:: FINSOP_1 semantic presentation

definition

let D be non empty set ;
let n be Nat;
let d be Element of D;
:: original: |->
redefine func n |-> d -> FinSequence of D;
coherence by FINSEQ_2:77;

end;

definition

let D be non empty set ;
let F be FinSequence of D;
let g be BinOp of D;
assume A1: ( g has_a_unity or len F >= 1 ) ;
func g "**" F -> Element of D means :Def1: :: FINSOP_1:def 1
it = the_unity_wrt g if ( g has_a_unity & len F = 0 )
otherwise ex f being Function of NAT ,D st
( f . 1 = F . 1 & ( for n being Nat st 0 <> n & n < len F holds
f . (n + 1) = g . (f . n),(F . (n + 1)) ) & it = f . (len F) );
existence

proof end;

uniqueness

proof end;

consistency ;

end;

:: deftheorem Def1 defines "**" FINSOP_1:def 1 :

theorem :: FINSOP_1:1

canceled;

theorem Th2: :: FINSOP_1:2

for D being non empty set
for F being FinSequence of D
for g being BinOp of D st len F >= 1 holds
ex f being Function of NAT ,D st
( f . 1 = F . 1 & ( for n being Nat st 0 <> n & n < len F holds
f . (n + 1) = g . (f . n),(F . (n + 1)) ) & g "**" F = f . (len F) ) by Def1;

theorem :: FINSOP_1:3

for D being non empty set
for d being Element of D
for F being FinSequence of D
for g being BinOp of D st len F >= 1 & ex f being Function of NAT ,D st
( f . 1 = F . 1 & ( for n being Nat st 0 <> n & n < len F holds
f . (n + 1) = g . (f . n),(F . (n + 1)) ) & d = f . (len F) ) holds
d = g "**" F by Def1;

definition

let B, A be non empty set ;
let b be Element of B;
:: original: -->
redefine func A --> b -> Function of A,B;
coherence by FUNCOP_1:58;

end;

definition

let A be non empty set ;
let F be Function of NAT ,A;
let p be FinSequence of A;
:: original: +*
redefine func F +* p -> Function of NAT ,A;
coherence

proof end;

end;

definition

let f be FinSequence;
:: original: dom
redefine func dom f -> Element of Fin NAT ;
coherence

proof end;

end;

Lm1: for D being non empty set
for F being FinSequence of D
for g being BinOp of D st len F >= 1 & g is associative & g is commutative holds
g "**" F = g $$ (dom F),((NAT --> (the_unity_wrt g)) +* F)

proof end;

Lm2: for D being non empty set
for F being FinSequence of D
for g being BinOp of D st len F = 0 & g has_a_unity & g is associative & g is commutative holds
g "**" F = g $$ (dom F),((NAT --> (the_unity_wrt g)) +* F)

proof end;

theorem :: FINSOP_1:4

for D being non empty set
for F being FinSequence of D
for g being BinOp of D st ( g has_a_unity or len F >= 1 ) & g is associative & g is commutative holds
g "**" F = g $$ (dom F),((NAT --> (the_unity_wrt g)) +* F)

proof end;

Lm3: for D being non empty set
for g being BinOp of D st g has_a_unity holds
g "**" (<*> D) = the_unity_wrt g

proof end;

Lm4: for D being non empty set
for d being Element of D
for g being BinOp of D holds g "**" <*d*> = d

proof end;

Lm5: for D being non empty set
for d being Element of D
for F being FinSequence of D
for g being BinOp of D st len F >= 1 holds
g "**" (F ^ <*d*>) = g . (g "**" F),d

proof end;

Lm6: for D being non empty set
for d being Element of D
for F being FinSequence of D
for g being BinOp of D st g has_a_unity & len F = 0 holds
g "**" (F ^ <*d*>) = g . (g "**" F),d

proof end;

theorem Th5: :: FINSOP_1:5

for D being non empty set
for d being Element of D
for F being FinSequence of D
for g being BinOp of D st ( g has_a_unity or len F >= 1 ) holds
g "**" (F ^ <*d*>) = g . (g "**" F),d

proof end;

theorem Th6: :: FINSOP_1:6

for D being non empty set
for F, G being FinSequence of D
for g being BinOp of D st g is associative & ( g has_a_unity or ( len F >= 1 & len G >= 1 ) ) holds
g "**" (F ^ G) = g . (g "**" F),(g "**" G)

proof end;

theorem Th7: :: FINSOP_1:7

for D being non empty set
for d being Element of D
for F being FinSequence of D
for g being BinOp of D st g is associative & ( g has_a_unity or len F >= 1 ) holds
g "**" (<*d*> ^ F) = g . d,(g "**" F)

proof end;

Lm7: for D being non empty set
for F, G being FinSequence of D
for g being BinOp of D st g is associative & g is commutative holds
for f being Permutation of dom F st len F >= 1 & len F = len G & ( for i being Nat st i in dom G holds
G . i = F . (f . i) ) holds
g "**" F = g "**" G

proof end;

Lm8: for D being non empty set
for F, G being FinSequence of D
for g being BinOp of D
for P being Permutation of dom F st g has_a_unity & len F = 0 & G = F * P holds
g "**" F = g "**" G

proof end;

theorem Th8: :: FINSOP_1:8

for D being non empty set
for F, G being FinSequence of D
for g being BinOp of D
for P being Permutation of dom F st g is commutative & g is associative & ( g has_a_unity or len F >= 1 ) & G = F * P holds
g "**" F = g "**" G

proof end;

Lm9: for D being non empty set
for F, G being FinSequence of D
for g being BinOp of D st g is associative & g is commutative & F is one-to-one & G is one-to-one & rng F = rng G & len F >= 1 holds
g "**" F = g "**" G

proof end;

Lm10: for D being non empty set
for F, G being FinSequence of D
for g being BinOp of D st len F = 0 & g has_a_unity & F is one-to-one & G is one-to-one & rng F = rng G holds
g "**" F = g "**" G

proof end;

theorem :: FINSOP_1:9

for D being non empty set
for F, G being FinSequence of D
for g being BinOp of D st ( g has_a_unity or len F >= 1 ) & g is associative & g is commutative & F is one-to-one & G is one-to-one & rng F = rng G holds
g "**" F = g "**" G

proof end;

Lm11: for D being non empty set
for F being FinSequence of D
for g being BinOp of D st len F = 1 holds
g "**" F = F . 1

proof end;

Lm12: for D being non empty set
for F, G, H being FinSequence of D
for g being BinOp of D st g is associative & g is commutative & len F >= 1 & len F = len G & len F = len H & ( for k being Nat st k in dom F holds
H . k = g . (F . k),(G . k) ) holds
g "**" H = g . (g "**" F),(g "**" G)

proof end;

Lm13: for D being non empty set
for F, G, H being FinSequence of D
for g being BinOp of D st g has_a_unity & len F = 0 & len F = len G & len F = len H holds
g "**" F = g . (g "**" G),(g "**" H)

proof end;

theorem :: FINSOP_1:10

for D being non empty set
for F, G, H being FinSequence of D
for g being BinOp of D st g is associative & g is commutative & ( g has_a_unity or len F >= 1 ) & len F = len G & len F = len H & ( for k being Nat st k in dom F holds
F . k = g . (G . k),(H . k) ) holds
g "**" F = g . (g "**" G),(g "**" H)

proof end;

theorem :: FINSOP_1:11

for D being non empty set
for g being BinOp of D st g has_a_unity holds
g "**" (<*> D) = the_unity_wrt g by Lm3;

theorem :: FINSOP_1:12

for D being non empty set
for d being Element of D
for g being BinOp of D holds g "**" <*d*> = d by Lm4;

theorem Th13: :: FINSOP_1:13

for D being non empty set
for d1, d2 being Element of D
for g being BinOp of D holds g "**" <*d1,d2*> = g . d1,d2

proof end;

theorem :: FINSOP_1:14

for D being non empty set
for d1, d2 being Element of D
for g being BinOp of D st g is commutative holds
g "**" <*d1,d2*> = g "**" <*d2,d1*>

proof end;

theorem Th15: :: FINSOP_1:15

for D being non empty set
for d1, d2, d3 being Element of D
for g being BinOp of D holds g "**" <*d1,d2,d3*> = g . (g . d1,d2),d3

proof end;

theorem :: FINSOP_1:16

for D being non empty set
for d1, d2, d3 being Element of D
for g being BinOp of D st g is commutative holds
g "**" <*d1,d2,d3*> = g "**" <*d2,d1,d3*>

proof end;

theorem Th17: :: FINSOP_1:17

for D being non empty set
for d being Element of D
for g being BinOp of D holds g "**" (1 |-> d) = d

proof end;

theorem :: FINSOP_1:18

for D being non empty set
for d being Element of D
for g being BinOp of D holds g "**" (2 |-> d) = g . d,d

proof end;

theorem Th19: :: FINSOP_1:19

for D being non empty set
for d being Element of D
for g being BinOp of D
for k, l being Nat st g is associative & ( g has_a_unity or ( k <> 0 & l <> 0 ) ) holds
g "**" ((k + l) |-> d) = g . (g "**" (k |-> d)),(g "**" (l |-> d))

proof end;

theorem :: FINSOP_1:20

for D being non empty set
for d being Element of D
for g being BinOp of D
for k, l being Nat st g is associative & ( g has_a_unity or ( k <> 0 & l <> 0 ) ) holds
g "**" ((k * l) |-> d) = g "**" (l |-> (g "**" (k |-> d)))

proof end;

theorem :: FINSOP_1:21

for D being non empty set
for F being FinSequence of D
for g being BinOp of D st len F = 1 holds
g "**" F = F . 1 by Lm11;

theorem :: FINSOP_1:22

for D being non empty set
for F being FinSequence of D
for g being BinOp of D st len F = 2 holds
g "**" F = g . (F . 1),(F . 2)

proof end;