Article ARROW, MML version 4.99.1005
:: ARROW:funcnot 1 => ARROW:func 1
definition
let a1, a2 be non empty set;
let a3 be non empty Element of bool a2;
let a4 be Function-like quasi_total Relation of a1,a3;
let a5 be Element of a1;
redefine func a4 . a5 -> Element of a3;
end;
:: ARROW:th 1
theorem
for b1 being finite set
st 2 <= card b1
for b2 being Element of b1 holds
ex b3 being Element of b1 st
b3 <> b2;
:: ARROW:th 2
theorem
for b1 being finite set
st 3 <= card b1
for b2, b3 being Element of b1 holds
ex b4 being Element of b1 st
b4 <> b2 & b4 <> b3;
:: ARROW:funcnot 2 => ARROW:func 2
definition
let a1 be non empty set;
func LinPreorders A1 -> set means
for b1 being set holds
b1 in it
iff
b1 is Relation of a1,a1 &
(for b2, b3 being Element of a1
st not [b2,b3] in b1
holds [b3,b2] in b1) &
(for b2, b3, b4 being Element of a1
st [b2,b3] in b1 & [b3,b4] in b1
holds [b2,b4] in b1);
end;
:: ARROW:def 1
theorem
for b1 being non empty set
for b2 being set holds
b2 = LinPreorders b1
iff
for b3 being set holds
b3 in b2
iff
b3 is Relation of b1,b1 &
(for b4, b5 being Element of b1
st not [b4,b5] in b3
holds [b5,b4] in b3) &
(for b4, b5, b6 being Element of b1
st [b4,b5] in b3 & [b5,b6] in b3
holds [b4,b6] in b3);
:: ARROW:funcreg 1
registration
let a1 be non empty set;
cluster LinPreorders a1 -> non empty;
end;
:: ARROW:funcnot 3 => ARROW:func 3
definition
let a1 be non empty set;
func LinOrders A1 -> Element of bool LinPreorders a1 means
for b1 being Element of LinPreorders a1 holds
b1 in it
iff
for b2, b3 being Element of a1
st [b2,b3] in b1 & [b3,b2] in b1
holds b2 = b3;
end;
:: ARROW:def 2
theorem
for b1 being non empty set
for b2 being Element of bool LinPreorders b1 holds
b2 = LinOrders b1
iff
for b3 being Element of LinPreorders b1 holds
b3 in b2
iff
for b4, b5 being Element of b1
st [b4,b5] in b3 & [b5,b4] in b3
holds b4 = b5;
:: ARROW:exreg 1
registration
let a1 be set;
cluster Relation-like reflexive antisymmetric connected transitive total Relation of a1,a1;
end;
:: ARROW:funcnot 4 => ARROW:func 3
definition
let a1 be non empty set;
func LinOrders A1 -> Element of bool LinPreorders a1 means
for b1 being set holds
b1 in it
iff
b1 is reflexive antisymmetric connected transitive total Relation of a1,a1;
end;
:: ARROW:def 3
theorem
for b1 being non empty set
for b2 being Element of bool LinPreorders b1 holds
b2 = LinOrders b1
iff
for b3 being set holds
b3 in b2
iff
b3 is reflexive antisymmetric connected transitive total Relation of b1,b1;
:: ARROW:funcreg 2
registration
let a1 be non empty set;
cluster LinOrders a1 -> non empty;
end;
:: ARROW:prednot 1 => ARROW:pred 1
definition
let a1 be non empty set;
let a2 be Element of LinPreorders a1;
let a3, a4 be Element of a1;
pred A3 <=_ A2,A4 means
[a3,a4] in a2;
end;
:: ARROW:dfs 4
definiens
let a1 be non empty set;
let a2 be Element of LinPreorders a1;
let a3, a4 be Element of a1;
To prove
a3 <=_ a2,a4
it is sufficient to prove
thus [a3,a4] in a2;
:: ARROW:def 4
theorem
for b1 being non empty set
for b2 being Element of LinPreorders b1
for b3, b4 being Element of b1 holds
b3 <=_ b2,b4
iff
[b3,b4] in b2;
:: ARROW:prednot 2 => ARROW:pred 1
notation
let a1 be non empty set;
let a2 be Element of LinPreorders a1;
let a3, a4 be Element of a1;
synonym a4 >=_ a2,a3 for a3 <=_ a2,a4;
end;
:: ARROW:prednot 3 => not ARROW:pred 1
notation
let a1 be non empty set;
let a2 be Element of LinPreorders a1;
let a3, a4 be Element of a1;
antonym a4 <_ a2,a3 for a3 <=_ a2,a4;
end;
:: ARROW:prednot 4 => not ARROW:pred 1
notation
let a1 be non empty set;
let a2 be Element of LinPreorders a1;
let a3, a4 be Element of a1;
antonym a3 >_ a2,a4 for a3 <=_ a2,a4;
end;
:: ARROW:th 3
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being Element of LinPreorders b1 holds
b2 <=_ b3,b2;
:: ARROW:th 4
theorem
for b1 being non empty set
for b2, b3 being Element of b1
for b4 being Element of LinPreorders b1
st b3 <_ b4,b2
holds b3 <=_ b4,b2;
:: ARROW:th 5
theorem
for b1 being non empty set
for b2, b3, b4 being Element of b1
for b5 being Element of LinPreorders b1
st (b3 <_ b5,b2 implies b2 <_ b5,b3) & (b4 <_ b5,b3 implies b3 <_ b5,b4)
holds b2 <=_ b5,b4;
:: ARROW:th 6
theorem
for b1 being non empty set
for b2, b3 being Element of b1
for b4 being Element of LinOrders b1
st b2 <=_ b4,b3 & b3 <=_ b4,b2
holds b2 = b3;
:: ARROW:th 7
theorem
for b1 being non empty set
for b2, b3, b4 being Element of b1
st b2 <> b3 & b3 <> b4 & b2 <> b4
holds ex b5 being Element of LinPreorders b1 st
b2 <_ b5,b3 & b3 <_ b5,b4;
:: ARROW:th 8
theorem
for b1 being non empty set
for b2 being Element of b1 holds
ex b3 being Element of LinPreorders b1 st
for b4 being Element of b1
st b4 <> b2
holds b2 <_ b3,b4;
:: ARROW:th 9
theorem
for b1 being non empty set
for b2 being Element of b1 holds
ex b3 being Element of LinPreorders b1 st
for b4 being Element of b1
st b4 <> b2
holds b4 <_ b3,b2;
:: ARROW:th 10
theorem
for b1 being non empty set
for b2, b3, b4 being Element of b1
for b5 being Element of LinPreorders b1
st b2 <> b3 & b2 <> b4
holds ex b6 being Element of LinPreorders b1 st
b2 <_ b6,b3 & b2 <_ b6,b4 & (b3 <_ b6,b4 implies b3 <_ b5,b4) & (b3 <_ b5,b4 implies b3 <_ b6,b4) & (b4 <_ b6,b3 implies b4 <_ b5,b3) & (b4 <_ b5,b3 implies b4 <_ b6,b3);
:: ARROW:th 11
theorem
for b1 being non empty set
for b2, b3, b4 being Element of b1
for b5 being Element of LinPreorders b1
st b2 <> b3 & b2 <> b4
holds ex b6 being Element of LinPreorders b1 st
b3 <_ b6,b2 & b4 <_ b6,b2 & (b3 <_ b6,b4 implies b3 <_ b5,b4) & (b3 <_ b5,b4 implies b3 <_ b6,b4) & (b4 <_ b6,b3 implies b4 <_ b5,b3) & (b4 <_ b5,b3 implies b4 <_ b6,b3);
:: ARROW:th 12
theorem
for b1 being non empty set
for b2, b3 being Element of b1
for b4, b5 being Element of LinOrders b1 holds
(b2 <_ b4,b3 implies b2 <_ b5,b3) & (b2 <_ b5,b3 implies b2 <_ b4,b3) & (b3 <_ b4,b2 implies b3 <_ b5,b2) & (b3 <_ b5,b2 implies b3 <_ b4,b2)
iff
b2 <_ b4,b3
iff
b2 <_ b5,b3;
:: ARROW:th 13
theorem
for b1 being non empty set
for b2 being Element of LinOrders b1
for b3 being Element of LinPreorders b1 holds
for b4, b5 being Element of b1
st b4 <_ b2,b5
holds b4 <_ b3,b5
iff
for b4, b5 being Element of b1 holds
b4 <_ b2,b5
iff
b4 <_ b3,b5;
:: ARROW:th 14
theorem
for b1, b2 being non empty finite set
for b3 being Function-like quasi_total Relation of Funcs(b2,LinPreorders b1),LinPreorders b1
st (for b4 being Element of Funcs(b2,LinPreorders b1)
for b5, b6 being Element of b1
st for b7 being Element of b2 holds
b5 <_ b4 . b7,b6
holds b5 <_ b3 . b4,b6) &
(for b4, b5 being Element of Funcs(b2,LinPreorders b1)
for b6, b7 being Element of b1
st for b8 being Element of b2 holds
(b6 <_ b4 . b8,b7 implies b6 <_ b5 . b8,b7) &
(b6 <_ b5 . b8,b7 implies b6 <_ b4 . b8,b7) &
(b7 <_ b4 . b8,b6 implies b7 <_ b5 . b8,b6) &
(b7 <_ b5 . b8,b6 implies b7 <_ b4 . b8,b6)
holds b6 <_ b3 . b4,b7
iff
b6 <_ b3 . b5,b7) &
3 <= card b1
holds ex b4 being Element of b2 st
for b5 being Element of Funcs(b2,LinPreorders b1)
for b6, b7 being Element of b1
st b6 <_ b5 . b4,b7
holds b6 <_ b3 . b5,b7;
:: ARROW:th 15
theorem
for b1, b2 being non empty finite set
for b3 being Function-like quasi_total Relation of Funcs(b2,LinOrders b1),LinPreorders b1
st (for b4 being Element of Funcs(b2,LinOrders b1)
for b5, b6 being Element of b1
st for b7 being Element of b2 holds
b5 <_ b4 . b7,b6
holds b5 <_ b3 . b4,b6) &
(for b4, b5 being Element of Funcs(b2,LinOrders b1)
for b6, b7 being Element of b1
st for b8 being Element of b2 holds
b6 <_ b4 . b8,b7
iff
b6 <_ b5 . b8,b7
holds b6 <_ b3 . b4,b7
iff
b6 <_ b3 . b5,b7) &
3 <= card b1
holds ex b4 being Element of b2 st
for b5 being Element of Funcs(b2,LinOrders b1)
for b6, b7 being Element of b1 holds
b6 <_ b5 . b4,b7
iff
b6 <_ b3 . b5,b7;