Article CONAFFM, MML version 4.99.1005

:: CONAFFM:attrnot 1 => CONAFFM:attr 1
definition
  let a1 be non empty OrtAfPl-like ParOrtStr;
  attr a1 is satisfying_DES means
    for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
          st b1 <> b2 & b1 <> b3 & b1 <> b4 & b1 <> b5 & b1 <> b6 & b1 <> b7 & not LIN b4,b5,b2 & not LIN b2,b3,b6 & LIN b1,b2,b3 & LIN b1,b4,b5 & LIN b1,b6,b7 & b2,b4 // b3,b5 & b2,b6 // b3,b7
       holds b4,b6 // b5,b7;
end;

:: CONAFFM:dfs 1
definiens
  let a1 be non empty OrtAfPl-like ParOrtStr;
To prove
     a1 is satisfying_DES
it is sufficient to prove
  thus for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
          st b1 <> b2 & b1 <> b3 & b1 <> b4 & b1 <> b5 & b1 <> b6 & b1 <> b7 & not LIN b4,b5,b2 & not LIN b2,b3,b6 & LIN b1,b2,b3 & LIN b1,b4,b5 & LIN b1,b6,b7 & b2,b4 // b3,b5 & b2,b6 // b3,b7
       holds b4,b6 // b5,b7;

:: CONAFFM:def 1
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
      b1 is satisfying_DES
   iff
      for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
            st b2 <> b3 & b2 <> b4 & b2 <> b5 & b2 <> b6 & b2 <> b7 & b2 <> b8 & not LIN b5,b6,b3 & not LIN b3,b4,b7 & LIN b2,b3,b4 & LIN b2,b5,b6 & LIN b2,b7,b8 & b3,b5 // b4,b6 & b3,b7 // b4,b8
         holds b5,b7 // b6,b8;

:: CONAFFM:attrnot 2 => CONAFFM:attr 2
definition
  let a1 be non empty OrtAfPl-like ParOrtStr;
  attr a1 is satisfying_AH means
    for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
          st b1,b2 _|_ b1,b3 & b1,b4 _|_ b1,b5 & b1,b6 _|_ b1,b7 & b2,b4 _|_ b3,b5 & b1,b2 // b4,b6 & b2,b6 _|_ b3,b7 & not b1,b6 // b1,b2 & not b1,b2 // b1,b4
       holds b4,b6 _|_ b5,b7;
end;

:: CONAFFM:dfs 2
definiens
  let a1 be non empty OrtAfPl-like ParOrtStr;
To prove
     a1 is satisfying_AH
it is sufficient to prove
  thus for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
          st b1,b2 _|_ b1,b3 & b1,b4 _|_ b1,b5 & b1,b6 _|_ b1,b7 & b2,b4 _|_ b3,b5 & b1,b2 // b4,b6 & b2,b6 _|_ b3,b7 & not b1,b6 // b1,b2 & not b1,b2 // b1,b4
       holds b4,b6 _|_ b5,b7;

:: CONAFFM:def 2
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
      b1 is satisfying_AH
   iff
      for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
            st b2,b3 _|_ b2,b4 & b2,b5 _|_ b2,b6 & b2,b7 _|_ b2,b8 & b3,b5 _|_ b4,b6 & b2,b3 // b5,b7 & b3,b7 _|_ b4,b8 & not b2,b7 // b2,b3 & not b2,b3 // b2,b5
         holds b5,b7 _|_ b6,b8;

:: CONAFFM:attrnot 3 => CONAFFM:attr 3
definition
  let a1 be non empty OrtAfPl-like ParOrtStr;
  attr a1 is satisfying_3H means
    for b1, b2, b3 being Element of the carrier of a1
          st not LIN b1,b2,b3
       holds ex b4 being Element of the carrier of a1 st
          b4,b1 _|_ b2,b3 & b4,b2 _|_ b1,b3 & b4,b3 _|_ b1,b2;
end;

:: CONAFFM:dfs 3
definiens
  let a1 be non empty OrtAfPl-like ParOrtStr;
To prove
     a1 is satisfying_3H
it is sufficient to prove
  thus for b1, b2, b3 being Element of the carrier of a1
          st not LIN b1,b2,b3
       holds ex b4 being Element of the carrier of a1 st
          b4,b1 _|_ b2,b3 & b4,b2 _|_ b1,b3 & b4,b3 _|_ b1,b2;

:: CONAFFM:def 3
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
      b1 is satisfying_3H
   iff
      for b2, b3, b4 being Element of the carrier of b1
            st not LIN b2,b3,b4
         holds ex b5 being Element of the carrier of b1 st
            b5,b2 _|_ b3,b4 & b5,b3 _|_ b2,b4 & b5,b4 _|_ b2,b3;

:: CONAFFM:attrnot 4 => CONAFFM:attr 4
definition
  let a1 be non empty OrtAfPl-like ParOrtStr;
  attr a1 is satisfying_ODES means
    for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
          st b1,b2 _|_ b1,b3 & b1,b4 _|_ b1,b5 & b1,b6 _|_ b1,b7 & b2,b4 _|_ b3,b5 & b2,b6 _|_ b3,b7 & not b1,b6 // b1,b2 & not b1,b2 // b1,b4
       holds b4,b6 _|_ b5,b7;
end;

:: CONAFFM:dfs 4
definiens
  let a1 be non empty OrtAfPl-like ParOrtStr;
To prove
     a1 is satisfying_ODES
it is sufficient to prove
  thus for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
          st b1,b2 _|_ b1,b3 & b1,b4 _|_ b1,b5 & b1,b6 _|_ b1,b7 & b2,b4 _|_ b3,b5 & b2,b6 _|_ b3,b7 & not b1,b6 // b1,b2 & not b1,b2 // b1,b4
       holds b4,b6 _|_ b5,b7;

:: CONAFFM:def 4
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
      b1 is satisfying_ODES
   iff
      for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
            st b2,b3 _|_ b2,b4 & b2,b5 _|_ b2,b6 & b2,b7 _|_ b2,b8 & b3,b5 _|_ b4,b6 & b3,b7 _|_ b4,b8 & not b2,b7 // b2,b3 & not b2,b3 // b2,b5
         holds b5,b7 _|_ b6,b8;

:: CONAFFM:attrnot 5 => CONAFFM:attr 5
definition
  let a1 be non empty OrtAfPl-like ParOrtStr;
  attr a1 is satisfying_LIN means
    for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
          st b1 <> b2 & b1 <> b3 & b1 <> b4 & b1 <> b5 & b1 <> b6 & b1 <> b7 & b2 <> b4 & b1,b6 _|_ b1,b7 & b1,b2 _|_ b1,b3 & b1,b4 _|_ b1,b5 & not LIN b1,b6,b2 & LIN b1,b2,b4 & LIN b1,b3,b5 & b2,b6 _|_ b3,b7 & b4,b6 _|_ b5,b7
       holds b2,b3 // b4,b5;
end;

:: CONAFFM:dfs 5
definiens
  let a1 be non empty OrtAfPl-like ParOrtStr;
To prove
     a1 is satisfying_LIN
it is sufficient to prove
  thus for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
          st b1 <> b2 & b1 <> b3 & b1 <> b4 & b1 <> b5 & b1 <> b6 & b1 <> b7 & b2 <> b4 & b1,b6 _|_ b1,b7 & b1,b2 _|_ b1,b3 & b1,b4 _|_ b1,b5 & not LIN b1,b6,b2 & LIN b1,b2,b4 & LIN b1,b3,b5 & b2,b6 _|_ b3,b7 & b4,b6 _|_ b5,b7
       holds b2,b3 // b4,b5;

:: CONAFFM:def 5
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
      b1 is satisfying_LIN
   iff
      for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
            st b2 <> b3 & b2 <> b4 & b2 <> b5 & b2 <> b6 & b2 <> b7 & b2 <> b8 & b3 <> b5 & b2,b7 _|_ b2,b8 & b2,b3 _|_ b2,b4 & b2,b5 _|_ b2,b6 & not LIN b2,b7,b3 & LIN b2,b3,b5 & LIN b2,b4,b6 & b3,b7 _|_ b4,b8 & b5,b7 _|_ b6,b8
         holds b3,b4 // b5,b6;

:: CONAFFM:attrnot 6 => CONAFFM:attr 6
definition
  let a1 be non empty OrtAfPl-like ParOrtStr;
  attr a1 is satisfying_LIN1 means
    for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
          st b1 <> b2 & b1 <> b3 & b1 <> b4 & b1 <> b5 & b1 <> b6 & b1 <> b7 & b2 <> b4 & b1,b6 _|_ b1,b7 & b1,b2 _|_ b1,b3 & b1,b4 _|_ b1,b5 & not LIN b1,b6,b2 & LIN b1,b2,b4 & LIN b1,b3,b5 & b2,b6 _|_ b3,b7 & b2,b3 // b4,b5
       holds b4,b6 _|_ b5,b7;
end;

:: CONAFFM:dfs 6
definiens
  let a1 be non empty OrtAfPl-like ParOrtStr;
To prove
     a1 is satisfying_LIN1
it is sufficient to prove
  thus for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
          st b1 <> b2 & b1 <> b3 & b1 <> b4 & b1 <> b5 & b1 <> b6 & b1 <> b7 & b2 <> b4 & b1,b6 _|_ b1,b7 & b1,b2 _|_ b1,b3 & b1,b4 _|_ b1,b5 & not LIN b1,b6,b2 & LIN b1,b2,b4 & LIN b1,b3,b5 & b2,b6 _|_ b3,b7 & b2,b3 // b4,b5
       holds b4,b6 _|_ b5,b7;

:: CONAFFM:def 6
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
      b1 is satisfying_LIN1
   iff
      for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
            st b2 <> b3 & b2 <> b4 & b2 <> b5 & b2 <> b6 & b2 <> b7 & b2 <> b8 & b3 <> b5 & b2,b7 _|_ b2,b8 & b2,b3 _|_ b2,b4 & b2,b5 _|_ b2,b6 & not LIN b2,b7,b3 & LIN b2,b3,b5 & LIN b2,b4,b6 & b3,b7 _|_ b4,b8 & b3,b4 // b5,b6
         holds b5,b7 _|_ b6,b8;

:: CONAFFM:attrnot 7 => CONAFFM:attr 7
definition
  let a1 be non empty OrtAfPl-like ParOrtStr;
  attr a1 is satisfying_LIN2 means
    for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
          st b1 <> b2 & b1 <> b3 & b1 <> b4 & b1 <> b5 & b1 <> b6 & b1 <> b7 & b2 <> b4 & b2,b3 // b4,b5 & b1,b2 _|_ b1,b3 & b1,b4 _|_ b1,b5 & not LIN b1,b6,b2 & LIN b1,b2,b4 & LIN b1,b3,b5 & b2,b6 _|_ b3,b7 & b4,b6 _|_ b5,b7
       holds b1,b6 _|_ b1,b7;
end;

:: CONAFFM:dfs 7
definiens
  let a1 be non empty OrtAfPl-like ParOrtStr;
To prove
     a1 is satisfying_LIN2
it is sufficient to prove
  thus for b1, b2, b3, b4, b5, b6, b7 being Element of the carrier of a1
          st b1 <> b2 & b1 <> b3 & b1 <> b4 & b1 <> b5 & b1 <> b6 & b1 <> b7 & b2 <> b4 & b2,b3 // b4,b5 & b1,b2 _|_ b1,b3 & b1,b4 _|_ b1,b5 & not LIN b1,b6,b2 & LIN b1,b2,b4 & LIN b1,b3,b5 & b2,b6 _|_ b3,b7 & b4,b6 _|_ b5,b7
       holds b1,b6 _|_ b1,b7;

:: CONAFFM:def 7
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
      b1 is satisfying_LIN2
   iff
      for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
            st b2 <> b3 & b2 <> b4 & b2 <> b5 & b2 <> b6 & b2 <> b7 & b2 <> b8 & b3 <> b5 & b3,b4 // b5,b6 & b2,b3 _|_ b2,b4 & b2,b5 _|_ b2,b6 & not LIN b2,b7,b3 & LIN b2,b3,b5 & LIN b2,b4,b6 & b3,b7 _|_ b4,b8 & b5,b7 _|_ b6,b8
         holds b2,b7 _|_ b2,b8;

:: CONAFFM:prednot 1 => CONAFFM:attr 1
notation
  let a1 be non empty OrtAfPl-like ParOrtStr;
  synonym DES_holds_in a1 for satisfying_DES;
end;

:: CONAFFM:prednot 2 => CONAFFM:attr 2
notation
  let a1 be non empty OrtAfPl-like ParOrtStr;
  synonym AH_holds_in a1 for satisfying_AH;
end;

:: CONAFFM:prednot 3 => CONAFFM:attr 3
notation
  let a1 be non empty OrtAfPl-like ParOrtStr;
  synonym 3H_holds_in a1 for satisfying_3H;
end;

:: CONAFFM:prednot 4 => CONAFFM:attr 4
notation
  let a1 be non empty OrtAfPl-like ParOrtStr;
  synonym ODES_holds_in a1 for satisfying_ODES;
end;

:: CONAFFM:prednot 5 => CONAFFM:attr 5
notation
  let a1 be non empty OrtAfPl-like ParOrtStr;
  synonym LIN_holds_in a1 for satisfying_LIN;
end;

:: CONAFFM:prednot 6 => CONAFFM:attr 6
notation
  let a1 be non empty OrtAfPl-like ParOrtStr;
  synonym LIN1_holds_in a1 for satisfying_LIN1;
end;

:: CONAFFM:prednot 7 => CONAFFM:attr 7
notation
  let a1 be non empty OrtAfPl-like ParOrtStr;
  synonym LIN2_holds_in a1 for satisfying_LIN2;
end;

:: CONAFFM:th 1
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
      st b1 is satisfying_ODES
   holds b1 is satisfying_DES;

:: CONAFFM:th 2
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
      st b1 is satisfying_ODES
   holds b1 is satisfying_AH;

:: CONAFFM:th 3
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
      st b1 is satisfying_LIN
   holds b1 is satisfying_LIN1;

:: CONAFFM:th 4
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
      st b1 is satisfying_LIN1
   holds b1 is satisfying_LIN2;

:: CONAFFM:th 5
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
      st b1 is satisfying_LIN
   holds b1 is satisfying_ODES;

:: CONAFFM:th 6
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
      st b1 is satisfying_LIN
   holds b1 is satisfying_3H;