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;