:: {E}uler's {P}artition {T}heorem
:: by Karol P\kak
::
:: Received March 26, 2015
:: Copyright (c) 2015-2021 Association of Mizar Users

registration

let r be ext-real number ;

cluster <*r*> -> ext-real-valued ;

coherence
<*r*> is ext-real-valued

proof end;

cluster <*r*> -> increasing decreasing non-decreasing non-increasing ;

coherence
( <*r*> is decreasing & <*r*> is increasing & <*r*> is non-decreasing & <*r*> is non-increasing ) ;

end;

theorem Th1: :: EULRPART:1

for f, g being ext-real-valued non-decreasing FinSequence st f . (len f) <= g . 1 holds
f ^ g is non-decreasing

proof end;

definition

let R be Relation;

attr R is odd-valued means :Def1: :: EULRPART:def 1
rng R c= OddNAT ;

end;

:: deftheorem Def1 defines odd-valued EULRPART:def 1 :
for R being Relation holds
( R is odd-valued iff rng R c= OddNAT );

theorem Th2: :: EULRPART:2

for n being Nat holds
( n in OddNAT iff n is odd )

proof end;

registration

cluster Relation-like odd-valued -> non-zero natural-valued for set ;

coherence
for b₁ being Relation st b₁ is odd-valued holds
( b₁ is non-zero & b₁ is natural-valued )

proof end;

end;

definition

let F be Function;

redefine attr F is odd-valued means :Def2: :: EULRPART:def 2
for x being object st x in dom F holds
F . x is odd Nat;

compatibility
( F is odd-valued iff for x being object st x in dom F holds
F . x is odd Nat )

proof end;

end;

:: deftheorem Def2 defines odd-valued EULRPART:def 2 :
for F being Function holds
( F is odd-valued iff for x being object st x in dom F holds
F . x is odd Nat );

registration

cluster empty Relation-like -> odd-valued for set ;

coherence
for b₁ being Relation st b₁ is empty holds
b₁ is odd-valued

proof end;

let i be odd Nat;

cluster <*i*> -> odd-valued ;

coherence
<*i*> is odd-valued

proof end;

end;

registration

let f, g be odd-valued FinSequence;

cluster f ^ g -> odd-valued ;

coherence
f ^ g is odd-valued

proof end;

end;

registration

cluster Relation-like OddNAT -valued -> odd-valued for set ;

coherence
for b₁ being Relation st b₁ is OddNAT -valued holds
b₁ is odd-valued by RELAT_1:def 19;

end;

definition

let n be Nat;

mode a_partition of n -> non-zero natural-valued non-decreasing FinSequence means :Def3: :: EULRPART:def 3
Sum it = n;

existence
ex b₁ being non-zero natural-valued non-decreasing FinSequence st Sum b₁ = n

proof end;

end;

:: deftheorem Def3 defines a_partition EULRPART:def 3 :
for n being Nat
for b₂ being non-zero natural-valued non-decreasing FinSequence holds
( b₂ is a_partition of n iff Sum b₂ = n );

theorem :: EULRPART:3

{} is a_partition of 0 by RVSUM_1:72, Def3;

registration

let n be Nat;

cluster Relation-like omega -defined RAT -valued non-zero Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued non-decreasing V294() odd-valued for a_partition of n;

existence
ex b₁ being a_partition of n st b₁ is odd-valued

proof end;

cluster Relation-like omega -defined RAT -valued non-zero Function-like one-to-one finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued non-decreasing V294() for a_partition of n;

existence
ex b₁ being a_partition of n st b₁ is one-to-one

proof end;

end;

registration

let n be Nat;
let a₁ be set ;
a_partition of a₁ ex b₁ being set st
for b₂ being a_partition of n holds b₂ in b₁

proof end;

end;

definition

let f be odd-valued FinSequence;

mode odd_organization of f -> valued_reorganization of f means :Def4: :: EULRPART:def 4
for n being Nat holds (2 * n) - 1 = f . (it _ (n,1)) & ... & (2 * n) - 1 = f . (it _ (n,(len (it . n))));

existence
ex b₁ being valued_reorganization of f st
for n being Nat holds (2 * n) - 1 = f . (b₁ _ (n,1)) & ... & (2 * n) - 1 = f . (b₁ _ (n,(len (b₁ . n))))

proof end;

end;

:: deftheorem Def4 defines odd_organization EULRPART:def 4 :
for f being odd-valued FinSequence
for b₂ being valued_reorganization of f holds
( b₂ is odd_organization of f iff for n being Nat holds (2 * n) - 1 = f . (b₂ _ (n,1)) & ... & (2 * n) - 1 = f . (b₂ _ (n,(len (b₂ . n)))) );

theorem Th4: :: EULRPART:4

for f being odd-valued FinSequence
for o being DoubleReorganization of dom f st ( for n being Nat holds (2 * n) - 1 = f . (o _ (n,1)) & ... & (2 * n) - 1 = f . (o _ (n,(len (o . n)))) ) holds
o is odd_organization of f

proof end;

theorem Th5: :: EULRPART:5

for i being Nat
for f being odd-valued FinSequence
for g being complex-valued FinSequence
for o1, o2 being DoubleReorganization of dom g st o1 is odd_organization of f & o2 is odd_organization of f holds
(Sum (g *. o1)) . i = (Sum (g *. o2)) . i

proof end;

theorem Th6: :: EULRPART:6

for n being Nat
for p being a_partition of n ex O being odd-valued FinSequence ex a being natural-valued FinSequence st
( len O = len p & len p = len a & p = O (#) (2 |^ a) & p . 1 = (O . 1) * (2 |^ (a . 1)) & ... & p . (len p) = (O . (len p)) * (2 |^ (a . (len p))) )

proof end;

theorem Th7: :: EULRPART:7

for D being finite set
for f being Function of D,NAT ex h being FinSequence of D st
for d being Element of D holds card (Coim (h,d)) = f . d

proof end;

Lm1: for f1, f2, g1, g2 being complex-valued FinSequence st len f1 = len g1 & len f2 <= len g2 holds
(f1 ^ f2) (#) (g1 ^ g2) = (f1 (#) g1) ^ (f2 (#) g2)

proof end;

theorem Th8: :: EULRPART:8

for f1, f2, g1, g2 being complex-valued FinSequence st len f1 = len g1 holds
(f1 ^ f2) (#) (g1 ^ g2) = (f1 (#) g1) ^ (f2 (#) g2)

proof end;

Lm2: for i being Nat
for f being complex-valued FinSequence holds ((id (dom f)) (#) f) . i = i * (f . i)

proof end;

theorem Th9: :: EULRPART:9

for f, h being natural-valued FinSequence st ( for i being Nat holds card (Coim (h,i)) = f . i ) holds
Sum h = ((1 * (f . 1)) + (2 * (f . 2))) + ((((id (dom f)) (#) f),3) +...)

proof end;

theorem Th10: :: EULRPART:10

for g being natural-valued FinSequence
for sort being DoubleReorganization of dom g ex h being 2 * (len b₂) -element FinSequence of NAT st
for j being Nat holds
( h . (2 * j) = 0 & h . ((2 * j) - 1) = (g . (sort _ (j,1))) + ((((g *. sort) . j),2) +...) )

proof end;

Lm3: for mu being natural-valued FinSequence st ( for j being Nat holds mu . (2 * j) = 0 ) holds
ex h being odd-valued FinSequence st
( h is non-decreasing & ( for j being Nat holds card (Coim (h,j)) = mu . j ) )

proof end;

theorem Th11: :: EULRPART:11

for n being Nat
for d being one-to-one a_partition of n ex e being odd-valued a_partition of n st
for j being Nat
for O1 being odd-valued FinSequence
for a1 being natural-valued FinSequence st len O1 = len d & len d = len a1 & d = O1 (#) (2 |^ a1) holds
for sort being DoubleReorganization of dom d st 1 = O1 . (sort _ (1,1)) & ... & 1 = O1 . (sort _ (1,(len (sort . 1)))) & 3 = O1 . (sort _ (2,1)) & ... & 3 = O1 . (sort _ (2,(len (sort . 2)))) & 5 = O1 . (sort _ (3,1)) & ... & 5 = O1 . (sort _ (3,(len (sort . 3)))) & ( for i being Nat holds (2 * i) - 1 = O1 . (sort _ (i,1)) & ... & (2 * i) - 1 = O1 . (sort _ (i,(len (sort . i)))) ) holds
( card (Coim (e,1)) = ((2 |^ a1) . (sort _ (1,1))) + (((((2 |^ a1) *. sort) . 1),2) +...) & card (Coim (e,3)) = ((2 |^ a1) . (sort _ (2,1))) + (((((2 |^ a1) *. sort) . 2),2) +...) & card (Coim (e,5)) = ((2 |^ a1) . (sort _ (3,1))) + (((((2 |^ a1) *. sort) . 3),2) +...) & card (Coim (e,((j * 2) - 1))) = ((2 |^ a1) . (sort _ (j,1))) + (((((2 |^ a1) *. sort) . j),2) +...) )

proof end;

definition

let n be Nat;
let p be one-to-one a_partition of n;

func Euler_transformation p -> odd-valued a_partition of n means :Def5: :: EULRPART:def 5
for j being Nat
for O being odd-valued FinSequence
for a being natural-valued FinSequence st len O = len p & len p = len a & p = O (#) (2 |^ a) holds
for sort being DoubleReorganization of dom p st 1 = O . (sort _ (1,1)) & ... & 1 = O . (sort _ (1,(len (sort . 1)))) & 3 = O . (sort _ (2,1)) & ... & 3 = O . (sort _ (2,(len (sort . 2)))) & 5 = O . (sort _ (3,1)) & ... & 5 = O . (sort _ (3,(len (sort . 3)))) & ( for i being Nat holds (2 * i) - 1 = O . (sort _ (i,1)) & ... & (2 * i) - 1 = O . (sort _ (i,(len (sort . i)))) ) holds
( card (Coim (it,1)) = ((2 |^ a) . (sort _ (1,1))) + (((((2 |^ a) *. sort) . 1),2) +...) & card (Coim (it,3)) = ((2 |^ a) . (sort _ (2,1))) + (((((2 |^ a) *. sort) . 2),2) +...) & card (Coim (it,5)) = ((2 |^ a) . (sort _ (3,1))) + (((((2 |^ a) *. sort) . 3),2) +...) & card (Coim (it,((j * 2) - 1))) = ((2 |^ a) . (sort _ (j,1))) + (((((2 |^ a) *. sort) . j),2) +...) );

existence
ex b₁ being odd-valued a_partition of n st
for j being Nat
for O being odd-valued FinSequence
for a being natural-valued FinSequence st len O = len p & len p = len a & p = O (#) (2 |^ a) holds
for sort being DoubleReorganization of dom p st 1 = O . (sort _ (1,1)) & ... & 1 = O . (sort _ (1,(len (sort . 1)))) & 3 = O . (sort _ (2,1)) & ... & 3 = O . (sort _ (2,(len (sort . 2)))) & 5 = O . (sort _ (3,1)) & ... & 5 = O . (sort _ (3,(len (sort . 3)))) & ( for i being Nat holds (2 * i) - 1 = O . (sort _ (i,1)) & ... & (2 * i) - 1 = O . (sort _ (i,(len (sort . i)))) ) holds
( card (Coim (b₁,1)) = ((2 |^ a) . (sort _ (1,1))) + (((((2 |^ a) *. sort) . 1),2) +...) & card (Coim (b₁,3)) = ((2 |^ a) . (sort _ (2,1))) + (((((2 |^ a) *. sort) . 2),2) +...) & card (Coim (b₁,5)) = ((2 |^ a) . (sort _ (3,1))) + (((((2 |^ a) *. sort) . 3),2) +...) & card (Coim (b₁,((j * 2) - 1))) = ((2 |^ a) . (sort _ (j,1))) + (((((2 |^ a) *. sort) . j),2) +...) ) by Th11;
uniqueness
for b₁, b₂ being odd-valued a_partition of n st ( for j being Nat
for O being odd-valued FinSequence
for a being natural-valued FinSequence st len O = len p & len p = len a & p = O (#) (2 |^ a) holds
for sort being DoubleReorganization of dom p st 1 = O . (sort _ (1,1)) & ... & 1 = O . (sort _ (1,(len (sort . 1)))) & 3 = O . (sort _ (2,1)) & ... & 3 = O . (sort _ (2,(len (sort . 2)))) & 5 = O . (sort _ (3,1)) & ... & 5 = O . (sort _ (3,(len (sort . 3)))) & ( for i being Nat holds (2 * i) - 1 = O . (sort _ (i,1)) & ... & (2 * i) - 1 = O . (sort _ (i,(len (sort . i)))) ) holds
( card (Coim (b₁,1)) = ((2 |^ a) . (sort _ (1,1))) + (((((2 |^ a) *. sort) . 1),2) +...) & card (Coim (b₁,3)) = ((2 |^ a) . (sort _ (2,1))) + (((((2 |^ a) *. sort) . 2),2) +...) & card (Coim (b₁,5)) = ((2 |^ a) . (sort _ (3,1))) + (((((2 |^ a) *. sort) . 3),2) +...) & card (Coim (b₁,((j * 2) - 1))) = ((2 |^ a) . (sort _ (j,1))) + (((((2 |^ a) *. sort) . j),2) +...) ) ) & ( for j being Nat
for O being odd-valued FinSequence
for a being natural-valued FinSequence st len O = len p & len p = len a & p = O (#) (2 |^ a) holds
for sort being DoubleReorganization of dom p st 1 = O . (sort _ (1,1)) & ... & 1 = O . (sort _ (1,(len (sort . 1)))) & 3 = O . (sort _ (2,1)) & ... & 3 = O . (sort _ (2,(len (sort . 2)))) & 5 = O . (sort _ (3,1)) & ... & 5 = O . (sort _ (3,(len (sort . 3)))) & ( for i being Nat holds (2 * i) - 1 = O . (sort _ (i,1)) & ... & (2 * i) - 1 = O . (sort _ (i,(len (sort . i)))) ) holds
( card (Coim (b₂,1)) = ((2 |^ a) . (sort _ (1,1))) + (((((2 |^ a) *. sort) . 1),2) +...) & card (Coim (b₂,3)) = ((2 |^ a) . (sort _ (2,1))) + (((((2 |^ a) *. sort) . 2),2) +...) & card (Coim (b₂,5)) = ((2 |^ a) . (sort _ (3,1))) + (((((2 |^ a) *. sort) . 3),2) +...) & card (Coim (b₂,((j * 2) - 1))) = ((2 |^ a) . (sort _ (j,1))) + (((((2 |^ a) *. sort) . j),2) +...) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def5 defines Euler_transformation EULRPART:def 5 :
for n being Nat
for p being one-to-one a_partition of n
for b₃ being odd-valued a_partition of n holds
( b₃ = Euler_transformation p iff for j being Nat
for O being odd-valued FinSequence
for a being natural-valued FinSequence st len O = len p & len p = len a & p = O (#) (2 |^ a) holds
for sort being DoubleReorganization of dom p st 1 = O . (sort _ (1,1)) & ... & 1 = O . (sort _ (1,(len (sort . 1)))) & 3 = O . (sort _ (2,1)) & ... & 3 = O . (sort _ (2,(len (sort . 2)))) & 5 = O . (sort _ (3,1)) & ... & 5 = O . (sort _ (3,(len (sort . 3)))) & ( for i being Nat holds (2 * i) - 1 = O . (sort _ (i,1)) & ... & (2 * i) - 1 = O . (sort _ (i,(len (sort . i)))) ) holds
( card (Coim (b₃,1)) = ((2 |^ a) . (sort _ (1,1))) + (((((2 |^ a) *. sort) . 1),2) +...) & card (Coim (b₃,3)) = ((2 |^ a) . (sort _ (2,1))) + (((((2 |^ a) *. sort) . 2),2) +...) & card (Coim (b₃,5)) = ((2 |^ a) . (sort _ (3,1))) + (((((2 |^ a) *. sort) . 3),2) +...) & card (Coim (b₃,((j * 2) - 1))) = ((2 |^ a) . (sort _ (j,1))) + (((((2 |^ a) *. sort) . j),2) +...) ) );

theorem Th12: :: EULRPART:12

for n being Nat
for p being one-to-one a_partition of n
for e being odd-valued a_partition of n holds
( e = Euler_transformation p iff for O being odd-valued FinSequence
for a being natural-valued FinSequence
for sort being odd_organization of O st len O = len p & len p = len a & p = O (#) (2 |^ a) holds
for j being Nat holds card (Coim (e,((j * 2) - 1))) = ((((2 |^ a) *. sort) . j),1) +... )

proof end;

registration

cluster Relation-like Function-like one-to-one real-valued non-decreasing -> real-valued increasing for set ;

coherence
for b₁ being real-valued Function st b₁ is one-to-one & b₁ is non-decreasing holds
b₁ is increasing

proof end;

end;

theorem Th13: :: EULRPART:13

for i being Nat
for O being odd-valued FinSequence
for a being natural-valued FinSequence
for s being odd_organization of O st len O = len a & O (#) (2 |^ a) is one-to-one holds
(a *. s) . i is one-to-one

proof end;

Lm4: for n being Nat
for p1, p2 being one-to-one a_partition of n st Euler_transformation p1 = Euler_transformation p2 holds
rng p1 c= rng p2

proof end;

theorem Th14: :: EULRPART:14

for n being Nat
for p1, p2 being one-to-one a_partition of n st Euler_transformation p1 = Euler_transformation p2 holds
p1 = p2

proof end;

theorem Th15: :: EULRPART:15

for n being Nat
for e being odd-valued a_partition of n ex p being one-to-one a_partition of n st e = Euler_transformation p

proof end;

:: Euler's partition theorem

theorem :: EULRPART:16

for n being Nat holds card { p where p is odd-valued a_partition of n : verum } = card { p where p is one-to-one a_partition of n : verum }

proof end;