:: ABIAN semantic presentation

definition

let c₁ be number ;
attr a₁ is even means :Def1: :: ABIAN:def 1
ex b₁ being Integer st a₁ = 2 * b₁;

end;

:: deftheorem Def1 defines even ABIAN:def 1 :

notation

let c₁ be number ;
antonym odd c₁ for even c₁;

end;

notation

let c₁ be Nat;
antonym odd c₁ for even c₁;

end;

definition

let c₁ be Nat;
redefine attr a₁ is even means :: ABIAN:def 2
ex b₁ being Nat st a₁ = 2 * b₁;
compatibility

proof end;

end;

:: deftheorem Def2 defines even ABIAN:def 2 :

registration

cluster even Element of NAT ;
existence

proof end;

cluster odd Element of NAT ;
existence

proof end;

cluster even set ;
existence

proof end;

cluster odd set ;
existence

proof end;

end;

theorem Th1: :: ABIAN:1

for b₁ being Integer holds
( not b₁ is even iff ex b₂ being Integer st b₁ = (2 * b₂) + 1 )

proof end;

registration

let c₁ be Integer;
cluster 2 * a₁ -> even ;
coherence by Def1;

end;

registration

let c₁ be even Integer;
cluster a₁ + 1 -> odd ;
coherence

proof end;

end;

registration

let c₁ be odd Integer;
cluster a₁ + 1 -> even ;
coherence

proof end;

end;

registration

let c₁ be even Integer;
cluster a₁ - 1 -> odd ;
coherence

proof end;

end;

registration

let c₁ be odd Integer;
cluster a₁ - 1 -> even ;
coherence

proof end;

end;

registration

let c₁ be even Integer;
let c₂ be Integer;
cluster a₁ * a₂ -> even ;
coherence

proof end;

cluster a₂ * a₁ -> even ;
coherence ;

end;

registration

let c₁, c₂ be odd Integer;
cluster a₁ * a₂ -> odd ;
coherence

proof end;

end;

registration

let c₁, c₂ be even Integer;
cluster a₁ + a₂ -> even ;
coherence

proof end;

end;

registration

let c₁ be even Integer;
let c₂ be odd Integer;
cluster a₁ + a₂ -> odd ;
coherence

proof end;

cluster a₂ + a₁ -> odd ;
coherence ;

end;

registration

let c₁, c₂ be odd Integer;
cluster a₁ + a₂ -> even ;
coherence

proof end;

end;

registration

let c₁ be even Integer;
let c₂ be odd Integer;
cluster a₁ - a₂ -> odd ;
coherence

proof end;

cluster a₂ - a₁ -> odd ;
coherence

proof end;

end;

registration

let c₁, c₂ be odd Integer;
cluster a₁ - a₂ -> even ;
coherence

proof end;

end;

definition

let c₁ be set ;
let c₂ be Function of c₁,c₁;
let c₃ be Nat;
redefine func iter as iter c₂,c₃ -> Function of a₁,a₁;
coherence by FUNCT_7:85;

end;

theorem Th2: :: ABIAN:2

for b₁ being non empty Subset of NAT st 0 in b₁ holds
min b₁ = 0

proof end;

theorem Th3: :: ABIAN:3

for b₁ being non empty set
for b₂ being Function of b₁,b₁
for b₃ being Element of b₁ holds (iter b₂,0) . b₃ = b₃

proof end;

definition

let c₁ be set ;
let c₂ be Function;
pred c₁ is_a_fixpoint_of c₂ means :Def3: :: ABIAN:def 3
( a₁ in dom a₂ & a₁ = a₂ . a₁ );

end;

:: deftheorem Def3 defines is_a_fixpoint_of ABIAN:def 3 :

definition

let c₁ be non empty set ;
let c₂ be Element of c₁;
let c₃ be Function of c₁,c₁;
redefine pred is_a_fixpoint_of as c₂ is_a_fixpoint_of c₃ means :Def4: :: ABIAN:def 4
a₂ = a₃ . a₂;
compatibility

proof end;

end;

:: deftheorem Def4 defines is_a_fixpoint_of ABIAN:def 4 :

definition

let c₁ be Function;
pred c₁ has_a_fixpoint means :Def5: :: ABIAN:def 5
ex b₁ being set st b₁ is_a_fixpoint_of a₁;

end;

:: deftheorem Def5 defines has_a_fixpoint ABIAN:def 5 :

notation

let c₁ be Function;
antonym c₁ has_no_fixpoint for c₁ has_a_fixpoint ;

end;

definition

let c₁ be set ;
let c₂ be Element of c₁;
attr a₂ is covering means :Def6: :: ABIAN:def 6
union a₂ = union (union a₁);

end;

:: deftheorem Def6 defines covering ABIAN:def 6 :

theorem Th4: :: ABIAN:4

for b₁ being set
for b₂ being Subset-Family of b₁ holds
( b₂ is covering iff union b₂ = b₁ )

proof end;

registration

let c₁ be set ;
cluster non empty finite covering Element of bool (bool a₁);
existence

proof end;

end;

theorem Th5: :: ABIAN:5

for b₁ being set
for b₂ being Function of b₁,b₁
for b₃ being non empty covering Subset-Family of b₁ st ( for b₄ being Element of b₃ holds b₄ misses b₂ .: b₄ ) holds
b₂ has_no_fixpoint

proof end;

definition

let c₁ be set ;
let c₂ be Function of c₁,c₁;
func =_ c₂ -> Equivalence_Relation of a₁ means :Def7: :: ABIAN:def 7
for b₁, b₂ being set st b₁ in a₁ & b₂ in a₁ holds
( [b₁,b₂] in a₃ iff ex b₃, b₄ being Nat st (iter a₂,b₃) . b₁ = (iter a₂,b₄) . b₂ );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def7 defines =_ ABIAN:def 7 :

theorem Th6: :: ABIAN:6

for b₁ being non empty set
for b₂ being Function of b₁,b₁
for b₃ being Element of Class (=_ b₂)
for b₄ being Element of b₃ holds b₂ . b₄ in b₃

proof end;

theorem Th7: :: ABIAN:7

for b₁ being non empty set
for b₂ being Function of b₁,b₁
for b₃ being Element of Class (=_ b₂)
for b₄ being Element of b₃
for b₅ being Nat holds (iter b₂,b₅) . b₄ in b₃

proof end;

registration

cluster empty-membered -> trivial set ;
coherence

proof end;

end;

registration

let c₁ be set ;
let c₂ be with_non-empty_element set ;
cluster non-empty M5(a₁,a₂);
existence

proof end;

end;

registration

let c₁ be non empty set ;
let c₂ be with_non-empty_element set ;
let c₃ be non-empty Function of c₁,c₂;
let c₄ be Element of c₁;
cluster a₃ . a₄ -> non empty ;
coherence

proof end;

end;

registration

let c₁ be non empty set ;
cluster bool a₁ -> with_non-empty_element ;
coherence

proof end;

end;

theorem Th8: :: ABIAN:8

for b₁ being non empty set
for b₂ being Function 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 )

proof end;

theorem Th9: :: ABIAN:9

for b₁ being natural number holds
( not b₁ is even iff ex b₂ being Nat st b₁ = (2 * b₂) + 1 )

proof end;