ {I, S} by FUNCT_2:def 1
,RELSET_1:11;
       id product f in Funcs(product f, product f) by FUNCT_2:12;
    then reconsider E = I --> id product f as
     Function of I,Funcs(product f, product f) by FUNCOP_1:57;
   take AMI-Struct(#{0,1},y,S,I,f,E #);
 thus thesis;
end;
 uniqueness;
end;

definition let N be set; let S be AMI-Struct over N;
 attr S is void means
:Def3: the Instruction-Locations of S is empty;
end;

registration let N be set;
 cluster Trivial-AMI N -> non empty non void;
 coherence
 proof
   thus the carrier of Trivial-AMI N is non empty by Def2;
   thus the Instruction-Locations of Trivial-AMI N is non empty by Def2;
 end;
end;

registration let N be set;
 cluster non empty non void AMI-Struct over N;
 existence
 proof
   take Trivial-AMI N;
   thus thesis;
 end;
end;

registration let N be set; let S be non void AMI-Struct over N;
 cluster the Instruction-Locations of S -> non empty;
 coherence by Def3;
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
:Def4:  not contradiction;
 existence;
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
  the Instruction-Counter of S;
 correctness;
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
  (the Object-Kind of S).o;
 correctness;
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
    ((the Execution of S).I).s;
 coherence
  proof
    consider f being Function such that
A1:   (the Execution of S).I = f &
      dom f = product the Object-Kind of S &
      rng f c= product the Object-Kind of S by FUNCT_2:def 2;
       (the Execution of S).I.s in rng f by A1,FUNCT_1:def 5;
    hence thesis by A1;
  end;
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
:Def8:  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
:Def9:  ex I being Instruction of S st I is halting;
end;

reserve E for set;

canceled 5;

theorem Th6:
 Trivial-AMI N is halting
  proof
   set T = Trivial-AMI N;
{[0,{}]} = the Instructions of T by Def2;
   then reconsider I = [0,{}] as Instruction of T by TARSKI:def 1;
   take I;
   thus I is halting
   proof
    let s be State of T;
A1: product the Object-Kind of T
      = product (0,1) --> ({1},{[0,{}]}) by Def2
     .= { (0,1) --> (1,[0,{}]) } by CARD_3:63;
    hence Exec(I,s) = (0,1) --> (1,[0,{}]) by TARSKI:def 1
     .= s by A1,TARSKI:def 1;
   end;
  end;

registration let N;
 cluster Trivial-AMI N -> halting;
coherence by Th6;
end;

registration let N;
 cluster halting (non void AMI-Struct over N);
existence
  proof
    take Trivial-AMI N;
    thus thesis;
  end;
end;

registration let N; let S be halting (non void AMI-Struct over N);
 cluster halting Instruction of S;
 existence by Def9;
end;

definition let N; let S be halting (non void AMI-Struct over N);
 func halt S -> Instruction of S equals
  choose { I where I is Instruction of S: I is halting };
 coherence
  proof set X = { I where I is Instruction of S: I is halting };
    consider I being Instruction of S such that
A1:   I is halting by Def9;
    I in X by A1;
    then choose X in X;
    then ex I being Instruction of S st choose X = I & I is halting;
   hence choose X is Instruction of S;
  end;
end;

registration let N; let S be halting (non void AMI-Struct over N);
 cluster halt S -> halting;
 coherence
  proof set X = { I where I is Instruction of S: I is halting };
    consider I being Instruction of S such that
A1:   I is halting by Def9;
    I in X by A1;
    then choose X in X;
    then ex I being Instruction of S st choose X = I & I is halting;
   hence thesis;
  end;
end;

definition let N be set; let IT be non empty AMI-Struct over N;
 attr IT is IC-Ins-separated means
:Def11:  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
:Def13:  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
:Def14:  for l being Instruction-Location of IT holds
   ObjectKind l = the Instructions of IT;
end;

theorem Th7:
 Trivial-AMI E is IC-Ins-separated
  proof
A1:  IC Trivial-AMI E = 0 by Def2;
   thus ObjectKind IC Trivial-AMI E
           = (0,1) --> ({1},{[0,{}]}).0 by A1,Def2
          .= {1} by FUNCT_4:66
          .= the Instruction-Locations of Trivial-AMI E by Def2;
  end;

canceled;

theorem Th9:
 for s1, s2 being State of Trivial-AMI E holds s1=s2
 proof let s1,s2 be State of Trivial-AMI E;
A1:  product the Object-Kind of Trivial-AMI E
   = product (0,1) --> ({1},{[0,{}]}) by Def2
  .= { (0,1) --> (1,[0,{}]) } by CARD_3:63;
  hence s1 = (0,1) --> (1,[0,{}]) by TARSKI:def 1
     .= s2 by A1,TARSKI:def 1;
 end;

theorem Th10:
 Trivial-AMI N is steady-programmed
  proof
   let s be State of Trivial-AMI N, i be Instruction of Trivial-AMI N,
       l be Instruction-Location of Trivial-AMI N;
   thus Exec(i,s).l = s.l by Th9;
  end;

theorem Th11:
 Trivial-AMI E is definite
  proof let l be Instruction-Location of Trivial-AMI E;
       l in the Instruction-Locations of Trivial-AMI E;
    then l in {1} by Def2;
then A1:    l = 1 by TARSKI:def 1;
   thus ObjectKind l
         = (0,1) --> ({1},{[0,{}]}).1 by A1,Def2
        .= {[0,{}]} by FUNCT_4:66
        .= the Instructions of Trivial-AMI E by Def2;
  end;

registration let E be set;
  cluster Trivial-AMI E -> IC-Ins-separated definite;
  coherence by Th7,Th11;
end;

registration let N be with_non-empty_elements set;
  cluster Trivial-AMI N -> steady-programmed;
  coherence by Th10;
end;

registration let E be set;
 cluster strict AMI-Struct over E;
 existence
  proof
    take Trivial-AMI E;
    thus thesis;
  end;
end;

registration let M be set;
 cluster IC-Ins-separated definite strict
              (non empty non void AMI-Struct over M);
 existence
  proof
    take Trivial-AMI M;
    thus thesis;
  end;
end;

registration let N;
 cluster IC-Ins-separated halting steady-programmed definite
         strict (non empty non void AMI-Struct over N);
 existence
  proof
    take Trivial-AMI N;
    thus thesis;
  end;
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
  s.IC S;
 coherence
  proof
       dom the Object-Kind of S = the carrier of S by FUNCT_2:def 1;
    then pi(product the Object-Kind of S,IC S) = ObjectKind IC S by CARD_3:22
      .= the Instruction-Locations of S by Def11;
    then s.IC S in the Instruction-Locations of S by CARD_3:def 6;
   hence thesis by Def4;
  end;
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
 f tolerates g & [x,y] in f & [x,z] in g implies y = z by CARD_3:65;

theorem
 (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 by CARD_3:66;

definition
 canceled;
end;

canceled 9;

theorem
 g in sproduct f implies dom g c= dom f &
   for x st x in dom g holds g.x in f.x by  CARD_3:67;

theorem
 {} in sproduct f by CARD_3:68;

theorem
 product f c= sproduct f by CARD_3:69;

theorem
 x in sproduct f implies x is PartFunc of dom f, union rng f by CARD_3:70;

theorem
 g in product f & h in sproduct f implies g +* h in product f by CARD_3:71;

theorem
 product f <> {} implies
 (g in sproduct f iff ex h st h in product f & g <= h) by CARD_3:72;

theorem
 sproduct f c= PFuncs(dom f,union rng f) by CARD_3:73;

theorem
 f c= g implies sproduct f c= sproduct g by CARD_3:74;

theorem
  sproduct {} = {{}} by CARD_3:75;

theorem
     PFuncs(A,B) = sproduct (A --> B) by CARD_3:76;

theorem
    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 <> {} })
                               by CARD_3:77;

theorem
 x in dom f & y in f.x implies x .--> y in sproduct f by CARD_3:78;

theorem
    sproduct f = {{}} iff for x st x in dom f holds f.x = {} by CARD_3:79;

theorem
 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 by CARD_3:80;

theorem
    g tolerates h & g in sproduct f & h in sproduct f
  implies g \/ h in sproduct f by CARD_3:81;

theorem
 g c= h & h in sproduct f implies g in sproduct f by CARD_3:82;

theorem
 g in sproduct f implies g|A in sproduct f by CARD_3:83;

theorem
 g in sproduct f implies g|A in sproduct f|A by CARD_3:84;

theorem
    h in sproduct(f+*g) implies
  ex f',g' being Function st f' in sproduct f & g' in sproduct g & h = f'+*g'
                        by CARD_3:85;

theorem
 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) by CARD_3:86;

theorem
    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) by CARD_3:87;

theorem
 g in sproduct f & h in sproduct f
  implies g +* h in sproduct f by CARD_3:88;

theorem
    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 by CARD_3:89;

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
 s.IC s;
 coherence
  proof
     dom the Object-Kind of S = the carrier of S by FUNCT_2:def 1;
    then pi(product the Object-Kind of S,IC s) = ObjectKind IC s by CARD_3:22
      .= the Instructions of S by Def14;
    hence thesis by CARD_3:def 6;
  end;
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
 Exec(CurInstr s,s);
 correctness;
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
:Def19:
  it.0 = s &
  for i holds it.(i+1) = Following(it.i);
 existence
   proof
     deffunc F(set, Element of product the Object-Kind of S) = Following $2;
     consider f being Function of NAT, product the Object-Kind of S
      such that
A1:    f.0 = s & for n being Element of NAT holds f.(n+1) = F(n,f.n)
          from RECDEF_1:sch 4;
      take f;
      thus thesis by A1;
   end;
 uniqueness
  proof let F1,F2 be Function of NAT, product the Object-Kind of S such that
A2: F1.0 = s &
   for i holds F1.(i+1) = Following(F1.i) and
A3: F2.0 = s &
   for i holds F2.(i+1) = Following(F2.i);
   deffunc F(set, Element of product the Object-Kind of S) = Following $2;
A4: F1.0 = s & for i holds F1.(i+1) = F(i,F1.i) by A2;
A5: F2.0 = s & for i holds F2.(i+1) = F(i,F2.i) by A3;
   thus F1 = F2 from RECDEF_1:sch 12(A4,A5);
  end;
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;
coherence by FUNCT_2:7;
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
:Def20: 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
:Def21:
  not the Instruction-Counter of IT in the Instruction-Locations of IT;
end;

theorem
 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
proof
 let S be IC-Ins-separated definite (non empty non void AMI-Struct over E);
 assume
Z: S is realistic;
 let l be Instruction-Location of S;
 assume IC S = l;
   then IC S in the Instruction-Locations of S;
  hence contradiction by Z,Def21;
end;

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

canceled 2;

theorem Th51:
 for k holds
 (Computation s).(i+k) = (Computation (Computation s).i).k
 proof
    defpred P[Element of NAT] means
     (Computation s).(i+$1) = (Computation (Computation s).i).$1;
A1: P[0] by Def19;
A2: now let k; assume
A3:  P[k];
      (Computation s).(i+(k+1)) = (Computation s).(i+k+1)
         .= Following (Computation s).(i+k) by Def19
         .= (Computation (Computation s).i).(k+1) by A3,Def19;
     hence P[k+1];
   end;
  thus for k holds P[k] from NAT_1:sch 1(A1,A2);
 end;

theorem Th52:
 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
 proof assume i <= j;
   then consider k being Nat such that
A1:   j = i + k by NAT_1:10;
      reconsider k as Element of NAT by ORDINAL1:def 13;
A2:   j = i + k by A1;
  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
A3: CurInstr((Computation s).i) = halt S;
    defpred P[Element of NAT] means
     (Computation s).(i+$1) = (Computation s).i;
A4: P[0];
A5: now let k; assume
A6:  P[k];
      (Computation s).(i+(k+1)) = (Computation s).(i+k+1)
         .= Following (Computation s).(i+k) by Def19
         .= (Computation s).i by  A3,A6,Def8;
     hence P[k+1];
   end;
      for k holds P[k] from NAT_1:sch 1(A4,A5);
  hence (Computation s).j = (Computation s).i by A2;
 end;

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
A1: s is halting;
 func Result s -> State of S means
:Def22:  ex k st it = (Computation s).k & CurInstr(it) = halt S;
 uniqueness
  proof let s1,s2 be State of S;
   given k1 being Element of NAT such that
A2: s1 = (Computation s).k1 & CurInstr(s1) = halt S;
   given k2 being Element of NAT such that
A3: s2 = (Computation s).k2 & CurInstr(s2) = halt S;
       k1 <= k2 or k2 <= k1;
   hence s1 = s2 by A2,A3,Th52;
  end;
 correctness
  proof
       ex k st CurInstr((Computation s).k) = halt S by A1,Def20;
   hence thesis;
  end;
end;

theorem
    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 by Def13;

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;
 coherence
  proof
     dom the Object-Kind of S = the carrier of S by FUNCT_2:def 1;
    then pi(product the Object-Kind of S,l) = ObjectKind l by CARD_3:22
      .= the Instructions of S by Def14;
   hence s.l is Instruction of S by CARD_3:def 6;
  end;
end;

theorem Th54:
 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
 proof
  let S be steady-programmed
    IC-Ins-separated definite (non empty non void AMI-Struct over N);
  let s be State of S, i be Instruction-Location of S;
    defpred P[Element of NAT] means s.i = (Computation s).$1.i;
A1: P[0] by Def19;
A2: now let k;
    assume P[k];
    then s.i = (Following (Computation s).k).i by Def13
          .= (Computation s).(k+1).i by Def19;
    hence P[k+1];
   end;
  thus for k holds P[k] from NAT_1:sch 1(A1,A2);
 end;

theorem
    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)
 proof
  let S be steady-programmed
   IC-Ins-separated definite (non empty non void AMI-Struct over N);
  let s be State of S;
  thus (Computation s).(k+1)
    = Following (Computation s).k by Def19
   .= Exec(s.(IC (Computation s).k),(Computation s).k) by Th54;
 end;

theorem Th56:
 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
proof
 let S be steady-programmed IC-Ins-separated
 halting definite (non empty non void AMI-Struct over N);
 let s be State of S, k such that
A1:  s.IC (Computation s).k = halt S;
A2:CurInstr((Computation s).k)
      = halt S by A1,Th54;
  then s is halting by Def20;
 hence Result s = (Computation s).k by A2,Def22;
end;

theorem
    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
 proof
  let S be steady-programmed IC-Ins-separated
 halting definite (non empty non void AMI-Struct over N);
  let s be State of S;
  given k such that
A1:   s.IC (Computation s).k = halt S;
   set s' = (Computation s).k;
A2: CurInstr s' = halt S by A1,Th54;
  let i;
      now per cases;
    suppose i <= k;
      then consider j being Nat such that
A3:     k = i + j by NAT_1:10;
      reconsider j as Element of NAT by ORDINAL1:def 13;
A4:    (Computation s).k = (Computation (Computation s).i).j by A3,Th51;
      then A5:    (Computation s).i is halting by A2,Def20;
     thus Result s = s' by A1,Th56
       .= Result (Computation s).i by A2,A4,A5,Def22;
    end;
    suppose i >= k;
then A6:     (Computation s).i = s' by A2,Th52;
A7:     (Computation (Computation s).k).0 = (Computation s).k by Def19;
      then A8:    (Computation s).i is halting by A2,A6,Def20;
     thus Result s = s' by A1,Th56
      .= Result (Computation s).i by A2,A6,A7,A8,Def22;
    end;
    end;
  hence Result s = Result (Computation s).i;
 end;

registration let N;
  let S be non empty non void AMI-Struct over N, o be Object of S;
  cluster ObjectKind o -> non empty;
coherence;
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
     { p where p is Element of sproduct the Object-Kind of S: p is finite };
   :: Fin sproduct the Object-Kind of S !!!
 coherence
  proof defpred P[set] means $1 is finite;
       { p where p is Element of sproduct the Object-Kind of S: P[p] } c=
    sproduct the Object-Kind of S from FRAENKEL:sch 10;
   hence thesis;
  end;
end;

registration let N be set; let S be AMI-Struct over N;
 cluster finite Element of sproduct the Object-Kind of S;
 existence
  proof
    {} in sproduct the Object-Kind of S & {} is finite by CARD_3:68;
   hence thesis;
  end;
end;

Lem: 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;

registration let N be set; let S be AMI-Struct over N;
 cluster FinPartSt S -> non empty functional;
 coherence
  proof
    {} in sproduct the Object-Kind of S by CARD_3:68;
    then {} in FinPartSt S;
   hence FinPartSt S is non empty;
   thus FinPartSt S is functional;
  end;
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;
 coherence
  proof let q be FinPartState of S;
    q in FinPartSt S;
    then
    ex p being Element of sproduct the Object-Kind of S st q = p & p is finite;
   hence thesis;
  end;
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
:Def25:  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
:Def26:  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
:Def27: ex s being FinPartState of IT st s is non empty autonomic;
end;

theorem Th58:
 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
proof
 let S be IC-Ins-separated definite (non empty non void AMI-Struct over N);
 let A,B be set, la,lb be Object of S such that
A1: ObjectKind la = A & ObjectKind lb = B;
 let a be Element of A, b be Element of B;
  set p = (la,lb) --> (a,b);
A2:dom p = {la,lb} by FUNCT_4:65;
A3: dom the Object-Kind of S = the carrier of S by FUNCT_2:def 1;
     now let x be set such that
A4:   x in dom p;
       now per cases by A2,A4,TARSKI:def 2;
     suppose
A5:    la <> lb & x = la;
       then p.x = a by FUNCT_4:66;
      hence p.x in (the Object-Kind of S).x by A5,A1;
     end;
     suppose
A6:    la <> lb & x = lb;
       then p.x = b by FUNCT_4:66;
      hence p.x in (the Object-Kind of S).x by A6,A1;
     end;
     suppose
A7:    la = lb & x = la;
       then p = la .--> b by CQC_LANG:44;
       then p.x = b by A7,CQC_LANG:6;
      hence p.x in (the Object-Kind of S).x by A7,A1;
    end;
    end;
   hence p.x in (the Object-Kind of S).x;
  end;
  then reconsider p as Element of sproduct the Object-Kind of S
                       by A2,A3,CARD_3:def 9;
  dom p = {la,lb} by FUNCT_4:65;
  then p is finite by FINSET_1:29;
 hence thesis by Lem;
end;

theorem Th59:
 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
proof let S be IC-Ins-separated definite
 (non empty non void AMI-Struct over N);
 let A be set, la be Object of S such that
A1: ObjectKind la = A;
 let a be Element of A;
  set p = la .--> a;
A2:dom p = {la} by CQC_LANG:5;
A3:dom the Object-Kind of S = the carrier of S by FUNCT_2:def 1;
     now let x be set;
   assume x in dom p;
    then A4:   x = la by A2,TARSKI:def 1;
    then p.x = a by CQC_LANG:6;
   hence p.x in (the Object-Kind of S).x by A4,A1;
  end;
 then reconsider p as Element of sproduct the Object-Kind of S
       by A2,A3,CARD_3:def 9;
     p is FinPartState of S by Lem;
 hence thesis;
end;

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;
 coherence by Th59;
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;
 coherence by Th58;
end;

theorem Th60:
 Trivial-AMI E is realistic
 proof
A1: the Instruction-Counter of Trivial-AMI E = 0 &
    the Instruction-Locations of Trivial-AMI E = {1} by Def2;
  assume the Instruction-Counter of Trivial-AMI E
      in the Instruction-Locations of Trivial-AMI E;
  hence thesis by TARSKI:def 1,A1;
 end;

theorem Th61:
 Trivial-AMI N is programmable
proof
  reconsider la = 0 as Object of Trivial-AMI N by Def2;
     ObjectKind la = ((0,1) --> ({1},{[0,{}]})).0 by Def2
    .= {1} by FUNCT_4:66;
  then reconsider a = 1 as Element of ObjectKind la by TARSKI:def 1;
 take la .--> a;
 thus la .--> a is non empty;
 let s1,s2 be State of Trivial-AMI N such that
     la .--> a c= s1 & la .--> a c= s2;
 let i;
 thus (Computation s1).i|dom(la .--> a) = (Computation s2).i|dom(la .--> a)
  by Th9;
end;

registration let E;
  cluster Trivial-AMI E -> realistic;
  coherence by Th60;
end;

registration let N;
  cluster Trivial-AMI N -> programmable;
  coherence by Th61;
end;

registration let E;
 cluster realistic strict AMI-Struct over E;
 existence
  proof
   take Trivial-AMI E;
   thus thesis;
  end;
end;

registration let M be set;
 cluster realistic strict IC-Ins-separated
 definite (non empty non void AMI-Struct over M);
 existence
  proof
   take Trivial-AMI M;
   thus thesis;
  end;
end;

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

theorem Th62:
 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
proof let S be non void AMI-Struct over N,
    s be State of S, p be FinPartState of S;
A1:  product the Object-Kind of S c= sproduct the Object-Kind of S
  & s in product the Object-Kind of S by CARD_3:69;
  dom(s|dom p) = dom s /\ dom p by RELAT_1:90;
  then s|dom p is finite Element of sproduct the Object-Kind of S
     by A1,FINSET_1:29,CARD_3:83;
 hence s|dom p is FinPartState of S by Lem;
end;

theorem
for N being set for S being AMI-Struct over N holds {} is FinPartState of S
proof let N be set, S be AMI-Struct over N;
  {} is finite Element of sproduct the Object-Kind of S by CARD_3:68;
 hence thesis by Lem;
end;

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;
 existence by Def27;
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;
 coherence
  proof
    f +* g is Element of sproduct the Object-Kind of S by CARD_3:88;
   hence thesis by Lem;
  end;
end;

begin :: Preprograms

theorem Th64:
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
proof
 let S be halting realistic
  IC-Ins-separated definite (non empty non void AMI-Struct over N);
 let loc be Instruction-Location of S;
 let l be Element of ObjectKind IC S such that
A1: l = loc;
 let h be Element of ObjectKind loc such that
A2: h = halt S;
   let s be State of S such that
A3: (IC S,loc) --> (l, h) c= s;
    loc in the Instruction-Locations of S;
    then
A4:  IC S <> loc by Def21;
       dom((IC S,loc) --> (l, h)) = {IC S,loc} by FUNCT_4:65;
then A5: IC S in dom((IC S,loc) --> (l, h)) & loc in dom((IC S,loc) --> (l, h))
     by TARSKI:def 2;
then A6: ((IC S,loc) --> (l, h)).IC S in dom((IC S,loc) --> (l, h))
        by A1,A4,FUNCT_4:66;
   thus CurInstr s = s.(((IC S,loc) --> (l, h)).IC S) by A3,A5,GRFUNC_1:8
     .= ((IC S,loc) --> (l, h)).(((IC S,loc) --> (l, h)).IC S)
           by A3,A6,GRFUNC_1:8
     .= ((IC S,loc) --> (l, h)).loc by A1,A4,FUNCT_4:66
     .= halt S by A2,A4,FUNCT_4:66;
end;

theorem Th65:
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
proof
 let S be halting realistic
  IC-Ins-separated definite (non empty non void AMI-Struct over N);
 let loc be Instruction-Location of S;
 let l be Element of ObjectKind IC S such that
A1: l = loc;
 let h be Element of ObjectKind loc such that
A2: h = halt S;
 thus (IC S,loc) --> (l, h) is halting
  proof let s be State of S such that
A3: (IC S,loc) --> (l, h) c= s;
   take 0;
   thus CurInstr((Computation s).0)
      = CurInstr s by Def19
     .= halt S by A1,A2,A3,Th64;
  end;
end;

theorem Th66:
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
proof
 let S be realistic halting
  IC-Ins-separated definite (non empty non void AMI-Struct over N);
 let loc be Instruction-Location of S;
 let l be Element of ObjectKind IC S such that
A1: l = loc;
 let h be Element of ObjectKind loc such that
A2: h = halt S;
   let s be State of S such that
A3: (IC S,loc) --> (l, h) c= s;
    defpred P[Element of NAT] means (Computation s).$1 = s;
A4: P[0] by Def19;
A5: now let i;
    assume
A6:  P[i];
      (Computation s).(i+1) = Following (Computation s).i by Def19
     .= Exec(halt S,s) by  A6,A1,A2,A3,Th64
     .= s by Def8;
    hence P[i+1];
   end;
 thus for i holds P[i] from NAT_1:sch 1(A4,A5);
end;

theorem Th67:
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
proof
 let S be realistic halting IC-Ins-separated definite
  (non empty non void AMI-Struct over N);
 let loc be Instruction-Location of S;
 let l be Element of ObjectKind IC S such that
A1: l = loc;
 let h be Element of ObjectKind loc such that
A2: h = halt S;
 thus (IC S,loc) --> (l, h) is autonomic
  proof let s1,s2 be State of S;
   assume
A3:    (IC S,loc) --> (l, h) c= s1 & (IC S,loc) --> (l, h) c= s2;
    then A4:   s1|dom((IC S,loc) --> (l, h)) = (IC S,loc) --> (l, h) &
     s2|dom((IC S,loc) --> (l, h)) = (IC S,loc) --> (l, h) by GRFUNC_1:64;
   let i;
       (Computation s1).i = s1 & (Computation s2).i = s2 by A1,A2,A3,Th66;
   hence thesis by A4;
  end;
end;

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;
 existence
  proof
    consider loc being Instruction-Location of S;
    reconsider l = loc as Element of ObjectKind IC S by Def11;
    reconsider h = halt S as Element of ObjectKind loc by Def14;
       (IC S,loc) --> (l, h) is autonomic halting by Th65,Th67;
   hence thesis;
  end;
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
A1: s is pre-program of S;
 func Result(s) -> FinPartState of S means
    for s' being State of S st s c= s' holds it = (Result s')|dom s;
 existence
  proof
    consider h being Function such that
A2:  h in product the Object-Kind of S and
A3:  s <= h by CARD_3:72;
    reconsider h as State of S by A2;
    reconsider R = (Result h)|dom s as FinPartState of S by Th62;
   take R;
   let s' be State of S such that
A4:  s c= s';
       h is halting by A1,A3,Def26;
    then consider k1 being Element of NAT such that
A5:  Result h = (Computation h).k1 & CurInstr(Result h) = halt S by Def22;
       s' is halting by A1,A4,Def26;
    then consider k2 being Element of NAT such that
A6:  Result s' = (Computation s').k2 & CurInstr(Result s') = halt S by Def22;
       now per cases;
     suppose k1 <= k2;
       then Result h = (Computation h).k2 by A5,Th52;
      hence R = (Result s')|dom s by A1,A3,A4,A6,Def25;
     end;
     suppose k1 >= k2;
       then Result s' = (Computation s').k1 by A6,Th52;
      hence R = (Result s')|dom s by A1,A3,A4,A5,Def25;
    end;
    end;
   hence R = (Result s')|dom s;
  end;
 correctness
  proof let p1,p2 be FinPartState of S such that
A7: for s' being State of S st s c= s' holds p1 = (Result s')|dom s and
A8: for s' being State of S st s c= s' holds p2 = (Result s')|dom s;
    consider h being Function such that
A9:  h in product the Object-Kind of S and
A10:  s <= h by CARD_3:72;
    reconsider h as State of S by A9;
   thus p1 = (Result h)|dom s by A7,A10 .= p2 by A8,A10;
  end;
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
:Def29:
  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 Th68:
 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 {}
  proof
   let S be realistic halting IC-Ins-separated definite
    (non empty non void AMI-Struct over N);
   let p be FinPartState of S;
   let x be set;
   assume
A1: x in dom {};
    then reconsider x as FinPartState of S;
   take x;
   thus thesis by A1;
  end;

theorem Th69:
 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)
  proof
   let S be realistic halting IC-Ins-separated definite
     (non empty non void AMI-Struct over N);
   let p be FinPartState of S;
   thus p is pre-program of S implies p computes {} .--> Result(p)
    proof assume
A1:   p is pre-program of S;
     let x be set such that
A2:    x in dom({} .--> Result(p));
         dom({} .--> Result(p)) = {{}} by CQC_LANG:5;
then A3:    x = {} by A2,TARSKI:def 1;
      then x is Element of sproduct the Object-Kind of S by CARD_3:68;
      then reconsider s = x as FinPartState of S by A3,Lem;
     take s;
     thus x = s;
     thus p +* s is pre-program of S by A1,A3,FUNCT_4:22;
         ({} .--> Result(p)).s = Result(p) by A3,CQC_LANG:6;
     hence ({} .--> Result(p)).s c= Result(p +* s) by A3,FUNCT_4:22;
    end;
      dom({} .--> Result(p)) = {{}} by CQC_LANG:5;
then A4: {} in dom({} .--> Result(p)) by TARSKI:def 1;
   assume p computes {} .--> Result(p);
    then consider s being FinPartState of S such that
A5:  s = {} and
A6:  p +* s is pre-program of S and
        ({} .--> Result(p)).s c= Result(p +* s) by A4,Def29;
   thus thesis by A5,A6,FUNCT_4:22;
  end;

theorem Th70:
 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 {} .--> {}
  proof
   let S be realistic halting IC-Ins-separated definite
    (non empty non void AMI-Struct over N);
   let p be FinPartState of S;
   thus p is pre-program of S implies p computes {} .--> {}
    proof assume
A1:   p is pre-program of S;
     let x be set such that
A2:    x in dom({} .--> {});
         dom({} .--> {}) = {{}} by CQC_LANG:5;
then A3:    x = {} by A2,TARSKI:def 1;
      then x is Element of sproduct the Object-Kind of S by CARD_3:68;
      then reconsider s = x as FinPartState of S by A3,Lem;
     take s;
     thus x = s;
     thus p +* s is pre-program of S by A1,A3,FUNCT_4:22;
         ({} .--> {}).s = {} by A3,CQC_LANG:6;
     hence ({} .--> {}).s c= Result(p +* s) by XBOOLE_1:2;
    end;
      dom({} .--> {}) = {{}} by CQC_LANG:5;
then A4: {} in dom({} .--> {}) by TARSKI:def 1;
   assume p computes {} .--> {};
    then consider s being FinPartState of S such that
A5:  s = {} and
A6:  p +* s is pre-program of S and
        ({} .--> {}).s c= Result(p +* s) by A4,Def29;
   thus thesis by A5,A6,FUNCT_4:22;
  end;

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
:Def30: ex p being FinPartState of S st p computes IT;
end;

theorem Th71:
 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
proof
 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;
  consider p being FinPartState of S;
 assume
A1: F = {};
 take p;
 thus thesis by A1,Th68;
end;

theorem Th72:
 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
proof
 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;
  consider p being pre-program of S;
 assume
A1: F = {} .--> {};
 take p;
 thus thesis by A1,Th70;
end;

theorem Th73:
 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
proof
 let S be realistic halting IC-Ins-separated definite
  (non empty non void AMI-Struct over N);
 let p be pre-program of S;
 let F be PartFunc of FinPartSt S, FinPartSt S;
 assume
A1: F = {} .--> Result(p);
 take p;
 thus thesis by A1,Th69;
end;

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
A1: F is computable;
  mode Program of F -> FinPartState of S means  it computes F;
  existence by A1,Def30;
end;

theorem
    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
proof
 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
A1: F = {};
 let p be FinPartState of S;
 thus F is computable by A1,Th71;
 thus p computes F by A1,Th68;
end;

theorem
    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
proof
 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
A1: F = {} .--> {};
 let p be pre-program of S;
 thus F is computable by A1,Th72;
 thus p computes F by A1,Th70;
end;

theorem
    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
proof
 let S be realistic halting IC-Ins-separated definite
  (non empty non void AMI-Struct over N);
 let p be pre-program of S;
 let F be PartFunc of FinPartSt S, FinPartSt S; assume
A1: F = {} .--> Result p;
 hence F is computable by Th73;
 thus p computes F by A1,Th69;
end;

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
:Def32: the Instructions of S c= [: NAT, ((union N) \/ the carrier of S)* :];
end;

registration let N be set;
 cluster Trivial-AMI N -> standard-ins;
 coherence
  proof
    {} in ((union N) \/ the carrier of Trivial-AMI N)* by FINSEQ_1:66;
    then
A1:  {{}} c= ((union N) \/ the carrier of Trivial-AMI N)* by ZFMISC_1:37;
    the Instructions of Trivial-AMI N = {[0,{}]} by Def2
           .= [:{0},{{}}:] by ZFMISC_1:35;
   hence the Instructions of Trivial-AMI N
     c= [: NAT, ((union N) \/ the carrier of Trivial-AMI N)* :]
                  by A1,ZFMISC_1:119;
  end;
end;

registration let N be set;
 cluster standard-ins non empty non void AMI-Struct over N;
 existence proof take Trivial-AMI N; thus thesis; end;
end;

registration
 let N be set, S be standard-ins AMI-Struct over N;
 cluster the Instructions of S -> Relation-like;
 coherence
  proof
    the Instructions of S c= [: NAT, ((union N) \/ the carrier of S)* :]
                     by Def32;
    hence thesis by RELAT_1:3;
  end;
end;

definition
 let N be set, S be standard-ins AMI-Struct over N;
 func InsCodes S equals
  dom the Instructions of S;
 correctness;
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;
 coherence by MCART_1:91;
end;

theorem
 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
:Defx: not contradiction;
 existence;
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
   f/.x;
 coherence
  proof
   thus f/.x is Instruction-Location of S by Def4;
  end;
end;

definition let N,S; let l1 be Instruction-Location of S;
 redefine func <*l1*> -> IL-FinSequence of S;
 coherence
  proof <*l1*> is FinSequence of the Instruction-Locations of S;
   hence thesis by Defx;
  end;
 let l2 be Instruction-Location of S;
 redefine func <*l1,l2*> -> IL-FinSequence of S;
 coherence
  proof <*l1,l2*> is FinSequence of the Instruction-Locations of S;
   hence thesis by Defx;
  end;
end;

registration let N,S;
 cluster non empty IL-FinSequence of S;
 existence
  proof consider l being Instruction-Location of S;
   take a = <*l*>;
   thus thesis;
  end;
end;

definition let N,S;
 let f1,f2 be IL-FinSequence of S;
 redefine func f1^'f2 -> IL-FinSequence of S;
 coherence
  proof
    f1^'f2 is FinSequence of the Instruction-Locations of S;
   hence thesis by Defx;
  end;
end;

definition let D be set;
 let N, S;
 mode IL-Function of D,S -> Function of D, the Instruction-Locations of S means
  not contradiction;
 existence;
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;
 correctness
  proof
    f.d is Element of the Instruction-Locations of S;
   hence thesis by Def4;
  end;
end;

definition let N,S;
 mode IL-DecoratedTree of S -> DecoratedTree of the Instruction-Locations of S
      means
  not contradiction;
 existence;
end;

definition let N,S; let T be IL-DecoratedTree of S;
 let x be set such that
A: x in dom T;
 func T.x -> Instruction-Location of S equals
  T.x;
 coherence
  proof reconsider x as Element of dom T by A;
    T.x is Element of the Instruction-Locations of S;
   hence thesis by Def4;
  end;
end;

scheme 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
 A1: X() is finite
proof
  set M = { F(w) where w is Instruction-Location of S(): w in X() };
  consider f being Function such that
A2: dom f = X() /\ the Instruction-Locations of S() and
A3: for y being set st y in X() /\ the Instruction-Locations of S()
      holds f.y = F(y) from FUNCT_1:sch 3;
     M = f.:X()
   proof
    thus M c= f.:X()
     proof let x be set;
      assume x in M;
       then consider u being Instruction-Location of S()
       such that
A4:     x = F(u) and
A5:     u in X();
A6:    u in dom f by A2,A5,XBOOLE_0:def 3;
       then f.u = F(u) by A2,A3;
      hence x in f.:X() by A4,A5,A6,FUNCT_1:def 12;
     end;
    let x be set;
    assume x in f.:X();
     then consider y being set such that
A7:   y in dom f and
A8:   y in X() and
A9:   x = f.y by FUNCT_1:def 12;
     y is Element of the Instruction-Locations of S() by A2,A7,XBOOLE_0:def 3;
     then reconsider y as Instruction-Location of S() by Def4;
        x = F(y) by A2,A3,A7,A9;
    hence x in M by A8;
   end;
 hence M is finite by A1,FINSET_1:17;
end;

scheme {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()] }
proof
 set X = { F(i) where i is Element of D():
    ex l being Instruction-Location of S() st l = T() & P[i,l] },
     Y = { F(j) where j is Element of D(): P[j,T()] };
 thus X c= Y
  proof let x be set;
   assume x in X;
    then ex i being Element of D() st x = F(i) &
       ex l being Instruction-Location of S() st l = T() & P[i,l];
   hence x in Y;
  end;
 let x be set;
 assume x in Y;
  then ex j being Element of D() st x = F(j) & P[j,T()];
 hence x in X;
end;