Article ABIAN, MML version 4.99.1005
:: ABIAN:attrnot 1 => ABIAN:attr 1
definition
let a1 be set;
attr a1 is even means
ex b1 being integer set st
a1 = 2 * b1;
end;
:: ABIAN:dfs 1
definiens
let a1 be set;
To prove
a1 is even
it is sufficient to prove
thus ex b1 being integer set st
a1 = 2 * b1;
:: ABIAN:def 1
theorem
for b1 being set holds
b1 is even
iff
ex b2 being integer set st
b1 = 2 * b2;
:: ABIAN:attrnot 2 => ABIAN:attr 1
notation
let a1 be set;
antonym odd for even;
end;
:: ABIAN:attrnot 3 => ABIAN:attr 1
definition
let a1 be set;
attr a1 is even means
ex b1 being Element of NAT st
a1 = 2 * b1;
end;
:: ABIAN:dfs 2
definiens
let a1 be Element of NAT;
To prove
a1 is even
it is sufficient to prove
thus ex b1 being Element of NAT st
a1 = 2 * b1;
:: ABIAN:def 2
theorem
for b1 being Element of NAT holds
b1 is even
iff
ex b2 being Element of NAT st
b1 = 2 * b2;
:: ABIAN:exreg 1
registration
cluster ext-real epsilon-transitive epsilon-connected ordinal natural complex real integer complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even Element of NAT;
end;
:: ABIAN:exreg 2
registration
cluster ext-real epsilon-transitive epsilon-connected ordinal natural complex real integer complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered odd Element of NAT;
end;
:: ABIAN:exreg 3
registration
cluster ext-real complex real integer even set;
end;
:: ABIAN:exreg 4
registration
cluster ext-real complex real integer odd set;
end;
:: ABIAN:th 1
theorem
for b1 being integer set holds
b1 is odd
iff
ex b2 being integer set st
b1 = (2 * b2) + 1;
:: ABIAN:funcreg 1
registration
let a1 be integer set;
cluster 2 * a1 -> even;
end;
:: ABIAN:funcreg 2
registration
let a1 be integer even set;
cluster a1 + 1 -> odd;
end;
:: ABIAN:funcreg 3
registration
let a1 be integer odd set;
cluster a1 + 1 -> even;
end;
:: ABIAN:funcreg 4
registration
let a1 be integer even set;
cluster a1 - 1 -> odd;
end;
:: ABIAN:funcreg 5
registration
let a1 be integer odd set;
cluster a1 - 1 -> even;
end;
:: ABIAN:funcreg 6
registration
let a1 be integer even set;
let a2 be integer set;
cluster a1 * a2 -> even;
end;
:: ABIAN:funcreg 7
registration
let a1 be integer even set;
let a2 be integer set;
cluster a2 * a1 -> even;
end;
:: ABIAN:funcreg 8
registration
let a1, a2 be integer odd set;
cluster a1 * a2 -> odd;
end;
:: ABIAN:funcreg 9
registration
let a1, a2 be integer even set;
cluster a1 + a2 -> even;
end;
:: ABIAN:funcreg 10
registration
let a1 be integer even set;
let a2 be integer odd set;
cluster a1 + a2 -> odd;
end;
:: ABIAN:funcreg 11
registration
let a1 be integer even set;
let a2 be integer odd set;
cluster a2 + a1 -> odd;
end;
:: ABIAN:funcreg 12
registration
let a1, a2 be integer odd set;
cluster a1 + a2 -> even;
end;
:: ABIAN:funcreg 13
registration
let a1 be integer even set;
let a2 be integer odd set;
cluster a1 - a2 -> odd;
end;
:: ABIAN:funcreg 14
registration
let a1 be integer even set;
let a2 be integer odd set;
cluster a2 - a1 -> odd;
end;
:: ABIAN:funcreg 15
registration
let a1, a2 be integer odd set;
cluster a1 - a2 -> even;
end;
:: ABIAN:funcreg 16
registration
let a1 be integer even set;
cluster a1 + 2 -> even;
end;
:: ABIAN:funcreg 17
registration
let a1 be integer odd set;
cluster a1 + 2 -> odd;
end;
:: ABIAN:funcnot 1 => ABIAN:func 1
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of a1,a1;
let a3 be Element of NAT;
redefine func iter(a2,a3) -> Function-like quasi_total Relation of a1,a1;
end;
:: ABIAN:th 2
theorem
for b1 being non empty Element of bool NAT
st 0 in b1
holds min b1 = 0;
:: ABIAN:th 3
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of b1,b1
for b3 being Element of b1 holds
(iter(b2,0)) . b3 = b3;
:: ABIAN:prednot 1 => ABIAN:pred 1
definition
let a1 be set;
let a2 be Relation-like Function-like set;
pred A1 is_a_fixpoint_of A2 means
a1 in proj1 a2 & a1 = a2 . a1;
end;
:: ABIAN:dfs 3
definiens
let a1 be set;
let a2 be Relation-like Function-like set;
To prove
a1 is_a_fixpoint_of a2
it is sufficient to prove
thus a1 in proj1 a2 & a1 = a2 . a1;
:: ABIAN:def 3
theorem
for b1 being set
for b2 being Relation-like Function-like set holds
b1 is_a_fixpoint_of b2
iff
b1 in proj1 b2 & b1 = b2 . b1;
:: ABIAN:prednot 2 => ABIAN:pred 2
definition
let a1 be non empty set;
let a2 be Element of a1;
let a3 be Function-like quasi_total Relation of a1,a1;
redefine pred A2 is_a_fixpoint_of A3 means
a2 = a3 . a2;
end;
:: ABIAN:dfs 4
definiens
let a1 be non empty set;
let a2 be Element of a1;
let a3 be Function-like quasi_total Relation of a1,a1;
To prove
a2 is_a_fixpoint_of a3
it is sufficient to prove
thus a2 = a3 . a2;
:: ABIAN:def 4
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being Function-like quasi_total Relation of b1,b1 holds
b2 is_a_fixpoint_of b3
iff
b2 = b3 . b2;
:: ABIAN:prednot 3 => ABIAN:pred 3
definition
let a1 be Relation-like Function-like set;
pred A1 has_a_fixpoint means
ex b1 being set st
b1 is_a_fixpoint_of a1;
end;
:: ABIAN:dfs 5
definiens
let a1 be Relation-like Function-like set;
To prove
a1 has_a_fixpoint
it is sufficient to prove
thus ex b1 being set st
b1 is_a_fixpoint_of a1;
:: ABIAN:def 5
theorem
for b1 being Relation-like Function-like set holds
b1 has_a_fixpoint
iff
ex b2 being set st
b2 is_a_fixpoint_of b1;
:: ABIAN:prednot 4 => not ABIAN:pred 3
notation
let a1 be Relation-like Function-like set;
antonym a1 has_no_fixpoint for a1 has_a_fixpoint;
end;
:: ABIAN:attrnot 4 => ABIAN:attr 2
definition
let a1 be set;
let a2 be Element of a1;
attr a2 is covering means
union a2 = union union a1;
end;
:: ABIAN:dfs 6
definiens
let a1 be set;
let a2 be Element of a1;
To prove
a2 is covering
it is sufficient to prove
thus union a2 = union union a1;
:: ABIAN:def 6
theorem
for b1 being set
for b2 being Element of b1 holds
b2 is covering(b1)
iff
union b2 = union union b1;
:: ABIAN:th 4
theorem
for b1 being set
for b2 being Element of bool bool b1 holds
b2 is covering(bool bool b1)
iff
union b2 = b1;
:: ABIAN:exreg 5
registration
let a1 be set;
cluster non empty finite covering Element of bool bool a1;
end;
:: ABIAN:th 5
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of b1,b1
for b3 being non empty covering Element of bool bool b1
st for b4 being Element of b3 holds
b4 misses b2 .: b4
holds b2 has_no_fixpoint;
:: ABIAN:funcnot 2 => ABIAN:func 2
definition
let a1 be set;
let a2 be Function-like quasi_total Relation of a1,a1;
func =_ A2 -> symmetric transitive total Relation of a1,a1 means
for b1, b2 being set
st b1 in a1 & b2 in a1
holds [b1,b2] in it
iff
ex b3, b4 being Element of NAT st
(iter(a2,b3)) . b1 = (iter(a2,b4)) . b2;
end;
:: ABIAN:def 7
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of b1,b1
for b3 being symmetric transitive total Relation of b1,b1 holds
b3 = =_ b2
iff
for b4, b5 being set
st b4 in b1 & b5 in b1
holds [b4,b5] in b3
iff
ex b6, b7 being Element of NAT st
(iter(b2,b6)) . b4 = (iter(b2,b7)) . b5;
:: ABIAN:th 6
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of b1,b1
for b3 being Element of Class =_ b2
for b4 being Element of b3 holds
b2 . b4 in b3;
:: ABIAN:th 7
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of b1,b1
for b3 being Element of Class =_ b2
for b4 being Element of b3
for b5 being Element of NAT holds
(iter(b2,b5)) . b4 in b3;
:: ABIAN:condreg 1
registration
cluster empty-membered -> trivial (set);
end;
:: ABIAN:exreg 6
registration
let a1 be set;
let a2 be with_non-empty_element set;
cluster Relation-like non-empty Function-like quasi_total Relation of a1,a2;
end;
:: ABIAN:funcreg 18
registration
let a1 be non empty set;
let a2 be with_non-empty_element set;
let a3 be non-empty Function-like quasi_total Relation of a1,a2;
let a4 be Element of a1;
cluster a3 . a4 -> non empty;
end;
:: ABIAN:funcreg 19
registration
let a1 be non empty set;
cluster bool a1 -> with_non-empty_element;
end;
:: ABIAN:th 8
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of b1,b1
st b2 has_no_fixpoint
holds ex b3, b4, b5 being set st
(b3 \/ b4) \/ b5 = b1 & b2 .: b3 misses b3 & b2 .: b4 misses b4 & b2 .: b5 misses b5;
:: ABIAN:th 9
theorem
for b1 being natural set holds
b1 is odd
iff
ex b2 being Element of NAT st
b1 = (2 * b2) + 1;