Article ALGSEQ_1, MML version 4.99.1005

:: ALGSEQ_1:th 10
theorem
for b1, b2 being natural set holds
   b1 in Segm b2
iff
   b1 < b2;

:: ALGSEQ_1:th 11
theorem
Segm 0 = {} & Segm 1 = {0} & Segm 2 = {0,1};

:: ALGSEQ_1:th 12
theorem
for b1 being natural set holds
   b1 in Segm (b1 + 1);

:: ALGSEQ_1:th 13
theorem
for b1, b2 being natural set holds
   b1 <= b2
iff
   Segm b1 c= Segm b2;

:: ALGSEQ_1:th 14
theorem
for b1, b2 being natural set
      st Segm b1 = Segm b2
   holds b1 = b2;

:: ALGSEQ_1:th 15
theorem
for b1, b2 being natural set
      st b1 <= b2
   holds Segm b1 = (Segm b1) /\ Segm b2 & Segm b1 = (Segm b2) /\ Segm b1;

:: ALGSEQ_1:th 16
theorem
for b1, b2 being natural set
      st (Segm b1 = (Segm b1) /\ Segm b2 or Segm b1 = (Segm b2) /\ Segm b1)
   holds b1 <= b2;

:: ALGSEQ_1:attrnot 1 => ALGSEQ_1:attr 1
definition
  let a1 be non empty ZeroStr;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
  attr a2 is finite-Support means
    ex b1 being natural set st
       for b2 being natural set
             st b1 <= b2
          holds a2 . b2 = 0. a1;
end;

:: ALGSEQ_1:dfs 1
definiens
  let a1 be non empty ZeroStr;
  let a2 be Function-like quasi_total Relation of NAT,the carrier of a1;
To prove
     a2 is finite-Support
it is sufficient to prove
  thus ex b1 being natural set st
       for b2 being natural set
             st b1 <= b2
          holds a2 . b2 = 0. a1;

:: ALGSEQ_1:def 2
theorem
for b1 being non empty ZeroStr
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
      b2 is finite-Support(b1)
   iff
      ex b3 being natural set st
         for b4 being natural set
               st b3 <= b4
            holds b2 . b4 = 0. b1;

:: ALGSEQ_1:exreg 1
registration
  let a1 be non empty ZeroStr;
  cluster Relation-like Function-like quasi_total finite-Support Relation of NAT,the carrier of a1;
end;

:: ALGSEQ_1:modenot 1
definition
  let a1 be non empty ZeroStr;
  mode AlgSequence of a1 is Function-like quasi_total finite-Support Relation of NAT,the carrier of a1;
end;

:: ALGSEQ_1:prednot 1 => ALGSEQ_1:pred 1
definition
  let a1 be non empty ZeroStr;
  let a2 be Function-like quasi_total finite-Support Relation of NAT,the carrier of a1;
  let a3 be natural set;
  pred A3 is_at_least_length_of A2 means
    for b1 being natural set
          st a3 <= b1
       holds a2 . b1 = 0. a1;
end;

:: ALGSEQ_1:dfs 2
definiens
  let a1 be non empty ZeroStr;
  let a2 be Function-like quasi_total finite-Support Relation of NAT,the carrier of a1;
  let a3 be natural set;
To prove
     a3 is_at_least_length_of a2
it is sufficient to prove
  thus for b1 being natural set
          st a3 <= b1
       holds a2 . b1 = 0. a1;

:: ALGSEQ_1:def 3
theorem
for b1 being non empty ZeroStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
for b3 being natural set holds
      b3 is_at_least_length_of b2
   iff
      for b4 being natural set
            st b3 <= b4
         holds b2 . b4 = 0. b1;

:: ALGSEQ_1:funcnot 1 => ALGSEQ_1:func 1
definition
  let a1 be non empty ZeroStr;
  let a2 be Function-like quasi_total finite-Support Relation of NAT,the carrier of a1;
  func len A2 -> Element of NAT means
    it is_at_least_length_of a2 &
     (for b1 being natural set
           st b1 is_at_least_length_of a2
        holds it <= b1);
end;

:: ALGSEQ_1:def 4
theorem
for b1 being non empty ZeroStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
for b3 being Element of NAT holds
      b3 = len b2
   iff
      b3 is_at_least_length_of b2 &
       (for b4 being natural set
             st b4 is_at_least_length_of b2
          holds b3 <= b4);

:: ALGSEQ_1:th 22
theorem
for b1 being natural set
for b2 being non empty ZeroStr
for b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b2
      st len b3 <= b1
   holds b3 . b1 = 0. b2;

:: ALGSEQ_1:th 24
theorem
for b1 being natural set
for b2 being non empty ZeroStr
for b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b2
      st for b4 being natural set
              st b4 < b1
           holds b3 . b4 <> 0. b2
   holds b1 <= len b3;

:: ALGSEQ_1:th 25
theorem
for b1 being natural set
for b2 being non empty ZeroStr
for b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b2
      st len b3 = b1 + 1
   holds b3 . b1 <> 0. b2;

:: ALGSEQ_1:funcnot 2 => ALGSEQ_1:func 2
definition
  let a1 be non empty ZeroStr;
  let a2 be Function-like quasi_total finite-Support Relation of NAT,the carrier of a1;
  func support A2 -> Element of bool NAT equals
    Segm len a2;
end;

:: ALGSEQ_1:def 5
theorem
for b1 being non empty ZeroStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
   support b2 = Segm len b2;

:: ALGSEQ_1:th 27
theorem
for b1 being natural set
for b2 being non empty ZeroStr
for b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b2 holds
      b1 = len b3
   iff
      Segm b1 = support b3;

:: ALGSEQ_1:sch 1
scheme ALGSEQ_1:sch 1
{F1 -> non empty ZeroStr,
  F2 -> natural set,
  F3 -> Element of the carrier of F1()}:
ex b1 being Function-like quasi_total finite-Support Relation of NAT,the carrier of F1() st
   len b1 <= F2() &
    (for b2 being natural set
          st b2 < F2()
       holds b1 . b2 = F3(b2))


:: ALGSEQ_1:th 28
theorem
for b1 being non empty ZeroStr
for b2, b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1
      st len b2 = len b3 &
         (for b4 being natural set
               st b4 < len b2
            holds b2 . b4 = b3 . b4)
   holds b2 = b3;

:: ALGSEQ_1:th 29
theorem
for b1 being non empty ZeroStr
   st the carrier of b1 <> {0. b1}
for b2 being natural set holds
   ex b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 st
      len b3 = b2;

:: ALGSEQ_1:funcnot 3 => ALGSEQ_1:func 3
definition
  let a1 be non empty ZeroStr;
  let a2 be Element of the carrier of a1;
  func <%A2%> -> Function-like quasi_total finite-Support Relation of NAT,the carrier of a1 means
    len it <= 1 & it . 0 = a2;
end;

:: ALGSEQ_1:def 6
theorem
for b1 being non empty ZeroStr
for b2 being Element of the carrier of b1
for b3 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
      b3 = <%b2%>
   iff
      len b3 <= 1 & b3 . 0 = b2;

:: ALGSEQ_1:th 31
theorem
for b1 being non empty ZeroStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
      b2 = <%0. b1%>
   iff
      len b2 = 0;

:: ALGSEQ_1:th 32
theorem
for b1 being non empty ZeroStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
      b2 = <%0. b1%>
   iff
      support b2 = {};

:: ALGSEQ_1:th 33
theorem
for b1 being natural set
for b2 being non empty ZeroStr holds
   <%0. b2%> . b1 = 0. b2;

:: ALGSEQ_1:th 34
theorem
for b1 being non empty ZeroStr
for b2 being Function-like quasi_total finite-Support Relation of NAT,the carrier of b1 holds
      b2 = <%0. b1%>
   iff
      proj2 b2 = {0. b1};