:: Ternary Fields
:: by Micha{\l} Muzalewski and Wojciech Skaba
::
:: Received October 15, 1990
:: Copyright (c) 1990-2021 Association of Mizar Users

:: TERNARY FIELDS
:: This few lines define the basic algebraic structure (F, 0, 1, T)
:: used in the whole article.

definition

attr c₁ is strict ;
struct TernaryFieldStr -> ZeroOneStr ;
aggr TernaryFieldStr(# carrier, ZeroF, OneF, TernOp #) -> TernaryFieldStr ;
sel TernOp c₁ -> TriOp of the carrier of c₁;

end;

registration

cluster non empty for TernaryFieldStr ;

existence
not for b₁ being TernaryFieldStr holds b₁ is empty

proof end;

end;

:: The following definitions let us simplify the notation

definition

let F be non empty TernaryFieldStr ;
mode Scalar of F is Element of F;

end;

definition

let F be non empty TernaryFieldStr ;
let a, b, c be Scalar of F;

func Tern (a,b,c) -> Scalar of F equals :: ALGSTR_3:def 1
the TernOp of F . (a,b,c);

correctness
coherence
the TernOp of F . (a,b,c) is Scalar of F;
;

end;

:: deftheorem defines Tern ALGSTR_3:def 1 :
for F being non empty TernaryFieldStr
for a, b, c being Scalar of F holds Tern (a,b,c) = the TernOp of F . (a,b,c);

:: The following definition specifies a ternary operation that
:: will be used in the forthcoming example: TernaryFieldEx
:: as its TriOp function.

definition

func ternaryreal -> TriOp of REAL means :Def2: :: ALGSTR_3:def 2
for a, b, c being Real holds it . (a,b,c) = (a * b) + c;

existence
ex b₁ being TriOp of REAL st
for a, b, c being Real holds b₁ . (a,b,c) = (a * b) + c

proof end;

uniqueness
for b₁, b₂ being TriOp of REAL st ( for a, b, c being Real holds b₁ . (a,b,c) = (a * b) + c ) & ( for a, b, c being Real holds b₂ . (a,b,c) = (a * b) + c ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def2 defines ternaryreal ALGSTR_3:def 2 :
for b₁ being TriOp of REAL holds
( b₁ = ternaryreal iff for a, b, c being Real holds b₁ . (a,b,c) = (a * b) + c );

:: Now comes the definition of example structure called: TernaryFieldEx.
:: This example will be used to prove the existence of the Ternary-Field mode.

definition

func TernaryFieldEx -> strict TernaryFieldStr equals :: ALGSTR_3:def 3
TernaryFieldStr(# REAL,(In (0,REAL)),(In (1,REAL)),ternaryreal #);

correctness
coherence
TernaryFieldStr(# REAL,(In (0,REAL)),(In (1,REAL)),ternaryreal #) is strict TernaryFieldStr ;
;

end;

:: deftheorem defines TernaryFieldEx ALGSTR_3:def 3 :
TernaryFieldEx = TernaryFieldStr(# REAL,(In (0,REAL)),(In (1,REAL)),ternaryreal #);

registration

cluster TernaryFieldEx -> non empty strict ;

coherence
not TernaryFieldEx is empty ;

end;

:: On the contrary to the Tern() (starting with uppercase), this definition
:: allows for the use of the currently specified example ternary field.

definition

let a, b, c be Scalar of TernaryFieldEx;

func tern (a,b,c) -> Scalar of TernaryFieldEx equals :: ALGSTR_3:def 4
the TernOp of TernaryFieldEx . (a,b,c);

correctness
coherence
the TernOp of TernaryFieldEx . (a,b,c) is Scalar of TernaryFieldEx;
;

end;

:: deftheorem defines tern ALGSTR_3:def 4 :
for a, b, c being Scalar of TernaryFieldEx holds tern (a,b,c) = the TernOp of TernaryFieldEx . (a,b,c);

theorem Th1: :: ALGSTR_3:1

for u, u9, v, v9 being Real st u <> u9 holds
ex x being Real st (u * x) + v = (u9 * x) + v9

proof end;

theorem :: ALGSTR_3:2

for u, a, v being Scalar of TernaryFieldEx
for z, x, y being Real st u = z & a = x & v = y holds
Tern (u,a,v) = (z * x) + y by Def2;

reconsider jj = 1, zz = 0 as Real ;

theorem :: ALGSTR_3:3

1 = 1. TernaryFieldEx ;

Lm1: for a being Scalar of TernaryFieldEx holds Tern (a,(1. TernaryFieldEx),(0. TernaryFieldEx)) = a

proof end;

Lm2: for a being Scalar of TernaryFieldEx holds Tern ((1. TernaryFieldEx),a,(0. TernaryFieldEx)) = a

proof end;

Lm3: for a, b being Scalar of TernaryFieldEx holds Tern (a,(0. TernaryFieldEx),b) = b

proof end;

Lm4: for a, b being Scalar of TernaryFieldEx holds Tern ((0. TernaryFieldEx),a,b) = b

proof end;

Lm5: for u, a, b being Scalar of TernaryFieldEx ex v being Scalar of TernaryFieldEx st Tern (u,a,v) = b

proof end;

Lm6: for u, a, v, v9 being Scalar of TernaryFieldEx st Tern (u,a,v) = Tern (u,a,v9) holds
v = v9

proof end;

Lm7: for a, a9 being Scalar of TernaryFieldEx st a <> a9 holds
for b, b9 being Scalar of TernaryFieldEx ex u, v being Scalar of TernaryFieldEx st
( Tern (u,a,v) = b & Tern (u,a9,v) = b9 )

proof end;

Lm8: for u, u9 being Scalar of TernaryFieldEx st u <> u9 holds
for v, v9 being Scalar of TernaryFieldEx ex a being Scalar of TernaryFieldEx st Tern (u,a,v) = Tern (u9,a,v9)

proof end;

Lm9: for a, a9, u, u9, v, v9 being Scalar of TernaryFieldEx st Tern (u,a,v) = Tern (u9,a,v9) & Tern (u,a9,v) = Tern (u9,a9,v9) & not a = a9 holds
u = u9

proof end;

definition

let IT be non empty TernaryFieldStr ;

attr IT is Ternary-Field-like means :Def5: :: ALGSTR_3:def 5
( 0. IT <> 1. IT & ( for a being Scalar of IT holds Tern (a,(1. IT),(0. IT)) = a ) & ( for a being Scalar of IT holds Tern ((1. IT),a,(0. IT)) = a ) & ( for a, b being Scalar of IT holds Tern (a,(0. IT),b) = b ) & ( for a, b being Scalar of IT holds Tern ((0. IT),a,b) = b ) & ( for u, a, b being Scalar of IT ex v being Scalar of IT st Tern (u,a,v) = b ) & ( for u, a, v, v9 being Scalar of IT st Tern (u,a,v) = Tern (u,a,v9) holds
v = v9 ) & ( for a, a9 being Scalar of IT st a <> a9 holds
for b, b9 being Scalar of IT ex u, v being Scalar of IT st
( Tern (u,a,v) = b & Tern (u,a9,v) = b9 ) ) & ( for u, u9 being Scalar of IT st u <> u9 holds
for v, v9 being Scalar of IT ex a being Scalar of IT st Tern (u,a,v) = Tern (u9,a,v9) ) & ( for a, a9, u, u9, v, v9 being Scalar of IT st Tern (u,a,v) = Tern (u9,a,v9) & Tern (u,a9,v) = Tern (u9,a9,v9) & not a = a9 holds
u = u9 ) );

end;

:: deftheorem Def5 defines Ternary-Field-like ALGSTR_3:def 5 :
for IT being non empty TernaryFieldStr holds
( IT is Ternary-Field-like iff ( 0. IT <> 1. IT & ( for a being Scalar of IT holds Tern (a,(1. IT),(0. IT)) = a ) & ( for a being Scalar of IT holds Tern ((1. IT),a,(0. IT)) = a ) & ( for a, b being Scalar of IT holds Tern (a,(0. IT),b) = b ) & ( for a, b being Scalar of IT holds Tern ((0. IT),a,b) = b ) & ( for u, a, b being Scalar of IT ex v being Scalar of IT st Tern (u,a,v) = b ) & ( for u, a, v, v9 being Scalar of IT st Tern (u,a,v) = Tern (u,a,v9) holds
v = v9 ) & ( for a, a9 being Scalar of IT st a <> a9 holds
for b, b9 being Scalar of IT ex u, v being Scalar of IT st
( Tern (u,a,v) = b & Tern (u,a9,v) = b9 ) ) & ( for u, u9 being Scalar of IT st u <> u9 holds
for v, v9 being Scalar of IT ex a being Scalar of IT st Tern (u,a,v) = Tern (u9,a,v9) ) & ( for a, a9, u, u9, v, v9 being Scalar of IT st Tern (u,a,v) = Tern (u9,a,v9) & Tern (u,a9,v) = Tern (u9,a9,v9) & not a = a9 holds
u = u9 ) ) );

registration

cluster non empty strict Ternary-Field-like for TernaryFieldStr ;

existence
ex b₁ being non empty TernaryFieldStr st
( b₁ is strict & b₁ is Ternary-Field-like ) by Def5, Lm1, Lm2, Lm3, Lm4, Lm5, Lm6, Lm7, Lm8, Lm9;

end;

definition

mode Ternary-Field is non empty Ternary-Field-like TernaryFieldStr ;

end;

theorem :: ALGSTR_3:4

for F being Ternary-Field
for a, a9, u, u9, v, v9 being Scalar of F st a <> a9 & Tern (u,a,v) = Tern (u9,a,v9) & Tern (u,a9,v) = Tern (u9,a9,v9) holds
( u = u9 & v = v9 )

proof end;

theorem :: ALGSTR_3:5

for F being Ternary-Field
for a being Scalar of F st a <> 0. F holds
for b, c being Scalar of F ex x being Scalar of F st Tern (a,x,b) = c

proof end;

theorem :: ALGSTR_3:6

for F being Ternary-Field
for a, b, x, x9 being Scalar of F st a <> 0. F & Tern (a,x,b) = Tern (a,x9,b) holds
x = x9

proof end;

theorem :: ALGSTR_3:7

for F being Ternary-Field
for a being Scalar of F st a <> 0. F holds
for b, c being Scalar of F ex x being Scalar of F st Tern (x,a,b) = c

proof end;

theorem :: ALGSTR_3:8

for F being Ternary-Field
for a, b, x, x9 being Scalar of F st a <> 0. F & Tern (x,a,b) = Tern (x9,a,b) holds
x = x9

proof end;